General Law of iterated logarithm for Markov processes
GGENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES
SOOBIN CHO, PANKI KIM, JAEHUN LEE
Abstract.
In this paper, we discuss general criteria and forms of both liminf and limsup laws ofiterated logarithm (LIL) for continuous-time Markov processes. We consider minimal assumptionsfor both LILs to hold at zero (at infinity, respectively) in general metric measure spaces. We establishLILs under local assumptions near zero (near infinity, respectively) on uniform bounds of the firstexit time from balls in terms of a function φ and uniform bounds on the tails of the jumping measurein terms of a function ψ . One of the main results is that a simple ratio test in terms of the functions φ and ψ completely determines whether there exists a positive nondecreasing function Ψ such thatlim sup | X t | / Ψ( t ) is positive and finite a.s., or not. We also provide a general formulation of liminfLIL, which covers jump processes whose jumping measures have logarithmic tails. Our results covera large class of subordinated diffusions, jump processes with mixed polynomial local growths, jumpprocesses with singular jumping kernels and random conductance models with long range jumps. Keywords: jump processes; law of the iterated logarithm; sample path;
MSC 2020: Introduction and general result
Law of the iterated logarithm for a stochastic process describes the magnitude of the fluctuationsof its sample path behaviors. With the law of large numbers and the central limit theorem, thelaw of the iterated logarithm (LIL) is considered as the fundamental limit theorem in Probabilitytheory. See [37] and the references therein.Let Y := ( Y t ) t ≥ be a strictly β -stable process on R d with 0 < β ≤
2, in the sense of [79, Definition13.1]. Assume that Y has a non-trivial jumping measure unless β = 2. Then Y satisfies the followingwell-known limsup LIL: If β = 2, then there exist constants c , c ∈ (0 , ∞ ) such thatlim sup t → t →∞ ) sup
2, then for every positive nondecreasing function Ψ, (cid:90) Ψ( t ) − β dt < ∞ or = ∞ (cid:16) resp . (cid:90) ∞ Ψ( t ) − β dt < ∞ or = ∞ (cid:17) ⇔ lim sup t → t →∞ ) sup Q ) How do we determine that a limsup law (either at zero or at infinity) of a given process Y isof type (1.1), or else of type (1.2)? More precisely, what is a necessary and sufficient conditionfor the existence of Ψ ∈ M + such that lim sup of | Y t | / Ψ( t ) converges to a positive, finite anddeterministic value as t tends to zero or infinity?Feller [38] studied the above question for symmetric random walks which belong to the domain ofattraction of the normal distribution, and established an integral test as an answer. He also foundthe exact form of Ψ when it exists. We also refer to the Kesten’s work [57]. For continuous timeprocesses, Fristedt [40] partially answered the above question for one-dimensional symmetric L´evyprocesses. Then Wee and Kim [88] establish a ratio test for entire one-dimensional L´evy processeswhich determine whether lim sup | Y t | / Ψ( t ) ∈ (0 , ∞ ).The purpose of this paper is to understand asymptotic behaviors of a given Markov processby establishing liminf and limsup laws of iterated logarithms for both near zero and near infinityunder some minimal assumptions. The main contribution of this paper is that we answer the abovequestion ( Q ) completely for a considerable class of Markov processes including random conductancemodels with stable-like jumps (see Theorems 1.17–1.18 and Example 6.6 below).Formally, our answer is( A ) There is a dichotomous classification on continuous time Markov processes (satisfying a neardiagonal lower bound estimate for the Dirichlet heat kernel): Based on whether the tails ofjumping measure and the mean exit time from balls are comparable or not, we can categorize ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 3
Markov processes into two non-overlaping classes A ∞ and A ∞ : A ∞ = (cid:110) For all Ψ ∈ M + , lim sup t →∞ d ( x, X t )Ψ( t ) is either 0 or ∞ for all x . (cid:111) A ∞ = (cid:110) There is Ψ ∈ M + such that lim sup t →∞ d ( x, X t )Ψ( t ) is deterministic and in (0 , ∞ ) for all x . (cid:111) If a given Markov process X is in the class A ∞ , then there is an integral test which determineswhether the above lim sup is zero or infinite. If a given Markov process X is in the class A ∞ ,then there are a natural explicit function Ψ and another integral test to determine when onecan take such specific Ψ. An analogous local result holds for lim sup LIL at zero.The reader will see that processes near the borderline of the above classes are so-called Brownian-like jump processes, i.e., jump processes with high intensity of small jumps and ones with lowintensity of large jumps.Regarding the liminf LIL, we introduce new but general version of it. See Theorems 1.13 and1.15 below. Our result covers processes with small jumps of slowly varying intensity.Assumptions in this paper are motivated by the second named author’s previous paper [63].In [63], the authors established liminf and limsup LILs (of type (1.2)) for Feller processes on a generalmetric measure space enjoying mixed stable-like heat kernel estimates (see Assumption 2.1 therein).Recently, a relationship between those heat kernel estimates, and certain conditions on the jumpingkernel and the first exit time from open balls is extensively studied. See, e.g. [3, 4, 9, 21–23, 46, 48].We adopt this framework and consider localized and relaxed conditions not only on the heat kernelbut also on the jumping kernel and the first exit time. We emphasize that, unlike the referencesmentioned above, we do not assume a weak lower scaling property of the scale function φ in severalstatements. (See Definition 1.11 for the definition of weak lower scaling properties.) However, westill need the weak lower scaling properties in some other statements, especially ones concerninglong time behaviors.Our assumptions are weak enough so that our results cover a lot of Markov processes includingjump processes with diffusion part, jump processes with low intensity of small jumps, some non-symmetric processes, processes with singular jumping kernels and random conductance models withlong range jumps. See the examples in Sections 2 and 6, and the references therein. In particular,the class of Markov processes considered in this paper extends the results of [63]. Moreover, metricmeasure spaces in this paper can be random, disconnected and highly space-inhomogeneous (seeDefinition 1.1).Let us now describe the main result of this paper precisely and, at the same time, fix the setupand notation of the paper.Throughout this paper (except Section 6), we assume that ( M, d ) is a locally compact separablemetric space, and µ is a positive Radon measure on M with full support. Denote by B ( x, r ) := { y ∈ M : d ( x, y ) < r } and V ( x, r ) := µ ( B ( x, r )) an open ball in M and its volume, respectively. We adda cemetery point ∂ to M and define M ∂ := M ∪ { ∂ } . For S ⊂ M and x ∈ M , we denote by δ S ( x )the distance between x and M \ S , namely, δ S ( x ) := inf { d ( x, y ) : y ∈ M \ S} . Note that δ S ( x ) = 0 if x ∈ M \ S . We also denote by d S ( x ) := δ M \S ( x ) + 1 , the distance between x and S added by 1. Note that d { z } ( x ) = d ( x, z ) + 1. Since d S ( x ) ≥
1, themap υ → d S ( x ) υ is nondecreasing. We write a ∧ b for min { a, b } and a ∨ b for max { a, b } .We introduce versions of weak volume doubling and reverse doubling property for the metricmeasure space ( M, d, µ ). SOOBIN CHO, PANKI KIM, JAEHUN LEE
Definition 1.1. (i) For an open set U ⊂ M and R ∈ (0 , ∞ ], we say that the interior volumedoubling and reverse doubling property near zero VRD R ( U ) holds (with C V ) if there exist constants C V ∈ (0 , d ≥ d > C µ ≥ ≥ c µ > x ∈ U and 0 < s ≤ r
If (
M, d ) is geodesic, then Ch ∞ ( M ) holds with A = 1.Let X = (Ω , F t , X t , θ t , t ≥ P x , x ∈ M ∂ ) be a Borel standard Markov process on M . Here( θ t , t ≥
0) is the shift operator with respect to the process X , which is defined as X s ( θ t ω ) = X s + t ( ω )for all t, s > ω ∈ Ω. A family of [0 , ∞ ]-valued random variables A = ( A t , t ≥
0) is calleda (perfect) positive continuous additive functional (PCAF) of the process X , if it satisfies that (i) A = 0 and t (cid:55)→ A t is continuous in [0 , ∞ ) a.s., (ii) A t ( · ) is F t -measurable for all t ≥
0, and (iii) A t + s ( ω ) = A t ( ω ) + A s ( θ t ω ) for all s, t ≥ ω ∈ Ω. See [43] for details.Since X is a Borel standard process on M , it is known that X admits a L´evy system ( N, H ). Here H is a PCAF of X with bounded 1-potential and N ( x, dy ) is a kernel on M with N ( x, { x } ) = 0 forall x ∈ M . See [12]. We assume that there exists a version of L´evy system ( N, H ) for X where theRevuz measure of H is absolutely continuous with respect to the reference measure µ so that theRevuz measure of H is ν H ( x ) µ ( dx ) for some nonnegative measurable function ν H ( x ).Let J ( x, dy ) = N ( x, dy ) ν H ( x ). Then, we see from the L´evy system formula that for any nonneg-ative Borel function F on M × M ∂ vanishing on the diagonal and any z ∈ M and t > E z (cid:20) (cid:88) s ≤ t F ( X s − , X s ) (cid:21) = E z (cid:20)(cid:90) t (cid:90) M ∂ F ( X s , y ) J ( X s , dy ) ds (cid:21) . (1.7)The measure J ( x, dy ) on M ∂ is called the L´evy measure of the process X in the literature. See [86].Here, we note that the killing term J ( x, ∂ ) is included in the L´evy measure. Also, we emphasizethat J ( x, dy ) can be identically zero and J ( x, dy ) may not be absolutely continuous with respect tothe reference measure µ (see Example 6.4).For an open set D ⊂ M , denote by τ D := inf { t > X t ∈ M ∂ \ D } the first exit time of X from D . The subprocess X D , defined by X Dt := X t { τ D >t } + ∂ { τ D ≤ t } , is called the killed process of X upon leaving D . We call a measurable function p D : (0 , ∞ ) × D × D → [0 , ∞ ] the heat kernel (orthe transition density) of X D if the followings hold: ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 5 (H1) E x [ f ( X Dt )] = (cid:82) D p D ( t, x, y ) f ( y ) µ ( dy ) for all t > x ∈ D and f ∈ L ∞ ( D ; µ );(H2) p D ( t + s, x, y ) = (cid:82) D p D ( t, x, z ) p D ( t, z, y ) µ ( dz ) for all t, s > x, y ∈ D .Let us extend p D ( t, x, y ) to all x, y ∈ M by setting p D ( t, x, y ) = 0 if x ∈ M \ D or y ∈ M \ D .We introduce a number of (local) conditions about the process X . Throughout this paper (exceptSection 6), we always assume that φ, ψ : (0 , ∞ ) → (0 , ∞ ) are increasing and continuous functionssuch that lim r → φ ( r ) = 0 and φ ( r ) ≤ ψ ( r ) for all r >
0. (1.8)Hereinafter, we say that f is increasing (resp. decreasing) if f ( s ) < f ( r ) ( f ( s ) > f ( r )) for all s < r in the domain of f . If f ( s ) ≤ f ( r ) ( f ( s ) ≥ f ( r )) for all s < r , we say that f is nondecreasing (resp.nonincreasing). Remark 1.4.
Since we will always impose local weak upper scaling conditions (see Definition 1.11)on the functions φ and ψ , the assumption that φ and ψ are increasing and continuous in (1.8) canbe relaxed to just being nondecreasing (see the construction (A.5)). However, for brevity, we assume(1.8) in this paper.The first condition is about tails of the L´evy measure J ( x, dy ). Definition 1.5. (i) For an open set U ⊂ M and R ∈ (0 , ∞ ], we say that Tail R ( ψ, U ) holds (with C ) if there exist constants C ∈ (0 , c J > x ∈ U and 0 < r < R ∧ ( C δ U ( x )),1 c J ψ ( r ) ≤ J ( x, M ∂ \ B ( x, r )) ≤ c J ψ ( r ) . (1.9)We say that Tail R ( ψ, U, ≤ ) (resp. Tail R ( ψ, U, ≥ )) holds (with C ) if the upper bound (resp. lowerbound) in (1.9) holds for all x ∈ U and 0 < r < R ∧ ( C δ U ( x )).(ii) For S ⊂ M , R ∞ ∈ [1 , ∞ ) and υ ∈ (0 , R ∞ ( ψ, S , υ ) holds if there exists aconstant c J > x ∈ M and r > R ∞ d S ( x ) υ . We say that Tail R ∞ ( ψ, S , ≤ , υ ) (resp. Tail R ∞ ( ψ, S , ≥ , υ )) holds if the upper bound (resp. lower bound) in (1.9) holds for all x ∈ M and r > R ∞ d S ( x ) υ . Remark 1.6. (i) Under (1.8), it is clear that the condition Tail R ( ψ, U, ≤ ) (resp. Tail R ∞ ( ψ, S , ≤ , υ ))implies Tail R ( φ, U, ≤ ) (resp. Tail R ∞ ( φ, S , ≤ , υ )).(ii) If X is a conservative diffusion process, then Tail ∞ ( φ, M, ≤ ) obviously holds. Here, the conser-vativeness is required since the killing term J ( x, ∂ ) is included in (1.9).(iii) For α ∈ (0 ,
1) and any open set D ⊂ R d , it is easy to see that a killed isotropic 2 α -stableprocess in D (whose generator is the Dirichlet Laplacian − ( − ∆) α | D ) satisfies Tail ∞ ( r α , D ) with1 /
2. Moreover, by [65, (2.22),(2.23) and (3.7)], for β, δ ∈ (0 ,
1) and a bounded open set D ⊂ R d , asubordinated killed stable process in D with the generator − (cid:0) ( − ∆) δ | D (cid:1) β satisfies Tail ∞ ( r βδ , D ).Here, we emphasize again that the killing term J ( x, ∂ ) is included in (1.9).The next two conditions consider estimates on the first exit time from open balls. Definition 1.7. (i) For an open set U ⊂ M and R ∈ (0 , ∞ ], we say that SP R ( φ, U ) holds (with C and C ) if there exist constants C ∈ (0 , C > a ∗ , a ∗ ∈ (0 ,
1) such that for all x ∈ U and 0 < r < R ∧ ( C δ U ( x )),( a ∗ ) n ≤ P x (cid:0) τ B ( x,r ) ≥ nφ ( C r ) (cid:1) ≤ ( a ∗ ) n for all n ∈ N . (1.10)(ii) For S ⊂ M , R ∞ ∈ [1 , ∞ ) and υ ∈ (0 , R ∞ ( φ, S , υ ) holds (with C ) if thereexist constants C > a ∗ , a ∗ ∈ (0 ,
1) such that (1.10) holds for all x ∈ M and r > R ∞ d S ( x ) υ . SOOBIN CHO, PANKI KIM, JAEHUN LEE
Remark 1.8. (i) When a diffusion admits a sub-Gaussian heat kernel estimates, every subordinateddiffusion satisfies SP ∞ ( φ, M ) for a certain scale function φ . See Lemma A.2 in Appendix.(ii) Any isotropic L´evy process X satisfies SP ∞ ( ϕ ( r − ) − , ϕ ( r − ) , R d ) where ξ → ϕ ( | ξ | ) is thecharacteristic exponent of X . See [50, Proposition 5.2]. Definition 1.9. (i) For an open set U ⊂ M and R ∈ (0 , ∞ ], we say that E R ( φ, U ) holds (with C and C ) if there exist constants C ∈ (0 , C > a ≥ x ∈ U and0 < r < R ∧ ( C δ U ( x )), a − φ ( C r ) ≤ E x [ τ B ( x,r ) ] ≤ aφ ( C r ) . (1.11)(ii) For S ⊂ M , R ∞ ∈ [1 , ∞ ) and υ ∈ (0 , R ∞ ( φ, S , υ ) holds (with C ) if there existconstants υ ∈ (0 , C > a ≥ x ∈ M and r > R ∞ d S ( x ) υ .The inequality φ ( r ) ≤ ψ ( r ) in (1.8) is quite natural under the assumptions VRD R ( U ), Tail R ( ψ, U, ≥ ) and E R ( φ, U ). See [3, Remark 2.7] and the paragraphs below it.The last condition concerns weak versions of near diagonal lower estimates of heat kernel (NDL)on open balls (see, e.g., [24] for a global version of NDL). Definition 1.10. (i) For an open set U ⊂ M and R ∈ (0 , ∞ ], we say that NDL R ( φ, U ) holds(with C ) if there exist constants C , η ∈ (0 ,
1) and c l > x ∈ U and 0 < r 1) and c l > x ∈ M and r > R ∞ d S ( x ) υ , the heat kernel p B ( x,r ) ( t, y, z ) of X B ( x,r ) exists and satisfies (1.12).As you see from the above Definitions 1.1(ii) 1.2(ii) 1.5(ii) 1.7(ii) 1.9(ii) and 1.10(ii), our conditionsat infinity are weaker by adding the restriction r > R ∞ d S ( x ) υ with υ ∈ (0 , Definition 1.11. For a given function g : (0 , ∞ ) → (0 , ∞ ) and constants a ∈ (0 , ∞ ], β , β > < c ≤ < c , we say that L a ( g, β , c ) (resp. L a ( g, β , c )) holds if g ( r ) g ( s ) ≥ c (cid:16) rs (cid:17) β for all s ≤ r < a (resp. a < s ≤ r ) . and we say that U a ( g, β , c ) (resp. U a ( g, β , c )) holds if g ( r ) g ( s ) ≤ c (cid:16) rs (cid:17) β for all s ≤ r < a (resp. a < s ≤ r ) . We say that L( g, β , c ) holds if L ∞ ( g, β , c ) holds, and that U( g, β , c ) holds if U ∞ ( g, β , c ) holds. Remark 1.12. For a positive function g , if both L a ( g, β , c ) and U a ( g, β , c ) hold, then thefunction g is known to be a O -regularly varying function at infinity (see [15, Section 2]), which is ageneralization of regularly varying functions.We will use the following implications several times:NDL R ( φ, U ) (resp. NDL R ∞ ( φ, S , υ )) Proposition 3 . ========= ⇒ VRD+U( φ, · , · ) SP c R ( φ, U ) (resp. SP c ( φ, S , υ )) Lemma 3 . ========= ⇒ E c R ( φ, U ) (resp. E c ( φ, S , υ )). (1.13)Now, we are ready to present our results in full generality. We first give liminf LILs. ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 7 Theorem 1.13. Suppose that there exist a constant R > and an open set U ⊂ M such that SP R ( φ, U ) , Tail R ( φ, U, ≤ ) and U R ( φ, β , C U ) hold. Then, there exist constants < a ≤ a < ∞ such that for all x ∈ U , there exists a x ∈ [ a , a ] satisfying lim inf t → φ (cid:0) sup Suppose that the assumptions of Theorem 1.13 are satisfied and that L R ( φ, β , C L ) holds. Then, there exist constants < (cid:101) a ≤ (cid:101) a < ∞ such that for all x ∈ U , there exists (cid:101) a x ∈ [ (cid:101) a , (cid:101) a ] satisfying lim inf t → sup Suppose that there exist constants R ∞ ≥ , υ ∈ (0 , and z ∈ M such that SP R ∞ ( φ, { z } , υ ) , Tail R ∞ ( φ, { z } , ≤ , υ ) , L R ∞ ( φ, β , C L ) and U R ∞ ( φ, β , C U ) hold. Then, there existconstants < b ≤ b < ∞ such that lim inf t →∞ φ (cid:0) sup Suppose that there exist constants R ∞ ≥ , υ ∈ (0 , and z ∈ M such that VRD R ∞ ( { z } , υ ) , NDL R ∞ ( φ, { z } , υ ) , Tail R ∞ ( φ, { z } , ≤ , υ ) , L R ∞ ( φ, β , C L ) and U R ∞ ( φ, β , C U ) hold.Then, there exists a constant b ∞ ∈ (0 , ∞ ) such that lim inf t →∞ sup Suppose that there exist a constant R > and an open set U ⊂ M such that Tail R ( ψ, U ) , U R ( φ, β , C U ) and U R ( ψ, β , C (cid:48) U ) hold.(i) Assume that E R ( φ, U ) holds. If lim sup r → ψ ( r ) /φ ( r ) < ∞ , then for any Ψ ∈ M + , (cid:90) dtφ (Ψ( t )) = ∞ (resp . < ∞ ) ⇔ lim sup t → sup Suppose that there exist constants R ∞ ≥ , υ ∈ (0 , and z ∈ M such that Tail R ∞ ( ψ, { z } , υ ) , L R ∞ ( φ, β , C L ) , U R ∞ ( φ, β , C U ) and U R ∞ ( ψ, β , C (cid:48) U ) hold. SOOBIN CHO, PANKI KIM, JAEHUN LEE (i) Assume that E R ∞ ( φ, { z } , υ ) hold. If lim sup r →∞ ψ ( r ) /φ ( r ) < ∞ , then for any Ψ ∈ M + , (cid:90) ∞ dtφ (Ψ( t )) = ∞ (resp . < ∞ ) ⇔ lim sup t →∞ sup 0. Thus, to push the function Ψ for a candidateof Ψ, it is reasonable to assume that a local upper scaling index of φ − is not greater than 1, orequivalently, the local lower scaling index β of φ is not less than 1. Under this assumption onthe local scaling property of φ , we give integral tests which fully determine whether one can putΨ = Ψ into (1.19) and (1.21), respectively. When these integral tests fail, we will see that for everyΨ ∈ M + satisfying (1.18) (resp. (1.20)), lim inf Ψ( t ) / Ψ ( t ) < ∞ but lim sup Ψ( t ) / Ψ ( t ) = ∞ , as t tends to zero (resp. infinity). See (1.24) and (1.27) below.Here, we note that in case of lim sup r → ψ ( r ) /φ ( r ) < ∞ , under U R ( φ, β , C U ), the integral(1.22) below is always infinite. Indeed, in such case, one can see that the integrand in (1.22)is larger than some constant multiple of r − (log | log r | ) − β . Similarly, under U R ∞ ( φ, β , C U ), iflim sup r →∞ ψ ( r ) /φ ( r ) < ∞ , then the integral (1.25) below is always infinite. Theorem 1.19. Suppose that there exist a constant R > and an open set U ⊂ M such that VRD R ( U ) , Ch R ( U ) , NDL R ( φ, U ) , Tail R ( ψ, U, ≤ ) , U R ( φ, β , C U ) , U R ( ψ, β , C (cid:48) U ) and L R ( φ, β , C L ) hold with β ≥ .(i) If the integral (cid:90) φ ( r ) log | log r | rψ ( r log | log r | ) dr (1.22) is finite, then there exist constants < a ≤ a < ∞ such that for all x ∈ U , there exists a x ∈ [ a , a ] satisfying lim sup t → d ( x, X t ) φ − ( t/ log | log t | ) log | log t | = a x , P x -a.s. , (1.23) (ii) Assume that (1.22) is infinite and Tail R ( ψ, U, ≥ ) also holds. Then for all x ∈ U , the left handside of (1.23) is infinite, P x -a.s. Moreover, if we also have lim sup r → ψ ( r ) /φ ( r ) = ∞ , then any Ψ ∈ M + satisfying (1.19) is factorized as Ψ( t ) = f ( t ) φ − ( t/ log | log t | ) log | log t | , (1.24) for a function f such that lim inf t → f ( t ) < ∞ , lim sup t → f ( t ) = ∞ . Theorem 1.20. Suppose that there exist constants R ∞ ≥ , υ ∈ (0 , and z ∈ M such that VRD R ∞ ( { z } , υ ) , Ch R ∞ ( { z } , υ ) , NDL R ∞ ( φ, { z } , υ ) , Tail R ∞ ( ψ, { z } , ≤ , υ ) , U R ∞ ( φ, β , C U ) , U R ∞ ( ψ, β , C (cid:48) U ) and L R ∞ ( φ, β , C L ) hold with β ≥ . ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 9 (i) If the integral (cid:90) ∞ φ ( r ) log log rrψ ( r log log r ) dr (1.25) is finite, then there exists a constant b ∞ ∈ (0 , ∞ ) such that for all x, y ∈ M , lim sup t →∞ d ( x, X t ) φ − ( t/ log log t ) log log t = b ∞ , P y -a.s., (1.26) (ii) Assume that (1.25) is infinite and Tail R ∞ ( ψ, { z } , ≥ , υ ) also holds. Then for all x, y ∈ M , theleft hand side of (1.26) is infinite, P y -a.s. Moreover, if we also have lim sup r →∞ ψ ( r ) /φ ( r ) = ∞ ,then any Ψ ∈ M + satisfying (1.21) is factorized as Ψ( t ) = f ( t ) φ − ( t/ log log t ) log log t, (1.27) for a function f such that lim inf t →∞ f ( t ) < ∞ , lim sup t →∞ f ( t ) = ∞ . Remark 1.21. (i) (1.23) and (1.26) are also valid (with possibly different constants a x and b ∞ )with the numerator sup 0, one can verifythat Ψ ( t ) (cid:16) sup s ∈ (0 , t ∧ (1 / Ψ ( s ) =: Ψ ( t ) ∈ M + for t ∈ (0 , / as the denominator instead of Ψ .However, for brevity, we simply used the function Ψ in (1.23), instead of Ψ . Remark 1.22. In this remark, we only write for the condition VRD and clearly, similar resultshold under other conditions Ch, Tail, SP, E and NDL.(i) For any nonempty set S ⊂ M and a point z ∈ M , we have d S ( x ) ≤ d { z } ( x ) + δ M \S ( z ) ≤ (1 + δ M \S ( z )) d { z } ( x ) , x ∈ M. Thus, VRD R ∞ ( S , υ ) implies VRD R (cid:48)∞ ( { z } , υ ) with R (cid:48)∞ = (1 + δ M \S ( z )) υ R ∞ ∈ [ R ∞ , ∞ ). Hence,the assumption VRD R ∞ ( { z } , υ ) in Theorems 1.15, 1.18 and 1.20 is the weakest one among theconditions of the form VRD R ∞ ( S , υ ) and it is appreciable to random conductance models with longrange jumps.(ii) VRD ∞ ( M ) implies VRD R ∞ ( S , υ ) for any S ⊂ M , R ∞ ∈ [1 , ∞ ) and υ ∈ (0 , 1) since (1.4) holdsfor all x ∈ M and 0 < s ≤ r .(iii) Since d M ( x ) ≡ 1, the condition VRD R ∞ ( M, υ ) is independent of υ .To prove of Corollary 1.16 and Theorems 1.18(ii) and 1.20(i), we need a proper zero-one law.Note that Corollary 1.16 requires the weakest assumptions among those three statements. We alsonote that unless R ∞ = 0, there is no assumption near zero in Corollary 1.16. Hence, under thesetting of Corollary 1.16, it is not possible to prove the continuity for parabolic functions in M .However, under that setting, we are able to establish an oscillation result of parabolic functions forlarge distances in Proposition 3.15. Then using this result, we show that a zero-one law holds forshift-invariant events without assuming that M is connected. Cf. [8, Theorem 8.4], [11, Proposition2.3] and [63, Theorem 2.10].The rest of the paper is organized as follows. In Section 2, we apply our main theorems to twoclasses of Markov processes: subordinated processes and symmetric Hunt processes. The proofs ofassertions in Section 2 are given in Appendix. Section 3 consists of preliminary results. In Sections 4and 5, we prove our liminf and limsup LILs, respectively. Proofs given Section 5 are the most delicatepart of the paper. We give some further examples in Section 6 including random conductance modelswith stable-like jumps, which was studied in [19] and [20]. Notations : Throughout this paper (except Section 6 and Appendix), the constants ¯ R, R , R ∈ (0 , ∞ ], κ ∈ (1 , ∞ ), R , R , γ , γ , c L , c U , κ ∈ (0 , ∞ ) will remain the same. Constants R ∈ (0 , ∞ ] and R ∞ ∈ [0 , ∞ ) may differ in each statement. We use same fixed positive real constants d , d , β , β , β , a ∗ , a ∗ , η , C L , C U , C (cid:48) U and C i , i = 0 , , , ... on conditions and statements at zeroand infinity, respectively. On the other hand, lower case letters ε , δ , c , b ∞ , a x , a x,i , a i , b i and c i , i = 0 , , , ... denote positive real constants and are fixed in each statement and proof, and thelabeling of these constants starts anew in each proof. We use the symbol “:=” to denote a definition,which is read as “is defined to be.” Recall that a ∧ b := min { a, b } and a ∨ b := max { a, b } . We set (cid:100) a (cid:101) := min { n ∈ Z : n ≥ a } and diag := { ( x, x ) : x ∈ M } , and denote by A the closure of A . Weextend a function f defined on M to M ∂ by setting f ( ∂ ) = 0. The notation f ( x ) (cid:16) g ( x ) means thatthere exist constants c ≥ c > c g ( x ) ≤ f ( x ) ≤ c g ( x ) for a specified range of x . For D ⊂ M , denote by C c ( D ) the space of all continuous functions with compact support in D .2. LILs for subordinated processes and symmetric jump processes Recall that ( M, d ) is a locally compact separable metric space, and µ is a positive Radon measureon M with full support. Let ¯ R := sup y,z ∈ M d ( y, z ). Throughout this section and Appendices A and B, we assume that VRD ¯ R ( M ) and Ch ¯ R ( M ) hold. In particular, since VRD ¯ R ( M ) holds, there exists a constant (cid:96) > V ( x, r ) ≥ V ( x, r/(cid:96) ) for all x ∈ M, r ∈ (0 , ¯ R ) . (2.1)Let F be an increasing continuous function on (0 , ∞ ) such thatL( F, γ , c L ) and U( F, γ , c U ) hold with some constants γ ≥ γ > 1. (2.2) Throughout this section and Appendices A and B, we also assume that there is a conservativeHunt process Z = (Ω , F , F t , Z t , t ≥ P x , x ∈ M ) on M which admits a heat kernel q ( t, x, y ) (withrespect to µ ) enjoying the following estimates: There exist constants R ∈ (0 , ¯ R ] , c , c , c > suchthat for all t ∈ (0 , F ( R )) and x, y ∈ M , c V ( x, F − ( t )) { F ( d ( x,y )) ≤ t } ≤ q ( t, x, y ) ≤ c V ( x, F − ( t )) exp (cid:0) − c F ( d ( x, y ) , t ) (cid:1) , (2.3) where the function F is defined as F ( r, t ) := sup s> (cid:16) rs − tF ( s ) (cid:17) . (2.4)The above function F is widely used in heat kernel estimates for diffusions on metric measurespaces including the Sierpinski gasket or carpet, nested fractals and affine nested fractals. See[7, 8, 10, 49, 52, 85]. Note that if F ( r ) = r γ for some γ > 1, then F ( t, r ) = c γ ( r γ /t ) / ( γ − for aconstant c γ > Z may not be µ -symmetric. For example, Z can be a Brownianmotion with drift on R d , which has ∆ + p · ∇ as the infinitesimal generator where the function p belongs to some suitable Kato class (see, e.g. [64, 91]). In this case, µ is the Lebesgue measure so V ( x, r ) = ω d r d for a constant ω d > F ( r ) = r .2.1. General subordinated processes. Let S = ( S t ) t ≥ be a subordinator on the probabilityspace (Ω , F , P ) independent of Z . Here, a subordinator is a nonnegative L´evy process on R suchthat S = 0, P -a.s. It is known that there exist a unique constant b ≥ ν on(0 , ∞ ), called the L´evy measure of S , satisfying (cid:82) (0 , ∞ ) (1 ∧ u ) ν ( du ) < ∞ such that − t − log E [ e − λS t ] =: φ ( λ ) = bλ + φ ( λ ) = bλ + (cid:90) (0 , ∞ ) (1 − e − λu ) ν ( du ) for all t > , λ ≥ . The function φ is called the Laplace exponent of the subordinator S . See [13]. In this subsection and Appendix A, we always assume either b (cid:54) = 0 or ν ((0 , ∞ )) = ∞ so that S is not a compound Poisson process. ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 11 Our goal in this subsection is to establish the LILs for a subordinated process X t := Z S t . It isknown that X is a Hunt process on M with the heat kernel p ( t, x, y ) := p M ( t, x, y ) and the L´evymeasure J ( x, dy ) = J ( x, y ) µ ( dy ) given by (see [16, p. 67, 73–75]) p ( t, x, y ) = (cid:90) ∞ q ( s, x, y ) P ( S t ∈ ds ) and J ( x, y ) = (cid:90) ∞ q ( s, x, y ) ν ( ds ) . (2.5)Following [56], let H ( λ ) := φ ( λ ) − λφ (cid:48) ( λ ), which is nonnegative and increasing. Denote by w ( r ) = ν (( r, ∞ )) the tail of the L´evy measure of S , and defineΦ( r ) := 1 φ ( F ( r ) − ) , Θ( r ) := 1 H ( F ( r ) − ) and Π( r ) := 2 ew ( F ( r )) for r > . (2.6)It is easy to verify that Φ , Θ are increasing, and Π is nondecreasing. Since S is not a compoundPoisson process, lim r → Φ( r ) = 0. Moreover, it is known that U( φ , , 1) and U( H , , 1) hold. See,e.g. [73, Lemma 2.1(a)]. Hence, since U( F, γ , c U ) holds, we have thatU(Φ , γ , c U ) and U(Θ , γ , c U ) hold. (2.7)We also have that, in view of [73, Lemma 2.6],Φ( r ) ≤ φ ( F ( r ) − ) − ≤ Θ( r ) ≤ Π( r ) for all r > . (2.8)Now, we state the liminf and limsup LILs for X t = Z S t . The proofs of next two theorems aregiven in Appendix A. Theorem 2.1. (Liminf LIL) (i) There exist constants a ≥ a > such that for all x ∈ M , thereexists a x ∈ [ a , a ] satisfying (1.14) with φ = Φ .(ii) Suppose that R = ∞ . Then there exist constants a ≥ a > such that for all x, y ∈ M , (1.16) holds with φ = Φ . Moreover, if L ( φ , α , c ) also holds, then there exists a constant b ∞ ∈ (0 , ∞ ) such that for all x, y ∈ M , (1.17) holds with φ = Φ . Recall that L( F, γ , c L ) holds with some constant γ > Theorem 2.2. (Limsup LILs at zero) (i) Suppose that lim sup s →∞ φ ( s ) /H ( s ) < ∞ (or, equiv-alently, lim sup r → φ ( r − ) /w ( r ) < ∞ ). Then for every Ψ ∈ M + and x ∈ M , (1.18) holds with φ = Φ .(ii) Suppose that lim sup s →∞ φ ( s ) /H ( s ) = ∞ . Then the following two statements are true.(a) Assume that L ( φ , α , c ) and U F ( R ) ( w − , α , c ) hold with R > . Then we can find afunction Ψ ∈ M + and constants a ≥ a > such that for all x ∈ M , Then we can find a function Ψ ∈ M + and constants a ≥ a > such that for all x ∈ M , Then there exist a function Ψ ∈ M + and constants a ≥ a > such that for all x ∈ M , (1.19) holds. Moreover, if α ≥ /γ , thenTheorem 1.19 holds with φ = Φ and ψ = Π .(b) Assume that L ( φ , α , c ) holds with α ≥ /γ . If the integral (1.22) is finite with φ = Φ and ψ = Θ , then there exist constants a ≥ a > such that for all x ∈ M , there exists a x ∈ [ a , a ] satisfying (1.23) with φ = Φ . Theorem 2.3. (Limsup LILs at infinity) Assume that R = ∞ .(i) Suppose that lim sup s → φ ( s ) /H ( s ) < ∞ (or, equivalently, lim sup r →∞ φ ( r − ) /w ( r ) < ∞ ).Then for every Ψ ∈ M + , (1.20) holds with φ = Φ .(ii) Suppose that lim sup s → φ ( s ) /H ( s ) = ∞ . Then the following two statements are true.(a) Assume that L ( φ , α , c ) and U F ( R ) ( w − , α , c ) hold with R > . Then there exist afunction Ψ ∈ M + and constants b , b ∈ (0 , ∞ ) such that for all x, y ∈ M , (1.21) holds. Moreover,if L ( φ , α , c ) holds with α ≥ /γ , then Theorem 1.20 holds with φ = Φ and ψ = Π .(b) Assume that L ( φ , α , c ) holds with α ≥ /γ . If the integral (1.25) is finite with φ = Φ and ψ = Θ , then there exists a constant b ∞ ∈ (0 , ∞ ) such that for all x, y ∈ M , (1.26) holds with φ = Φ . Remark 2.4. (i) According to [27, Remark 1.3(1)], for all α > 0, we haveU F ( R ) ( w − , α, c ) (resp. U F ( R ) ( w − , α, c )) ⇒ U ( φ , α ∧ , c ) (resp. U ( φ , α ∧ , c )). (2.9)Moreover, if α < 1, then there exists r > F ( r ) ( w − , α, c ) (resp. U F ( r ) ( w − , α, c )) ⇐ U ( φ , α, c ) (resp. U ( φ , α, c )).We note that the constant α in (2.9) can be larger than 1. By imposing a weak scaling property on w instead of φ , our results cover not only mixed α -stable-like subordinators but also a large classof subordinators whose Laplace exponent is regularly varying at infinity of index 1.Below, we give some concrete LILs for the subordinated process X t := Z S t . Recall that S t is thesubordinator with the Laplace exponent φ . In the remainder of this subsection, we assume that F ( r ) = r γ for some γ > Example 2.5. (Non-vanishing diffusion term) Suppose that φ ( λ ) = bλ + φ ( λ ) for b > 0. Thensince lim λ →∞ φ ( λ ) /λ = 0, we have that φ ( r − ) (cid:16) br − for r ∈ (0 , r → φ ( r − ) /w ( r ) ≥ b (2 e ) − lim r → r − /φ ( r − ) = ∞ . Moreover, by the change of the variables u = r − γ (log | log r | ) − γ and Tonelli’s theorem, since γ > 1, we see that (cid:90) Φ( r ) log | log r | r Θ( r log | log r | ) dr = (cid:90) H ( r − γ (log | log r | ) − γ ) log | log r | rφ ( r − γ ) dr ≤ c (cid:90) r γ − H ( r − γ (log | log r | ) − γ ) log | log r | dr ≤ c (cid:90) ∞ u − H ( u )(log log u ) − γ du (2.10)= c (cid:90) ∞ (cid:90) ∞ u − (1 − e − us − use − us )(log log u ) − γ duν ( ds ) ≤ c (cid:90) ∞ (cid:90) ∞ u − (( us ) ∧ duν ( ds ) ≤ c (cid:90) ∞ ( s ∧ ν ( ds ) < ∞ . In the third inequality above, we used the fact that 1 − e − λ − λe − λ ≤ λ ∧ λ ≥ 0. Therefore,since γ > 1, according to Theorems 2.1(i) and 2.2(ii-b), there exist constants a ≥ a > x ∈ M , there are constants a x, , a x, ∈ [ a , a ] satisfyinglim inf t → sup 0, we see that the underlying process Z t satisfies the aboveLILs. We remark here that limsup LIL of d ( x, Z t ) at zero is already studied in [10, Theorem 4.7]when Z is µ -symmetric.On the other hand, note that if φ (cid:54)≡ 0, then lim λ → φ ( λ ) /λ ∈ (0 , ∞ ] and hence φ ( λ ) (cid:16) φ ( λ )for all λ ∈ (0 , (cid:101) S t be a subordinator without drift having the Laplace exponent φ ( λ ). Thenwe can see that the liminf LIL at infinity concerning sup Let φ ( λ ) = (log(1 + λ α )) δ for α, δ ∈ (0 , αδ < 1. Here, φ is indeed the Laplace exponent of a subordinator accordingto [81, Theorem 5.2, Proposition 7.13 and Example 16.4.26]. A prototype of such X t = Z S t is ageometric 2 α -stable process on R d (0 < α < R d with the characteristicexponent log(1 + | ξ | α ). In view of [27, Lemma 2.1(iii) and (iv)], we have (cid:0) log(1 + r − α ) (cid:1) δ = φ ( r − ) (cid:16) w ( r ) (cid:16) (cid:40) | log r | δ , for r ∈ (0 , e − ) ,r − αδ , for r ∈ ( e, ∞ ) . (2.12) ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 13 We first study the LILs at zero. According to (2.12) and Theorems 2.1(i) and 2.2(i), there existconstants a ≥ a > x ∈ M , there exists a x ∈ [ a , a ] satisfyinglim inf t → | log sup 0. Thus, byTheorems 2.1(ii) and 2.3(i), we deduce that the liminf LIL (1.17) and limsup LIL (1.20) hold with φ ( r ) = r γαδ .Let T t be a αδ -stable subordinator, that is, a subordinator with the Laplace exponent λ αδ . Byusing Theorems 2.1(ii) and 2.3(i) again, one can see that a subordinated process Z T t also enjoys theliminf LIL (1.17) and limsup LIL (1.20) with φ ( r ) = r γαδ . However, we show in the below that if δ < 1, then estimates on the heat kernel p ( t, x, y ) of X t are different from ones on the heat kernel (cid:101) p ( t, x, y ) of Z T t , even for large t . Precisely, we will see that p ( t, x, x ) = ∞ for all t > x ∈ M ,but (cid:101) p ( t, x, y ) ≤ C/V ( x, t / ( γαδ ) ) for all t > x, y ∈ M for a constant C > (cid:101) p ( t, x, y ) from [21, Example 6.1]. On the other hand,observe that φ (cid:48) ( λ ) (cid:16) λ − (log λ ) δ − for all λ > e so that ( φ (cid:48) ) − ( r ) (cid:16) r − | log r | δ − and ( H ◦ ( φ (cid:48) ) − )( r ) (cid:16) | log r | δ for r ∈ (0 , e − ). Thus, by (A.1) in Appendix, the integration by parts and [56,Lemma 5.2(ii)], since VRD ¯ R ( M ) holds, we get that for all t > x ∈ M , p ( t, x, x ) ≥ (cid:90) ∞ q B ( x, ( s, x, x ) P ( S t ∈ ds ) ≥ c lim sup ε → (cid:90) ε V ( x, s /γ ) − P ( S t ∈ ds ) ≥ c lim sup ε → ε − c P ( S t ≤ ε ) ≥ c lim sup ε → exp (cid:0) c | log ε | − c t | log( ε/t ) | δ (cid:1) = ∞ . In particular, we note that the above calculation shows that NDL c ( r γαδ , M ) for X may not besharp (which holds for some c > X .We mention that our results also cover the cases when φ ( λ ) = φ ( λ ) = ( φ lg ◦ ... ◦ φ lg )( λ ), where φ lg ( λ ) := log(1 + λ ), for arbitrary finite number of compositions. (These functions are the Laplaceexponent of a subordinator according to [81, Theorem 5.2, Corollary 7.9 and Example 16.4.26].) (cid:50) Example 2.7. (Jump processes with mixed polynomial growths) In this example, we workwith pure jump processes enjoying the limsup LIL of type (1.19) (or (1.21)). Throughout thisexample, we assume b = 0 so that φ = φ . Then according to [27, Lemma 2.1],Φ( r ) = φ ( r − γ ) − (cid:16) er γ (cid:82) r γ w ( s ) ds = r γ γ (cid:82) r s γ − Π( s ) − ds for r > , (2.15)where comparison constants are independent of S .(i) limsup LIL at zero. Here, we give examples of w which make the integral (1.22) be finite orinfinite with φ = Φ and ψ = Π. Based on that, we study a small time limsup LIL for X .(a) Let p > w ( r ) (cid:16) r − | log r | − p for r ∈ (0 , e − ). Here, p must be larger than 1because it should holds that (cid:82) w ( s ) ds < ∞ . Then according to (2.15), we see that for r, t ∈ (0 , e − ),Π( r ) (cid:16) r γ | log r | p and Φ( r ) (cid:16) r γ | log r | p − (2.16)and Φ − ( t/ log | log t | ) log | log t | (cid:16) t /γ | log t | (1 − p ) /γ (log | log t | ) ( γ − /γ =: Ψ ( t ) . (2.17)Since lim r → Π( r ) / Φ( r ) = ∞ , in view of Theorem 2.2(ii-a) and (2.6), we can deduce that thereexists Ψ ∈ M + satisfying (1.19). Moreover, one can see from (2.16) that the integral (1.22) is finiteif γ > 2, and is infinite if γ ≤ 2. Hence, by Theorem 2.2(ii-a) and (2.17), if γ > 2, then there existconstants a ≥ a > x ∈ M , there is a x ∈ [ a , a ] satisfyinglim sup t → d ( x, X t ) t /γ | log t | (1 − p ) /γ (log | log t | ) ( γ − /γ = a x , P x -a.s. , (2.18)and if γ ≤ 2, then for all x ∈ M , P x -a.s., the left hand side of (2.18) is infinite. Thus, for example,if M is a Sierpinski gasket or carpet, then the left hand side of (2.18) is a deterministic constanta.s. while it is infinite a.s. if M is a Euclidean space.Now, we give an example of a proper rate function Ψ satisfying (1.19) in case of γ ≤ δ > t ) := ∞ (cid:88) n =4 (log n ) exp (cid:0) − γ − n (log n ) − γ + δ (cid:1) · ( t n +1 ,t n ] ( t ) (cid:16) ∞ (cid:88) n =4 Ψ ( t n ) · ( t n +1 ,t n ] ( t )where t n := n p − (log n ) (2 − γ + δ ) p + γ − δ − exp (cid:0) − n (log n ) − γ + δ (cid:1) . For the above Ψ and t n , one can see that since 2 − γ + δ > n →∞ Ψ(2 t n +1 ) / Ψ (2 t n +1 ) (cid:16) lim n →∞ Ψ ( t n ) / Ψ ( t n +1 ) ≥ c lim n →∞ ( t n /t n +1 ) / (2 γ ) = ∞ . Thus, the above Ψ can be factorized as (1.24) with φ = Φ.(b) Let p > w ( r ) (cid:16) r − | log r | − (log | log r | ) − p for r ∈ (0 , e − ). (As in (a), p must be larger than 1 since (cid:82) w ( s ) ds < ∞ .) In this case, by (2.15), we see thatΠ( r ) (cid:16) r γ | log r | (log | log r | ) p and Φ( r ) (cid:16) r γ (log | log r | ) p − for r ∈ (0 , e − ) . (2.19)Since lim r → Π( r ) / Φ( r ), as in (a), there exists Ψ ∈ M + which satisfies (1.19). Moreover, we seefrom (2.19) that the integral (1.22) is always finite. According to Theorem 2.2(ii-a), it follows thatthere exist constants a ≥ a > x ∈ M , there is a x ∈ [ a , a ] satisfyinglim sup t → d ( x, X t ) t /γ (log | log t | ) ( γ − p ) /γ = a x , P x -a.s. (2.20)(ii) limsup LIL at infinity. Let k, q > w ( r ) (cid:16) r − (log r ) − exp (cid:0) k (log log r ) q (cid:1) for r ∈ ( e , ∞ ). Then we see that for r, t large enough,Π( r ) (cid:16) r γ (log r ) exp (cid:0) − k (log log r γ ) q (cid:1) and Φ( r ) (cid:16) r γ (log log r ) q − exp (cid:0) − k (log log r γ ) q (cid:1) (2.21) ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 15 and Φ − ( t/ log log t ) log log t (cid:16) t /γ (log log t ) ( γ − q ) /γ exp (cid:0) γ − k (log log t ) q (cid:1) =: Ψ ∞ ( t ) . (2.22)Since lim r →∞ Π( r ) / Φ( r ) = ∞ , by Theorem 2.3(ii-a) and (2.6), we can see that there existsΨ ∈ M + satisfying (1.21). Besides, in view of (2.21), one can check that the integral (1.25) is finiteif q ∈ (0 , γ − q ≥ γ − 1. By Theorem 2.3(ii-a) and (2.22), it follows that if q ∈ (0 , γ − b ∞ ∈ (0 , ∞ ) such thatlim sup t →∞ d ( x, X t ) t /γ (log log t ) ( γ − q ) /γ exp (cid:0) γ − k (log log t ) q (cid:1) = b ∞ for all x, y ∈ M, P y -a.s.,and if q ≥ γ − 1, then the left hand side of the above equality is infinite for all x, y ∈ M , P y -a.s.Moreover, in case of q ≥ γ − 1, one can find an example of a proper rate function Ψ satisfying (1.21):Choose any δ > t ) := ∞ (cid:88) n =5 (log n ) exp (cid:0) γ − n (log n ) q − γ +1+ δ (cid:1) · ( t n − ,t n ] ( t ) (cid:16) ∞ (cid:88) n =5 Ψ ∞ ( t n ) · ( t n − ,t n ] ( t ) (2.23)where t n := (log n ) q exp (cid:0) n (log n ) q − γ +1+ δ − k (cid:0) log n + ( q − γ + 1 + δ ) log log n (cid:1) q (cid:1) . With the above Ψ and t n , one can see that lim n →∞ Ψ(2 t n − ) / Ψ ∞ (2 t n − ) = ∞ so that Ψ admits afactorization (1.27) with φ = Φ.By putting q = 1 and k = 1 − p , we obtain the following analogous result to (2.18): Let p < w ( r ) (cid:16) r − (log r ) − p for r ∈ ( e , ∞ ). If γ > 2, then there exists a constant b (cid:48)∞ ∈ (0 , ∞ )such that lim sup t →∞ d ( x, X t ) t /γ (log t ) (1 − p ) /γ (log log t ) ( γ − /γ = b (cid:48)∞ for all x, y ∈ M, P y -a.s., (2.24)and if γ ≤ 2, then the left hand side in (2.24) is infinite for all x, y ∈ M , P y -a.s. and an example ofΨ satisfying (1.21) is given as (2.23) (with q = 1, k = 1 − p and any δ > (cid:50) Symmetric Hunt processes. Recall that we have assumed that the conditions VRD ¯ R ( M )and Ch ¯ R ( M ) hold, and there exists a conservative Hunt process Z in ( M, d, µ ) having a heat kernel q ( t, x, y ) which satisfies (2.3). In this subsection, we also assume that the underlying process Z is µ -symmetric and associated with a regular strongly local Dirichlet form ( E Z , F Z ) on L ( M ; µ ).See [43] for the definitions of a regular Dirichlet form and the strongly local property.Let ( E X , F X ) be a regular Dirichlet form on L ( M ; µ ) having the following expression: Thereexist a constant Λ ≥ J X ( dx, dy ) = J X ( x, dy ) µ ( dx )on M × M \ diag such that E X ( f, g ) = Λ E X, ( c ) ( f, g ) + (cid:90) M × M \ diag ( f ( x ) − f ( y ))( g ( x ) − g ( y )) J X ( x, dy ) µ ( dx ) , f, g ∈ F X , F X = (cid:8) f ∈ C c ( M ) : E X ( f, f ) < ∞ (cid:9) E X , (2.25)where E X, ( c ) ( f, g ) is the strongly local part of ( E X , F X ) (that is, E X, ( c ) ( f, g ) = 0 for all f, g ∈ F X such that ( f − c ) g = 0 µ -a.e. on M for some c ∈ R ) and E X ( f, f ) := E X ( f, f ) + (cid:107) f (cid:107) L ( M ; µ ) .See [43, Theorem 3.2.1] for a general representation theorem on regular Dirichlet forms.Since ( E X , F X ) is regular, it is known that there exists an associated µ -symmetric Hunt process X = ( X t , t ≥ P x , x ∈ M \ N ) where N is a properly exceptional set in the sense that N isnearly Borel, µ ( N ) = 0 and M ∂ \ N is X -invariant. This Hunt process is unique up to a properlyexceptional set. See [43, Chapter 7 and Theorem 4.2.8] for details. We fix X and N , and write M := M \ N . Since ( E Z , F Z ) is a regular Dirichlet form, it is well known that for any bounded f ∈ F Z , thereexists a unique positive Radon measure Γ Z ( f, f ) on M such that (cid:82) M gd Γ Z ( f, f ) = E Z ( f, f g ) − E Z ( f , g ) for every g ∈ F Z ∩ C c ( M ). See [18, Section 3]. One can see that the measure Γ Z ( f, f )can be uniquely extended to any f ∈ F Z as the nondecreasing limit of Γ Z ( f n , f n ), where f n :=(( − n ) ∨ f ) ∧ n . The measure Γ Z ( f, f ) is called the energy measure (which is also called the carr´edu champ ) of f for E Z . We write Γ X, ( c ) ( f, f ) for the energy measure of f ∈ F X for E X, ( c ) .We consider the following assumption with a convention that B ( x, ∞ ) = M . Assumption L. There exist an open set U ⊂ M , constants R ∈ (0 , ∞ ], κ ∈ (0 , 1) and anincreasing function ψ on (0 , R ) such that the followings hold.(L1) L R ( ψ , β , c ) and U R ( ψ , β , c ) hold with constants β ≥ β > (cid:82) ψ ( s ) − dF ( s ) < ∞ ;(L2) There exist a function J X ( · , · ) on U × M and a constant c ≥ x ∈ U , J X ( x, dy ) = J X ( x, y ) µ ( dy ) in B (cid:0) x, R ∧ ( κ δ U ( x )) (cid:1) and c − V ( x, d ( x, y )) ψ ( d ( x, y )) ≤ J X ( x, y ) ≤ c V ( x, d ( x, y )) ψ ( d ( x, y )) , y ∈ B (cid:0) x, R ∧ ( κ δ U ( x )) (cid:1) ;(L3) There exists a constant c > x ∈ U , J X (cid:0) x, M ∂ \ B (cid:0) x, R ∧ ( κ δ U ( x )) (cid:1)(cid:1) ≤ c ψ ( R ∧ δ U ( x )) ;(L4) If Λ > 0, then F X ∩ C c ( U ) = F Z ∩ C c ( U ), and d Γ X, ( c ) ( f, f ) (cid:16) d Γ Z ( f, f ) in U for all f ∈ F X .Under Assumption L, we define (cf. (2.15) and [3, (2.20)])Φ ( r ) := F ( r ) (cid:82) r ψ ( s ) − dF ( s ) + Λ , r ∈ (0 , R ) , (2.26)which is well-defined due to (L1). Note that Φ ( r ) ≤ ψ ( r ) F ( r ) / ( (cid:82) r dF ( s )) = ψ ( r ) for r ∈ (0 , R ).Now, we establish the following liminf and limsup LILs for X . The proof is given in Appendix B. Theorem 2.8. Suppose that Assumption L holds.(i) The liminf and limsup LILs at zero given in Theorems 1.13, 1.17 and 1.19 hold for all x ∈ U ∩ M with φ = Φ and ψ = ψ .(ii) Assume that U = M and R = R = ∞ . Then the liminf and limsup LILs at infinity given inCorollary 1.16, Theorems 1.18 and 1.20 hold for all x ∈ M with φ = Φ and ψ = ψ . As consequences of Theorem 2.8, we provide the following two explicit examples, which are similarto Examples 2.5 and 2.7(i). In the following two examples, we assume that F ( r ) = r γ for some γ > F is the function in (2.3)), X is a symmetric Hunt process associated with the Dirichletform (2.25), and Assumption L holds. Example 2.9. Suppose that Λ > E X , F X ) has non-zero strongly localpart. Then one can see that Φ ( r ) (cid:16) Λ − r γ for r ∈ (0 , (cid:90) Φ ( r ) log | log r | rψ ( r log | log r | ) dr ≤ c (cid:90) s γ − (log | log s | ) − γ ψ ( s ) ds ≤ cγ − (cid:90) dF ( s ) ψ ( s ) < ∞ . Thus, by Theorem 2.8, we deduce that the LILs at zero (2.11) holds for all x ∈ U ∩ M with b = Λ. (cid:50) ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 17 Example 2.10. Suppose that Λ = 0, and Assumption L holds with R = e − .(i) Assume that (L3) holds with ψ ( r ) = r γ | log r | p . Then one can see that Φ ( r ) (cid:16) r γ | log r | p − for r ∈ (0 , e − ) (cf. (2.16)). Hence, according to Theorem 2.8(i), by similar arguments to the ones givenin Example 2.7(i-a), we deduce the following liminf and limsup LILs hold: There exist constants c ≥ c > x ∈ U ∩ M there is a constant c x ∈ [ c , c ] satisfyinglim inf t → d ( x, X t ) t /γ | log t | (1 − p ) /γ (log | log t | ) − /γ = c x , P x -a.s.Moreover, if γ > 2, then the limsup LIL at zero (2.18) holds for all x ∈ U , and if γ ≤ 2, then theleft hand side of (2.18) is infinite for all x ∈ U ∩ M , P x -a.s.(ii) Assume that (L3) holds with ψ ( r ) = r γ | log r | (log | log r | ) p . Then by Theorem 2.8(i) and similararguments to the ones given in Example 2.7(i-b), we deduce that there exist constants c ≥ c > x ∈ U ∩ M there is a constant c (cid:48) x ∈ [ c , c ] satisfyinglim inf t → d ( x, X t ) t /γ (log | log t | ) − p/γ = c (cid:48) x , P x -a.s.and the limsup LIL at zero (2.20) holds for all x ∈ U ∩ M . (cid:50) Analogously, by Theorem 2.8(ii), one can construct concrete examples of the LILs at infinity forsymmetric pure-jump process from Example 2.7(ii).3. Preliminary Auxiliary functions Υ , Υ , ϑ and ϑ . In this subsection, we introduce some auxiliaryfunctions which will be used in tail probability estimates on the first exit times from open balls(see Propositions 3.10 and 3.13). Recall that we always assume (1.8). Hence, φ − ( t ) ≥ ψ − ( t ) forall t > 0. DefineΥ ( t ) := min (cid:8) tρφ ( ρ ) : ψ − ( t ) ≤ ρ ≤ φ − ( t ) (cid:9) and Υ ( t ) := max (cid:8) tρφ ( ρ ) : ψ − ( t ) ≤ ρ ≤ φ − ( t ) (cid:9) . Then one can easily see that for all 0 < t < lim r →∞ φ ( r ), it holdsΥ ( t ) ≤ tφ − ( t ) φ ( φ − ( t )) = φ − ( t ) ≤ Υ ( t ) . (3.1)We also define functions ϑ , ϑ : (0 , ∞ ) × [0 , ∞ ) → (0 , ∞ ) as (cf. [28, (1.13)]) ϑ ( t, r ) := φ − ( t ) , if r ∈ [0 , Υ ( t )) , min (cid:110) ρ ∈ [ ψ − ( t ) , φ − ( t )] : tρφ ( ρ ) ≤ r (cid:111) , if r ∈ [Υ ( t ) , Υ ( t )] ,ψ − ( t ) , if r ∈ (Υ ( t ) , ∞ ) (3.2)and ϑ ( t, r ) := φ − ( t ) , if r ∈ [0 , Υ ( t )) , max (cid:110) ρ ∈ [ ψ − ( t ) , φ − ( t )] : tρφ ( ρ ) ≥ r (cid:111) , if r ∈ [Υ ( t ) , Υ ( t )] ,ψ − ( t ) , if r ∈ (Υ ( t ) , ∞ ) . See Figure 1. Since φ and ψ are continuous, the above ϑ and ϑ are well-defined. Note that foreach fixed t > r (cid:55)→ ϑ ( t, r ) and r (cid:55)→ ϑ ( t, r ) are nonincreasing. Intuitively, Υ ( t ) represents themaximal distance reachable by a standard chaining argument at time t , and ϑ ( t, r ) and ϑ ( t, r )represent the number of minimal and maximal number of chains to reach the distance r , respectively. Lemma 3.1. For any ε ∈ (0 , , it holds that for all < t < lim r →∞ φ ( r ) and Υ ( t ) ≤ εr , εrϑ ( t, εr ) ≥ tφ ( ϑ ( t, εr )) . Figure 1. Auxiliary functions Proof. If εr ≤ Υ ( t ), then tϑ ( t, εr ) /φ ( ϑ ( t, εr )) ≤ εr and hence rϑ ( t, εr ) − tφ ( ϑ ( t, εr )) = 1 ϑ ( t, εr ) (cid:16) r − tϑ ( t, εr ) φ ( ϑ ( t, εr )) (cid:17) ≥ (1 − ε ) rϑ ( t, εr ) . Otherwise, if εr > Υ ( t ), then since tϑ ( t, εr ) /φ ( ϑ ( t, εr )) ≤ Υ ( t ), we also obtain rϑ ( t, εr ) − tφ ( ϑ ( t, εr )) = 1 ϑ ( t, εr ) (cid:16) r − tϑ ( t, εr ) φ ( ϑ ( t, εr )) (cid:17) ≥ r − Υ ( t ) ϑ ( t, εr ) > (1 − ε ) rϑ ( t, εr ) . (cid:50) Lemma 3.2. (i) Suppose that U R ( ψ, β , C (cid:48) U ) holds. Then for all k > and < ψ − ( t ) ≤ r < R , exp (cid:16) − krψ − ( t ) (cid:17) ≤ C (cid:48) U β β k − β tψ ( r ) . (3.3) (ii) Suppose that U R ∞ ( ψ, β , C (cid:48) U ) holds. Then (3.3) holds for all k > and R ∞ < ψ − ( t ) ≤ r . Proof. (i) Since e − x ≤ β β x − β for all x, β > 0, and U R ( ψ, β , C (cid:48) U ) holds, we obtainexp (cid:16) − krψ − ( t ) (cid:17) ≤ β β (cid:16) ψ − ( t ) kr (cid:17) β ≤ C (cid:48) U ( β /k ) β tψ ( r ) . (3.4)(ii) Since U R ∞ ( ψ, β , C (cid:48) U ) holds, (3.4) holds for all k > R ∞ < ψ − ( t ) ≤ r . (cid:50) Estimates on the first exit times from open balls. In this subsection, we study the firstexit time from open balls. The main results are Propositions 3.10 and 3.13. Note that a similarresult appears in [3, (3.25)] (see also [23, (3.3)]). However, in this paper, we not only give localversions (in the variable r ) of that result, but also remove an extra assumption that the local lowerscaling index of φ is strictly larger than 1 therein, by using auxiliary functions defined in subsection3.1 and adopting the ideas from the first and second named authors’ previous paper [28, Theorem3.10].We begin with a simple observation on the relation between two conditions SP and E. Lemma 3.3. (i) SP R ( φ, U ) with C and C implies E R ( φ, U ) with the same C and C .(ii) SP R ∞ ( φ, S , υ ) with C implies E R ∞ ( φ, S , υ ) with the same C . ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 19 Proof. (i) For all x ∈ U and 0 < r < R ∧ ( C δ U ( x )), by Markov inequality, we obtain E x [ τ B ( x,r ) ] ≥ φ ( C r ) P x ( τ B ( x,r ) ≥ φ ( C r )) ≥ a ∗ φ ( C r )and E x [ τ B ( x,r ) ] ≤ ∞ (cid:88) n =1 nφ ( C r ) P x (cid:0) τ B ( x,r ) ∈ [( n − φ ( C r ) , nφ ( C r )) (cid:1) ≤ φ ( C r ) ∞ (cid:88) n =1 n ( a ∗ ) n − = c φ ( C r ) . (ii) It can be proved by the same way as (i). (cid:50) In the following proposition, we let (cid:96) > r := η R / (6 (cid:96) ) where η ∈ (0 , 1) is the constant in NDL R ( φ, U ) (3.5)in the first statement, and r ∞ := (4 (cid:96)R ∞ ) / (1 − υ ) ∨ (2 R ∞ /η ) where υ, η ∈ (0 , 1) are the constants in NDL R ∞ ( φ, S , υ ) (3.6)in the latter statement. Note that r = ∞ if R = ∞ . Proposition 3.4. (i) Suppose that VRD R ( U ) , NDL R ( φ, U ) and U R ( φ, β , C U ) hold with C V , C .Then, SP r ( φ, U ) holds with C := η ( C V ∧ C ) / (5 (cid:96) ) .(ii) Suppose that VRD R ∞ ( S , υ ) , NDL R ∞ ( φ, S , υ ) and U R ∞ ( φ, β , C U ) hold. Then SP r ∞ ( φ, S , υ ) holds. Proof. We adopt the idea from the proof for [24, Proposition 3.5(ii)].(i) Choose any x ∈ U and 0 < r < r ∧ ( C δ U ( x )). Let t r := φ (3 (cid:96)η − r ). Then φ − ( t r ) /η = 3 (cid:96)r/η 0, by (3.6) we have R ∞ d S ( w ) υ ≤ R ∞ ( d S ( x ) + 2 (cid:96)r ) υ ≤ R ∞ ( d S ( x ) υ + (2 (cid:96)r ) υ ) < (2 − + 2 (cid:96)R ∞ r υ − ∞ ) r ≤ r. Hence, by similar arguments, we can deduce that (3.9) holds (with possibly different constants c , c ). Besides, we can obtain (3.10) from NDL R ∞ ( φ, S , υ ), and there exists c > x and r such that V ( x, η r ) ≥ c V ( x, r ) because VRD R ∞ ( S , υ ) holds. Therefore, we can follow(3.11) line by line, and finish the proof. (cid:50) Recall (1.13). Here, we give three families of conditions. Assumption 3.5. E R ( φ, U ), Tail R ( φ, U, ≤ ) and U R ( φ, β , C U ) hold (with C ∈ (0 , R ∈ (0 , ∞ ] and an open set U ⊂ M . Assumption 3.6. E R ∞ ( φ, S , υ ), Tail R ∞ ( φ, S , ≤ , υ ) and U R ∞ ( φ, β , C U ) hold for some R ∞ ∈ [1 , ∞ ), υ ∈ (0 , 1) and nonempty S ⊂ M . Assumption 3.7. VRD R ∞ ( S , υ ), NDL R ∞ ( φ, S , υ ), Tail R ∞ ( φ, S , ≤ , υ ), L R ∞ ( φ, β , C L ) andU R ∞ ( φ, β , C U ) hold for some R ∞ ∈ [1 , ∞ ), υ ∈ (0 , 1) and nonempty S ⊂ M .Observe that Assumption 3.5 is assumed in our all LILs at zero (Theorems 1.13, 1.17, 1.19, andCorollary 1.14), and Assumption 3.6 (where S = { z } for some z ∈ M ) is assumed in our allLILs at infinity (Theorems 1.15, 1.18, 1.20, and Corollary 1.16). By Proposition 3.4(ii) and Lemma3.3(ii), if Assumption 3.7 holds, then Assumption 3.6 holds.For ρ > 0, let X ( ρ ) be a Borel standard Markov process on M with the ρ -truncated L´evy measure J ( ρ ) ( x, dy ) which is given by J ( ρ ) ( x, A ) := J ( x, A ∩ B ( x, ρ )) for every measurable set A ⊂ M .Then the process X ( ρ ) satisfies the following L´evy system: for any nonnegative Borel function F on M × M ∂ vanishing on the diagonal, z ∈ M and t > E z (cid:20) (cid:88) s ≤ t F ( X ( ρ ) s − , X ( ρ ) s ) (cid:21) = E z (cid:20)(cid:90) t (cid:90) M ∂ F ( X ( ρ ) s , y ) J ( ρ ) ( X ( ρ ) s , dy ) ds (cid:21) . By the Meyer’s construction (see [72], and also [9, Section 3]), the original process X can beconstructed from X ( ρ ) by attaching large jumps whose sizes are bigger than ρ .Our first goal is obtaining Proposition 3.10 below. For this, we prepare two lemmas. The mainstrategies of the proofs for the next two lemmas are similar to those for [21, Lemmas 4.20 and 4.21].Under Assumption 3.6, for each υ ∈ ( υ, R (cid:48)∞ = R (cid:48)∞ ( υ ) := (7 R ∞ ) υ / ( υ − υυ ) ∈ [7 R ∞ , ∞ ) . (3.12)For ρ > D ⊂ M , let τ ( ρ ) D := inf { t > X ( ρ ) t ∈ M ∂ \ D } . Lemma 3.8. (i) Suppose that Assumption 3.5 holds. Then, there exist constants δ ∈ (0 , and C ≥ such that for all x ∈ U , < ρ ≤ r < − (cid:0) R ∧ ( C δ U ( x )) (cid:1) and λ ≥ C /φ ( ρ ) , E x (cid:104) exp (cid:0) − λτ ( ρ ) B ( x,r ) (cid:1)(cid:105) ≤ − δ. (3.13) (ii) Suppose that Assumption 3.6 holds. Then, there exist constants δ ∈ (0 , and C ≥ such that (3.13) holds for any υ ∈ ( υ, , x ∈ M , r > R (cid:48)∞ d S ( x ) υ , x ∈ B ( x , r ) , r υ/υ ≤ ρ ≤ r and λ ≥ C /φ ( ρ ) , where R (cid:48)∞ = R (cid:48)∞ ( υ ) is defined by (3.12) . ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 21 Proof. (i) Choose any x ∈ U and 0 < r < − (cid:0) R ∧ ( C δ U ( x )) (cid:1) . Set B := B ( x, r ). Then we have C δ U ( z ) ≥ C ( δ U ( x ) − r ) > r for all z ∈ B. (3.14)Hence, since E R ( φ, U ) holds, by using the inequality τ B ≤ t + ( τ B − t )1 { τ B >t } and the Markovproperty, we get that for all t > c φ ( r ) ≤ E x τ B ≤ t + E x (cid:2) { τ B >t } E X t τ B (cid:3) ≤ t + P x ( τ B > t ) sup z ∈ B E z τ B ≤ t + P x ( τ B > t ) sup z ∈ B E z τ B ( z, r ) ≤ t + c P x ( τ B > t ) φ (2 r ) . Since U R ( φ, β , C U ) holds and φ is increasing, it follows that for c := c ( c β C U ) − ∈ (0 , P x ( τ B > t ) ≥ c φ ( r ) c φ (2 r ) − tc φ (2 r ) ≥ c − tc φ ( r ) for all t > . (3.15)We claim that there exists a constant c > x and r such that (cid:12)(cid:12)(cid:12) P x (cid:0) τ B > t (cid:1) − P x (cid:0) τ ( ρ ) B > t (cid:1)(cid:12)(cid:12)(cid:12) ≤ c tφ ( ρ ) for all ρ ∈ (0 , r ] , t > . (3.16)To prove (3.16), we need some preparations. Choose any ρ ∈ (0 , r ]. Let ξ be an exponential randomvariable with rate parameter 1 independent of X ( ρ ) . Define J t := (cid:82) t J (cid:0) X ( ρ ) s , M ∂ \ B ( X ( ρ ) s , ρ ) (cid:1) ds and T := inf { t ≥ J t ≥ ξ } . In view of the Meyer’s construction, we may identify T with T := inf { t > X t (cid:54) = X ( ρ ) t } , the first attached jump time for X ( ρ ) (see [9, Section 3.1]). SinceTail R ( φ, U, ≤ ) holds and X ( ρ ) s ∈ B for all 0 ≤ s < t ∧ τ ( ρ ) B , by (3.14), we have that P x (cid:0) T ≤ t ∧ τ ( ρ ) B (cid:1) = P x (cid:0) J t ∧ τ ( ρ ) B ≥ ξ (cid:1) ≤ P x (cid:0) c J tφ ( ρ ) ≥ ξ (cid:1) ≤ c J tφ ( ρ ) . (3.17)Besides, since X s = X ( ρ ) s for s < T , we have { t < T ∧ τ B } = { t < T ∧ τ ( ρ ) B } and { T ≤ t < τ B } ⊂{ T ≤ t ∧ τ ( ρ ) B } for all t > 0. Thus, we can see from (3.17) that for all t > (cid:12)(cid:12)(cid:12) P x (cid:0) t < τ B (cid:1) − P x (cid:0) t < τ ( ρ ) B (cid:1)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) P x (cid:0) T ≤ t < τ B (cid:1) − P x (cid:0) T ≤ t < τ ( ρ ) B (cid:1)(cid:12)(cid:12)(cid:12) ≤ P x (cid:0) T ≤ t < τ B (cid:1) + P x (cid:0) T ≤ t < τ ( ρ ) B (cid:1) ≤ P x (cid:0) T ≤ t ∧ τ ( ρ ) B (cid:1) ≤ c J tφ ( ρ ) . This yields (3.16) with c = 2 c J .Now, set δ = c / ∈ (0 , / 3) and C = ( c − + c ) δ − log( δ − ). In view of (3.15) and (3.16), bytaking t ρ = δφ ( ρ ) / ( c − + c ), we conclude that for all ρ ∈ (0 , r ] and λ ≥ C /φ ( ρ ), E x (cid:104) exp (cid:0) − λτ ( ρ ) B (cid:1)(cid:105) ≤ E x (cid:104) exp (cid:0) − λτ ( ρ ) B (cid:1) ; τ ( ρ ) B ≤ t ρ (cid:105) + e − λt ρ ≤ P x ( τ ( ρ ) B ≤ t ρ ) + e − λt ρ ≤ − c + ( c − + c ) t ρ φ ( ρ ) + exp (cid:16) − C t ρ φ ( ρ ) (cid:17) ≤ − c + 2 δ ≤ − δ. (3.18)Since δ and C are independent of x and r , this completes the proof.(ii) Choose any x ∈ M , r > R (cid:48)∞ d S ( x ) υ , x ∈ B ( x , r ) and 2 r υ/υ ≤ ρ ≤ r . Then bye:infty’ itholds that for any z ∈ B ( x, r ) ⊂ B ( x , r ), R ∞ d S ( z ) υ ≤ R ∞ d S ( x ) υ + (7 r ) υ R ∞ < ( R ∞ R (cid:48)− υ/υ ∞ + 7 υ R ∞ R (cid:48) υ − υ/υ ∞ ) r υ/υ < r υ/υ ≤ ρ. (3.19)Hence, since E R ∞ ( φ, S , υ ), Tail R ∞ ( φ, S , ≤ , υ ) and U R ∞ ( φ, β , C U ) hold, we see that (3.15) and (3.16)hold by following the proof for (i). By repeating the calculations (3.18), we obtain the result. (cid:50) Lemma 3.9. (i) Suppose that Assumption 3.5 holds. Let C be the constant in Lemma 3.8(i). Thereexist constants C > , C ∈ (0 , such that for all x ∈ U and < ρ ≤ r < − (cid:0) R ∧ ( C δ U ( x )) (cid:1) , E x (cid:20) exp (cid:16) − C φ ( ρ ) τ ( ρ ) B ( x,r ) (cid:17)(cid:21) ≤ C exp (cid:0) − C r/ρ (cid:1) . (3.20) (ii) Suppose that Assumption 3.6 holds. Let C be the constant in Lemma 3.8(ii). There existconstants C > , C ∈ (0 , such that (3.20) holds for any υ ∈ ( υ, , x ∈ M , r > R (cid:48)∞ d S ( x ) υ , x ∈ B ( x , r ) and r υ/υ ≤ ρ ≤ r where R (cid:48)∞ is defined by (3.12) . Proof. (i) Choose any x ∈ U and 0 < ρ ≤ r < − (cid:0) R ∧ ( C δ U ( x )) (cid:1) . If 6 ρ > r , then by taking C larger than e , we are done. Hence, we assume that 6 ρ ≤ r .Set λ ρ := C /φ ( ρ ), n ρ := (cid:100) r/ (6 ρ ) (cid:101) , τ := τ ( ρ ) B ( x,r ) , u ( y ) := E y [ e − λ ρ τ ] and m k := (cid:107) u (cid:107) L ∞ ( B ( x,kρ ); µ ) .Let δ ∈ (0 , 1) be the constant in Lemma 3.8(i). Fix any δ (cid:48) ∈ (0 , δ ). Then for each 1 ≤ k ≤ n ρ ,choose any x k ∈ B ( x, kρ ) such that (1 − δ (cid:48) ) m k ≤ u ( x k ) ≤ m k . Since (2 n ρ + 2) ρ < r/ ρ ≤ r , wesee that B ( x k , ρ ) ⊂ B ( x, ( k + 2) ρ ) ⊂ B ( x, r ) for all 1 ≤ k ≤ n ρ .Set τ k := τ ( ρ ) B ( x k ,ρ ) and v k ( y ) := E y [ e − λ ρ τ k ]. Note that 3 − C δ U ( y ) > − C ( δ U ( x ) − r ) > − r ≥ ρ for all y ∈ B ( x, r ). Hence, by Lemma 3.8(i), we obtain v k ( x k ) ≤ − δ . Then by the strong Markovproperty, it holds that for all 1 ≤ k ≤ n ρ − − δ (cid:48) ) m k ≤ u ( x k ) = E x k (cid:2) e − λ ρ τ ; τ k ≤ τ < ∞ (cid:3) = E x k (cid:2) e − λ ρ τ k e − λ ρ ( τ − τ k ) ; τ k ≤ τ < ∞ (cid:3) = E x k (cid:2) e − λ ρ τ k E X ( ρ ) τk ( e − λ ρ τ ) (cid:3) = E x k (cid:2) e − λ ρ τ k u ( X ( ρ ) τ k ) (cid:3) ≤ v k ( x k ) (cid:107) u (cid:107) L ∞ ( B ( x k , ρ ); µ ) ≤ v k ( x k ) m k +2 ≤ (1 − δ ) m k +2 . (3.21)In the first inequality above, we used the fact that X ( ρ ) τ k ∈ B ( x k , ρ ) since the jump size of X ( ρ ) cannot be larger than ρ . The second inequality above holds since B ( x k , ρ ) ⊂ B ( x, ( k + 2) ρ ).In the end, by (3.21), we conclude that E x (cid:104) exp (cid:16) − C φ ( ρ ) τ ( ρ ) B ( x,r ) (cid:17)(cid:105) = u ( x ) ≤ m ≤ (cid:16) − δ − δ (cid:48) (cid:17) n ρ − m n ρ − ≤ − δ (cid:48) − δ exp (cid:16) − r ρ log 1 − δ (cid:48) − δ (cid:17) . In the last inequality above, we used the fact that m n ρ − ≤ (cid:107) u (cid:107) L ∞ ( M ; µ ) ≤ 1. This proves (3.20).(ii) By following the proof for (i), using Lemma 3.8(ii), one can obtain (3.20). Note that since x ∈ B ( x , r ), the point x k in (3.21) satisfies x k ∈ B ( x, kρ ) ⊂ B ( x , r ) so that we can applyLemma 3.8(ii) in the counterpart of (3.21). (cid:50) Recall that Υ , Υ , ϑ and ϑ are auxiliary functions discussed in Subsection 3.1. Proposition 3.10. (i) Suppose that Assumption 3.5 holds. Then, there exists a constant C ≥ such that for all x ∈ U , < r < − (cid:0) R ∧ ( C δ U ( x )) (cid:1) and t > , P x ( τ B ( x,r ) ≤ t ) ≤ C tφ ( r ) . (3.22) If Tail R ( ψ, U, ≤ ) and U R ( ψ, β , C (cid:48) U ) also hold, then there exist constants C , C , C > such thatfor all x ∈ U , < r < − ( R ∧ ( C δ U ( x ))) and t > , P x ( τ B ( x,r ) ≤ t ) ≤ C (cid:18) tψ ( r ) + exp (cid:16) − C rϑ ( t, C r ) (cid:17)(cid:19) (3.23) (ii) Suppose that Assumption 3.6 holds. Then, there exists a constant C ≥ such that (3.22) holdsfor any υ ∈ ( υ, , x ∈ M , r > R (cid:48)∞ d S ( x ) υ and t ≥ φ (2 r υ/υ ) where R (cid:48)∞ = R (cid:48)∞ ( v ) is defined by (3.12) . If Tail R ∞ ( ψ, S , ≤ , υ ) and U R ∞ ( ψ, β , C (cid:48) U ) also hold, then there exist constants C , C , C > such that (3.23) holds for all x ∈ M , r > R (cid:48)∞ d S ( x ) υ and t ≥ ψ (2 r υ/υ ) . ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 23 Proof. (i) First, assuming that (3.23) holds under the further assumptions Tail R ( ψ, U, ≤ ) andU R ( ψ, β , C (cid:48) U ) for the moment. Then since Tail R ( φ, U, ≤ ) and U R ( φ, β , C U ) are contained inAssumption 3.5, we deduce that under Assumption 3.5 only, for all x ∈ U , 0 < r < − (cid:0) R ∧ ( C δ U ( x )) (cid:1) and t ∈ (0 , φ ( r )], since ϑ ( t, C r ) ≤ φ − ( t ) and U R ( φ, β , C U ) holds, by (3.3), P x ( τ B ≤ t ) ≤ C (cid:18) tφ ( r ) + exp (cid:16) − C rφ − ( t ) (cid:17)(cid:19) ≤ ( C + β β C U C − β ) tφ ( r ) . Hence, we obtain (3.22) since it holds trivially in case t > φ ( r ). Therefore, it suffices to prove (3.23)under the further assumptions.Now, we prove (3.23). Choose any x ∈ U , 0 < r < − (cid:0) R ∧ ( C δ U ( x )) (cid:1) and t > 0. Set B := B ( x, r ) and let C , C , C be the constants in Lemmas 3.8(i) and 3.9(i). Let C = C / (8 C ). If ϑ ( t, C r ) ≥ r/ 4, then by taking C larger than e C , we obtain (3.23). Besides, if C r < Υ ( t ), thenby (3.1) and (3.2), it holds exp (cid:0) − C r/ϑ ( t, C r ) (cid:1) ≥ exp (cid:0) − C C − (cid:1) . Thus, by taking C largerthan e C /C , we get (3.23). Therefore, we suppose that ϑ ( t, C r ) < r/ C r ≥ Υ ( t ).Let ρ ∈ [ ψ − ( t ) , r/ 4) be a constant chosen later. Let Y := X ( ρ ) be a ρ -truncated process. Recallthat the L´evy measure of Y is given by J ( x, dy ) := J ( x, dy ) {| x − y | <ρ } . Set J ( x, dy ) := J ( x, dy ) { ρ ≤| x − y | 2, thelaw of (cid:0) Y u : u ∈ [ (cid:101) τ B ∧ T ,k − , (cid:101) τ B ∧ T ,k ) (cid:1) is the same as the one for (cid:0) Y u : u ∈ [0 , (cid:101) τ B ∧ T ,k − (cid:101) τ B ∧ T ,k − ) (cid:1) starting from Y (cid:101) τ B ∧ T ,k − . Therefore, by (3.31), (3.32) and the strong Markov property, we obtain P = P x (cid:0) max { g , g } > r/ , (cid:101) τ B ≤ t, (cid:101) τ B < T , ∧ T , (cid:1) ≤ P x (cid:0) min ≤ k ≤ τ ( ρ ) B ( Y (cid:101) τB ∧ T ,k − ,r/ ◦ θ Y (cid:101) τ B ∧ T ,k − ≤ t, (cid:101) τ B ≤ t, (cid:101) τ B < T , ∧ T , (cid:1) ≤ z ∈ B ( x, r/ P z (cid:0) τ ( ρ ) B ( z,r/ ≤ t (cid:1) ≤ C exp (cid:16) − C r ρ + C tφ ( ρ ) (cid:17) , (3.33)where θ Y denotes the shift operator with respect to Y . In the second inequality above, we used thesubadditivity of probability measures and the fact that Y (cid:101) τ B ∧ T ,k − ∈ B ( x, r + r/ 4) for each k = 1 , P x ( τ B ≤ t ) ≤ c tψ ( r ) + (cid:16) c tψ ( ρ ) (cid:17) + 2 C exp (cid:16) − C r ρ + C tφ ( ρ ) (cid:17) . (3.34) ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 25 For the last step of the proof, we first assume tψ ( r/ ≤ ψ ( φ − ( t )) . Then by Lemma 3.1, we see C tφ ( ϑ ( t, C r )) ≤ C C rϑ ( t, C r ) = C r ϑ ( t, C r ) . Moreover, since U R ( ψ, β , C (cid:48) U ) holds, (cid:16) tψ ( ϑ ( t, C r )) (cid:17) ≤ (cid:16) tψ ( φ − ( t )) (cid:17) ≤ tψ ( r/ ≤ C (cid:48) U β tψ ( r ) . Thus, we get (3.23) by substituting ρ = ϑ ( t, C r ) ∈ [ ψ − ( t ) , r/ 4) in (3.34).Otherwise, if tψ ( r/ > ψ ( φ − ( t )) , then we substitute ρ = ψ − ( t / ψ ( r/ / ) ∈ ( φ − ( t ) , r/ 4) in(3.34). Since U R ( ψ, β , C (cid:48) U ) holds, we have ( t/ψ ( ρ )) = t/ψ ( r/ ≤ C (cid:48) U β t/ψ ( r ) andexp (cid:0) − C r ρ + C tφ ( ρ ) (cid:1) ≤ e C (2 β /C ) β (cid:16) ρr/ (cid:17) β ≤ c (cid:16) ψ ( ρ ) ψ ( r/ (cid:17) = c tψ ( r/ ≤ c tψ ( r ) . In the first inequality above, we used the fact that e − x ≤ (2 β ) β x − β for all x > 0. This completesthe proof for (i).(ii) We follow the proof of (i) and use the same notations. It still suffices to prove (3.23) underthe further assumptions. Choose any x ∈ M , r > R (cid:48)∞ d S ( x ) υ and t > ψ (2 r υ/υ ). Let C , C , C bethe constants chosen by Lemmas 3.8(ii) and 3.9(ii). We can still assume that ϑ ( t, C r ) < r/ C r ≥ Υ ( t ) where C = C / (8 C ).For each ρ ∈ [ ψ − ( t ) , r/ ⊂ [2 r υ/υ , r/ Y and Y froma ρ -truncated process Y := X ( ρ ) so that the joint law (cid:0) ( Y u , Y u ) : u ∈ [0 , ∞ ) (cid:1) is the same as (cid:0) ( X ( r/ u , X u ) : u ∈ [0 , lim n →∞ ( T ,n ∧ T ,n )) (cid:1) by the similar way to the one given in (i). In view of(3.19), we obtain (3.26) and (3.27) so that (3.29) and (3.30) are still valid. Moreover, we get (3.32)from Lemma 3.9(ii). Since U R ∞ ( ψ, β , C (cid:48) U ), we conclude (3.23) by the same arguments. (cid:50) Corollary 3.11. Suppose that Assumption 3.6 and L R ∞ ( φ, β , C L ) hold. Then X is conservative P x -a.s. for all x ∈ M . Proof. Denote by ζ the lifetime of X . Fix any υ ∈ ( υ, 1) and choose any x ∈ M and T > R ∞ ( φ, β , C L ) holds, we have that for all φ − ( t ) ≥ R (cid:48) υ/υ ∞ d S ( x ) υ + φ − ( T ), P x ( ζ ≤ T ) ≤ P x ( τ B ( x, ( φ − ( t ) / υ /υ ) ≤ t ) ≤ C tφ (cid:0) ( φ − ( t ) / υ /υ (cid:1) ≤ c ( φ − ( t )) − β ( υ − υ ) /υ . Hence, by taking t → ∞ , we obtain P x ( ζ ≤ T ) = 0 which yields the desired result. (cid:50) We turn to establish lower estimates on P x ( τ B ( x,r ) ≤ t ). We first obtain a priori pure jump typeestimates on that. Proposition 3.12. (i) Suppose that U R ( ψ, β , C (cid:48) U ) and Tail R ( ψ, U, ≥ ) hold with C . Then, forevery a ≥ , there exists c > such that for all x ∈ U and < r < − (cid:0) ( a − R ) ∧ ( C δ U ( x )) (cid:1) , P x ( τ B ( x,r ) ≤ t ) ≥ ctψ ( r ) , < t ≤ ψ ( ar ) . (3.35) (ii) Suppose that U R ∞ ( ψ, β , C (cid:48) U ) and Tail R ∞ ( ψ, S , ≥ , υ ) hold. Then, for every a ≥ , there exists c > such that (3.35) holds for all x ∈ M and r > (2 R ∞ ) / (1 − υ ) d S ( x ) υ . Proof. (i) Choose any x ∈ U , 0 < r < − (cid:0) ( a − R ) ∧ ( C δ U ( x )) (cid:1) and denote B := B ( x, r ). Set T r := inf { s > d ( X s − , X s ) ≥ r } . Then we see that T r ≥ τ B , P x -a.s. Hence, thanks to the Meyer’sconstruction, since Tail R ( ψ, U, ≥ ) holds, we have that, for all t > P x ( τ B > t ) = P x ( T r ∧ τ B > t ) ≤ P x ( c t/ψ (2 r ) < ξ ) = e − c t/ψ (2 r ) , where ξ is an exponential random variable with rate parameter 1 independent of X . Hence, sinceU R ( ψ, β , C (cid:48) U ) holds, we get that for all 0 < t ≤ ψ ( ar ), P x ( τ B ≤ t ) ≥ − e − c t/ψ (2 r ) ≥ c tψ (2 r ) e − c ψ ( ar ) /ψ (2 r ) ≥ c tψ ( r ) . (ii) By using a similar calculation to (3.19), it can be proved by the same way as the one for (i). (cid:50) Now, under the local chain condition Ch (see Definition 1.2), we obtain lower estimates on P x ( τ B ( x,r ) ≤ t ) which have similar forms to (3.23). Proposition 3.13. (i) Suppose that VRD R ( U ) , Ch R ( U ) , NDL R ( φ, U ) and U R ( φ, β , C U ) holdwith C V and C . Then, there exists a constant c (cid:48) > such that for each a > , there are constants C , C > such that for all x ∈ U , < r < c (cid:48) (cid:0) ( a − R ) ∧ (( C V ∧ C ) δ U ( x )) (cid:1) and < t ≤ φ ( ar ) satisfying C r ≤ Υ ( t ) , P x ( τ B ( x,r ) ≤ t ) ≥ exp (cid:16) − C rϑ ( t, C r ) (cid:17) . (3.36) If Tail R ( ψ, U, ≥ ) and U R ( ψ, β , C (cid:48) U ) also hold, then there exists a constant C > such that forall x ∈ U , < r < c (cid:48) (cid:0) ( a − R ) ∧ (( C V ∧ C ∧ C ) δ U ( x )) (cid:1) and < t ≤ φ ( ar ) , P x ( τ B ( x,r ) ≤ t ) ≥ C (cid:16) tψ ( r ) + exp (cid:16) − C rϑ ( t, C r ) (cid:17)(cid:17) . (3.37) (ii) Suppose that VRD R ∞ ( S , υ ) , Ch R ∞ ( S , υ ) , NDL R ∞ ( φ, S , υ ) and U R ∞ ( φ, β , C U ) hold. Let υ ∈ ( υ, . Then for every a > , there exist constants R (cid:48)(cid:48)∞ ∈ [ R ∞ , ∞ ) , C , C > such that (3.36) holds for all x ∈ M , r > R (cid:48)(cid:48)∞ d S ( x ) υ and ψ ( R (cid:48)(cid:48)∞ r υ/υ ) < t ≤ φ ( ar ) satisfying C r ≤ Υ ( t ) . If Tail R ∞ ( ψ, S , ≥ , υ ) and U R ∞ ( ψ, β , C (cid:48) U ) also hold, then there exists a constant C > such that (3.37) holds for all x ∈ M , r > R (cid:48)(cid:48)∞ d S ( x ) υ and ψ ( R (cid:48)(cid:48)∞ r υ/υ ) < t ≤ φ ( ar ) . Proof. (i) Let (cid:96) > A ≥ η ∈ (0 , 1) be the constants in (1.5), Ch R ( U ) and NDL R ( φ, U ),respectively. Then we set c (cid:48) := 3 − (cid:96) − A − η ∈ (0 , / 3) and C := 8 C U (1 + a ) β (cid:96)Aη − . Chooseany x ∈ U , 0 < r < c (cid:48) (cid:0) R ∧ (( C V ∧ C ) δ U ( x )) (cid:1) and t ∈ (0 , φ ( ar )]. Write B := B ( x, r ).We first prove (3.36). Write ϑ := ϑ ( t, C r ) ∈ [ ψ − ( t ) , φ − ( t )] and set n := (cid:100) C r/ (2 ϑ ) (cid:101) . Thenwe see that n ≥ (cid:100) C r/ (2 φ − ( t )) (cid:101) ≥ (cid:100) C / (2 a ) (cid:101) ≥ 4. Since Υ ( t ) ≥ C r ≥ ar ≥ φ − ( t ) ≥ Υ ( t ),by the definition of ϑ , we have tϑ/φ ( ϑ ) ≥ C r . Thus, since C r/ (2 ϑ ) ≤ n ≤ C r/ϑ , we obtain φ − ( t/n ) ≥ φ − ( tϑ/ ( C r )) ≥ ϑ ≥ − n − C r > n − (cid:96)Aη − r. (3.38)On the other hand, since U R ( φ, β , C U ) holds, we see that n φ ( r ) ≥ C rφ ( r )2 φ − ( t ) ≥ C φ ( r )2 a ≥ C U (1 + a ) β φ ( r ) ≥ φ ( ar ) ≥ t so that φ − ( t/n ) ≤ r. By (1.5), we can choose y ∈ B ( x, (cid:96)r ) \ B ( x, r ). Then since 2 (cid:96)r < R , by Ch R ( U ), there existsa sequence ( z i ) ≤ i ≤ n ⊂ M such that z = x , z n = y and d ( z i − , z i ) ≤ (cid:96)Ar/n for all 1 ≤ i ≤ n .Note that for all 1 ≤ i ≤ n , we have d ( x, z i ) ≤ (cid:80) ij =1 d ( z j − , z j ) ≤ (cid:96)Ar and hence( C V ∧ C ) δ U ( z i ) ≥ ( C V ∧ C )( δ U ( x ) − (cid:96)Ar ) ≥ η − (cid:96)Ar > η − r ≥ η − φ − ( t/n ) . Eventually, by the semigroup property, since NDL R ( φ, U ) and VRD R ( U ) hold, we obtain P x ( τ B ≤ t ) ≥ P x (cid:0) X t ∈ B ( y, r ) (cid:1) ≥ P x (cid:0) X t ∈ B ( y, φ − ( t/n )) (cid:1) ≥ (cid:90) B ( z , − ηφ − ( t/n )) ... (cid:90) B ( z n , − ηφ − ( t/n )) p B ( x,η − φ − ( t/n )) ( t/n , x, u ) × p B ( z ,η − φ − ( t/n )) ( t/n , u , u ) . . . p B ( z n − ,η − φ − ( t/n )) ( t/n , u n − , u n ) du n ...du ≥ n − (cid:89) i =0 (cid:18) c V ( z i , − ηφ − ( t/n )) V ( z i , η − φ − ( t/n )) (cid:19) ≥ e − c n ≥ e − c C r/ϑ . (3.39) ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 27 In the third inequality above, we used the fact that since (3.38) holds, for all 1 ≤ j ≤ n and u j ∈ B ( z j , − ηφ − ( t/n )), it holds that d ( u j , z j − ) ∨ d ( u j , z j ) ≤ d ( u j , z j ) + d ( z j − , z j ) ≤ − ηφ − ( t/n ) + 2 (cid:96)Ar/n < ηφ − ( t/n ) . This proves that (3.36) holds.Now, we assume that Tail R ( ψ, U, ≥ ) and U R ( ψ, β , C (cid:48) U ) hold, and then prove (3.37). By Propo-sition 3.12 and (3.36), since ψ ≥ φ , we can deduce that (3.37) holds when C r ≤ Υ ( t ). On theother hand, if C r > Υ ( t ), then by the definition, ϑ ( t, C r ) = ψ − ( t ). Thus, in view of Lemma3.2(i) and Proposition 3.12, since ψ − ≤ φ − , we can also deduce that (3.37) holds.(ii) We follow the proof of (i). Set C := 8 C U (1 + a ) β (cid:96)Aη − and R (cid:48)(cid:48)∞ := (2 (cid:96) ) υ C R ∞ with theconstants (cid:96), A, η in (1.5), Ch R ∞ ( S , υ ) and NDL R ∞ ( φ, S , υ ). Choose any x ∈ M , r > R (cid:48)(cid:48)∞ d S ( x ) υ and ψ ( R (cid:48)(cid:48)∞ r υ/υ ) < t ≤ φ ( ar ). Write ϑ := ϑ ( t, C r ) and n := (cid:100) C r/ (2 ϑ ) (cid:101) as before. By a similarargument to the one given in the last paragraph of the proof for (i), it suffices to show that (3.36)holds when C r ≤ Υ ( t ).By (1.5), there exists y ∈ B ( x, (cid:96)r ) \ B ( x, r ). Since Ch R ∞ ( S , υ ) holds and n R ∞ ( d S ( x ) ∨ d S ( y )) υ ≤ C R ∞ r ψ − ( t ) ( d S ( x ) + 2 (cid:96)r ) υ ≤ r − υ/υ υ (cid:96) υ ( r υ/υ + (2 (cid:96) ) υ r υ ) < r, there is a sequence ( z i ) ≤ i ≤ n ⊂ M such that z = x , z n = y and d ( z i − , z i ) ≤ (cid:96)Ar/n for all1 ≤ i ≤ n . Observe that (3.38) is still valid. Hence, we have that for all 0 ≤ i ≤ n ,2 − ηφ − ( t/n ) > n − (cid:96)Ar ≥ C − (cid:96)Aϑ ≥ C − (cid:96)Aψ − ( t ) > (cid:96)AR ∞ r υ/υ > R ∞ d S ( x ) υ + (2 (cid:96)A ) υ R ∞ r υ > R ∞ ( d S ( x ) + 2 (cid:96)Ar ) υ ≥ R ∞ d S ( z i ) υ . Then since NDL R ∞ ( φ, S , υ ) and VRD R ∞ ( S , υ ) hold, we get (3.36) by repeating (3.39). (cid:50) Estimates on oscillation of parabolic functions and zero-one law. In this subsection,we first establish estimates on oscillation of bounded parabolic functions for large distances underAssumption 3.7 (Proposition 3.15). Then, as an application, we obtain a zero-one law for shift-invariant events (Proposition 3.16) which will be used in the proofs for LILs at infinity (Corollary1.16 and Theorems 1.18(ii) and 1.20(i)). Throughout this subsection, we always assume that L R ∞ ( φ, β , C L ) and NDL R ∞ ( φ, S , υ ) hold. With the constants β > C L ∈ (0 , 1] in L R ∞ ( φ, β , C L ), we set κ := (2 /C L ) β ∈ [1 , ∞ ) . (3.40)Then since L R ∞ ( φ, β , C L ) holds, we have that φ ( κ r ) ≥ φ ( r ) for all r ∈ ( R ∞ , ∞ ) . (3.41)With the constant κ in (3.40), and η ∈ (0 , 1) in NDL R ∞ ( φ, S , υ ), we define open cylinders Q , Q − , Q + and I as follows: for x ∈ M , r > t ≥ φ (2 κ ηr ), Q ( t, x, r ) := ( t − φ (2 κ ηr ) , t ) × B ( x, κ r ) , Q − ( t, x, r ) := ( t − φ (2 κ ηr ) , t − φ ( κ ηr )) × B ( x, η r ) I ( t, x, r ) := ( t − φ (2 ηr ) , t ) × B ( x, κ r ) , Q + ( t, x, r ) := ( t − φ ( ηr ) , t ) × B ( x, η r ) . (3.42)Note that Q − ( t, x, r ), Q + ( t, x, r ), I ( t, x, r ) ⊂ Q ( t, x, r ) and Q + ( t, x, r ) ∩ Q − ( t, x, r ) = ∅ .Denote by Z = ( Z s ) s ≥ = ( V s , X s ) s ≥ a time-space stochastic process corresponding to X with V s := V − s . For D ∈ B ([0 , ∞ ) × M ), we write σ ZD := inf { t > Z t ∈ D } and τ ZD := inf { t > Z t ∈ [0 , ∞ ) × M ∂ \ D } .Let ( dt ⊗ µ ) be the product measure of the Lebesgue measure on [0 , ∞ ) and µ on M . Lemma 3.14. There exists a constant c > such that for all x ∈ M , r > η − R ∞ d S ( x ) υ , t ≥ φ (2 κ ηr ) and any compact set D ⊂ Q − ( t, x, r ) , inf w ∈Q + ( t,x,r ) P w (cid:0) σ ZD < τ Z Q ( t,x,r ) (cid:1) ≥ c ( dt ⊗ µ )( D ) (cid:0) φ (2 κ ηr ) − φ ( κ ηr ) (cid:1) V ( x, κ r ) , where the constant κ and cylinders Q − , Q + and Q are defined as (3.40) and (3.42) . Proof. For u > 0, we set D u := { y ∈ M : ( u, y ) ∈ D } . Write τ r := τ Z Q ( t,x,r ) . Observe that for any w = ( s, z ) ∈ Q + ( t, x, r ), (cid:0) φ (2 κ ηr ) − φ ( κ ηr ) (cid:1) P w ( σ ZD < τ r ) ≥ (cid:0) φ (2 κ ηr ) − φ ( κ ηr ) (cid:1) P w (cid:18) (cid:90) τ r D ( s − u, X u ) du > (cid:19) ≥ (cid:90) φ (2 κ ηr ) − φ ( κ ηr )0 P z (cid:18) (cid:90) τ r D ( s − u, X u ) du > a (cid:19) da = E z (cid:20) (cid:90) τ r D ( s − u, X u ) du (cid:21) = (cid:90) φ (2 κ ηr ) − ( t − s ) φ ( κ ηr ) − ( t − s ) P z (cid:0) X B ( x, κ r ) u ∈ D s − u (cid:1) du = (cid:90) φ (2 κ ηr ) φ ( κ ηr ) (cid:90) D t − u p B ( x, κ r ) (cid:0) u − ( t − s ) , z, y (cid:1) µ ( dy ) du. (3.43)In the first equality above, we used the fact that the integral (cid:82) τ r D ( s − u, X u ) du can not be largerthan φ (2 κ ηr ) − φ ( κ ηr ) since D ⊂ Q − ( t, x, r ).By the definition of Q + , we have t − s ∈ (0 , φ ( ηr )). Thus, for all u ∈ [ φ ( κ ηr ) , φ (2 κ ηr )], by (3.41),we see that 2 κ ηr ≥ φ − ( u − t + s ) ≥ φ − ( φ ( κ ηr ) − φ ( ηr )) ≥ ηr . Since NDL R ∞ ( φ, S , υ ) holds and z ∈ B ( x, η r ), it follows that for all u ∈ [ φ ( κ ηr ) , φ (2 κ ηr )] and y ∈ B ( x, η r ), p B ( x, κ r ) ( u − ( t − s ) , z, y ) ≥ p B ( x,η − φ − ( u − t + s )) ( u − t + s, z, y ) ≥ c V ( x, κ r ) . Then by combining with (3.43), we get (cid:0) φ (2 κ ηr ) − φ ( κ ηr ) (cid:1) P w ( σ ZD < τ r ) ≥ c V ( x, κ r ) (cid:90) φ (2 κ ηr ) φ ( κ ηr ) (cid:90) D t − u µ ( dy ) du = c ( dt ⊗ µ )( D ) V ( x, κ r ) . This yields the desired inequality. (cid:50) We say that a Borel measurable function q ( t, x ) on [0 , ∞ ) × M is parabolic on D = ( a, b ) × B ( x, r )for the process X , if for every open set U ⊂ D with U ⊂ D , it holds q ( t, x ) = E ( t,x ) [ q ( Z τ ZU )] forevery ( t, x ) ∈ U ∩ ([0 , ∞ ) × M ).In the following proposition, we let r ∞ be the constant defined as (3.6). Proposition 3.15. Suppose that Assumption 3.7 holds. Let υ ∈ ( υ, . Then, there exist con-stants c , b > such that for all x ∈ M , r > (8 κ η − r ∞ ) υ / ( υ − υ ) d S ( x ) υ , t ≥ φ (2 κ ηr ) , and anyfunction q which is nonnegative in [ t − φ (2 κ ηr ) , t ] × M and parabolic in Q ( t, x, r ) , it holds for all ( s , y ) , ( s , y ) ∈ I ( t, x, r ) that (cid:12)(cid:12) q ( s , y ) − q ( s , y ) (cid:12)(cid:12) ≤ c (cid:107) q (cid:107) L ∞ ([ t − φ (2 κ ηr ) , t ] × M ; dt ⊗ µ ) (cid:16) φ − ( | s − s | ) + d ( y , y ) + r υ/υ r (cid:17) b , (3.44) where the constant κ and cylinders Q and I are defined as (3.40) and (3.42) , respectively. Proof. Before giving the main argument for (3.44), we first set up some constants and functions.According to Lemma 3.14, since VRD R ∞ ( S , υ ) holds, there exists a constant c ∈ (0 , 1) such thatfor all z ∈ M , l > η − R ∞ d S ( z ) υ , u ≥ φ (2 κ ηl ) and any compact set D ⊂ Q − ( u, z, l ) satisfying( dt ⊗ µ )( D ) ≥ − ( dt ⊗ µ )( Q − ( u, z, l )), we haveinf w ∈Q + ( u,z,l ) P w (cid:0) σ D < τ Z Q ( u,z,l ) (cid:1) ≥ c . (3.45) ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 29 Besides, by the L´evy system (1.7), Proposition 3.4(ii) and Lemma 3.3(ii), since Tail R ∞ ( φ, S , ≤ , υ )and U R ∞ ( φ, β , C U ) hold, we see that for all z , z ∈ M and r ∞ ( d S ( z ) ∨ d S ( z )) υ < l ≤ (cid:101) l/ P z (cid:0) X τ B ( z,l ) ∈ M ∂ \ B ( z, (cid:101) l ) (cid:1) ≤ E z (cid:104) (cid:90) τ B ( z,l ) J (cid:0) X s , M ∂ \ B ( X s , (cid:101) l/ (cid:1) ds (cid:105) ≤ c E z [ τ B ( z , l + d ( z,z )) ] φ ( (cid:101) l/ ≤ c φ ( l + d ( z, z )) φ ( (cid:101) l ) , (3.46)for some constant c > 1. The first inequality in (3.46) is valid since for all y ∈ B ( z, l ), it holdsthat R ∞ d S ( y ) υ ≤ R ∞ d S ( z ) υ + R ∞ l υ < l/ l/ ≤ (cid:101) l/ l ≥ r ∞ > r − υ ∞ > R ∞ . Using theconstants c , c in (3.45) and (3.46), we define ξ = (cid:16) − c (cid:17) / , N := (cid:24) log(2 β /C L ) + log(( c + 4 c ) / ( c ξ )) β log(2 κ /η ) (cid:25) . Choose any x ∈ M , r > (8 κ η − r ∞ ) υ / ( υ − υ ) d S ( x ) υ and t ≥ φ (2 κ ηr ). Then we set l := r and l i := 2 − κ − η l i − for i ≥ . (3.47)Note that l > κ r ∞ r υ/υ . Let N := max { n ≥ l n > κ r ∞ r υ/υ } . Since L R ∞ ( φ, β , C L ) holds, φ (4 κ l i + N ) ≤ β C − L (cid:16) η κ (cid:17) N β φ (2 κ l i ) ≤ c ξc + 4 c φ (2 κ l i ) , ≤ i ≤ N − N . (3.48)For a given ( u, z ) ∈ I ( t, x, r ), we set for i ≥ Q i = Q i ( u, z ) := Q ( u, z, l i ) , Q + i := Q + ( u, z, l i ) and Q − i := Q − ( u, z, l i ) . Since φ is increasing, by (3.47) and (3.41), we have φ (2 κ ηl ) < φ (2 ηr ) ≤ φ (2 κ ηr ) − φ (2 ηr ) and2 κ l < κ r . Thus, we have Q ( u, z ) ⊂ Q ( t, x, r ) since ( u, z ) ∈ I ( t, x, r ). Moreover, by (3.47), we cansee that Q i +1 ⊂ Q + i for all i ≥ q , we assume that (cid:107) q (cid:107) L ∞ ([ t − φ (2 κ ηr ) ,t ] × M ; dt ⊗ µ ) = 1. We claim that for all ( u, z ) ∈ I ( t, x, r ),sup Q jN ( u,z ) q − inf Q jN ( u,z ) q ≤ ξ j , ≤ j < N /N . (3.49)In the followings, we prove (3.49) by induction. Let a j = inf Q jN ( u,z ) q and b j = sup Q jN ( u,z ) q .First, we have b − a ≤ (cid:107) q (cid:107) L ∞ ([ t − φ (2 κ ηr ) ,t ] × M ; dt ⊗ µ ) = 1.For the next step, we suppose that b j − a j ≤ ξ j for all 0 ≤ j ≤ k < (cid:100) N /N (cid:101) − 1. Set D (cid:48) := (cid:8) z ∈ Q − kN : q ( z ) ≤ − ( a k + b k ) (cid:9) . We may assume ( dt ⊗ µ )( D (cid:48) ) ≥ − ( dt ⊗ µ )( Q − kN ), or else using 1 − q instead of q . Let D be a compact subset of D (cid:48) such that ( dt ⊗ µ )( D ) ≥ − ( dt ⊗ µ )( Q − kN ). Note that we have l kN ≥ l N > κ r ∞ r υ/υ > κ η − R ∞ ( r /υ + r ) υ ≥ η − R ∞ ( d S ( x ) + κ r ) υ ≥ η − R ∞ d S ( z ) υ since r ∞ ≥ η − R ∞ . Write τ k := τ ZQ kN . Then by (3.45), since Q k +1) N ⊂ Q +1+ kN , we haveinf w ∈ Q k +1) N P w ( σ D < τ k ) ≥ c . (3.50)For a given arbitrary ε > 0, find w , w ∈ Q k +1) N such that q ( w ) ≥ b k +1 − ε/ q ( w ) ≤ a k +1 + ε/ 2. Since q is parabolic in Q ( t, x, r ) and Q kN \ D ⊂ Q ( t, x, r ), we obtain b k +1 − a k +1 − ε ≤ q ( w ) − q ( w ) = E w (cid:2) q ( Z σ D ∧ τ k ) − q ( w ) (cid:3) = E w (cid:2) q ( Z σ D ) − q ( w ); σ D < τ k (cid:3) + E w (cid:2) q ( Z τ k ) − q ( w ); σ D ≥ τ k , Z τ k ∈ Q k − N (cid:3) + k +1 (cid:88) j =2 E w (cid:2) q ( Z τ k ) − q ( w ); σ D ≥ τ k , Z τ k ∈ Q k − j ) N \ Q k +1 − j ) N (cid:3) =: K + K + K , where Q − N := ( t − φ (2 κ ηr ) , t ) × M ∂ .First, by (3.50) and the induction hypothesis, since D ⊂ D (cid:48) and ξ ∈ (0 , K + K ≤ (2 − ( a k + b k ) − a k ) P w ( σ D < τ k ) + ( b k − − a k − ) P w ( σ D ≥ τ k )= 2 − ( b k − a k ) P w ( σ D < τ k ) + ( b k − − a k − ) (cid:0) − P w ( σ D < τ k ) (cid:1) ≤ ξ k − (cid:0) − ξ P w ( σ D < τ k ) + 1 − P w ( σ D < τ k ) (cid:1) ≤ ξ k − (1 − − c ) . (3.51)On the other hand, observe that for each j ≥ 2, the process Z can enter the set Q k − j ) N \ Q k +1 − j ) N at time τ k only through a jump. Write w = ( u , z ) and observe that r ∞ ( d S ( z ) ∨ d S ( z )) υ ≤ r ∞ ( d S ( x )+2 κ r ) υ < r ∞ r υ/υ +2 κ r ∞ r υ < κ r ∞ r υ/υ < l N ≤ l kN . (3.52)By (3.46), (3.48) and (3.52), since (cid:107) q (cid:107) L ∞ ([ t − φ (2 κ ηr ) ,t ] × M ; dt ⊗ µ ) = 1 and d ( z, z ) ≤ κ l kN , we get K ≤ k − (cid:88) j =2 ( b k − j − a k − j ) P z (cid:0) X τ B ( z, κ l kN ∈ M ∂ \ B ( z, κ l k +1 − j ) N ) (cid:1) + P z (cid:0) X τ B ( z, κ l kN ∈ M ∂ \ B ( z, κ l N ) (cid:1) ≤ c k (cid:88) j =2 ξ k − j φ (4 κ l kN ) φ (2 κ l k +1 − j ) N ) ≤ c ξ k − k (cid:88) j =2 (cid:16) c c + 4 c (cid:17) j − ≤ c ξ k − ∞ (cid:88) j =1 (cid:16) c c + 4 c (cid:17) j = c ξ k − . Since we can choose ε arbitrarily small and ξ = 1 − − c , by combining with (3.51), it follows that b k +1 − a k +1 ≤ ( K + K ) + K ≤ ξ k − (cid:0) − − c + 4 − c (cid:1) = ξ k +1 , which shows that (3.49) holds for j = k + 1. Therefore, we conclude that the claim (3.49) holds.Let ( s , y ) , ( s , y ) ∈ I ( t, x, r ) be two different pairs such that s ≥ s . Fix any s ∈ ( s , t ) suchthat φ − ( s − s ) ≤ φ − ( s − s ) + d ( y , y ). Recall that Q − N = ( t − φ (2 κ ηr ) , t ) × M ∂ . We set j := max (cid:8) j ∈ Z : − ≤ j < N /N , ( s , y ) ∈ Q jN ( s , y ) (cid:9) . Then, by (3.49), since (cid:107) q (cid:107) L ∞ ([ t − φ (2 κ ηr ) ,t ] × M ; dt ⊗ µ ) = 1, we obtain | q ( s , y ) − q ( s , y ) | ≤ ξ j . (3.53)If j ≥ N /N − 1, then we see that l j +1) N ≤ l N ≤ κ r ∞ r υ/υ . Otherwise, if j < N /N − s , y ) / ∈ Q j +1) N ( s , y ), it must holds either φ − ( s − s ) ≥ κ ηl j +1) N or ηd ( y , y ) ≥ κ ηl j +1) N . Thus, by (3.47), whether j ≥ N /N − φ − ( s − s ) + d ( y , y ) + 4 κ ηr ∞ r υ/υ ≥ − (cid:0) φ − ( s − s ) + ηd ( y , y ) (cid:1) + 4 κ ηr ∞ r υ/υ ≥ κ ηl j +1) N = 2 − η r ( η / (2 κ )) ( j +1) N . Therefore, by combining with (3.53), we conclude that (cid:12)(cid:12) q ( s , y ) − q ( s , y ) (cid:12)(cid:12) ≤ ξ − exp (cid:32) log( ξ − ) N log(2 κ η − ) log (cid:16) φ − ( s − s ) + d ( y , y ) + 4 κ ηr ∞ r υ/υ − η r (cid:17)(cid:33) . This completes the proof. (cid:50) An event G is called shift-invariant if G is a tail event (i.e. ∩ ∞ t> σ ( X s : s > t )-measurable), and P y ( G ) = P y ( G ◦ θ t ) for all y ∈ M and t > Proposition 3.16. Suppose that Assumption 3.7 holds. Then, for every shift-invariant event G , itholds either P z ( G ) = 0 for all z ∈ M or else P z ( G ) = 1 for all z ∈ M . ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 31 Proof. Fix υ ∈ ( υ, x ∈ M and choose any ε ∈ (0 , T , C > t > T , P x (cid:0) sup s ≤ t d ( x , X s ) > φ − ( ε − C t ) (cid:1) = P x (cid:0) τ B ( x ,φ − ( ε − C t )) ≤ t (cid:1) ≤ ε. (3.54)Since L R ∞ ( φ, β , C L ) holds, we have lim t →∞ φ − ( t ) = ∞ . Note that the map ( t, x ) (cid:55)→ P t f ( x ) := E x f ( X t ) is parabolic in Q ( t + 1 , x , φ − ( t )) for all t > 0. Hence, by Proposition 3.15, for each t > T , by taking t > t large enough, one can get that for all nonnegative f ∈ L ∞ ( M ; µ ) and x ∈ M with d ( x, x ) ≤ φ − ( ε − C t ), it holds | P t f ( x ) − P t f ( x ) | ≤ c sup t> (cid:107) P t f (cid:107) L ∞ ( M ; µ ) (cid:16) φ − ( ε − C t ) + φ − ( t ) υ/υ φ − ( t ) (cid:17) b ≤ ε (cid:107) f (cid:107) L ∞ ( M ; µ ) . (3.55)Indeed, we see that ( t , x ) , ( t , x ) ∈ I ( t + 1 , x , φ − ( t )) for all t > P z ( G ) ∈ { , } for all z ∈ M .Now, let O = { z ∈ M : P z ( G ) = 1 } (which is Borel measurable) and suppose that both O and O c are nonempty. Since at least one of µ ( O ) and µ ( O c ) is positive, by considering the event G c insteadof G , we assume that µ ( O ) > x / ∈ O . Since µ ( O ) > R > R ∞ d S ( x ) υ such that µ ( B ( x , η R ) ∩ O ) > 0. Then since G is shift-invariant, weget from NDL R ∞ ( φ, S , υ ) and the Markov property that0 = P x ( G ) = P x ( G ◦ θ φ ( ηR ) ) ≥ P x (cid:0) P X φ ( ηR ) ( G ); X φ ( ηR ) ∈ B ( x , η R ) ∩ O (cid:1) = P x (cid:0) X φ ( ηR ) ∈ B ( x , η R ) ∩ O (cid:1) = (cid:90) B ( x ,η R ) ∩O p ( φ ( ηR ) , x , y ) µ ( dy ) ≥ c µ ( B ( x , η R ) ∩ O ) V ( x , R ) , which is a contradiction. Thus, we get that either O = ∅ or O c = ∅ . This completes the proof. (cid:50) Proofs of liminf LILs: Theorems 1.13, 1.15 and Corollaries 1.14, 1.16 Proof of Theorem 1.13. Recall that Assumption 3.5 holds under the setting of Theorem 1.13(see (1.13)). Let C , a ∗ , a ∗ ∈ (0 , 1) and C > q ≥ q > x ∈ U ,lim sup r → τ B ( x,r ) φ ( r ) log | log φ ( r ) | ∈ [ q , q ] , P x -a.s. (4.1)We follow the main idea of the proof in [63, Theorem 3.7] and will prove upper and lower bound ofthe limsup behavior in (4.1) separately. Choose any x ∈ U .First, we show the upper bound. For n ≥ 3, let l n := φ − ( e − n ) and B n := (cid:110) sup l n +1 ≤ r ≤ l n τ B ( x,r ) φ ( r ) log | log φ ( r ) | ≥ C U ( C + 1) β | log a ∗ | (cid:111) ⊂ (cid:110) τ B ( x,l n ) ≥ C U ( C + 1) β e − n − log n | log a ∗ | (cid:111) . The above inclusion holds since φ is increasing. Since SP R ( φ, U ) and U R ( φ, β , C U ) hold, we havethat, for all n large enough, P x ( B n ) ≤ ( a ∗ ) − exp (cid:0) − C U ( C + 1) β e − n − log n/φ ( C l n ) (cid:1) ≤ ( a ∗ ) − n − /e . Thus, (cid:80) ∞ n =3 P x ( B n ) < ∞ and hence by the Borel-Cantelli lemma, we deduce that the upper boundin (4.1) holds with q = 3 C U ( C + 1) β / | log a ∗ | .Now, we prove the lower bound. For n ≥ 3, let m n := φ − ( e − n ) and σ n := e − n log n C U ( C − β + 1) | log a ∗ | = φ ( m n ) log | log φ ( m n ) | C U ( C − β + 1) | log a ∗ | . (4.2) We also define for n ≥ F n := (cid:8) sup Now, by joining (4.3), (4.5), (4.7) and (4.8) all together, we obtain that for all n ≥ n , P x ( A n ) ≤ P x (cid:0) ∩ nk = n ( G k \ F n ) (cid:1) + n (cid:88) k = n P x ( F k ) + n (cid:88) k = n P x ( H k ) ≤ exp( − a ∗ − / n / ) + c n (cid:88) k = n e − k − log( j + 1) + c n (cid:88) k = n e − k ≤ c exp( − n / ) . Therefore, (cid:80) ∞ n =3 P x ( A n ) < ∞ and hence by the Borel-Cantelli lemma, P x (cid:0) lim sup k →∞ τ B ( x, m k ) σ k ≥ (cid:1) = 1 . (4.9)Since U R ( φ, β , C U ) holds, we see that for all k large enough, σ k ≥ − − β C − U ( C − β + 1) − ( | log a ∗ | ) − φ (2 m k ) log | log φ (2 m k ) | . (4.10)Thus, by (4.9), we conclude that the lower bound in (4.1) holds.To end the proof, we claim that for all x ∈ U , it holds thatlim inf t → φ (sup 0. For n ≥ 4, let r n := φ − ( e − ) e − n and t n ∈ (0 , e − ) bea constant satisfying φ ( r n ) = t n / log | log t n | . According to (4.1), P x -a.s. ω , there exists N = N ( ω )such that for all n ≥ N , τ B ( x,r n ) ≤ ( q + δ ) t n log( | log t n | + log log | log t n | )log | log t n | ≤ ( q + δ )(1 + δ ) t n =: t δ,n . Therefore, since U R ( φ, β , C U ) holds, we get thatlim inf t → φ (sup 1) is chosen to satisfy φ ( (cid:101) r n ) = (cid:101) t n / log | log (cid:101) t n | . It follows that P x -a.s.,lim inf t → φ (sup Since L R ( φ, β , C L ) and U R ( φ, β , C U ) hold, according to (4.11), thereexist constants c ≥ c > x ∈ U , lim inf t → sup We follow the arguments in the proof of Theorem 1.13. By an analogousargument to the one used for obtaining (4.11), it suffices to show that there exist constants q ≥ q > x, y ∈ M ,lim sup r →∞ τ B ( x,r ) φ ( r ) log | log φ ( r ) | ∈ [ q , q ] , P y -a.s. (4.12)Observe that for all x, y ∈ M , since d ( x, y ) < ∞ and U R ∞ ( φ, β , C U ) holds,lim sup r →∞ τ B ( x,r ) φ ( r ) log | log φ ( r ) | ≤ lim sup r →∞ τ B ( y,r + d ( x,y )) φ ( r ) log | log φ ( r ) | ≤ C U β lim sup r →∞ τ B ( y, r ) φ (2 r ) log | log φ (2 r ) | andlim sup r →∞ τ B ( x,r ) φ ( r ) log | log φ ( r ) | ≥ lim sup r →∞ τ B ( y,r − d ( x,y )) φ ( r ) log | log φ ( r ) | ≥ C − U − β lim sup r →∞ τ B ( y,r/ φ ( r/ 2) log | log φ ( r/ | . Thus, to prove (4.12), it suffices to show that for all y ∈ M ,lim sup r →∞ τ B ( y,r ) φ ( r ) log | log φ ( r ) | ∈ [ C U β q , C − U − β q ] , P y -a.s. (4.13)Here, we prove (4.13). Choose any y ∈ M . By SP R ∞ ( φ, { z } , υ ), the upper bound in (4.13) can beproved by following the proof of Theorem 1.13 after redefining the sequence l n := φ − ( e n ) therein.Next, set (cid:101) m n := φ − ( e n ) and (cid:101) σ n := e n log n/ [4 C U ( C − β + 1) | log a ∗ | ] where C and a ∗ are theconstants in (1.10). Then we define (cid:101) F n := (cid:8) sup By combining with (4.14), we obtain (cid:80) ∞ n =1 P y ( (cid:101) A n ) ≤ (cid:80) ∞ n =1 ( ∪ nk = n (cid:101) F k ) + (cid:80) ∞ n =1 ( ∩ nk = n ( (cid:101) G k \ (cid:101) F k )) < ∞ and hence P y (lim sup (cid:101) A n ) = 0 by the Borel-Cantelli lemma. Then by using an analogous bound to(4.10), we conclude that the lower bound in (4.13) holds. This completes the proof. (cid:50) Proof of Corollary 1.16. Under the current setting, the conditions of Theorem 1.15 and Assump-tion 3.6 hold (see (1.13)). In particular, by Corollary 3.11, X is conservative almost surely. Thus,by Theorem 1.15 and Proposition 3.16, it suffices to show that for any x ∈ M and λ > E = E ( x, λ ) := (cid:110) lim inf t →∞ φ (cid:0) sup 0, we have φ (cid:0) sup We give a version of conditional Borel-Cantelli lemma which will be used in the proofs of ourlimsup LILs. Proposition 5.1. Let ( ˜Ω , ˜ P , G , ( G t ) t ≥ ) be a filtered probability space.(i) Suppose that there exist a decreasing sequence ( t n ) n ≥ such that lim n →∞ t n = 0 and a sequenceof events ( A n ) n ≥ satisfying the following conditions:(Z1) A n ∈ G t n for all n ≥ .(Z2) There exist a sequence of nonnegative numbers ( a n ) n ≥ , a sequence of events ( G n ) n ≥ and aconstant p ∈ (0 , such that (cid:80) ∞ n =1 a n = ∞ and for all n ≥ , ˜ P ( G cn ) ≤ p and ˜ P ( A n | G t n +1 ) ≥ a n G n , ˜ P -a.s.Then, ˜ P (lim sup A n ) ≥ − p . In particular, if lim n →∞ ˜ P ( G cn ) = 0 , then ˜ P (lim sup A n ) = 1 .(ii) Suppose that there exist an increasing sequence ( s n ) n ≥ such that lim n →∞ s n = ∞ and a sequenceof events ( B n ) n ≥ satisfying the following conditions:(I1) B n ∈ G s n for all n ≥ . (I2) There exist a sequence of non-negative numbers ( b n ) n ≥ , a sequence of events ( H n ) n ≥ anda constant p ∈ (0 , such that (cid:80) ∞ n =1 b n = ∞ and for all n ≥ , ˜ P ( H cn ) ≤ p and ˜ P ( B n +1 | G s n ) ≥ b n +1 H n +1 , ˜ P -a.s.Then, ˜ P (lim sup B n ) ≥ − p . In particular, if lim n →∞ ˜ P ( H cn ) = 0 , then ˜ P (lim sup B n ) = 1 . Proof. Since the proofs are similar, we only give the proof for (i). Choose any n and ε ∈ (0 , − p ). Itsuffices to show that ˜ P ( ∪ k ≥ A n + k ) ≥ − p − ε . To prove this, we fix any N such that (cid:80) Nk =0 a n + k >ε − and observe that ˜ P ( ∪ k ≥ A n + k ) ≥ N (cid:88) k =0 ˜ P (cid:0) A n + k ∩ ( ∪ N − ki =1 A n + k + i ) c (cid:1) . (5.1)Let L n,k := ∪ N − ki =1 A n + k + i = ∪ Ni = k +1 A n + i . By (Z1) and (Z2), it holds that for all 0 ≤ k ≤ N ,˜ P ( A n + k ∩ L cn,k ) = ˜ E (cid:2) ˜ E [ A n + k L cn,k | G t n + k +1 ] (cid:3) = ˜ E (cid:2) L cn,k ˜ E [ A n + k | G t n + k +1 ] (cid:3) ≥ a n + k ˜ E [ L cn,k G n + k ] ≥ a n + k (˜ P ( L cn,k ) − p ) . (5.2)In the last inequality, we used the fact that ˜ P ( A ∩ B ) ≥ ˜ P ( A ) − ˜ P ( B c ).Suppose that ˜ P ( L cn, ) > p + ε . Then ˜ P ( L cn,k ) ≥ ˜ P ( L cn, ) > p + ε for all 0 ≤ k ≤ N . Thus, by (5.1)and (5.2), ˜ P ( ∪ k ≥ A n + k ) ≥ (cid:80) Nk =0 a n + k (˜ P ( L cn,k ) − p ) ≥ ε (cid:80) Nk =0 a n + k > 1, which is a contradiction.Therefore, we conclude that ˜ P ( L cn, ) ≤ p + ε and hence ˜ P ( ∪ k ≥ A n + k ) ≥ ˜ P ( L n, ) ≥ − p − ε . (cid:50) We know that a mixed stable-like process and a Brownian-like jump process have totally differentlimsup LILs. Indeed, the former enjoys a limsup LIL of type (1.2), while the latter enjoys a oneof type (1.1). (See [4, 63].) Under our setting, the process X may behave like a mixed stable-likeprocess in some time range, while it may behave like a Brownian-like jump process in some othertime range. See [28, Section 4.2] for an explicit example of a subordinator which behaves like this.Hence, one must overcome significant technical difficulties to obtain limsup LILs for our X . Thefollowing proofs of Theorems 1.17(ii) and 1.18(ii), together with Propositions 3.10 and 3.13, are themost delicate part of this paper. Proof of Theorem 1.17. Choose any x ∈ U and let ε x := 2 − ( δ U ( x ) ∧ ∈ (0 , r → ψ ( r ) /φ ( r ) < ∞ , by (1.8), we have φ ( r ) (cid:16) ψ ( r ) for r ∈ (0 , c ≥ z ∈ U , 0 < r < − (cid:0) R ∧ ( C δ U ( z )) (cid:1) and 0 < t ≤ φ ( r ), c − tφ ( r ) ≤ P z ( τ B ( z,r ) ≤ t ) ≤ c tφ ( r ) . (5.3)(i-a) First, assume that (cid:82) φ (Ψ( t )) − dt = ∞ . Choose any K ≥ A n := (cid:110) sup s ∈ [2 − ( n +1) , − n ] d ( X , X s ) ≥ K (cid:0) Ψ(2 − n ) ∨ φ − (2 − n ) (cid:1)(cid:111) and G n := (cid:8) τ B ( X ,ε x ) ≥ − n (cid:9) . Observe that by the Markov property and the triangle inequality, we have P x ( A n | F − ( n +1) ) ≥ inf y ∈ B ( x,ε x ) P y (cid:16) sup s ∈ [0 , − ( n +1) ] d ( X , X s ) ≥ K (cid:0) Ψ(2 − n ) ∨ φ − (2 − n ) (cid:1)(cid:17) G n =: a n G n . According to (5.3), since U R ( φ, β , C U ) holds, we see that for all n large enough, a n ≥ c − − ( n +1) φ (cid:0) K (Ψ(2 − n ) ∨ φ − (2 − n )) (cid:1) = (2 c ) − − n φ (2 K Ψ(2 − n )) ∨ φ (2 Kφ − (2 − n )) ≥ c K − β (cid:16) ∧ − n φ (Ψ(2 − n )) (cid:17) . (5.4)Since (cid:82) φ (Ψ( t )) − dt = ∞ , it follows (cid:80) ∞ n =1 a n = ∞ . Moreover, we also get from (5.3) thatlim n →∞ P x ( G cn ) = lim n →∞ P x ( τ B ( x,ε x ) < − n ) ≤ c lim n →∞ − n = 0 . ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 37 Thus, by Proposition 5.1(i), we obtain thatlim sup t → d ( x, X t ) / Ψ( t ) ≥ lim n →∞ sup s ∈ [2 − ( n +1) , − n ] d ( x, X s ) / Ψ(2 − n ) ≥ K, P x -a.s.Since the above holds with any K ≥ 1, we conclude that lim sup t → d ( x, X t ) / Ψ( t ) = ∞ , P x -a.s.(i-b) Next, assume that (cid:82) φ (Ψ( t )) − dt < ∞ . Choose any λ ∈ (0 , 1) and define B n := (cid:8) sup s ∈ [0 , − n +1 ] d ( X , X s ) ≥ λ Ψ(2 − n ) (cid:9) . By (5.3), since U R ( φ, β , C U ) holds, for all n large enough, we have P x ( B n ) ≤ c − n /φ ( λ Ψ(2 − n )) ≤ c − n /φ (Ψ(2 − n )). Since (cid:82) φ (Ψ( t )) − dt < ∞ , it follows (cid:80) ∞ n =1 P x ( B n ) < ∞ . Hence, by the Borel-Cantelli lemma, we get lim sup t → sup 1) such that nφ ( s n )(log n ) δ ≤ ψ ( s n log n ) and 4 φ ( s n +1 ) ≤ φ ( s n ) for all n ≥ . (5.5)Such sequence ( s n ) n ≥ exists because lim sup r → ψ ( r ) /φ ( r ) = ∞ . Then we define Ψ ∈ M + asΨ( t ) := ∞ (cid:88) n =4 ( s n log n ) ( t n +1 ,t n ] ( t ) + ( s log 4) · ( t , ∞ ) ( t ) where t n := φ ( s n ) log n. (5.6)We claim that there exist constants 0 < q ≤ q < ∞ such that for all x ∈ U , q ≤ lim sup t → d ( x, X t )Ψ( t ) ≤ lim sup t → sup 2) for all n large enough. (5.9)Hence, for all n large enough, since q C > t n s n /φ ( s n ) = s n log n ∈ [Υ ( t n ) , Υ ( t n )], we have ϑ ( t n , q C s n log n ) ≤ ϑ ( t n , s n log n ) ≤ s n . It follows that ∞ (cid:88) n =4 exp (cid:16) − q C s n log nϑ ( t n , q C s n log n ) (cid:17) ≤ c + c ∞ (cid:88) n =4 n − q C ≤ c + c ∞ (cid:88) n =4 n − < ∞ . Therefore, by the Borel-Cantelli lemma, we can see that the upper bound in (5.7) holds.Next, we prove the first inequality in (5.7). Choose C , C according to Proposition 3.13(i) with a = C and set q := 4 − ( C + C ) − . Here, C is the constant in SP r ( φ, U ). Define E n := (cid:110) sup s ∈ [ t n +1 ,t n ] d ( X , X s ) ≥ q Ψ( t n ) (cid:111) and F n := (cid:8) τ B ( X ,ε x ) ≥ t n (cid:9) . Then by the Markov property and the triangle inequality, since t n > t n +1 for all n ≥ P x ( E n | F t n +1 ) ≥ inf y ∈ B ( x,ε x ) P y (cid:0) τ B ( y, q Ψ( t n )) ≤ − t n (cid:1) · F n =: b n F n for all n ≥ . By (5.9), since ψ is increasing, for all n large enough, s n ∈ [ ψ − (2 − t n ) , φ − (2 − t n )]. Besides,for all n ≥ 4, since 2 − t n s n /φ ( s n ) = 2 − s n log n ≥ q C s n log n and ϑ ( t, · ) is nonincreasing, wehave ϑ (2 − t n , q C s n log n ) ≥ ϑ (2 − t n , − t n s n /φ ( s n )) ≥ s n . Thus, for all n large enough, sinceSP r ( φ, U ) holds, by Proposition 3.13(i), we get that whether 2 − t n ≤ φ (2 q C Ψ( t n )) or not, b n ≥ (1 − a ∗ ) ∧ (cid:0) C exp( − q C Ψ( t n ) /s n ) (cid:1) ≥ c n − q C ≥ c n − / . for large n satisfying 2 q Ψ( t n ) < − c (cid:48) (cid:0) C − R ∧ (cid:0) ( C V ∧ C ∧ C ) δ U ( x ) (cid:1)(cid:1) . Thus, (cid:80) ∞ n =4 b n = ∞ . Notethat since SP r ( φ, U ) holds, we have lim n →∞ P x ( F cn ) = 0. Therefore, by Proposition 5.1(i), since Ψis nondecreasing, we conclude that the first inequality in (5.7) holds.In the end, we deduce (1.19) from (5.7) and the Blumenthal’s zero-one law. (cid:50) Proof of Theorem 1.18. Choose any x, y ∈ M and put υ := υ / ∈ ( υ, φ ( r ) (cid:16) ψ ( r ) for r ≥ 1, according to (3.22) and (3.35), there exist c , c ≥ z ∈ M and r > c d S ( z ) υ , we have P z ( τ B ( z,r ) ≤ t ) ≥ c − tφ ( r ) for 0 < t ≤ φ ( r ) , P z ( τ B ( z,r ) ≤ t ) ≤ c tφ ( r ) for t ≥ φ (2 r υ/υ ) . (5.10)(i-a) First, assume that (cid:82) ∞ φ (Ψ( t )) − dt = ∞ . Choose any K ≥ A (cid:48) n := (cid:110) sup s ∈ [2 n − , n ] d ( X , X s ) ≥ K (cid:0) Ψ(2 n ) ∨ φ − (2 n ) (cid:1)(cid:111) and G (cid:48) n := (cid:8) τ B ( X ,φ − (2 n ) υ /υ ) ≥ C U β + n (cid:9) . Then by the Markov property and the triangle inequality, we get P y ( A (cid:48) n | F n − ) ≥ inf z ∈ B ( y,φ − (2 n ) υ /υ ) P z (cid:16) sup s ∈ [0 , n − ] d ( X , X s ) ≥ K (cid:0) Ψ(2 n ) ∨ φ − (2 n ) (cid:1)(cid:17) G (cid:48) n =: a (cid:48) n G (cid:48) n . Observe that for all n large enough and z ∈ B ( y, φ − (2 n ) υ /υ ), since υ /υ = υ / < 1, we have c d S ( z ) υ ≤ c ( d S ( y ) υ + φ − (2 n ) υ /υ ) < φ − (2 n ) < K (cid:0) Ψ(2 n ) ∨ φ − (2 n ) (cid:1) . Hence, by (5.10) and U R ∞ ( φ, β , C U ), we get that for all n large enough (cf. (5.4)), a (cid:48) n ≥ c − n − φ (cid:0) K (Ψ(2 n ) ∨ φ − (2 n )) (cid:1) ≥ c K − β (cid:16) ∧ n φ (Ψ(2 n )) (cid:17) so that (cid:80) ∞ n =1 a (cid:48) n = ∞ since (cid:82) ∞ φ (Ψ( t )) − dt = ∞ . We also see from (5.10), U R ∞ ( φ, β , C U ) andL R ∞ ( φ, β , C L ) thatlim n →∞ P y ( G (cid:48) cn ) ≤ lim n →∞ c C U β φ (cid:0) φ − (2 n ) (cid:1) φ (cid:0) φ − (2 n ) υ /υ (cid:1) ≤ c C U β C − L lim n →∞ φ − (2 n ) − β ( υ − υ ) /υ = 0 . Therefore, according to Proposition 5.1(ii), we obtain thatlim sup t →∞ d ( x, X t )Ψ( t ) ≥ lim sup n →∞ sup s ∈ [2 n − , n ] d ( y, X s )Ψ(2 n ) − lim sup t →∞ d ( x, y )Ψ( t ) ≥ K − lim sup t →∞ d ( x, y )Ψ( t ) , P y -a.s.Since we can choose K arbitrarily large, this yields that lim sup t →∞ d ( x, X t ) / Ψ( t ) = ∞ , P y -a.s.(i-b) Now, assume that (cid:82) ∞ φ (Ψ( t )) − dt < ∞ . Define Ψ ∧ ( t ) := Ψ( t ) ∧ φ − ( t ) υ /υ . Then we havethat, by L R ∞ ( φ, β , C L ) and U R ∞ ( φ, β , C U ), (cid:90) ∞ dtφ (Ψ ∧ ( t )) ≤ (cid:90) ∞ dtφ (Ψ( t )) + (cid:90) ∞ t − φ ( φ − ( t )) φ ( φ − ( t ) υ /υ ) dt ≤ c (cid:18) (cid:90) ∞ dtt β β − ( υ − υ ) /υ (cid:19) < ∞ . (5.11)For λ ∈ (0 , / B (cid:48) n := { sup s ∈ [0 , n +1 ] d ( X , X s ) ≥ λ Ψ ∧ (2 n ) } . Note that φ (2 λ Ψ ∧ (2 n ) υ/υ ) ≤ ( φ ◦ φ − )(2 n ) < n +1 . Hence, by (5.10), for all n large enough, we have P y ( B (cid:48) n ) ≤ c n +1 /φ ( λ Ψ ∧ (2 n )) ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 39 so that (cid:80) ∞ n =1 P y ( B (cid:48) n ) < ∞ by U R ∞ ( φ, β , C U ) and (5.11). Therefore, by the Borel-Cantelli lemma,since lim t →∞ Ψ ∧ ( t ) = ∞ in view of (5.11), we conclude thatlim sup t →∞ sup R > 0, by L R ∞ ( φ, β , C L ), U R ∞ ( φ, β , C U ) and similar calculations to (5.13), wehave that, for all n large enough and z ∈ B ( y, s υ /υn ), since υ /υ < R d S ( z ) υ < R d S ( y ) υ + Rs υ /υn < q s n log n = 2 q Ψ( t n ) and ψ ∧ (cid:0) R (2 q Ψ( t n )) υ/υ (cid:1) < − t n . Hence, according to (3.36), by following the proof of Theorem 1.17(ii), we get that (cid:80) ∞ n =5 P y ( E (cid:48) n ) = ∞ . Moreover, we also get from (3.22) and L R ∞ ( φ, β , C L ) thatlim n →∞ P y ( F (cid:48) n ) ≤ c lim n →∞ φ ( s n ) log s n /φ ( s υ /υn ) = 0 . It follows from Proposition 5.1(ii) that P y (lim sup E (cid:48) n ) = 1 so thatlim sup t →∞ d ( x, X t )Ψ( t ) ≥ lim sup t →∞ d ( y, X t ) − d ( x, y )Ψ( t ) = lim sup t →∞ d ( y, X t )Ψ( t ) ≥ q , P y -a.s. (5.15) Note that by a similar argument to the one given in the proof of Corollary 1.16, one can verify thatfor any λ > 0, events { lim sup t →∞ sup Since the proofs are similar, we only prove Theorem 1.19.Set p := 2(4 /C L ) /β and q := 2 C − + ( C C L ) − + 1. Since L R ( φ, β , C L ) holds, we see that φ ( λ ) ≥ φ ( λ/p ) for all λ ∈ (0 , R ). Define s n := p − n , t n := φ ( s n ) log n andΨ ( t ) := ∞ (cid:88) n =4 ( s n log n ) · ( t n +1 ,t n ] ( t ) . Then since L R ( φ, β , C L ) and U R ( φ, β , C U ) hold, one can see that for all sufficiently small t ,2 − C − U φ − ( t/ log | log t | ) log | log t | ≤ Ψ ( t ) ≤ C − L pφ − ( t/ log | log t | ) log | log t | . (5.16)(i) We claim that after replacing Ψ with Ψ, (5.7) is still valid with the redefined constant q . Indeed,we can see that (5.8) still works (with redefined s n and t n ), and since φ and ψ are increasing andU R ( φ, β , C U ) holds, it still holds that ∞ (cid:88) n =4 t n ψ ( q s n log n ) ≤ c ∞ (cid:88) n =4 φ ( p − n ) log | log p − n | ψ ( p − n log | log p n | ) ≤ c (cid:90) p − φ ( r ) log | log r | rψ ( r log | log r | ) < ∞ . (5.17)Note that s n ≤ φ − ( t n ) for all n ≥ 4. Thus, for all n large enough, if s n ≥ ψ − ( t n ), then ϑ ( t n , q C s n log n ) ≤ ϑ ( t n , s n log n ) = ϑ ( φ ( s n ) log n, s n log n ) ≤ s n . Otherwise, if s n < ψ − ( t n ),then since L R ( φ, β , C L ) holds with β ≥ 1, we get q C s n log n = q C t n s n φ ( s n ) ≥ q C C L t n ψ − ( t n ) φ ( ψ − ( t n )) > t n ψ − ( t n ) φ ( ψ − ( t n ))and hence ϑ ( t n , q C s n log n ) ≤ ϑ ( t n , t n ψ − ( t n ) /φ ( ψ − ( t n ))) = ψ − ( t n ). Thus, for all n largeenough, whether s n ≥ ψ − ( t n ) or not, we have that ϑ ( t n , q C s n log n ) ≤ s n ∨ ψ − ( t n ) . Therefore, since U R ( ψ, β , C (cid:48) U ) holds, by (5.17), we get that ∞ (cid:88) n =4 exp (cid:16) − q C s n log nϑ ( t n , q C s n log n ) (cid:17) ≤ c ∞ (cid:88) n =4 (cid:16) n − q C + exp (cid:16) − q C s n log nψ − ( t n ) (cid:17)(cid:17) ≤ c + c ∞ (cid:88) n =4 (cid:16) ψ − ( t n ) s n log n (cid:17) β ≤ c + c ∞ (cid:88) n =4 t n ψ ( s n log n ) < ∞ . (5.18)Hence, by the Borel-Cantelli lemma, we can deduce that the third inequality in (5.7) still holds.On the other hand, by using the fact that ϑ (2 − t n , r ) ≥ ψ − (2 − t n ) for all r , we can follow theproof for the first inequality in (5.7) line by line, whether s n ≥ ψ − (2 − t n ) or not. Here, we mentionthat the condition Tail R ( ψ, U, ≥ ) is unnecessary in that proof.Finally, in view of (5.7) (with Ψ ) and (5.16), by using the Blumenthal’s zero-one law again, weconclude the desired result.(ii) Since U R ( φ, β , C U ) and U R ( ψ, β , C (cid:48) U ) hold, for every K ≥ 1, we have ∞ (cid:88) n =4 t n ψ ( Ks n log n ) ≥ c ∞ (cid:88) n =4 φ ( s n ) log | log s n | ψ ( s n log | log s n | ) ≥ c (cid:90) / φ ( r ) log | log r | rψ ( r log | log r | ) dr = ∞ . Then by a similar proof to the one for Theorem 1.17(i) (the case when (cid:82) dt/φ (Ψ( t )) = ∞ ), using(3.35), one can deduce that for all x ∈ U , it holds lim sup t → d ( x, X t ) / Ψ ( t ) = ∞ , P x -a.s. In viewof (5.16), this shows that for all x ∈ U , the limsup in (1.23) is infinte, P x -a.s.Now, we prove (1.24). Let Ψ ∈ M + be a function satisfying (1.19). (According to Theorem1.17(ii), at least one such Ψ exists.) Then we define f ( t ) := Ψ( t ) / ( φ − ( t/ log | log t | ) log | log t | ). ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 41 From the definition, one can see that lim sup t → f ( t ) = ∞ since the limsup in (1.23) is infinite, P x -a.s for all x ∈ U . Thus, it remains to show that lim inf t → f ( t ) < ∞ .Suppose lim inf t → f ( t ) = ∞ . Then by (5.16), we have lim inf t → Ψ( t ) / Ψ ( t ) = ∞ . Fix any x ∈ U and let E n ( ε ) := P x ( d ( x, X t ) > ε Ψ( t ) for some t ∈ ( t n +1 , t n ]) for ε > 0. According to Proposition3.10(i), it holds that for all n large enough, P x ( E n ( ε )) ≤ c (cid:16) t n ψ ( ε Ψ( t n +1 )) + exp (cid:16) − εC Ψ( t n +1 ) ϑ ( t n , εC Ψ( t n +1 )) (cid:17)(cid:17) =: c ( I n, + I n, ) . Observe that (cid:82) dt/ψ (Ψ( t )) < ∞ . Indeed, if this integral is infinite, then by repeating the prooffor Theorem 1.17(i) again, one can see that for all x ∈ U , it holds lim sup t → d ( x, X t ) / Ψ( t ) = ∞ , P x -a.s., which is contradictory to (1.19). Since U R ( ψ, β , C (cid:48) U ) holds, it follows that (cid:82) dt/ψ ( ε Ψ( t )) < ∞ and hence (cid:80) ∞ n =1 I n, < ∞ .On the other hand, note that since L R ( φ, β , C L ) holds, in view of (5.16), one can check thatthere exists c > c Ψ ( t n +1 ) ≥ Ψ ( t n ) for all n ≥ 4. Since ϑ ( t, · ) is nonincreasing andlim inf t → Ψ( t ) / Ψ ( t ) = ∞ , there exists N ( ε ) depending on ε such that I n, ≤ exp (cid:16) − c q C Ψ ( t n +1 ) ϑ ( t n , c q C Ψ ( t n +1 )) (cid:17) ≤ exp (cid:16) − q C Ψ ( t n ) ϑ ( t n , q C Ψ ( t n )) (cid:17) for all n ≥ N ( ε ) . Therefore, by (5.18), we also obtain (cid:80) ∞ n =1 I n, < ∞ .In the end, by the Borel-Cantelli lemma, we deduce that lim sup t → d ( x, X t ) / Ψ( t ) ≤ ε , P x -a.s. forevery ε > 0. This contradicts (1.19). Hence, we conclude that lim inf t → f ( t ) < ∞ . (cid:50) Further examples We begin this section with observations on NDL. Recall the notion of the heat kernel from Section1. In the next two lemmas, we let ( M, d, µ ) be a metric measure space, φ be an increasing functionon (0 , ∞ ), and X be a strong Markov process on M having the heat kernel p ( t, x, y ) := p M ( t, x, y )such that p ( t, x, y ) < ∞ unless x = y . Then by the strong Markov property of X , one can see thatfor any open set D ⊂ M , the heat kernel p D ( t, x, y ) of X D exists and can be written as p D ( t, x, y ) = p ( t, x, y ) − E x (cid:104) E X τD (cid:2) p ( t − τ D , X τ D , y ); τ D < t (cid:3)(cid:105) . (6.1) Lemma 6.1. Suppose that there exist an open set U ⊂ M and constants R ∈ (0 , ∞ ] , C, C (cid:48) ≥ such that VRD R ( U ) holds (with C V ), and for all t ∈ (0 , φ ( R )) , p ( t, x, y ) ≤ CtV ( y, d ( x, y )) φ ( d ( x, y )) for all x ∈ M, y ∈ U with d ( x, y ) > C (cid:48) φ − ( t ) (6.2) and p ( t, x, y ) ≥ C − V ( x, φ − ( t )) for all x, y ∈ U with d ( x, y ) < C (cid:48)− φ − ( t ) . (6.3) Then NDL R ( φ, U ) holds with C = C V . Proof. Let d , d , c µ , C µ be the constants in (1.4), and set η := (2 C (cid:48) ) − (2 d +1 C C µ c − µ ) − /d .Choose any x ∈ U and 0 < r < R ∧ ( C V δ U ( x )). By (6.1), we have that, for any y, z ∈ B ( x, η r ), p B ( x,r ) ( φ ( ηr ) , y, z ) = p ( φ ( ηr ) , y, z ) − E y (cid:104) E X τB ( x,r ) (cid:2) p ( φ ( ηr ) − τ B ( x,r ) , X τ B ( x,r ) , z ); τ B ( x,r ) < φ ( ηr ) (cid:3)(cid:105) . (6.4)Note that B ( x, η r ) ⊂ B ( x, δ U ( x )) ⊂ U . Thus, by (6.3) and VRD R ( U ), we get that for any y, z ∈ B ( x, η r ), since d ( y, z ) ≤ η r < C (cid:48)− ηr , B ( y, ηr ) ⊂ B ( x, ηr ) and 2 ηr < r < R ∧ ( C V δ U ( x )), p ( φ ( ηr ) , y, z ) ≥ C − V ( y, ηr ) ≥ C − V ( x, ηr ) ≥ − d C − C − µ V ( x, ηr ) . (6.5) On the other hand, observe that for all z ∈ B ( x, η r ), since η < − , we have that, d ( X τ B ( x,r ) , z ) ≥ d ( X τ B ( x,r ) , x ) − d ( x, z ) ≥ (1 − η ) r > r/ > C (cid:48) ηr . Thus, by (6.2) and VRD R ( U ), we see that forall z ∈ B ( x, η r ), since φ is increasing, B ( z, r/ ⊃ B ( x, r/ 2) and r < R ∧ ( C V δ U ( x )), p ( φ ( ηr ) − τ B ( x,r ) , X τ B ( x,r ) , z ) ≤ Cφ ( ηr ) V ( z, d ( X τ B ( x,r ) , z )) φ ( d ( X τ B ( x,r ) , z )) ≤ Cφ ( ηr ) V ( z, r/ φ (3 r/ ≤ CV ( x, r/ ≤ Cc − µ (2 η ) d V ( x, ηr ) ≤ − d − C − C − µ V ( x, ηr ) . (6.6)Finally, by combining (6.4) with (6.5) and (6.6), we conclude that NDL R ( φ, U ) holds. (cid:50) Analogously, we obtain the following lemma which is a near infinity counterpart of Lemma 6.1.By the virtue of Remark 1.22(iii), without loss of generality, we write VRD R ∞ (resp. NDL R ∞ ( φ )and Tail R ∞ ( φ )) for VRD R ∞ ( M, υ ) (resp.NDL R ∞ ( φ, M, υ ) and Tail R ∞ ( φ, M, υ )) from now on. Lemma 6.2. Suppose that there exist constants R ∞ ∈ (0 , ∞ ) , C, C (cid:48) ≥ such that VRD R ∞ holds, p ( t, x, y ) ≤ CtV ( y, d ( x, y )) φ ( d ( x, y )) for all t > , x, y ∈ M with d ( x, y ) > C (cid:48) φ − ( t ) and p ( t, x, y ) ≥ C − V ( x, φ − ( t )) for all t > φ ( R ∞ ) , x, y ∈ M with d ( x, y ) < C (cid:48) φ − ( t ) . Then, there exists a constant R ∈ [ R ∞ , ∞ ) such that NDL R ( φ ) holds. Proof. We follow the proof of Lemma 6.1. Let d , d , c µ , C µ be the constants in (1.4), and set η := (2 C (cid:48) ) − (2 d +1 C C µ c − µ ) − /d . Define R = η − R ∞ . Then for all x ∈ M , r ∈ ( R, ∞ ) and y, z ∈ B ( x, η r ), since ηr > R ∞ , we see that (6.5) and (6.6) hold . Therefore, since (6.4) holds forall x ∈ M , r ∈ ( R, ∞ ) and y, z ∈ B ( x, η r ), we obtain the result. (cid:50) Example 6.3. (Non-symmetric Feller processes) Let ν be a nonincreasing function on (0 , ∞ )satisfying (cid:82) ∞ ( r d − ∧ r d +1 ) ν ( r ) dr < ∞ . We also let a ( x, z ) be a Borel function on R d × R d such thatfor some constants a , a , a > β ∈ (0 , a ≤ a ( x, z ) ≤ a and | a ( x, z ) − a ( y, z ) | ≤ a | x − y | β for all x, y, z ∈ R d . (6.7)In this example, we deal with a non-symmetric operator L a which takes one of the following forms: L a f ( x ) := (cid:90) R d (cid:0) f ( x + z ) − f ( x ) − | z | < (cid:104) z, ∇ f ( x ) (cid:105) (cid:1) a ( x, z ) ν ( | z | ) dz, (6.8) L a f ( x ) := (cid:90) R d (cid:0) f ( x + z ) − f ( x ) (cid:1) a ( x, z ) ν ( | z | ) dz, (6.9) L a f ( x ) := 12 (cid:90) R d (cid:0) f ( x + z ) + f ( x − z ) − f ( x ) (cid:1) a ( x, z ) ν ( | z | ) dz. (6.10)Following [76], set for r > g ( r ) := (cid:90) | x |≥ r ν ( | x | ) dx, h ( r ) := (cid:90) R d (cid:16) | x | r ∧ (cid:17) ν ( | x | ) dx, G ( r ) := g ( r ) − and H ( r ) := h ( r ) − . Then it is easy to see that H ≤ G , H is increasing and U( H , , 1) holds.We say that a condition (P) holds if one of the following three conditions (P1)–(P3) is satisfied:(P1) L a is given by (6.8), and L ( H , α, c ) holds with 1 < α ≤ L a is given by (6.9), and L ( H , α, c ) and U ( H , β, c ) hold with 0 < α ≤ β < L a is given by (6.10), L ( H , α, c ) holds with 0 < α < a ( x, z ) = a ( x, − z ) , x, z ∈ R d . ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 43 By [51, Theorem 1.3], if (6.7) and (P) hold, then there exists a Feller process X on R d whoseinfinitesimal generator ( A a , D ( A a )) satisfies that C c ( R d ) ⊂ D ( A a ), A a = L a on C c ( R d ), and( A a , D ( A a )) is the closure of ( L a , C ∞ c ( R d )). Such Feller process is the unique solution to the mar-tingale problem for ( L a , C ∞ c ( R d )) (see [51, Remark 1.5]). Note that the L´evy measure of X is givenas J ( x, dy ) = a ( x, y − x ) ν ( | y − x | ) dy , which is typically non-symmetric.Since a ( x, z ) is bounded above and below by positive constants, one can see that Tail ∞ ( G , R d )and Tail ∞ ( H , R d , ≤ ) hold. Moreover, by [51, Theorem 1.2(4) and Lemma 4.11] and our Lemma 6.1,NDL r ( H , R d ) holds for some r > 0. Therefore, by Proposition 3.4, we deduce that the liminf LILat zero given in Corollary 1.14 hold with U = R d and φ = H .To obtain a limsup LIL at zero, in addition to (6.7) and (P), we also assume U r ( G , γ, c ) holdsfor some r , c ∈ (0 , 1) and γ > 0. Here, we emphasize that γ may not be smaller than 2. Underthis further assumption, since Tail ∞ ( G , R d ) and NDL r ( H , R d ) hold, we deduce that for all x ∈ R d ,the limsup LILs at zero given in Theorems 1.17 and 1.19 hold with φ = H and ψ = G . (cid:50) Example 6.4. (Singular L´evy measure) In this example, we consider a Hunt process on R d whose L´evy measure is singular with respect to the Lebesgue measure. We refer to [90].Define a kernel on R d × R d \ diag as J ( x, y ) = (cid:40) b ( x, y ) | x − y | − − α , if y − x ∈ ∪ di =1 R i \ { } , , otherwise , (6.11)where α ∈ (0 , 2) is a constant and b ( x, y ) is a symmetric function on R d × R d that is boundedbetween two positive constants. Using this kernel, define a symmetric form ( E , F ) on L ( R d ; dx ) as E ( u, v ) = (cid:90) R d (cid:16) d (cid:88) i =1 (cid:90) R ( u ( x + e i τ ) − u ( x ))( v ( x + e i τ ) − v ( x )) J ( x, x + e i τ ) dτ (cid:17) dx, F = { u ∈ L ( R d ; dx ) | E ( u, u ) < ∞} . According to [90, Theorem 3.9], the above form ( E , F ) is a regular Dirichlet form. Hence, by thegeneral theory (see, e.g. [43]), there exists a Hunt process X in R d associated with ( E , F ). Below,we get LILs for X , both at zero and at infinity, from our main theorems. To do this, it suffices tocheck Tail ∞ ( r α , R d ) and SP ∞ ( r α , R d ) by Remark 1.22(ii).First, we see from the definition (6.11) that Tail ∞ ( r α , R d ) holds. Indeed, for all x ∈ R d and r > (cid:82) B ( x,r ) c J ( x, dy ) = (cid:80) di =1 (cid:82) | τ |≥ r J ( x, x + τ e i ) dτ (cid:16) d (cid:82) | τ |≥ r τ − − α dτ = c r − α . Next, by [90, Proposition4.4 and the proof of Theorem 4.6], there exist constants C , c > x ∈ R d and r > 0, the upper bound in (1.10) holds for all n ∈ N with φ ( r ) = r α and a ∗ = 2 − , and P x ( τ B ( x,r ) < t ) ≤ c tr − α for all t > . (6.12)Moreover, in view of [90, Proposition 4.18], for all x ∈ R d \ { } and t > P x (cid:0) (cid:104) X t − x, x (cid:105) ≤ −| X t − x || x | (cid:1) ≥ c t − d/α (cid:90) (cid:104) y − x,x (cid:105)≤−| y − x || x | , | y − x |≤ c t /α dy ≥ c c ( d ) , (6.13)for some constants c , c > c ( d ) > d .Now, by combining (6.12) with (6.13), one can follow the proof of [50, Proposition 5.2] and deducethat the lower bound in (1.10) holds for all n ∈ N with φ ( r ) = r α .Eventually, by Corollaries 1.14 and 1.16, Lemma 3.3, and Theorems 1.17 and 1.18, we concludethat for all x, y ∈ R d , the liminf LILs (1.15) and (1.17), and the limsup LILs (1.18) and (1.20) holdwith φ ( r ) = r α . (cid:50) Example 6.5. (Long range random walks on graphs) Let ( M, d ) be a connected countableinfinite graph with its natural graph metric. Let µ be a measure on M such that µ x := µ ( { x } ) (cid:16) x ∈ M . We suppose that a uniform volume doubling condition holds, that is, there exist annondecreasing function V on (0 , ∞ ) and constants d > C µ , c ≥ V, d , C µ ) holds and c − V ( x, r ) ≤ V ( r ) ≤ c V ( x, r ) for all x ∈ M, r > . (6.14)Under (6.14), by [47, Proposition 5.2], we see that VRD holds since ( M, d ) is connected and infinite.Let β ∈ (0 , p ∈ R , C > η xy : x, y ∈ M ) be a family of nonnegative realnumbers such that for all x, y ∈ M , η xx = 0 , η xy = η yx and C − V ( d ( x, y )) F β,p ( d ( x, y )) ≤ η xy ≤ CV ( d ( x, y )) F β,p ( d ( x, y )) , (6.15)where F β,p ( r ) := r β (log( e + r )) p . Define ν x := (cid:80) y ∈ M η xy for x ∈ M . Then by [21, Lemma 2.1], since(6.14) and (6.15) hold, we have sup x ∈ M ν x < ∞ .In this example, we consider a constant speed continuous time random walk associated with the conductance ( η xy ) satisfying (6.15). Define L C f ( x ) = ν − x (cid:88) y η xy ( f ( y ) − f ( x )) (6.16)and let X be the simple random walks on M whose generator in L ( M ; ν ) is L C . Note that X is asymmetric Markov process which waits at each vertex x for an exponential time with mean 1 andjumps according to the transitions η xy /ν x . X is called the constant speed random walk (CSRW).Since (6.14) and (6.15) hold, by using [21, Lemma 2.1] again, we can see that Tail ( F β,p ) holds.Moreover, by [74, Theorem 1.1 and Remark 2] and our Lemma 6.2, NDL r ( F β,p ) holds with aconstant r > 1. Thus, by our Corollary 1.16, Theorem 1.18(i) and (1.13), we conclude that thereexists a constant b ∞ ∈ (0 , ∞ ) such that for all x, y ∈ M ,lim inf t →∞ sup 1. Thus, since VRD with respect to the counting measure still holds, by our theorems, we conclude that Y enjoys thesame forms of liminf and limsup LILs at infinity as the ones for X . (cid:50) Example 6.6. (Random conductance models) Let ( Ω, F , P ) be a probability space. For ω ∈ Ω ,let ( η xy ( ω ) : x, y ∈ M ) be a family of nonnegative random variables and ν x ( ω ) := (cid:80) y ∈ M η xy ( ω ).For ω ∈ Ω , we denote by X ω (resp. Y ω ) the CSRW (resp. VSRW) associated with the randomconductance ( η xy ( ω )), that is, having the generator L C in (6.16) (resp. L V in (6.17)).An important example of random conductance models is the following bond percolation in Z d (see [58]): Let p ∈ [0 , 1] and E d := {{ x, y } : | x − y | = 1 } be the set of non-oriented edges in Z d .Let ( η e : e ∈ E d ) be i.i.d. Bernoulli random variables in ( Ω, F , P ) with P ( η e = 1) = p ∈ [0 , x, y ∈ Z d , η xy = η e if { x, y } = e ∈ E d and η xy ≡ e iscalled open if η e = 1. A set C ⊂ Z d is called an open cluster if every x, y ∈ C are connected ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 45 by an open path. It is known that there exists a critical probability p c ∈ (0 , 1) depends on d such that if p > p c , then P -a.s., there exists a unique infinite open cluster in each configuration,which we denote C ∞ = C ∞ ( ω ). When p > p c , the following limsup LIL for CSRW X ω on C ∞ is obtained in [32] using results in [6]: For P -a.s. ω , CSRW X ω started from any x ∈ C ∞ satisfieslim sup t →∞ | X ωt | / ( t log log t ) / = c ( p, d ) ∈ (0 , ∞ ), almost surely. This result was extended in [25,26]to cover more general models such that enjoys weak sub-Gaussian heat kernel estimates (see [25,Assumption 1.1]). Note that liminf LIL at infinity was also treated in [25, 26].In this example, we study liminf and limsup LILs for random conductance models with longjumps that do not enjoy weak sub-Gaussian heat kernel estimates. Let α ∈ (0 , d , d ≥ d = d + d > − α and L := Z d × Z d + . Let ( η xy ( ω ) : x, y ∈ L ) be a sequence of independentnon-negative random variables in ( Ω, F , P ) such that η xx = 0, η xy = η yx for all x, y andsup x (cid:54) = y P ( η xy = 0) < − and sup x (cid:54) = y (cid:0) E [ η pxy ] + E [ η − qxy { η xy (cid:54) =0 } ] (cid:1) < ∞ , for some p, q ∈ Z + with p > d + 2 d ∨ d + 12(2 − α ) and q > d + 2 d , (6.18)where E is the expectation with respect to P . When we deal with CSRW X ω , we also assume fu that η xy ( ω ) > x (cid:54) = y and there are constants µ ≥ µ > µ ≤ (cid:80) y ∈ L η xy ( ω ) | x − y | d + α ≤ µ for all ω ∈ Ω and x ∈ L . Below, we show that X ω and Y ω enjoy the limsup law at infinity (1.20)with φ ( r ) = r α , and there exist constants 0 < a ≤ a < ∞ such that P a.s. ω , there exist a ( ω ) , a ( ω ) ∈ [ a , a ] so thatlim inf t →∞ sup 1) independent of ω . In particular, by [20, (1.4) and (HK1) (ii)] and [19,Lemma 3.3], we see that P -a.s. ω , there is a constant r = r ( ω ) ≥ r ( { } , υ ) andTail r ( r α , { } , υ ) hold.Next, we claim that there exist υ, θ (cid:48) ∈ ( υ , 1) and c i > i = 1 , , P -a.s. ω , there is r = r ( ω ) ≥ x ∈ L , R ≥ r (1 + | x | ) υ and R θ (cid:48) α ≤ t ≤ c R α , p B ( x ,R ) ( t, y, z ) ≥ c t − d/α , y, z ∈ B ( x , R/ 4) with | y − z | ≤ c t /α . (6.20)Note that (6.20) implies NDL r ( r α , { } , υ ). Hence once we obtain (6.20), we can conclude the liminfand limsup LILs for X ω and Y ω in (6.19) hold from Corollary 1.16, Theorem 1.18 and (1.13). Now,we prove (6.20) by following Step (1) in the proof of [20, Proposition 2.11]. According to [20, (2.10)], P -a.s. ω , there exist c > R ≥ δ, θ (cid:48) ∈ ( υ , 1) such that for any s ≥ R and s δ ≤ r ≤ s sup y ∈ B (0 , s ) P y ( τ B ( x,r ) ≤ t ) ≤ c t / r − α/ , t ≥ r θ (cid:48) α . (6.21)Without loss of generality, we may assume θ (cid:48) > δ . Let υ := δ/θ (cid:48) ∈ ( δ, x ∈ L and R ≥ R (1 + | x | ) υ . Put s = ( R + | x | ) / R ≥ (1 + | x | ) υ , we have R /υ ≥ s ≥ R/ ≥ R .Thus, by (6.21), for any x ∈ B ( x , R ), R θ (cid:48) = R δ/υ ≤ r ≤ R/ t ≥ R θ (cid:48) α , (cid:88) y ∈ B ( x,r ) c p B ( x ,R ) ( t/ , x, y ) µ y ≤ P x ( τ B ( x,r ) ≤ t/ ≤ sup y ∈ B (0 , | x | + R ) P y ( τ B ( y,r ) ≤ t ) ≤ c t / r − α/ , where µ y is either (cid:80) z ∈ L η yz ( ω ) / | y − z | d + α (for X ω ) or 1 (for Y ω ). Notice that the above inequalitiesare similar to [20, (2.30)]. Now, by using [20, Theorem 2.10] (without loss of generality, one canassume that [20, Theorem 2.10] holds with δ ∈ ( υ , 1) above), we can repeat the rest of Step (1) in the proof of [20, Proposition 2.11] and deduce (6.20).Here, we emphasize that the assumption (6.18) is weaker than the one in [20, Theorems 4.1and 4.3]. Hence, we do not know whether X ω and Y ω enjoy pure-jump type heat kernel estimates (see [20, (4.4)]). However, our LILs only require NDL and, without any information on off-diagonalheat kernel estimates, we succeeded in obtaining LILs for X ω and Y ω in (6.19). (cid:50) Appendix A. Proofs of Theorems 2.1, 2.2 and 2.3 Recall that the function F is increasing continuous on (0 , ∞ ) satisfying (2.2) and the function F is defined in (2.4). Lemma A.1. (i) For any constants k , k > , there exists a constant c ≥ such that exp (cid:0) − k F ( r, s ) (cid:1) ≤ c (cid:0) s/F ( r ) (cid:1) k for all r, s > . (ii) F ( ar, s ) ≥ aF ( r, s ) for all a ≥ and r, s > . Proof. (i) It suffices to consider the case when F ( r ) > s . In such case, according to [49, Lemma3.19], we get that for all r, s > (cid:0) − k F ( r, s ) (cid:1) ≤ exp (cid:0) − c k ( F ( r ) /s ) / ( γ − (cid:1) ≤ c (cid:0) s/F ( r ) (cid:1) k . (ii) See [49, (3.42)]. (cid:50) By a similar proof to the one for Lemma 6.1, using Lemma A.1(i), one can see from (2.3) thatthere exist constants ε , η ∈ (0 , c > x ∈ M and r ∈ (0 , ε R ), q B ( x,r ) ( t, y, z ) ≥ c V ( x, F − ( t )) for all t ∈ (0 , F ( η r )] , y, z ∈ B ( x, η F − ( t )) . (A.1)Recall the definitions of Φ , Θ and Π from (2.6). Lemma A.2. There exists a constant ε ∈ (0 , such that SP εR (Φ , M ) holds. Proof. Let ε , η be the constants in (A.1). Since (A.1), (2.1) and L( F, γ , c L ) hold, by similarcalculations to the ones given in (3.8), we can see that there exist constants δ ∈ (0 , ε ], c ∈ (0 , k ∈ [1 , ∞ ) such that for all x ∈ M , r ∈ (0 , δ R ) and y ∈ B ( x, r ), it holds that P y (cid:0) sup 0. Thus,we get that for all x ∈ M and r ∈ (0 , δ R ),inf y ∈ B ( x,r ) P y (cid:0) τ B ( x,r ) ≤ Φ( k r ) (cid:1) ≥ inf y ∈ B ( x,r ) P y (cid:0) sup 1) such that P ( S t ≤ /φ − ( t − )) ≥ p for all t > 0. Since S is independent of Z and U( F, γ , c U ) holds, by theMarkov properties of S and Z , it follows that for all x ∈ M , r ∈ (0 , δ R ) and n ∈ N , P x (cid:0) τ B ( x,r ) ≥ n Φ( k r ) (cid:1) ≥ P x (cid:0) Z B ( x,r ) S n Φ( k r ) ∈ B ( x, r ) (cid:1) ≥ P (cid:0) S n Φ( k r ) ≤ nF ( k r ) (cid:1) P x (cid:0) Z B ( x,r ) nF ( k r ) ∈ B ( x, r ) (cid:1) ≥ P (cid:0) S Φ( k r ) ≤ F ( k r ) (cid:1) n (cid:2) inf y ∈ B ( x,η r ) P y (cid:0) Z B ( x,r ) F ( η r ) ∈ B ( x, η r ) (cid:1)(cid:3) (cid:100) nF ( k r ) /F ( η r ) (cid:101) ≥ ( p c c U ( k /η ) γ ) n . ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 47 We used the facts that S n Φ( k r ) − S ( n − k r ) has the same law as the one of S Φ( k r ) for all n ∈ N in the third inequality, and that F ( r ) = 1 /φ − (Φ( r ) − ) in the last inequality above. This finishesthe proof. (cid:50) Lemma A.3. (i) Tail R (Θ , M, ≤ ) holds. Therefore, by (2.8) , Tail R (Φ , M, ≤ ) also holds.(ii) If U F ( R ) ( w − , α , c ) holds, then there exists ε ∈ (0 , such that Tail ε ( R ∧ R ) (Π , M ) holds.(iii) If R = ∞ and U F ( R ) ( w − , α , c ) holds, then Tail R (Π , M ) holds. Proof. Since Z is conservative, and VRD ¯ R ( M ), U( F, γ , c U ) and L( F, γ , c L ) hold with γ > 1, by(2.3) and Lemma A.1, we see that for all x ∈ M , r ∈ (0 , R ) and s ∈ (0 , F ( r )) (cf. [5, Lemma 3.9]), (cid:90) M ∂ \ B ( x,r ) q ( s, x, y ) µ ( dy ) = ∞ (cid:88) k =0 (cid:90) B ( x, k +1 r ) \ B ( x, k r ) q ( s, x, y ) µ ( dy ) ≤ c exp (cid:0) − − c F ( r, s ) (cid:1) V ( x, F − ( s )) ∞ (cid:88) k =0 µ (cid:0) B ( x, k +1 r ) \ B ( x, k r ) (cid:1) exp (cid:0) − − c F (2 k r, s ) (cid:1) ≤ c V ( x, r ) exp (cid:0) − − c F ( r, s ) (cid:1) V ( x, F − ( s )) ∞ (cid:88) k =0 kd (cid:16) s k F ( r ) (cid:17) d ≤ c (cid:16) F ( r ) s (cid:17) d /γ (cid:16) sF ( r ) (cid:17) d exp (cid:0) − − c F ( r, s ) (cid:1) ≤ c exp (cid:0) − − c F ( r, s ) (cid:1) . Besides, by [3, Lemma 3.9(ii)], there exists a constant c > F ( r, s ) ≤ c for all r ∈ (0 , R ) and s ≥ F ( r ). Hence, by taking c larger than e c c / , we get that (cid:82) B ( x,r ) c q ( s, x, y ) µ ( dy ) ≤ c exp( − − c F ( r, s )) for all x ∈ M , r ∈ (0 , R ) and s > 0. In view of (2.5) and Tonelli’s theorem,it follows that for all x ∈ M and r ∈ (0 , R ), J (cid:0) x, M ∂ \ B ( x, r ) (cid:1) = (cid:90) ∞ (cid:90) M ∂ \ B ( x,r ) q ( s, x, y ) µ ( dy ) ν ( ds ) ≤ c (cid:90) ∞ exp (cid:0) − − c F ( r, s ) (cid:1) ν ( ds ) ≤ c (cid:90) (0 ,F ( r )] exp (cid:0) − − c F ( r, s ) (cid:1) ν ( ds ) + c w ( F ( r )) =: J + c w ( F ( r )) . (i) By (2.8) and Lemma A.1(i), using the fact that 1 − e − λ − λe − λ ≥ (2 e ) − ( λ ∧ 1) for all λ ≥ J + c w ( F ( r )) ≤ c (cid:90) (0 ,F ( r )] (cid:16) sF ( r ) (cid:17) ν ( ds ) + c w ( F ( r )) ≤ ( c + c ) (cid:90) ∞ (cid:16)(cid:16) sF ( r ) (cid:17) ∧ (cid:17) ν ( ds ) ≤ e ( c + c ) (cid:90) ∞ (cid:0) − e − s/F ( r ) − sF ( r ) − e − s/F ( r ) (cid:1) ν ( ds ) = 2 e ( c + c )Θ( r ) . (ii) By Lemma A.1(i), U F ( R ) ( w − , α , c ) and the integration by parts, for all r ∈ (0 , R ), J ≤ c (cid:90) (0 ,F ( r )] (cid:16) sF ( r ) (cid:17) α +1 ν ( ds ) ≤ c ( α + 1) F ( r ) − α − (cid:90) (0 ,F ( r )] s α w ( s ) ds ≤ c F ( r ) − α − F ( r ) α w ( F ( r )) (cid:90) (0 ,F ( r )] ds = c w ( F ( r )) . (A.2)Besides, by (2.5), (2.3) and (2.1), since U( F, γ , c U ) and U F ( R ) ( w − , α , c ) hold, by taking c > w (cid:0) F ( c (cid:96) ( R ∧ R )) (cid:1) ≥ w (cid:0) F ( R ) (cid:1) , where (cid:96) is the constant in (2.1), we getthat for all x ∈ M and r ∈ (0 , c ( R ∧ R )), J (cid:0) x, M ∂ \ B ( x, r ) (cid:1) ≥ c (cid:90) F ( R ) F ( (cid:96) r ) (cid:90) B ( x,F − ( s )) \ B ( x,r ) µ ( dy ) V ( x, F − ( s )) ν ( ds ) ≥ c (cid:90) F ( R ) F ( (cid:96) r ) ν ( ds ) ≥ c w (cid:0) F ( (cid:96) r ) (cid:1) ≥ c w (cid:0) F ( r ) (cid:1) . (A.3)(iii) By similar calculations to (A.2), since U F ( R ) ( w − , α , c ) holds, we see that for all r > R , J ≤ c ( α + 1) F ( r ) − α − (cid:16) (cid:90) (0 ,F ( R )] s α w ( s ) ds + (cid:90) ( F ( R ) ,F ( r )] s α w ( s ) ds (cid:17) ≤ c F ( r ) − α − (cid:0) F ( r ) α +1 w ( F ( r )) (cid:1) ≤ c w ( F ( r )) . Moreover, by similar calculations to (A.3), we also have that for all r > R , J (cid:0) x, M ∂ \ B ( x, r ) (cid:1) ≥ c (cid:90) ∞ F ( (cid:96) r ) µ (cid:0) B ( x, F − ( s )) \ B ( x, r ) (cid:1) V ( x, F − ( s )) ν ( ds ) ≥ c w (cid:0) F ( (cid:96) r ) (cid:1) ≥ c w (cid:0) F ( r ) (cid:1) . This completes the proof. (cid:50) Lemma A.4. (cf. [3, Lemma 4.5]) (i) If L ( φ , α , c ) holds, then NDL ε ( R ∧ (Φ , M ) holds where ε ∈ (0 , is the constant in (A.1) .(ii) If R = ∞ and L ( φ , α , c ) holds, then there exists c > such that NDL c (Φ , M ) holds. Proof. (i) Since L ( φ , α , c ) holds, we see that U φ (1) ( φ − , /α , c − /α ) holds. Then since φ − isincreasing, for all a > 0, there exists c > a ( φ − , /α , c ) holds (see [4, Remark 2.1]).Thus, by [73, Proposition 2.4], there exist constants c > c > t ∈ (0 , Φ( R ∧ P (cid:0) S t ∈ [2 − φ − ( t − ) − , c φ − ( t − ) − ] (cid:1) ≥ c . (A.4)Let η := (2 c /c L ) /γ η . Here, c L , γ and η are the constants in L( F, γ , c L ) and (A.1). By (A.1)and (A.4), since VRD ¯ R ( M ) holds, we get that for all r ∈ (0 , ε ( R ∧ x ∈ M and y, z ∈ B ( x, η r ), p B ( x,r ) (Φ( η r ) , y, z ) ≥ (cid:90) ∞ q B ( x,r ) ( s, y, z ) P ( S Φ( η r ) ∈ ds ) ≥ c inf (cid:8) q B ( x,r ) ( s, y, z ) : 2 − φ − (Φ( η r ) − ) − ≤ s ≤ c φ − (Φ( η r ) − ) − (cid:9) ≥ c V (cid:0) x, F − ( c F ( η r )) (cid:1) ≥ c V ( x, r ) . Indeed, since φ − (Φ( η r ) − ) − = F ( η r ) and L( F, β , c L ) holds, we see that c φ − (Φ( η r ) − ) − ≤ c c − L ( η /η ) β F ( η r ) ≤ F ( η r ) and η F − (2 − F ( η r )) ≥ η η ( c L / /β r > η r .(ii) Similarly, since L ( φ , α , c ) holds, for all a > 0, there exists c > a ( φ − , /α , c )holds. Thus, by applying [73, Proposition 2.4] again, one can see that (A.4) holds for all t ∈ (Φ(1) , ∞ ). Define η as in (i). Then by the same argument as the one for (i), we can verify thatNDL /η (Φ , M ) holds. (cid:50) Now, before giving proofs of Theorems 2.1, 2.2 and 2.3, we mention that although Π is notan increasing continuous function in general, under U F ( R ) ( w − , α , c ) (resp. U F ( R ) ( w − , α , c )),there exists an increasing continuous function Π such that Π( r ) (cid:16) Π ( r ) for all r ∈ (0 , R / 2) (resp. r ∈ ( R , ∞ )). Since our tail condition is invariant even if we change the scale function ψ to someother function comparable with ψ , this observation allows us to assume that Π is increasing andcontinuous without loss of generality. Indeed, for example, define an function Π on (0 , ∞ ) asΠ ( r ) := (cid:40)(cid:0) (cid:82) ∞ e − u w ( F ( ru )) du + e − r w ( F (2 − R )) (cid:1) − , if U F ( R ) ( w − , α , c ) holds , (cid:0) (cid:82) ∞ e − u w ( F ( ru )) du (cid:1) − , if U F ( R ) ( w − , α , c ) holds . (A.5)Since w is nonincreasing, we have that (cid:82) ∞ e − u w ( F ( ru )) du ≤ w ( F ( r )) and (cid:82) ∞ e − u w ( F ( ru )) du ≥ w ( F (2 r )) (cid:82) e − u du for all r > 0. Note that under U F ( R ) ( w − , α , c ) (resp. U F ( R ) ( w − , α , c )), ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 49 w ( F ( r )) (cid:16) w ( F (2 r )) for all r ∈ (0 , R / 2) (resp. r ∈ ( R , ∞ )). Thus, we see that Π ( r ) (cid:16) Π( r ) forall r ∈ (0 , R / 2) (resp. r ∈ ( R , ∞ )) under U F ( R ) ( w − , α , c ) (resp. U F ( R ) ( w − , α , c )). Proof of Theorem 2.1. By (2.7) and Lemmas A.2, A.3(i) and A.4(ii), the results follow from Theorems1.13, 1.15 and Corollary 1.16. (cid:50) Proof of Theorem 2.2. (i) Suppose that lim sup r → φ ( r − ) /H ( r − ) < ∞ . Then φ ( λ ) (cid:16) H ( λ ) forall λ ≥ 1. Since lim r → r − /H ( r − ) ≥ lim r → r − /φ ( r − ) = ∞ , it follows that b = 0 and hence φ ( λ ) = φ ( λ ). Note that ( λ − φ ( λ )) (cid:48) = − λ − H ( λ ). Thus, we can see that for all λ ≥ λ − H ( λ ) (cid:16) λ − φ ( λ ) = λ − φ ( λ ) = (cid:90) ∞ λ u − H ( u ) du, (A.6)According to [15, Corollary 2.6.4], (A.6) implies that U ( H , − ε, c ) holds with some constants ε, c > 0. Hence, by [73, Lemma 2.6], there exists c ∈ (0 , 1) such that w ( r ) (cid:16) H ( r − ) (cid:16) φ ( r − )for all r ∈ (0 , c ). In particular, lim sup r → φ ( r − ) /w ( r ) < ∞ , and U c ( w − , , c ) holds. Finally,by Lemmas A.2 and A.3(ii), we conclude the result from Theorem 1.17(i).(ii) By (2.7) and Lemmas A.2, A.3(i,ii) and A.4(i), we obtain the result from Theorems 1.17(ii) and1.19 (see the paragraph below Lemma A.4). (cid:50) Proof of Theorem 2.3. (i) Suppose that lim sup r →∞ φ ( r − ) /H ( r − ) < ∞ . Since lim r →∞ rφ ( r − )= lim λ → φ (cid:48) ( λ ) ∈ (0 , ∞ ] and φ ( λ ) ≥ H ( λ ) for all λ , we see that (A.6) holds for all λ ∈ (0 , f ( s ) := sH (1 /s ). By the change of the variables, we obtain (cid:82) s u − f ( u ) du = sφ (1 /s ) (cid:16) f ( s ) forall s ≥ 1. By [15, Corollary 2.6.2], it follows that L ( f, ε, c ) holds with some constants ε, c > ( H , − ε, c − ) holds. Thus, by [27, Lemma 2.1(iii) and (iv)], we get that w ( r ) (cid:16) H ( r − ) (cid:16) φ ( r − ) for all r ≥ 1. Therefore, lim sup r →∞ φ ( r − ) /w ( r ) < ∞ and U ( w − , , c ) holds. Thanksto Lemmas A.2 and A.3(iii), the result follows from Theorem 1.18(i).(ii) By Lemmas A.2, A.3(i,iii) and A.4(ii), the result follows from Theorems 1.18(ii) and 1.20 (seethe paragraph below Lemma A.4). (cid:50) Appendix B. Proof of Theorem 2.8 Throughout this subsection, we assume that Assumption L holds, and that, without loss ofgenerality, β ≤ γ . Here, β is the constant in (L1), and γ is the one in (2.2). When R < ∞ , weextend ψ to all r ∈ (0 , ∞ ) by setting ψ ( r ) = ψ ( R − ) R − γ r γ for r ≥ R . Note that,L( ψ , β , c ) and U( ψ , β + 2 γ , c ) hold. (B.1) Lemma B.1. Tail ( ¯ R ∧ R ) /(cid:96) ( ψ , U ) holds where (cid:96) is the constant in (2.1) . Proof. Choose any x ∈ U and 0 < (cid:96) r < ¯ R ∧ R ∧ ( κ δ U ( x )). Then, by (L2), (L3), (B.1) and [21,Lemma 2.1], since VRD ¯ R ( M ) holds, we see that J X (cid:0) x, M ∂ \ B ( x, r ) (cid:1) ≤ (cid:90) { r ≤ d ( x,y ) 0; ¯ P x , x ∈ M \ ¯ N } be a Hunt process associated with ( E , F ) where ¯ N is the properly exceptional set for ¯ X . Write the strongly part of ( E , F ) as E ( c ) , and let Γ and Γ ( c ) be the energy measures for ( E , F ) and ( E ( c ) , F ), respectively. Set F b := F ∩ L ∞ ( M ; µ ). For an open set D ⊂ M , denote by F D the E -closure of F ∩ C c ( D ) in F .Here, we introduce local (and interior) versions of a cut-off Sobolev inequality , the Faber-Krahninequality and the (weak) Poincar´e inequality for ( E , F ). See [22] for global versions.For open sets D ⊂ U ⊂ M with D ⊂ U , we say a measurable function ϕ on M is a cut-offfunction for D ⊂ U , if ϕ = 1 on D , ϕ = 0 on M \ U and 0 ≤ ϕ ≤ M . Definition B.2. For an open set U ⊂ M and R ∈ (0 , ∞ ], we say that a local cut-off Sobolevinequality CS R ( φ, U ) (for ( E , F )) holds (with C ) if there exist constants C ∈ (0 , a ∈ (0 , 1] and c > x ∈ U , all 0 < r ≤ R < R ∧ ( C δ U ( x )) and any f ∈ F , there exists acut-off function ϕ ∈ F b for B ( x , R ) ⊂ B ( x , R + r ) satisfying the following inequality: (cid:90) B ( x ,R +(1+ a ) r ) f d Γ( ϕ, ϕ ) ≤ cφ ( r ) (cid:90) B ( x ,R +(1+ a ) r ) f dµ + c (cid:90) B ( x ,R + r ) ϕ d Γ ( c ) ( f, f )+ c (cid:90) B ( x ,R + r ) × B ( x ,R +(1+ a ) r ) ϕ ( x )( f ( x ) − f ( y )) J ( dx, dy ) . Definition B.3. For an open set U ⊂ M and R ∈ (0 , ∞ ], we say that a local Faber-Krahn inequality FK R ( φ, U ) (for ( E , F )) holds (with C and ν ) if there exist constants C, ν ∈ (0 , 1) and c > x ∈ U , 0 < r < R ∧ ( C δ U ( x )) and any open set D ⊂ B ( x, r ), λ ( D ) ≥ cφ ( r ) (cid:18) V ( x, r ) µ ( D ) (cid:19) ν , where λ ( D ) := inf {E ( f, f ) : f ∈ F D with (cid:107) f (cid:107) = 1 } . Definition B.4. For an open set U ⊂ M and R ∈ (0 , ∞ ], we say that a (weak) Poincar´e inequality PI R ( φ, U ) (for ( E , F )) holds (with C ) if there exist constants C ∈ (0 , 1) and c > x ∈ U , 0 < r < R ∧ ( C δ U ( x )) and any f ∈ F b , (cid:90) B ( x,r ) ( f − ¯ f B ( x,r ) ) dµ ≤ cφ ( r ) (cid:16) (cid:90) B ( x,r/C ) d Γ ( c ) ( f, f ) + (cid:90) B ( x,r/C ) × B ( x,r/C ) ( f ( y ) − f ( x )) J ( dx, dy ) (cid:17) , where ¯ f B ( x,r ) := V ( x,r ) (cid:82) B ( x,r ) f dµ is the average value of f on B ( x, r ).We first show that FK R ( φ, U ) is equivalent to an existence and upper bound (B.3) of Dirichletheat kernel for small balls contained in U (and a local Nash inequality). When U = M and R = ∞ ,the following result was established in [47, Section 5] (when φ ( r ) = r β for some β > 0) and [21,Proposition 7.3]. Even though the proof is similar, we give the full proof for the reader’s convenience.For an open set D ⊂ M , we write ¯ τ D , and ¯ p D ( t, x, y ) for the first exit time of ¯ X from D and heatkernel of killed process ¯ X D , respectively. Lemma B.5. Let U ⊂ M be an open set, and R ∈ (0 , ∞ ] and ν, κ ∈ (0 , be constants. Then, thefollowings are equivalent. (1) FK R ( φ, U ) holds with κ and ν . (2) There exists a constant K > such that for all x ∈ U and < r < R ∧ ( κ δ U ( x )) , V ( x, r ) ν φ ( r ) (cid:107) u (cid:107) ν (cid:107) u (cid:107) − ν ≤ K E ( u, u ) , ∀ u ∈ F B ( x,r ) . (B.2)(3) There exists a constant K > such that for all x ∈ U and < r < R ∧ ( κ δ U ( x )) , the(Dirichlet) heat kernel ¯ p B ( x,r ) ( t, · , · ) exists and satisfies that ess sup y,z ∈ B ( x,r ) ¯ p B ( x,r ) ( t, y, z ) ≤ K V ( x, r ) (cid:16) φ ( r ) t (cid:17) /ν , ∀ t > . (B.3) ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 51 Proof. Choose any x ∈ U and 0 < r < R ∧ ( κ δ U ( x )), and denote by B = B ( x, r ).(1) ⇒ (2). We first assume that 0 ≤ u ∈ F ∩ C c ( B ). Set B s := { y ∈ B : u ( y ) > s } for s > B s ⊂ B is open because u is continuous. By the Markovian property of E , since FK R ( φ, U )holds, we have that, for any t > s > E ( u, u ) ≥ E (cid:0) ( u − t ) + , ( u − t ) + (cid:1) ≥ λ ( B s ) (cid:90) B s (( u − t ) + ) dµ ≥ cφ ( r ) (cid:16) V ( x, r ) µ ( B s ) (cid:17) ν (cid:90) B s (( u − t ) + ) dµ. (B.4)Observe that since u ≥ 0, we have (( u − t ) + ) ≥ u − tu . Thus, we get that (cid:82) B s (( u − t ) + ) dµ = (cid:82) M (( u − t ) + ) dµ ≥ (cid:82) M ( u − tu ) dµ = (cid:107) u (cid:107) − t (cid:107) u (cid:107) . Moreover, by the definition of B s , it holdsthat µ ( B s ) ≤ s − (cid:82) B s udµ ≤ s − (cid:107) u (cid:107) . Therefore, we get from (B.4) that for any t > s > E ( u, u ) ≥ cV ( x, r ) ν φ ( r ) (cid:16) s (cid:107) u (cid:107) (cid:17) ν ( (cid:107) u (cid:107) − t (cid:107) u (cid:107) ) . By letting t ↓ s and taking s = (cid:107) u (cid:107) / (4 (cid:107) u (cid:107) ) in the above inequality, we obtain that E ( u, u ) ≥ cV ( x, r ) ν φ ( r ) (cid:16) (cid:107) u (cid:107) (cid:107) u (cid:107) (cid:17) ν (cid:107) u (cid:107) c − ν − V ( x, r ) ν φ ( r ) (cid:107) u (cid:107) ν (cid:107) u (cid:107) − ν . Thus, (B.2) holds for any 0 ≤ u ∈ F ∩ C c ( B ). Then since E ( | u | , | u | ) ≤ E ( u, u ) by the Markovianproperty, we can see that (B.2) holds for any signed u ∈ F ∩ C c ( B ).Now, we consider a general u ∈ F B . By the definition of F B , there exists a sequence ( u n ) n ≥ ⊂F ∩ C c ( B ) such that E ( u n − u, u n − u ) + (cid:107) u n − u (cid:107) → 0. Since µ ( B ) < ∞ , by Cauchy inequality, italso holds (cid:107) u n − u (cid:107) ≤ (cid:112) µ ( B ) (cid:107) u n − u (cid:107) → 0. Thus, since (B.2) holds for each u n ∈ F ∩ C c ( B ), bypassing to the limit as n → ∞ , we conclude that (B.2) holds for any u ∈ F B .(2) ⇒ (3). Let 0 ≤ f ∈ L ( B ; µ ) with (cid:107) f (cid:107) = 1. Set u t ( · ) = ¯ P Bt f ( · ) := E · f ( ¯ X Bt ) and denote m ( t ) = (cid:107) u t (cid:107) for t > 0. Then since ( E , F ) is regular, we have u t ∈ F B for every t > 0. Hence, by(2), because ¯ P Bt is a L -contraction, it holds that − K dmdt = K E ( u t , u t ) ≥ V ( x, r ) ν φ ( r ) m ( t ) ν (cid:107) u (cid:107) − ν ≥ V ( x, r ) ν φ ( r ) m ( t ) ν =: a ( r ) m ( t ) ν . It follows ddt ( m ( t ) − ν ) ≥ ν − K − a ( r ) so that m ( t ) ≤ − /ν ν /ν K /ν a ( r ) − /ν t − /ν . Hence, we obtain (cid:107) ¯ P Bt (cid:107) L → L ≤ c a ( r ) − / (2 ν ) t − / (2 ν ) . Thus, by [47, Lemma 3.7], we conclude that (3) holds.(3) ⇒ (1) Let D ⊂ B be an open set. By (3), we see that for µ -almost every y ∈ D , P y (¯ τ D > t ) = (cid:90) D ¯ p D ( t, y, z ) µ ( dz ) ≤ (cid:90) D ¯ p B ( t, y, z ) µ ( dz ) ≤ K µ ( D ) V ( x, r ) (cid:16) φ ( r ) t (cid:17) /ν . (B.5)Set T := (cid:0) µ ( D ) /V ( x, r ) (cid:1) ν φ ( r ). It follows from (B.5) that for µ -almost every y ∈ D ,¯ E y ¯ τ D ≤ T + (cid:90) ∞ T ¯ P y (¯ τ D > t ) dt ≤ T + K T /ν (cid:90) ∞ T t − /ν dt = (cid:0) ν (1 − ν ) − K (cid:1) T. (B.6)Thus, by [47, Lemma 6.2], we obtain that λ ( D ) ≥ (ess sup y ∈ D ¯ E y ¯ τ D ) − ≥ c T − . (cid:50) Corollary B.6. Suppose that one of the equivalent conditions of Lemma B.5 is satisfied. Then thereexists a constant c > such that ¯ E x [¯ τ B ( x,r ) ] ≤ cφ ( r ) for all x ∈ U \ ¯ N and < r < R ∧ ( κ δ U ( x )) . Proof. Choose any x ∈ U \ ¯ N and 0 < r < R ∧ ( κ δ U ( x )). By taking D = B ( x, r ) in (B.6), we seethat ess sup y ∈ B ( x,r ) ¯ E y ¯ τ B ( x,r ) ≤ c φ ( r ) for a constant c > x and r . Hence, by thestrong Markov property, since the heat kernel ¯ p B ( x,r ) ( t, · , · ) exists due to Lemma B.5, we obtain¯ E x [¯ τ B ( x,r ) ] ≤ φ ( r ) + ¯ E x [¯ τ B ( x,r ) − φ ( r ); ¯ τ B ( x,r ) ≤ φ ( r )] + ¯ E x [¯ τ B ( x,r ) − φ ( r ); ¯ τ B ( x,r ) > φ ( r )] ≤ φ ( r ) + (cid:90) B ( x,r ) ¯ E z [¯ τ B ( x,r ) ]¯ p B ( x,r ) ( φ ( r ) , x, z ) µ ( dz ) ≤ (1 + c ) φ ( r ) . (cid:50) A global version of the following lemma can be founded in [21, Proposition 7.6]. Lemma B.7. Let U ⊂ M be an open set, R ∈ (0 , ∞ ] a constant, and φ an increasing functionsatisfying U( φ, γ, c ) for some γ > . Suppose that PI R ( φ, U ) holds. Then there exists a constant ε ∈ (0 , such that FK ε ( R ∧ ¯ R ) ( φ, U ) holds. Proof. Without loss of generality, we may assume that R ≤ ¯ R . By Lemma B.5, it suffices toprove that PI R ( φ, U ) implies (2) therein with R = εR for some ε ∈ (0 , C V ∈ (0 , 1) be theconstant that VRD ¯ R ( M ) holds with.By following the proof of [78, Theorem 2.1] or [31, Proposition 2.3], since PI R ( φ, U ) holds, onecan see that the following Sobolev-type inequality holds: There is constants ν ∈ (0 , κ ∈ (0 , C V )and c > x ∈ U and 0 < r < R ∧ ( κ δ U ( x )), (cid:107) u (cid:107) ν ≤ c V ( x, r ) ν (cid:107) u (cid:107) ν (cid:0) φ ( r ) E ( u, u ) + (cid:107) u (cid:107) (cid:1) , ∀ u ∈ F B ( x,r ) . (B.7)Indeed, even though they only proved (B.7) when φ ( r ) = r , with simple modifications, one caneasily follow their proof with general φ satisfying U( φ, γ, c ).Now, we adopt a method in the proof of [31, Proposition 2.3]. Choose a constant C > c V ( x, r ) ν ≤ V ( x, Cr ) ν for all x ∈ U, < r < R ∧ ( κ δ U ( x )) . Since VRD ¯ R ( M ) holds, R ≤ ¯ R and κ < C V , such constant C exists. By (B.7) and Cauchyinequality, we get that for all x ∈ U , 0 < r < C − (cid:0) R ∧ ( κ δ U ( x )) (cid:1) and u ∈ F B ( x,r ) ⊂ F B ( x,Cr ) , (cid:107) u (cid:107) ν ≤ c V ( x, Cr ) ν (cid:107) u (cid:107) ν (cid:0) φ ( Cr ) E ( u, u ) + (cid:107) u (cid:107) (cid:1) ≤ φ ( Cr )2 V ( x, r ) ν (cid:107) u (cid:107) ν E ( u, u ) + (cid:107) u (cid:107) ν V ( x, r ) ν (cid:107) u (cid:107) ≤ φ ( Cr )2 V ( x, r ) ν (cid:107) u (cid:107) ν E ( u, u ) + 12 (cid:107) u (cid:107) ν . Thus, since U( φ, γ, c ) holds, we conclude that (2) in Lemma B.5 holds with R = C − R . (cid:50) Now, let us define a L´evy measure ν and a Bernstein function φ as (cf. [3, (4.6)]) ν ( ds ) = ν ( ds ; ψ ) := dssψ ( F − ( s )) , φ ( λ ) := Λ λ + (cid:90) ∞ (1 − e − λs ) ν ( ds ) . (B.8)Since (L1) and U( F, γ , c U ) hold, one can see ν ((0 , ∞ )) = ∞ and (cid:82) ∞ (1 ∧ s ) ν ( ds ) < ∞ . Hence, theabove φ is well-defined.Let S t be a subordinator with the Laplace exponent φ , and set Y t := Z S t . Define J Y ( x, y ) as theright hand side of the second equality in (2.5) with ν = ν . Then one can see that Y t is a symmetricHunt process associated with the following regular Dirichlet form ( E Y , F Y ) on L ( M ; µ ): E Y ( f, f ) = Λ E Z ( f, f ) + (cid:90) M × M \ diag ( f ( x ) − f ( y )) J Y ( x, y ) µ ( dx ) µ ( dy ) , f ∈ F Y , (B.9) F Y = (cid:8) f ∈ C c ( M ) : E Y ( f, f ) < ∞ (cid:9) E Y . Moreover, we have that, F Z ⊂ F Y , and F Z = F Y if Λ > 0. See [1].Recall the definition of Φ from (2.26). ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 53 Lemma B.8. (i) It holds that ν (( r, ∞ )) (cid:16) ψ ( F − ( r )) − on (0 , ∞ ) .(ii) L( φ , α , c ) holds for some α , c > .(iii) It holds that Φ ( r ) (cid:16) φ ( F ( r ) − ) − on (0 , ∞ ) . Proof. (i) Observe that L( ψ ◦ F − , γ − β , c ) and U( ψ ◦ F − , γ − ( β + 2 γ ) , c ) hold for someconstants c ≥ ≥ c > 0. Thus, by [28, Lemma 2.3(1)], we get the result.(ii) Let φ ( λ ) := φ ( λ ) − Λ λ . Since L( ψ ◦ F − , γ − β , c ) holds, by a similar argument to the onefor [28, Lemma 2.3(3)], we see that L( φ , α , c ) holds for some α ∈ (0 , > 0. Then we see that φ ( λ ) (cid:16) φ ( λ ) on (0 , φ ( λ ) (cid:16) Λ λ on [1 , ∞ ) (see Example 2.5). Since Λ λ is linear, we deduce the result from L( φ , α , c ).(iii) By [27, Lemma 2.1(i)], the above (i) and the change of the variables, we obtain φ (cid:0) F ( r ) − (cid:1) − (cid:16) F ( r ) (cid:16) (cid:90) F ( r )0 ν (( s, ∞ )) ds + Λ (cid:17) − (cid:16) Φ ( r ) on (0 , ∞ ) . (cid:50) As a consequence of the above Lemma B.8(ii, iii), since U( φ , , 1) holds, we have thatL(Φ , α γ , c (cid:48) L ) and U(Φ , γ , c (cid:48) U ) hold with some constants c (cid:48) U ≥ ≥ c (cid:48) L > 0. (B.10) Lemma B.9. (i) There exists a ≥ such that for all x, y ∈ M with d ( x, y ) < F − (2 − ( F ( R )) , a − V ( x, d ( x, y )) ψ ( d ( x, y )) ≤ J Y ( x, y ) ≤ a V ( x, d ( x, y )) ψ ( d ( x, y )) . (B.11) (ii) There exists a ≥ such that for all x, y ∈ U with d ( x, y ) < F − (2 − F ( R )) ∧ R ∧ ( κ δ U ( x )) , a − J X ( x, y ) ≤ J Y ( x, y ) ≤ a J X ( x, y ) . (B.12) Proof. (i) Fix any x, y ∈ M with d ( x, y ) < F − (2 − ( F ( R )) and let l := d ( x, y ).First, we have that, by (2.2), (2.3), (B.1) and (B.8), since VRD ¯ R ( M ) holds, J Y ( x, y ) ≥ (cid:90) F ( l ) F ( l ) q ( s, x, y ) ν ( ds ) ≥ c V ( x, l ) F ( l ) ψ ( l ) (cid:90) F ( l ) F ( l ) ds = c V ( x, l ) ψ ( l ) . Next, to prove the upper bound in (B.11), we claim that there exists a constant c > x such that sup z ∈ M q ( t, x, z ) ≤ c V ( x, F − ( t ) ∧ R ) , ∀ t > . (B.13)Indeed, in view of (2.3), the above (B.13) holds for all t ∈ (0 , F ( R )). Hence, it suffices to considerthe case R < ∞ . In such case, by the semigroup property, since Z is µ -symmetric and VRD ¯ R ( M )holds, we see that for all t ≥ F (2 − R ) and z ∈ M , q ( t, x, z ) = (cid:90) M q ( F (2 − R ) , x, w ) q ( t − F (2 − R ) , w, z ) µ ( dw ) ≤ c V ( x, − R ) (cid:90) M q ( t − F (2 − R ) , z, w ) µ ( dw ) ≤ c V ( x, R ) . Thus, one can deduce that (B.13) holds.Now, by (2.2), (2.3), (B.1), (B.8), (B.13), and Lemmas A.1(i) and B.8(i), since VRD ¯ R ( M ) holds,we obtain J Y ( x, y ) = (cid:16) (cid:90) F ( l )0 + (cid:90) F ( R ) F ( l ) + (cid:90) ∞ F ( R ) (cid:17) q ( s, x, y ) ν ( ds ) ≤ (cid:90) F ( l )0 c exp (cid:0) − c F ( l, s ) (cid:1) sV ( x, F − ( s )) ψ ( F − ( s )) ds + (cid:90) F ( R ) F ( l ) c dssV ( x, F − ( s )) ψ ( F − ( s )) + c V ( x, R ) ψ ( R ) ≤ (cid:90) F ( l )0 c F ( l ) V ( x, F − ( s )) ψ ( l ) (cid:16) sF ( l ) (cid:17) d /γ ds + c V ( x, l ) ψ ( l ) + c V ( x, R ) ψ ( R ) ≤ c V ( x, l ) ψ ( l ) . We used conventions that V ( x, ∞ ) = ψ ( ∞ ) = ∞ in the above.(ii) The result follows from (B.11) and (L2). (cid:50) Proposition B.10. There exists a constant ε ∈ (0 , such that CS ε R (Φ , M ) , FK ε R (Φ , M ) and PI ε R (Φ , M ) for ( E Y , F Y ) hold. Proof. By Lemmas A.3(i), A.4(i,ii) and B.8(ii,iii), we see that Tail R (Φ , M, ≤ ) and NDL c R (Φ , M )for Y hold with some c ∈ (0 , Y satisfies Assumption 3.5with U = M . For an open set D ⊂ M , denote by τ YD the first exit time of Y from D , and ( P Y,Dt ) t ≥ the semigroup of the subprocess Y D in L ( M ; µ ). By Proposition 3.10(i) and (B.10), it follows thatthere exist constants c ∈ (0 , 1] and c > z ∈ M and r ∈ (0 , c R ),ess inf y ∈ B ( z, r ) P Y,B ( z,r ) t ( y ) ≥ ess inf y ∈ B ( z, r ) P y ( τ YB ( y, r ) > t ) ≥ / , ∀ < t ≤ c Φ ( r ) . (B.14)By (B.14), since Tail R (Φ , M, ≤ ) for Y holds, one can follow the proofs of [46, Lemma 2.8] and [22,Proposition 2.5] in turn to deduce that CS c R (Φ , M ) for ( E Y , F Y ) holds with some c ∈ (0 , c ).Next, since NDL c R (Φ , M ) for Y holds, by following the proof of [24, Proposition 3.5(i)], wecan deduce that PI c R (Φ , M ) for ( E Y , F Y ) holds with some c ∈ (0 , E given in the fifthline of the proof of [24, Proposition 3.5(i)] by¯ E ( u, v ) = Λ (cid:90) B ( x ,r ) d Γ Z ( u, v ) + (cid:90) B ( x ,r ) × B ( x ,r ) ( u ( x ) − u ( y ))( v ( x ) − v ( y )) J Y ( x, y ) µ ( dx ) µ ( dy ) , one can follow the rest of the proof.Lastly, since R ≤ ¯ R , by Lemma B.7, FK c R (Φ , M ) for ( E Y , F Y ) holds with some c ∈ (0 , c ).In the end, we finish the proof by taking ε = c ∧ c . (cid:50) Lemma B.11. It holds that F X U = F Y U . Proof. By (L3), (L4) and (B.12), since Tail R (Φ , M, ≤ ) for Y holds due to Lemmas A.3(i) andB.8(iii), one can see that for every f ∈ C c ( U ), E X ( f, f ) < ∞ if and only if E Y ( f, f ) < ∞ . Therefore,we have F X ∩ C c ( U ) = F Y ∩ C c ( U ).Let f ∈ F X U . Then there exists a sequence ( f n ) n ≥ ⊂ F X ∩ C c ( U ) such that E X ( f − f n , f − f n ) → S ⊂ U such that supp( f ) , supp( f n ) ⊂ S and let δ := inf y ∈ S,z ∈ M \U d ( y, z ) > R (Φ , M, ≤ ) for Y holds, we have that E Y ( f − f n , f − f n ) ≤ c E X ( f − f n , f − f n ) + (cid:90) U (cid:90) M \ B ( x,c δ ) (cid:0) ( f ( x ) − f n ( x )) − ( f ( y ) − f n ( y )) (cid:1) J Y ( x, y ) µ ( dx ) µ ( dy )+ (cid:90) M \U (cid:90) S (cid:0) ( f ( x ) − f n ( x )) − ( f ( y ) − f n ( y )) (cid:1) J Y ( x, y ) µ ( dx ) µ ( dy ) ≤ c E X ( f − f n , f − f n ) + c Φ (( c ∧ δ ) (cid:107) f − f n (cid:107) L ( M ; µ ) → n → ∞ . Hence, f ∈ F Y U . Conversely, by the same argument, one can see that F Y U ⊂ F X U . (cid:50) Proposition B.12. There exists a constant R ∈ (0 , ∞ ] such that CS R (Φ , U ) , FK R (Φ , U ) and PI R (Φ , U ) for ( E X , F X U ) hold. In particular, if U = M and R = R = ∞ , then CS ∞ (Φ , M ) , FK ∞ (Φ , M ) and PI ∞ (Φ , M ) for ( E X , F X ) hold. ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 55 Proof. By Proposition B.10, CS ε R (Φ , M ), FK ε R (Φ , M ) and PI ε R (Φ , M ) for ( E Y , F Y ) holdwith a constant κ ∈ (0 , 1) for some ε ∈ (0 , κ ∈ (0 , 1) be the constant in Assumption L. Let C ≥ (cid:101) κ := κ κ/ (3 C ) and R := ε (cid:101) κ ( F − (2 − F ( R )) ∧ R ).Choose any x ∈ U and r ∈ (0 , R ∧ ( (cid:101) κ δ U ( x ))).For all y, z ∈ B ( x, Cr/κ ), we have d ( y, z ) < Cr/κ < CR /κ < F − (2 − F ( R )) ∧ R and d ( y, z ) κ < Crκ κ < C (cid:101) κκ κ δ U ( x ) = 23 δ U ( x ) ≤ δ U ( x ) − C (cid:101) κκ δ U ( x ) < δ U ( x ) − Crκ < δ U ( y ) . Hence, by (L4) and (B.12), there exists c > x and r such that for any f ∈ F X U ,Λ (cid:90) B ( x,Cr/κ ) d Γ Z ( f, f ) + (cid:90) B ( x,Cr/κ ) × B ( x,Cr/κ ) ( f ( y ) − f ( z )) J Y ( y, z ) µ ( dy ) µ ( dz ) ≤ c (cid:18) Λ (cid:90) B ( x,r/ (cid:101) κ ) d Γ X, ( c ) ( f, f ) + (cid:90) B ( x,r/ (cid:101) κ ) × B ( x,r/ (cid:101) κ ) ( f ( y ) − f ( z )) J X ( y, z ) µ ( dy ) µ ( dz ) (cid:19) . (B.15)Thus, we can deduce that PI R (Φ , U ) for ( E X , F X U ) holds from PI ε R (Φ , M ) for ( E Y , F Y ) andLemma B.11.Next, by (B.9), (B.15), (L4), Lemmas A.3(i), B.8(iii) and B.11, (B.10) and FK ε R (Φ , M ) for( E Y , F Y ), we have that, for any open set D ⊂ B ( x, r ) and f ∈ F X U with (cid:107) f (cid:107) = 1, E X ( f, f ) ≥ c − E Y ( f, f ) − (cid:107) f (cid:107) sup y ∈ B ( x,r ) (cid:90) B ( x,Cr/κ ) c J Y ( y, z ) µ ( dz ) ≥ c − E Y ( f, f ) − c Φ (( C/κ − r ) ≥ c − c Φ ( r ) (cid:18) V ( x, r ) µ ( D ) (cid:19) ν − c c − L ( C/κ − α γ Φ ( r ) ≥ c − c − c c (cid:48)− L ( C/κ − − α γ Φ ( r ) (cid:18) V ( x, r ) µ ( D ) (cid:19) ν . (B.16)The last inequality above is valid since V ( x, r ) /µ ( D ) ≥ 1. Here, we point out that the constants c , c and c are independent of x and r . Now, we choose C = 1 + (2 c c / ( c c (cid:48) L )) / ( α γ ) . Then weget from (B.16) that FK R (Φ , U ) for ( E X , F X U ) holds.Now, we show that CS R (Φ , U ) for ( E X , F X U ) holds. Observe that for all x ∈ U , f ∈ F X U ,0 < u ≤ s < R ∧ ( (cid:101) κ δ U ( x )) and a ∈ (0 , ϕ for B ( x , s ) ⊂ B ( x , s + u ),by (L3), (L4), (B.12) and Lemma B.1, it holds that (cid:90) B ( x ,s +(1+ a ) u ) f d Γ X ( ϕ, ϕ )= (cid:90) B ( x ,s +(1+ a ) u ) f d Γ X, ( c ) ( ϕ, ϕ ) + (cid:90) B ( x ,s +(1+ a ) u ) × M f ( y ) ( ϕ ( y ) − ϕ ( z )) J X ( y, z ) µ ( dy ) µ ( dz ) ≤ c (cid:90) B ( x ,s +(1+ a ) u ) f d Γ Y ( ϕ, ϕ ) + (cid:90) B ( x ,s +(1+ a ) u ) f ( y ) J X (cid:16) y, M \ B (cid:0) y, ( ε − (cid:101) κ − R ) ∧ ( κ δ U ( y )) (cid:1)(cid:17) µ ( dy ) ≤ c (cid:90) B ( x ,s +(1+ a ) u ) f d Γ Y ( ϕ, ϕ ) + c ψ ( u ) (cid:90) B ( x ,s +(1+ a ) u ) f dµ. (B.17)Therefore, since CS ε R (Φ , M ) for ( E Y , F Y ) hold and Φ ( u ) ≤ ψ ( u ) for all u ∈ (0 , R ), we candeduce from Lemma B.11, (B.17), (L4) and (B.12) that CS R (Φ , U ) for ( E X , F X U ) holds.Lastly, since R = ∞ if R = R = ∞ , and F X U = F X if U = M , the latter assertion holds. (cid:50) Proposition B.13. There exists a constant R ∈ (0 , ∞ ] such that NDL R (Φ , U ∩ M ) for X holds.Moreover, if U = M and R = R = ∞ , then NDL ∞ (Φ , M ) for X holds. Proof. According to Lemma B.1 and Proposition B.12, since Φ ( r ) ≤ ψ ( r ) for r ∈ (0 , R ), we seethat Tail R (Φ , U , ≤ ), FK R (Φ , U ) and CS R (Φ , U ) hold with κ for some R ∈ (0 , ∞ ] and κ ∈ (0 , R ∈ (0 , R ] and c , c ∈ (0 , 1) such that for any x ∈ U ∩ M and r ∈ (0 , R ∧ ( κ δ U ( x ))), P x ( τ B ( x,r ) ≥ Φ (2 c r )) ≥ c . (B.18)Fix any x ∈ U ∩ M , 0 < r < R ∧ ( κ δ U ( x )) and let B = B ( x , r ). By Lemma B.5, the heatkernel p B ( t, x, y ) of X B = ( X U ) B exists. Moreover, since X B is symmetric, by (B.18) and Cauchyinequality, we have that for all x ∈ B ( x , r/ \ N and t ≤ Φ ( c r ), p B ( t, x, x ) = (cid:90) B p B ( t/ , x, y ) µ ( dy ) ≥ µ ( B ) (cid:18)(cid:90) B p B ( t/ , x, y ) µ ( dy ) (cid:19) ≥ V ( x , r ) P x (cid:0) τ B ( x,r/ ≥ Φ ( c r ) (cid:1) ≥ c V ( x , r ) . (B.19)We claim that after assuming that R and κ are sufficiently small, there are constants c , θ > η ∈ (0 , 1) independent of x and r such that for all t ≤ Φ ( c r ) and x, y ∈ B ( x , η Φ − ( t )) \ N , (cid:12)(cid:12) p B ( x ,c − Φ − ( t )) ( t, x, x ) − p B ( x ,c − Φ − ( t )) ( t, x, y ) (cid:12)(cid:12) ≤ c V ( x , Φ − ( t )) (cid:18) d ( x, y )Φ − ( t ) (cid:19) θ . (B.20)To prove (B.20), we first obtain a local version of elliptic H¨older regularity (EHR) of harmonicfunctions. First, since CS R (Φ , U ) holds, we can follow the proofs of [22, Proposition 2.9] and [24,Proposition 4.12] to get local versions of those two proposition, by applying Tail R (Φ , U , ≤ ) inthe third inequality in the display of the proof for [22, Proposition 2.9], and the last inequalityin [24, p.3787]. By using those two results, one can see that a local version of [22, Proposition5.1] holds by following its proof line by line, after redefining the notation Tail φ ( u ; x , r ) therein by φ ( r ) (cid:82) M ∂ \ B ( x ,r ) | u ( z ) | J X ( x , dz ). Besides, since FK R (Φ , U ) and CS R (Φ , U ) hold, by following theproofs of [21, Lemmas 4.6, 4.8 and 4.10] and using our Tail R (Φ , U , ≤ ) whenever the condition J φ, ≤ therein used, one can obtain their local versions hold (so-called Caccioppoli inequality, comparisoninequality over balls and L -mean value inequality for subharmonic functions). In the end, by usinglocal versions of [22, Proposition 5.1] and [21, Lemmas 4.8, 4.10], since Tail R (Φ , U , ≤ ), CS R (Φ , U )and PI R (Φ , U ) hold, one can follow the proofs of [24, Corollary 4.13 and Proposition 4.14] in turn toobtain a local EHR. Now, by using the local EHR, Corollary B.6, (B.3) and (B.10), since VRD ¯ R ( M )holds, we can follow the proof of [24, Lemma 4.8] and deduce that (B.20) holds.Eventually, by choosing (cid:101) η ∈ (0 , c η / 2) sufficiently small, since VRD ¯ R ( M ) holds, we get from(B.19) and (B.20) that for all y, z ∈ B ( x , (cid:101) η r ) \ N , p B (Φ ( (cid:101) ηr ) , y, z ) ≥ p B ( x ,c − (cid:101) ηr ) (Φ ( (cid:101) ηr ) , y, z ) ≥ p B ( x ,c − (cid:101) ηr ) (Φ ( (cid:101) ηr ) , y, y ) − c V ( x , (cid:101) ηr ) (cid:18) (cid:101) η r (cid:101) ηr (cid:19) θ ≥ V ( x , c − (cid:101) ηr ) (cid:18) c − θ (cid:101) η θ c V ( x , c − (cid:101) ηr ) V ( x , (cid:101) ηr ) (cid:19) ≥ − c V ( x , r ) . This shows that NDL R (Φ , U ∩ M ) for X holds. Then in view of the latter assertion in PropositionB.12, by the same proof, we can see that the second claim in the proposition also holds. (cid:50) Finally, the proof for Theorem 2.8 is straightforward. Proof of Theorem 2.8. By (B.1), (B.10), Lemma B.1, Proposition B.13 and (1.13), the theoremfollows from Theorems 1.13, 1.17–1.20 and Corollary 1.16 in Section 1. (cid:50) ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 57 References [1] S. Albeverio and B. R¨udiger. Subordination of symmetric quasi-regular Dirichlet forms. Random Oper. Stoch.Equ. , 13(1):17–38, 2005.[2] F. Aurzada, L. D¨oring, and M. Savov. Small time Chung-type LIL for L´evy processes. Bernoulli , 19(1):115–136,2013.[3] J. Bae, J. Kang, P. Kim, and J. Lee. Heat kernel estimates and their stabilities for symmetric jump processeswith general mixed polynomial growths on metric measure spaces. available at arXiv:1904.10189. , 2019.[4] J. Bae, J. Kang, P. Kim, and J. Lee. Heat kernel estimates for symmetric jump processes with mixed polynomialgrowths. Ann. Probab. , 47(5):2830–2868, 2019.[5] M. T. Barlow. Diffusions on fractals. In Lectures on probability theory and statistics (Saint-Flour, 1995) , volume1690 of Lecture Notes in Math. , pages 1–121. Springer, Berlin, 1998.[6] M. T. Barlow. Random walks on supercritical percolation clusters. Ann. Probab. , 32(4):3024–3084, 2004[7] M. T. Barlow and R. F. Bass. Transition densities for Brownian motion on the Sierpi´nski carpet. Probab. TheoryRelat. Fields , 91(3-4):307–330, 1992.[8] M. T. Barlow and R. F. Bass. Brownian motion and harmonic analysis on Sierpinski carpets. Can. J. Math. ,51(4):673–744, 1999.[9] M. T. Barlow, A. Grigor’yan, and T. Kumagai. Heat kernel upper bounds for jump processes and the first exittime. J. Reine Angew. Math. , 626:135–157, 2009.[10] M. T. Barlow and E. A. Perkins. Brownian motion on the Sierpi´nski gasket. Probab. Theory Relat. Fields ,79(4):543–623, 1988.[11] R. F. Bass and T. Kumagai. Laws of the iterated logarithm for some symmetric diffusion processes. Osaka J.Math. , 37(3):625–650, 2000.[12] A. Benveniste and J. Jacod. Syst`emes de L´evy des processus de Markov. Invent. Math. , 21:183–198, 1973.[13] J. Bertoin. Subordinators: examples and applications. In Lectures on probability theory and statistics (Saint-Flour,1997) , volume 1717 of Lecture Notes in Math. , pages 1–91. Springer, Berlin, 1999.[14] N. H. Bingham. Variants on the law of the iterated logarithm. Bull. Lond. Math. Soc. , 18(5):433–467, 1986.[15] N. H. Bingham, C. M. Goldie, and J. L. Teugels. Regular variation . Cambridge University Press, Cambridge,1987.[16] N. Bouleau. Quelques r´esultats probabilistes sur la subordination au sens de Bochner. Seminar on potentialtheory, Paris, No. 7 , 54–81, Lecture Notes in Math., 1061, Springer, Berlin, 1984.[17] B. Buchmann and R. Maller. The small-time Chung-Wichura law for L´evy processes with non-vanishing Browniancomponent. Probab. Theory Relat. Fields , 149(1-2):303–330, 2011.[18] E. A. Carlen, S. Kusuoka, and D. W. Stroock. Upper bounds for symmetric Markov transition functions. Ann.Inst. H. Poincar´e Probab. Statist. , 23(2, suppl.):245–287, 1987.[19] X. Chen, T. Kumagai, and J. Wang. Random conductance models with stable-like jumps: quenched invarianceprinciple. To appear in Ann. Appl. Probab., available at arXiv:1805.04344. , 2018.[20] X. Chen, T. Kumagai, and J. Wang. Random conductance models with stable-like jumps: heat kernel estimatesand Harnack inequalities. J. Funct. Anal. , 279(7): 108656, 51 pp, 2020.[21] Z.-Q. Chen, T. Kumagai, and J. Wang. Stability of heat kernel estimates for symmetric non-local Dirichlet forms. To appear in Memoirs Amer. Math. Soc., available at arXiv:1604.04035. , 2016.[22] Z.-Q. Chen, T. Kumagai, and J. Wang. Heat kernel estimates and parabolic Harnack inequalities for symmetricDirichlet forms. Adv. Math. , 374:107269, 2020.[23] Z.-Q. Chen, T. Kumagai, and J. Wang. Heat kernel estimates for general symmetric pure jump Dirichlet forms. available at arXiv:1908.07655. , 2020.[24] Z.-Q. Chen, T. Kumagai, and J. Wang. Stability of parabolic Harnack inequalities for symmetric non-localDirichlet forms. J. Eur. Math. Soc. , 22(11):3747–3803, 2020.[25] N. Chikara. Rate functions for random walks on random conductance models and related topics. Kodai Math. J.40 , 2:289–321, 2017.[26] N. Chikara, T. Takashi. Laws of the iterated logarithm for random walks on random conductance models. Stochas-tic analysis on large scale interacting systems, RIMS Kˆokyˆuroku Bessatsu, B59, Res. Inst. Math. Sci. (RIMS),Kyoto , 141–156, 2016.[27] S. Cho and P. Kim. Estimates on the tail probabilities of subordinators and applications to general time fractionalequations. Stoch. Process. Appl. , 130(7):4392–4443, 2020.[28] S. Cho and P. Kim. Estimates on transition densities of subordinators with jumping density decaying in mixedpolynomial orders. available at arXiv:1912.10565. , 2020.[29] J. Chover. On Strassen’s version of the log log law. Z. Wahrsch. und Verw. Gebiete , 8:83–90, 1967.[30] K.-L. Chung. On the maximum partial sums of sequences of independent random variables. Trans. Amer. Math.Soc. , 64:205–233, 1948. [31] T. Coulhon and A. Grigoryan. Random walks on graphs with regular volume growth. Geom. Funct. Anal. ,8(4):656–701, 1998.[32] H. Duminil-Copin. Law of the Iterated Logarithm for the random walk on the infinite percolation cluster. Masterthesis, available at arXiv:0809.4380. , 2008.[33] C. Dupuis. Mesure de Hausdorff de la trajectoire de certains processus `a accroissements ind´ependants et station-naires. In S´eminaire de Probabilit´es, VIII , pages 37–77. Lecture Notes in Math., Vol. 381. 1974.[34] U. Einmahl and D. Li. Some results on two-sided LIL behavior. Ann. Probab. , 33(4):1601–1624, 2005.[35] U. Einmahl and D. M. Mason. A universal Chung-type law of the iterated logarithm. Ann. Probab. , 22(4):1803–1825, 1994.[36] P. Erd¨os. On the law of the iterated logarithm. Ann. of Math. (2) , 43:419–436, 1942.[37] W. Feller. The fundamental limit theorems in probability. Bull. Amer. Math. Soc. , 51:800–832, 1945.[38] W. Feller. An extension of the law of the iterated logarithm to variables without variance. J. Math. Mech. ,18:343–355, 1968/69.[39] B. Fristedt. Sample function behavior of increasing processes with stationary, independent increments. Pacific J.Math. , 21:21–33, 1967.[40] B. Fristedt. Upper functions for symmetric processes with stationary, independent increments. Indiana Univ.Math. J. , 21:177–185, 1971/72.[41] B. Fristedt. Sample functions of stochastic processes with stationary, independent increments. In Advances inprobability and related topics, Vol. 3 , pages 241–396. 1974.[42] B. Fristedt and W. E. Pruitt. Lower functions for increasing random walks and subordinators. Z. Wahrsch. undVerw. Gebiete , 18:167–182, 1971.[43] M. Fukushima, Y. Oshima, and M. Takeda. Dirichlet forms and symmetric Markov processes , volume 19 of DeGruyter Studies in Mathematics . Walter de Gruyter & Co., Berlin, extended edition, 2011.[44] B. Gnedenko. Sur la croissance des processus stochastiques homog`enes `a accroissements ind´ependants. Bull. Acad.Sci. URSS. S´er. Math. Izv. Akad. Nauk SSSR , 7:89–110, 1943.[45] P. S. Griffin. Laws of the iterated logarithm for symmetric stable processes. Z. Wahrsch. und Verw. Gebiete ,68(3):271–285, 1985.[46] A. Grigor’yan, E. Hu, and J. Hu. Two-sided estimates of heat kernels of jump type Dirichlet forms. Adv. Math. ,330:433–515, 2018.[47] A. Grigor’yan and J. Hu. Upper bounds of heat kernels on doubling spaces. Mosc. Math. J. , 14(3):505–563, 2014.[48] A. Grigor’yan, J. Hu, and K.-S. Lau. Estimates of heat kernels for non-local regular Dirichlet forms. Trans. Amer.Math. Soc. , 366(12):6397–6441, 2014.[49] A. Grigor’yan and A. Telcs. Two-sided estimates of heat kernels on metric measure spaces. Ann. Probab. ,40(3):1212–1284, 2012.[50] T. Grzywny, M. Ryznar, and B. Trojan. Asymptotic behaviour and estimates of slowly varying convolutionsemigroups. Int. Math. Res. Not. IMRN , (23):7193–7258, 2019.[51] T. Grzywny and K. Szczypkowski. Heat kernels of non-symmetric L´evy-type operators. J. Differential Equations ,267(10):6004–6064, 2019.[52] B. M. Hambly and T. Kumagai. Transition density estimates for diffusion processes on post critically finiteself-similar fractals. Proc. Lond. Math. Soc. (3) , 78(2):431–458, 1999.[53] P. Hartman and A. Wintner. On the law of the iterated logarithm. Amer. J. Math. , 63:169–176, 1941.[54] C. C. Heyde. A note concerning behaviour of iterated logarithm type. Proc. Amer. Math. Soc. , 23:85–90, 1969.[55] N. C. Jain and W. E. Pruitt. The other law of the iterated logarithm. Ann. Probab. , 3(6):1046–1049, 1975.[56] N. C. Jain and W. E. Pruitt. Lower tail probability estimates for subordinators and nondecreasing random walks. Ann. Probab. , 15(1):75–101, 1987.[57] H. Kesten. Sums of independent random variables without moment conditions. Ann. Math. Statist. , 43:701–732,1972.[58] H. Kesten. Percolation theory for mathematicians. Boston: Birkhfiuser, 1982.[59] H. Kesten. A universal form of the Chung-type law of the iterated logarithm. Ann. Probab. , 25(4):1588–1620,1997.[60] A. Khintchine. ¨Uber dyadische br¨uche. Math. Z. , 18(1):109–116, 1923.[61] A. Khintchine. Asymptotische Gesetze der Wahrscheinlichkeitsrechnung . Springer, Berlin, 1933.[62] A. Khintchine. Zwei S¨atze ¨uber stochastische Prozesse mit stabilen Verteilungen. Rec. Math. Moscou, n. Ser. ,3:577–584, 1938.[63] P. Kim, T. Kumagai, and J. Wang. Laws of the iterated logarithm for symmetric jump processes. Bernoulli ,23(4A):2330–2379, 2017.[64] P. Kim and R. Song. Two-sided estimates on the density of Brownian motion with singular drift. Illinois J. Math. ,50(1-4):635–688, 2006.[65] P. Kim, R. Song, and Z. Vondraˇcek. On the boundary theory of subordinate killed L´evy processes. PotentialAnal. , 53(1):131–181, 2020. ENERAL LAW OF ITERATED LOGARITHM FOR MARKOV PROCESSES 59 [66] M. J. Klass. Toward a universal law of the iterated logarithm. I. Z. Wahrsch. und Verw. Gebiete , 36(2):165–178,1976.[67] M. J. Klass. Toward a universal law of the iterated logarithm. II. Z. Wahrsch. und Verw. Gebiete , 39(2):151–165,1977.[68] V. Knopova and R. L. Schilling. On the small-time behaviour of L´evy-type processes. Stoch. Process. Appl. ,124(6):2249–2265, 2014.[69] A. Kolmogoroff. ¨Uber das Gesetz des iterierten Logarithmus. Math. Ann. , 101(1):126–135, 1929.[70] P. L´evy. Processus Stochastiques et Mouvement Brownien. Suivi d’une note de M. Lo`eve . Gauthier-Villars, Paris,1948.[71] P. L´evy. Propri´et´es asymptotiques de la courbe du mouvement brownien `a N dimensions. C. R. Acad. Sci. Paris ,241:689–690, 1955.[72] P. A. Meyer. Renaissance, recollements, m´elanges, ralentissement de processus de Markov. Ann. Inst. Fourier(Grenoble) , 25(3-4):xxiii, 465–497, 1975.[73] A. Mimica. Heat kernel estimates for subordinate Brownian motions. Proc. Lond. Math. Soc. (3) , 113(5):627–648,2016.[74] M. Murugan and L. Saloff-Coste. Transition probability estimates for long range random walks. New York J.Math. , 21:723–757, 2015.[75] W. E. Pruitt. General one-sided laws of the iterated logarithm. Ann. Probab. , 9(1):1–48, 1981.[76] W. E. Pruitt. The growth of random walks and L´evy processes. Ann. Probab. , 9(6):948–956, 1981.[77] B. A. Rogozin. On the question of the existence of exact upper sequences. Teor. Verojatnost. i Primenen. ,13:701–707, 1968.[78] L. Saloff-Coste. A note on Poincar´e, Sobolev, and Harnack inequalities. Internat. Math. Res. Notices , (2):27–38,1992.[79] K.-I. Sato. L´evy processes and infinitely divisible distributions , volume 68 of Cambridge Studies in AdvancedMathematics . Cambridge University Press, Cambridge, 2013.[80] M. Savov. Small time two-sided LIL behavior for L´evy processes at zero. Probab. Theory Relat. Fields , 144(1-2):79–98, 2009.[81] R. L. Schilling, R. Song, and Z. Vondraˇcek. Bernstein functions , volume 37 of De Gruyter Studies in Mathematics .Walter de Gruyter & Co., Berlin, second edition, 2012. Theory and applications.[82] Y. Shiozawa and J. Wang. Long-time heat kernel estimates and upper rate functions of Brownian motion typefor symmetric jump processes. Bernoulli , 25(4B):3796–3831, 2019.[83] V. Strassen. An invariance principle for the law of the iterated logarithm. Z. Wahrsch. und Verw. Gebiete ,3:211–226 (1964), 1964.[84] S. J. Taylor. Sample path properties of a transient stable process. J. Math. Mech. , 16:1229–1246, 1967.[85] A. Telcs. The art of random walks , volume 1885 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, 2006.[86] S. Watanabe. On discontinuous additive functionals and L´evy measures of a Markov process. Jpn. J. Math. ,34:53–70, 1964.[87] I. S. Wee. Lower functions for processes with stationary independent increments. Probab. Theory Relat. Fields ,77(4):551–566, 1988.[88] I. S. Wee and Y. K. Kim. General laws of the iterated logarithm for L´evy processes. J. Kor. Statist. Soc. ,17(1):30–45, 1988.[89] M. J. Wichura. On the functional form of the law of the iterated logarithm for the partial maxima of independentidentically distributed random variables. Ann. Probab. , 2:202–230, 1974.[90] F. Xu. A class of singular symmetric Markov processes. Potential Anal. , 38(1):207–232, 2013.[91] Q. S. Zhang. Gaussian bounds for the fundamental solutions of ∇ ( A ∇ u ) + B ∇ u − u t = 0. Manuscripta Math. ,93(3):381–390, 1997.(Cho) Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic ofKorea Email address : [email protected] (Kim) Department of Mathematical Sciences and Research Institute of Mathematics, Seoul Na-tional University, Seoul 08826, Republic of Korea Email address : [email protected] (Lee) Korea Institute for Advanced Study, Seoul 02455, Republic of Korea Email address ::
Theorem 1.17.
a − /δx , where the constant c x may depend on x . We also note that the integral in (2.14) is finite (resp.infinite) if for sufficiently small t ,Ψ( t ) (cid:16) exp (cid:0) − kt − /δ | log t | − /δ (log | log t | ) − /δ ... (log ◦ ... ◦ log)( | log t | ) − p/δ (cid:1) with some constants k > p > p < ( φ , αδ, c ) holds for some c > / ( C δ U ( x )) such that4 m k ≤ k − < − (cid:0) R ∧ ( C ( δ U ( x ) − n − )) (cid:1) for all k ≥ n . (4.4)Below, we estimate probabilities of the events given in the right hand side of (4.3). First, byProposition 3.10(i), (4.4) and (4.2), we have that for all k ≥ n , P x ( F k ) = P x ( τ B ( x,m k ) ≤ σ k +1 ) ≤ C σ k +1 φ ( m k ) = c e − k log( k + 1) . (4.5)Next, we observe that for all z ∈ B ( x, n − ) and k ≥ n , by (4.4) and (4.2), since SP R ( φ, U ) andU R ( φ, β , C U ) hold, P z (cid:0) m k ≤ sup
E u := E ◦ θ u = (cid:110) lim inf t →∞ φ (cid:0) sup φ is positive, for all t, u > P y -a.s. ω ∈ E , it holds thatlim inf t →∞ φ (cid:0) sup P y -a.s. ω ∈ E u ,lim inf t →∞ φ (cid:0) sup
s n ) n ≥ ⊂ (0 , q Ψ( t ) for some t ∈ ( t n +1 , t n ] (cid:1) ≤ P x (cid:0) τ B ( x,q s n log n ) ≤ t n (cid:1) ≤ C (cid:18) t n ψ ( q s n log n ) + exp (cid:16) − q C s n log nϑ ( t n , q C s n log n ) (cid:17)(cid:19) . (5.8)Since ψ is increasing, by (5.5), we have that (cid:80) ∞ n =4 t n /ψ ( q s n log n ) ≤ (cid:80) ∞ n =4 n − (log n ) − − δ < ∞ .Besides, since ψ and φ are increasing, and U R ( ψ, β , C (cid:48) U ) holds, by (5.5), we also have that ψ − ( t n ) ≤ ψ − ( n − ψ ( s n log n )) ≤ s n < φ − ( t n / s n ) n ≥ such that s n ≥ e n , nφ ( s n )(log n ) δ ≤ ψ ∧ ( s n log n ) and φ ( s n +1 ) ≥ φ ( s n ) for all n . We putΨ( t ) := ∞ (cid:88) n =5 ( s n log n ) ( t n − ,t n ] ( t ) where t n := φ ( s n ) log n. Recall that SP r ∞ ( φ ) (with υ, C ) and Assumption 3.6 hold with R ∞ = r ∞ where the constant r ∞ is defined as (3.6) under the present setting (see (1.13)). Let C , C be the constants in Proposition3.10(ii) and choose C , C according to Proposition 3.13(ii) with a = C . Then we set q :=2 C − + C − + 1 and q := 4 − ( C + C ) − , and we fix x, y ∈ M .By Proposition 3.10(ii), for all n large enough, since we have B ( x, q s n log n ) ⊃ B ( y, q s n log n − d ( x, y )) ⊃ B ( y, q s n log n ), P y (cid:0) d ( x, X t ) > q Ψ( t ) for some t ∈ ( t n − , t n ] (cid:1) ≤ P y (cid:0) τ B ( x, q s n log n ) ≤ t n (cid:1) ≤ P y (cid:0) τ B ( y,q s n log n ) ≤ t n (cid:1) ≤ C (cid:18) t n ψ ∧ ( q s n log n ) + exp (cid:16) − q C s n log nϑ ( t n , q C s n log n ) (cid:17)(cid:19) . (5.12)Indeed, for all n large enough, since υ < υ , s n ≥ e n and φ is increasing, we see that ψ ∧ (cid:0) q s n log n ) υ/υ (cid:1) ≤ φ (cid:0) √ υ /υ ( q s n log n ) √ υ/υ (cid:1) ≤ φ ( s n ) < t n . (5.13)Using (5.12), by following the proof of Theorem 1.17(ii), we obtainlim sup t →∞ sup t n − for all n , we have P y ( E (cid:48) n | F t n − ) ≥ inf z ∈ B ( y,s υ /υn ) P z (cid:0) τ B ( y, q Ψ( t n )) ≤ − t n (cid:1) · F (cid:48) n =: b (cid:48) n F (cid:48) n for all n ≥ . For any constant
r (cid:1) ≥ c . Besides, by [73, Proposition 2.4], it holds that P ( S t ≥ − /φ − ( t − )) ≥ − e − / for all t > r (cid:1) ≥ c P (cid:0) S Φ( k r ) ≥ − F ( k r ) (cid:1) ≥ c (1 − e − / ) . Then by repeating similar calculations to the ones given in (3.9), we can deduce that the upperbound in (1.10) holds with C = k and a ∗ = 1 − c (1 − e − / ).On the other hand, by (A.1), since VRD ¯ R ( M ) holds, we see that for all x ∈ M and r ∈ (0 , δ R ), P y (cid:0) Z B ( x,r ) F ( η r ) ∈ B ( x, η r ) (cid:1) ≥ c V ( x, η r ) V ( x, η r ) ≥ c for all y ∈ B ( x, η r ) . Moreover, by [27, Lemmas 2.11 and 2.4(ii)], there exists a universal constant p ∈ (0 ,