Analysis of jump processes with nondegenerate jumping kernels
aa r X i v : . [ m a t h . P R ] D ec ANALYSIS OF JUMP PROCESSES WITH NONDEGENERATEJUMPING KERNELS
MORITZ KASSMANN AND ANTE MIMICA
Abstract.
We prove regularity estimates for functions which are harmonic withrespect to certain jump processes. The aim of this article is to extend the methodof Bass-Levin[BL02] and Bogdan-Sztonyk[BS05] to more general processes. Fur-thermore, we establish a new version of the Harnack inequality that implies regu-larity estimates for corresponding harmonic functions. Introduction
Let α ∈ (0 , L by L f ( x ) = ˆ R d \{ } ( f ( x + h ) − f ( x ) − h∇ f ( x ) , h i {| h |≤ } ) n ( x, h ) dh, (1.1)for f ∈ C b ( R d ). Here n : R d × (cid:0) R d \ { } (cid:1) → [0 , ∞ ) is a measurable function with c | h | − d − α ≤ n ( x, h ) ≤ c | h | − d − α (1.2)for every h ∈ R d \ { } , any x ∈ R d and fixed positive reals c < c . Note that n ( x, h ) = | h | − d − α for every h implies L f = − c ( α )( − ∆) α/ f with some appropriateconstant c ( α ).In [BL02] it is shown that harmonic functions with respect to L satisfy a Harnackinequality in the following sense: There is a constant c ≥ B R the following implication holds: f ≥ R d , f harmonic in B R ⇒ ∀ x, y ∈ B R/ : f ( x ) ≤ c f ( y ) . Date : June 12, 2018.2000
Mathematics Subject Classification.
Primary 60J75, Secondary 31B05, 31B10, 35B45,47G20, 60J45.
Key words and phrases.
Jump process, harmonic function, regularity estimate, Harnackinequality.
In [BL02] it is also shown that harmonic functions with respect to L satisfy thefollowing a-priori estimate: There are constants β ∈ (0 , c ≥ B R the following implication holds: f harmonic in B R ⇒ k f k C β ( B R/ ) ≤ c k f k ∞ . This result and its proof recently generated several research activities, see the shortdiscussion below. Our aim is to prove similar results under weaker assumptions onthe kernel n .Let us be more precise. We consider kernels n : R d × (cid:0) R d \ { } (cid:1) → [0 , ∞ ) that satisfyfor every x, h ∈ R d , h = 0 n ( x, h ) = n ( x, − h ) (1.3)and k (cid:0) h | h | (cid:1) j ( | h | ) ≤ n ( x, h ) ≤ k (cid:0) h | h | (cid:1) j ( | h | ) (1.4)where k , k : S d − → [0 , ∞ ) are measurable bounded symmetric functions on theunit sphere satisfying the following conditions: There are δ > , N ∈ N , ε , . . . , ε N > η , . . . , η N ∈ S d − such that for S i = S d − ∩ (cid:0) B ( η i , ε i ) ∪ B ( − η i , ε i ) (cid:1) k ( ξ ) ≥ k ( ξ ) ≥ δ if ξ ∈ N [ i =1 S i and k ( ξ ) = k ( ξ ) = 0 otherwise . (1.5)Let j : (0 , ∞ ) → [0 , ∞ ) be a function such that ´ R d ( | z | ∧ j ( | z | ) dz is finite. Weassume further:(J1) There exists α ∈ (0 ,
2) and a function ℓ : (0 , → (0 , ∞ ) which is slowlyvarying at 0 (i.e. lim r → ℓ ( λr ) ℓ ( r ) = 1 for any λ >
0) and bounded away from 0and ∞ on every compact interval such that j ( t ) = ℓ ( t ) t d + α for every 0 < t ≤ . (J2) There is a constant κ ≥ j ( t ) ≤ κj ( s ) whenever 1 ≤ s ≤ t . In order to establish regularity estimates we need an additional weak assumption.
ONDEGENERATE JUMP PROCESSES 3 (J3) There is σ > R →∞ R σ ˆ | z | >R j ( | z | ) dz ≤ . If this condition holds, then one can always choose σ ∈ (0 , α ). Remark 1.1.
The symmetry assumption (1.3) is used only in Proposition 2.4 andcan be dispensed with if α ∈ (0 , . Example 1:
If a kernel n satisfies condition (1.2), then it also satisfies (J1)-(J3).Choose N = 1, ε = 4, i.e. S = S d − , k ≡ δ = c , k ≡ c , j ( s ) = s − d − α in (1.4), ℓ ≡ κ = 1 in (J2) and σ ∈ (0 , α ) arbitrarily in (J3). In general, (J1)-(J3)hold for jumping kernels corresponding to stable processes, stable-like processes andtruncated versions. Sums of such jumping kernels can be considered, too. Example 2:
Let N ∈ N , η , . . . , η N ∈ S d − and ε , . . . , ε N be positive real numberssuch that the sets S i = S d − ∩ (cid:0) B ( η i , ε i ) ∪ B ( − η i , ε i ) (cid:1) are pairwise disjoint for i = 1 , . . . , N . Set B = N S i =1 S i . Let k = δ B for some δ > k = ck for some c >
1. Let j ( s ) = s − d − α for s >
0. Then our assumptions are satisfied if (1.4) and(1.3) hold true. For the particular choice where x n ( x, h ) is constant (case ofL´evy process), this class of examples is treated in [BS05, p.148], where it is shownthat for N = ∞ the Harnack inequality fails.Given a linear operator L as in (1.1) we assume that there exists a strong Markovprocess X = ( X t , P x ) with paths that are right-continuous with left limits such thatthe process (cid:26) f ( X t ) − f ( X ) − ˆ t L f ( X s ) ds (cid:27) t ≥ is a P x -martingale for all x ∈ R d and f ∈ C b ( R d ). We say that a bounded function f : R d → R is harmonic with respect to L in an open set Ω if f (cid:0) X min( t,τ Ω ′ ) (cid:1) is aright-continuous martingale for every open Ω ′ ⊂ R d with Ω ′ ⊂ Ω.We can prove the following version of the Harnack inequality.
Theorem 1.2.
Assume (J1) and (J2). There exist constants c , c ≥ such thatfor every x ∈ R d , r ∈ (0 , ) and every bounded function f : R d → R which isnon-negative in B ( x , r ) and harmonic in B ( x , r ) the following estimate holds f ( x ) ≤ c f ( y ) + c (cid:18) r α ℓ ( r ) (cid:19) sup v ∈ B ( x , r ) ˆ B ( x , r ) c f − ( z ) n ( v, z − v ) dz for all x, y ∈ B ( x , r ) . MORITZ KASSMANN AND ANTE MIMICA
Remark 1.3. If f is, in addition, non-negative in all of R d , then the classicalversion of the Harnack inequality follows, i.e. for all x, y ∈ B ( x , r ) : f ( x ) ≤ c f ( y ) . As a corollary to the Harnack inequality we obtain the following regularity result.
Theorem 1.4.
Assume (J1), (J2) and (J3). Then there exist β ∈ (0 , , c , c ≥ such that for every x ∈ R d , every R ∈ (0 , , every function f : R d → R which isharmonic in B ( x , R ) and every ρ ∈ (0 , R/ x,y ∈ B ( x ,ρ ) | f ( x ) − f ( y ) | ≤ c k f k ∞ ( ρ/R ) β , (1.6) in particular k f k C β ( B ( x ,R/ ≤ c k f k ∞ . (1.7)Let us comment on the differences between our results and those of [BL02]:(1) We can treat kernels n ( x, h ) for which the quantityinf x ∈ R d lim inf r → |{ h ∈ B (0 , r ); n ( x, h ) = 0 }|| B (0 , r ) | is arbitrarily close to 1, e.g. n ( x, h ) as in (1.9).(2) For fixed x ∈ R d , upper and lower bounds for n ( x, h ) may not allow for scaling.(3) Large jumps of the process might not be comparable, i.e. the quantitysup (cid:26) n ( x, h ) n ( y, h ) ; | x − y | ≤ , | h − h | ≤ , | h | + | h | ≥ (cid:27) might be infinite.(4) We establish a new version of the Harnack inequality and derive a-priori H¨olderregularity estimate as a consequence. In a different setting, this procedure wasrecently established in [Kas].The constants in the main results of our work and [BL02] depend on α . It would bedesirable to adopt the technique further such that results would be robust for α → n such thatsimilar results hold true? ONDEGENERATE JUMP PROCESSES 5
We call a kernel n of the above type nondegenerate if there is a function N : (0 , → (0 , ∞ ) with lim ρ → N ( ρ ) = + ∞ and λ, Λ > ρ ∈ (0 ,
1) and x ∈ R d the symmetric matrix [ A ρij ( x )] di,j =1 defined by A ρij ( x ) = N ( ρ ) ˆ { < | h |≤ ρ } h i h j n ( x, h ) dh . satisfies for every ξ ∈ R d λ | ξ | ≤ d X i,j =1 A ρi,j ( x ) ξ i ξ j ≤ Λ | ξ | . (1.8)If n depends only on h and N ( ρ ) = ρ α − , then this condition implies that the corre-sponding L´evy process has a smooth density, see [Pic96]. Note that condition (1.2)implies the nondegeneracy condition (1.8) with N ( ρ ) = ρ α − but is not necessary,just consider the example n ( x, h ) = | h | − d − α {| h |≥ . | h |} . (1.9)Note that (1.8) holds under our assumptions.Let us comment on other articles that generalize the results of [BL02]. Note thatwe do not include works on nonlocal Dirichlet forms. [SV04] gives conditions onL´evy processes and more general Markov jump processes such that the theory of[BL02] is applicable. In [BK05a] the theory is extended to the variable order caseand to situations where the lower and upper bound in (1.2) behave differently for | h | →
0. In these cases, regularity of harmonic functions does not hold. Regularity isestablished in [BK05b] for variable order cases under additional assumptions. Finepotential theoretic results are obtained in [BSS02, BS05] for stable processes. Thecase of L´evy processes with truncated stable L´evy densities is covered in [KS07]and generalized in [Mim10]. As mentioned above there is an independent approachwith analytic methods developed in [Sil06, CS09] covering linear and fully nonlinearintegro-differential operators.
Notation:
For two functions f and g we write f ( t ) ∼ g ( t ) if f ( t ) /g ( t ) →
1. For A ⊂ R d open or closed τ A denotes the first exit time of the Markov process underconsideration. T A denotes the the first hitting time of the set A . Acknowledgement:
The authors thank an anonymous referee for pointing outthat the previous version of assumptions (1.4), (1.5) was overly general. Example 2was added in order to motivate these assumptions.
MORITZ KASSMANN AND ANTE MIMICA Some probabilistic estimates
In this section we prove useful auxiliary results. We follow closely the ideas of [BL02].However, we need to provide several computations because of the appearance of aslowly varying function in (J1). The proofs of Proposition 2.7 and Proposition 2.9are significantly different from their counterparts in [BL02].The following proposition will be used often in obtaining probabilistic estimates.
Proposition 2.1.
Let
A, B ⊂ R d be disjoint Borel sets. Then for every boundedstopping time T E x "X s ≤ T { X s − ∈ A,X s ∈ B } = E x (cid:20) ˆ T ˆ B A ( X s ) n ( X s , u − X s ) du (cid:21) for every x ∈ R d .Proof. By [BL02, Proposition 2.3] it follows that the process (X s ≤ t { X s − ∈ A,X s ∈ B } − ˆ t ˆ B A ( X s ) n ( X s , u − X s ) du ) t ≥ is a P x -martingale. Therefore the result follows by the optional stopping theorem. (cid:3) The following result, taken from the theory of regular variation, will be repeatedlyused throughout the paper.
Proposition 2.2.
Assume that ℓ : (0 , → (0 , ∞ ) varies slowly at and let β > − and β > . Then the following is true: (i) ˆ r u β ℓ ( u ) du ∼ r β β ℓ ( r ) as r → , (ii) ˆ r u − β ℓ ( u ) du ∼ r − β β − ℓ ( r ) as r → .Proof. By a change of variables and using [BGT87, Proposition 1.5.10] we obtain ˆ r u β ℓ ( u ) du = ˆ ∞ r − u − β − ℓ ( u − ) du ∼ r β ℓ ( r )1 + β , ONDEGENERATE JUMP PROCESSES 7 since u ℓ ( u − ) varies slowly at infinity. This proves (i). Similarly, with the helpof [BGT87, Proposition 1.5.8] we obtain (ii). (cid:3) Remark 2.3.
Using [BGT87, Theorem 1.5.4] we conclude that for a function ℓ : (0 , → (0 , ∞ ) that varies slowly at there exists a non-increasing function φ : (0 , → (0 , ∞ ) such that lim r → r − d − α ℓ ( r ) φ ( r ) = 1 . Before proving our main probabilistic estimates, note that (1.5) implies that thereexists ϑ ∈ (0 , π/
2] such that for every i ∈ { , . . . , N } n ( x, h ) ≥ δ j ( | h | ) for all h ∈ R d , h = 0 , |h h, η i i|| h | ≥ cos ϑ. (2.1)2.1. Exit time estimates.Proposition 2.4.
There exists a constant C > such that for every x ∈ R d , r ∈ (0 , and t > P x ( τ B ( x ,r ) ≤ t ) ≤ C t ℓ ( r ) r α . Proof.
Again, we closely follow the ideas in [BL02]. Let x ∈ R d , r ∈ (0 ,
1) and let f ∈ C ( R d ) be a positive function such that f ( x ) = (cid:26) | x − x | , | x − x | ≤ r r , | x − x | ≥ r and | f ( x ) | ≤ c r , (cid:12)(cid:12)(cid:12)(cid:12) ∂f∂x i ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c r and (cid:12)(cid:12)(cid:12)(cid:12) ∂ f∂x i ∂x j ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c , for some constant c > x ∈ B ( x , r ). We estimate L f ( x ) in a few steps.First ˆ B ( x ,r ) (cid:0) f ( x + h ) − f ( x ) − h∇ f ( x ) , h i {| h |≤ } (cid:1) n ( x, h ) dh ≤ c ˆ B ( x ,r ) | h | n ( x, h ) dh ≤ c ˆ B ( x ,r ) | h | − d − α ℓ ( | h | ) dh ≤ c r − α ℓ ( r ) , MORITZ KASSMANN AND ANTE MIMICA where in the last line we have used Proposition 2.2 (i). Similarly, by Proposition 2.2(ii) on B ( x , r ) c we get ˆ B ( x ,r ) c ( f ( x + h ) − f ( x )) n ( x, h ) dh ≤ k f k ∞ ˆ B ( x ,r ) c n ( x, h ) dh ≤ k f k ∞ (cid:18) ˆ B ( x , \ B ( x ,r ) | h | − d − α ℓ ( | h | ) dh + ˆ B ( x , c n ( x, h ) dh (cid:19) ≤ c r (cid:0) c r − α ℓ ( r ) + c (cid:1) ≤ c r − α ℓ ( r ) . In the last inequality we have used the fact that lim r → r − α ℓ ( r ) = ∞ (cf. [BGT87,Proposition 1.3.6 (v)]). Finally, by symmetry of the kernel, we have ˆ B ( x , \ B ( x ,r ) h h, ∇ f ( x ) i n ( x, h ) dh = 0 . (2.2)Therefore, by preceding estimates, we conclude that there is a constant c > x ∈ R d and r ∈ (0 , L f ( x ) ≤ c r − α ℓ ( r ) . (2.3)It follows from the optional stopping theorem that E x f ( X t ∧ τ B ( x ,r ) ) − f ( x ) = E x ˆ t ∧ τ B ( x ,r ) L f ( X s ) ds ≤ c tr − α ℓ ( r ) , t > . (2.4)On { τ B ( x ,r ) ≤ t } one has X t ∧ τ B ( x ,r ) B ( x , r ) and so f ( X t ∧ τ B ( x ,r ) ) ≥ r . Then(2.4) gives P x ( τ B ( x ,r ) ≤ t ) ≤ c t r − α ℓ ( r ) . (cid:3) Proposition 2.5.
There exists a constant C > such that for every r ∈ (0 , and x ∈ R d inf y ∈ B ( x ,r/ E y τ B ( x ,r ) ≥ C r α ℓ ( r ) . Proof.
Let r ∈ (0 , x ∈ R d and y ∈ B ( x , r/ P y ( τ B ( x ,r ) ≤ t ) ≤ P y ( τ B ( y,r/ ≤ t ) ≤ C t r − α ℓ ( r ) for t > . Let t = r α C ℓ ( r ) . Then E y τ B ( x ,r ) ≥ t P y ( τ B ( x ,r ) ≥ t ) ≥ r α C ℓ ( r ) . (cid:3) ONDEGENERATE JUMP PROCESSES 9
Proposition 2.6.
There exists a constant C > such that for every r ∈ (0 , ) and x ∈ R d sup y ∈ B ( x ,r ) E y τ B ( x ,r ) ≤ C r α ℓ ( r ) . Proof.
Let r ∈ (0 , ), x ∈ R d and y ∈ B ( x , r ). Denote by S the first time whenprocess ( X t ) t ≥ has a jump larger than 2 r , i.e. S = inf { t > | X t − X t − | > r } . Assume first that P y ( S ≤ r α ℓ ( r ) ) ≤ . Then by Proposition 2.1 P y (cid:16) S ≤ r α ℓ ( r ) (cid:17) = E y X s ≤ rαℓ ( r ) ∧ S {| X s − X s − | > r } = E y " ˆ rαℓ ( r ) ∧ S ˆ B (0 , r ) c n ( X s , h ) dh ds (2.5)Choose arbitrary ξ ∈ { η , . . . , η N } and let ϑ be as in (2.1). Then ˆ B (0 , r ) c n ( X s , h ) dh ≥ ˆ n h ∈ R d : 2 r ≤| h | < , |h h,ξ i|| h | ≥ cos ϑ o n ( X s , h ) dh ≥ δ ˆ n h ∈ R d : 2 r ≤| h | < , |h h,ξ i|| h | ≥ cos ϑ o ℓ ( | h | ) | h | d + α dh ≥ c ˆ r ℓ ( t ) t α dt ≥ c ℓ ( r ) r α , where in the last inequality we have used Proposition 2.2 (ii). Using this estimatewe get from (2.5) the following estimate P y (cid:18) S ≤ r α ℓ ( r ) (cid:19) ≥ c ℓ ( r ) r α E y (cid:20) r α ℓ ( r ) ∧ S (cid:21) ≥ c P y (cid:18) S > r α ℓ ( r ) (cid:19) ≥ c . Therefore, in any case the following inequality holds: P y (cid:18) S ≤ r α ℓ ( r ) (cid:19) ≥ ∧ c . Since S ≥ τ B ( x ,r ) we conclude P y (cid:18) τ B ( x ,r ) ≤ r α ℓ ( r ) (cid:19) ≥ P y (cid:18) S ≤ r α ℓ ( r ) (cid:19) ≥ c , with c = ∧ c . By the Markov property, for m ∈ N we obtain P y (cid:18) τ B ( x ,r ) > ( m + 1) r α ℓ ( r ) (cid:19) ≤ P y (cid:18) τ B ( x ,r ) > m r α ℓ ( r ) , τ B ( x ,r ) ◦ θ m rαℓ ( r ) > r α ℓ ( r ) (cid:19) = E y (cid:20) P X m rαℓ ( r ) (cid:18) τ B ( x ,r ) > r α ℓ ( r ) (cid:19) ; τ B ( x ,r ) > m r α ℓ ( r ) (cid:21) ≤ (1 − c ) P y (cid:18) τ B ( x ,r ) > m r α ℓ ( r ) (cid:19) , where θ s denotes the usual shift operator. By iteration we obtain P y (cid:18) τ B ( x ,r ) > m r α ℓ ( r ) (cid:19) ≤ (1 − c ) m , m ∈ N . Finally, E y τ B ( x ,r ) ≤ r α ℓ ( r ) ∞ X m =0 ( m + 1) P y (cid:18) τ B ( x ,r ) > m r α ℓ ( r ) (cid:19) ≤ r α ℓ ( r ) ∞ X m =0 ( m + 1)(1 − c ) m ≤ c r α ℓ ( r ) . (cid:3) Krylov-Safonov type estimate.
Fix ϑ ∈ (0 , π/
2] such that (2.1) holds.
Proposition 2.7.
Let λ ∈ (cid:0) , sin ϑ (cid:3) . There exists a constant C = C ( λ ) > suchthat for every x ∈ R d , r ∈ (0 , ) , closed set A ⊂ B ( x , λr ) and x ∈ B ( x , λr ) , P x ( T A < τ B ( x ,r ) ) ≥ C | A || B ( x , r ) | . Proof.
Choose arbitrary ξ ∈ { η , . . . , η N } and set ˜ x = x − r ξ . The idea is tochoose λ ∈ (0 , ] such that |h u − v, ξ i|| u − v | ≥ cos ϑ (2.6)for all u ∈ B ( x , λr ) , v ∈ B (˜ x , λr ). Since for every u ∈ B ( x , λr ) and v ∈ B (˜ x , λr ) |h u − v, ξ i|| u − v | ≥ p ( r ) − (2 λr ) r = p − (8 λ ) . ONDEGENERATE JUMP PROCESSES 11 b v b x b e x ϑϑ B ( x , r ) Figure 1.
The choice of e x and λ .it is enough to choose λ ∈ (0 , ] such that p − (8 λ ) ≥ cos ϑ, or, more explicitly, λ ≤ sin ϑ . For s > B ( x , s ) and B (˜ x , s ) by B s and ˜ B s . Let r ∈ (0 , λ ∈ (0 , sin ϑ ], x ∈ B λr and let A ⊂ B λr be a closed subset. The strong Markov property nowimplies P x ( T A < τ B r ) ≥ P x (cid:16) X τ B λr ∈ ˜ B λr , X τ ˜ B λr ◦ θ τ B λr ∈ A (cid:17) = E x h P X τB λr ( X τ ˜ B λr ∈ A ); X τ B λr ∈ ˜ B λr i . (2.7)For every y ∈ ˜ B λr and t > P y ( X τ ˜ B λr ∧ t ∈ A ) = E y X s ≤ τ ˜ B λr ∧ t { X s − = X s ,X s ∈ A } = E y (cid:20) ˆ τ ˜ B λr ∧ t ˆ A n ( X s , z − X s ) dz ds (cid:21) ≥ δ E y (cid:20) ˆ τ ˜ B λr ∧ t ˆ A ℓ ( | z − X s | ) | z − X s | d + α dz ds (cid:21) . Letting t → ∞ and using the monotone convergence theorem we deduce P y ( X τ ˜ B λr ∈ A ) ≥ δ E y (cid:20) ˆ τ ˜ B λr ˆ A ℓ ( | z − X s | ) | z − X s | d + α dz ds (cid:21) . Since | z − X s | ≤ r/ λr ≤ r , by Remark 2.3 we conclude P y ( X τ ˜ B λr ∈ A ) ≥ c ℓ ( r ) r d + α | A | E y τ ˜ B λr ≥ c ℓ ( r ) | A || B r | r − α E y τ ˜ B λr . Using Proposition 2.5 we deduce P y ( X τ ˜ B λr ∈ A ) ≥ c ℓ ( r ) ℓ (2 λr ) λ α | A || B r | . (2.8)Since ℓ varies slowly at 0 we finally obtain P y ( X τ ˜ B λr ∈ A ) ≥ c | A || B r | for all y ∈ ˜ B λr , (2.9)for some constant c = c ( λ ) >
0. By symmetry and (2.9) we deduce P x ( X τ B λr ∈ ˜ B λr ) ≥ c | ˜ B λr || ˜ B r | for all x ∈ B λr . (2.10)Finally, by (2.7), (2.9) and (2.10) we get P x ( T A < τ B r ) ≥ c λ d | A || B r | . (cid:3) Restricted Harnack inequality.
The aim of this subsection is to establisha Harnack inequality for a restricted class of harmonic functions.The following lemma can be proved similarly as [Mim10, Lemma 2.7].
Lemma 2.8.
Let g : (0 , ∞ ) → [0 , ∞ ) be a function satisfying g ( s ) ≤ cg ( t ) for all < t ≤ s, for some constant c > . There is a constant c ′ > such that for any x ∈ R d and r > we have g ( | z − x | ) ≤ c ′ r − d ˆ B ( x ,r ) g ( | z − u | ) du, for all x ∈ B ( x , r/ and z ∈ B ( x , r ) c . ONDEGENERATE JUMP PROCESSES 13
Proposition 2.9.
There is a constant λ ∈ (0 , ) so that for every λ ∈ (0 , λ ] there exists a constant C = C ( λ ) ≥ such that for all x ∈ R d , r ∈ (0 , ) and x, y ∈ B ( x , λr ) E x [ H ( X τ B ( x ,λr ) )] ≤ C E y [ H ( X τ B ( x ,r ) )] , for every non-negative function H : R d → [0 , ∞ ) supported in B ( x , r/ c .Proof. Let x ∈ R d , r ∈ (0 , ) and let x, y ∈ B ( x , λr ), where λ ∈ (0 , λ ) and λ ∈ (0 , ) is chosen later. λ will depend only on constants in our main assumptions.Take z ∈ B ( x , r/ c . There are only two cases. Case 1:
There exists u ∈ B ( x , λr ) so that n ( u , z − u ) > Case 2: n ( u, z − u ) = 0 for all u ∈ B ( x , λr ).We consider Case 1. By (1.4) and (1.5) there exist ξ ′ ∈ {± η , . . . , ± η N } and ϑ ′ ∈ (0 , π ] with h z − u , ξ ′ i| z − u | ≥ cos ϑ ′ . Note that ξ ′ , ϑ ′ depend on u , z, x and r but ϑ ′ ≥ ϑ uniformly with ϑ as in (2.1).Set ˜ x = x − r ξ ′ and take λ ≤ sin ϑ . Let B s := B ( x , s ) and ˜ B s := B (˜ x , s ). As in(2.6), for λ ≤ λ we have |h u − v, ξ ′ i|| u − v | ≥ cos ξ ′ for all u ∈ B λr , v ∈ ˜ B λr . Choose ˜ z ∈ ∂B r/ so that the following conditions hold:2 | z − w | ≤ | z − u | for all u ∈ B λr , w ∈ B ( ˜ z , λr ) , h w − v, ξ ′ i| w − v | ≥ cos ϑ ′ for all v ∈ ˜ B λr , w ∈ B ( ˜ z , λr ) , h z − w, ξ ′ i| z − w | ≥ cos ϑ ′ for all w ∈ B ( ˜ z , λr ) . (2.11)In the appendix we briefly explain the geometric argument behind the choice of˜ z ∈ ∂B r/ . Let B ′ s = B ( ˜ z , s ). By the strong Markov property, E y (cid:20) ˆ τ Br n ( X s , z − X s ) ds (cid:21) ≥ E y " ˆ τ Br τ B λr n ( X s , z − X s ) ds ; X τ B λr ∈ ˜ B λr = E y (cid:20)(cid:26) ˆ τ Br n ( X s , z − X s ) ds (cid:27) ◦ θ τ B λr ; X τ B λr ∈ ˜ B λr (cid:21) = E y (cid:20) E X τB λr (cid:20) ˆ τ Br n ( X s , z − X s ) ds (cid:21) ; X τ B λr ∈ ˜ B λr (cid:21) . (2.12)Similarly, for v ∈ ˜ B λr we have E v (cid:20) ˆ τ Br n ( X s , z − X s ) ds (cid:21) ≥ E v (cid:20) E X τ ˜ B λr (cid:20) ˆ τ Br n ( X s , z − X s ) ds (cid:21) ; X τ ˜ B λr ∈ B ′ λr (cid:21) . (2.13)Let w ∈ B ′ λr . Then (J1), (J2), Proposition 2.5 and (2.11) yield E w (cid:20) ˆ τ Br n ( X s , z − X s ) ds (cid:21) ≥ E w (cid:20) ˆ τ B ′ λr n ( X s , z − X s ) ds (cid:21) ≥ c E w (cid:20) ˆ τ B ′ λr j ( | z − X s | ) ds (cid:21) ≥ c E w τ B ′ λr (4 λr ) − d ˆ B λr j ( | z − u | ) du ≥ c λ α − d r α − d ℓ ( λr ) ˆ B λr j ( | z − u | ) du . (2.14)Combining (2.12), (2.13) and (2.14) we obtain E y (cid:20) ˆ τ Br n ( X s , z − X s ) ds (cid:21) ≥ c λ α − d r α − d ℓ ( λr ) ˆ B λr j ( | z − u | ) du E y h P X τB λr ( X τ ˜ B λr ∈ B ′ λr ); X τ B λr ∈ ˜ B λr i . Similarly as in the proof of Proposition 2.7 we obtain, for some c = c ( λ ) > P v ( X τ ˜ B λr ∈ B ′ λr ) ≥ c for all v ∈ ˜ B λr and P u ( X τ B λr ∈ ˜ B λr ) ≥ c for all u ∈ B λr . ONDEGENERATE JUMP PROCESSES 15
Therefore, E y (cid:20) ˆ τ Br n ( X s , z − X s ) ds (cid:21) ≥ c r α − d ℓ ( λr ) ˆ B λr j ( | z − u | ) du . (2.15)On the other hand, by Proposition 2.6 and Lemma 2.8, E x (cid:20) ˆ τ Bλr n ( X s , z − X s ) ds (cid:21) ≤ c E x (cid:20) ˆ τ Bλr j ( | z − X s | ) ds (cid:21) ≤ c E x τ B λr (4 r ) − d ˆ B λr j ( | z − u | ) du ≤ c r α − d ℓ (2 λr ) ˆ B λr j ( | z − u | ) du. (2.16)It follows from (2.15) and (2.16) that E x (cid:20) ˆ τ Bλr n ( X s , z − X s ) ds (cid:21) ≤ c E y (cid:20) ˆ τ Br n ( X s , z − X s ) ds (cid:21) . (2.17)Next, we consider Case 2, i.e. n ( u, z − u ) = 0 for all u ∈ B ( x , λr ). Also in thiscase, assertion (2.17) holds true, because E y (cid:20) ˆ τ Br n ( X s , z − X s ) ds (cid:21) ≥ , E x (cid:20) ˆ τ Bλr n ( X s , z − X s ) ds (cid:21) = 0 . (2.18)We have shown that (2.17) always holds. It is enough to prove the proposition for H = A , where A ⊂ B ( x , r/ c . We conclude from Proposition 2.1 and (2.17)that P y ( X τ Br ∈ A ) = ˆ A E y (cid:20) ˆ τ Br n ( X s , z − X s ) ds (cid:21) dz ≥ c − ˆ A E x (cid:20) ˆ τ Bλr n ( X s , z − X s ) ds (cid:21) dz = c − P x ( X τ Bλr ∈ A ) . (cid:3) Harnack inequality
In this section we prove Theorem 1.2.
Proof of Theorem 1.2.
Since f is non-negative in B ( x , r ), we may assume thatinf x ∈ B ( x ,r ) f ( x ) is positive. If not, we would prove the claim for f ε = f + ε andthen consider ε → f we may further assumeinf x ∈ B ( x ,r ) f ( x ) = .Choose u ∈ B ( x , r ) such that f ( u ) ≤
1. By Proposition 2.6 and using propertiesof slowly varying functions we can find a constant c > u, v ∈ R d and s ∈ (0 , r ] E u τ B ( v, s ) ≤ c s α ℓ ( s ) and E u τ B ( v,s ) ≤ c r α ℓ ( r ) . (3.1)From Proposition 2.7 we deduce that there is a constant c > λ ∈ (0 , sin ϑ ]such that for all A ⊂ B ( x , λr ) and y ∈ B ( x , λr ) P y ( T A < τ B ( x , r ) ) ≥ c | A || B ( x , r ) | . (3.2)Similarly, by Proposition 2.7 we see that there exists a constant c ∈ (0 ,
1) such thatfor every x ∈ R d , s < r and C ⊂ B ( x, λs ) with | C | / | B ( x, λs ) | ≥ P x ( T C < τ B ( x,s ) ) ≥ c . The idea of the proof is to show that f is bounded from the above in B ( x , r ) by c r α ℓ ( r ) sup v ∈ B ( x , r ) ˆ B ( x , r ) c f − ( z ) n ( v, z − v ) dz ! , for some constant c > f . This will be proved bycontradiction.Define η = c ζ = η C , (3.3)where C is taken from Proposition 2.9.Assume that there exists x ∈ B ( x , r ) such that f ( x ) = K for some K > max (cid:26) K ζ , · d λ − d K c ζ (cid:27) , ONDEGENERATE JUMP PROCESSES 17 where K = 1 + c r α ℓ ( r ) sup v ∈ B ( x , r ) ˆ B ( x , r ) c f − ( z ) n ( v, z − v ) dz. (3.4)Let s = (cid:16) K c ζK (cid:17) /d λ − r . Then s < r and | B ( x, λs ) | = 2 K c ζ K | B ( x , r ) | . Set B s := B ( x, s ) and τ s := τ B ( x,s ) . Let A be a compact subset of A ′ = { w ∈ B ( x, λs ) : f ( w ) ≥ ζ K } . By the optional stopping theorem, (3.1), (3.2) and Proposition 2.11 ≥ f ( u ) = E u [ f ( X T A ∧ τ B ( x , r ) )] ≥ E u [ f ( X T A ∧ τ B ( x , r ) ); T A < τ B ( x , r ) ] − E u [ f − ( X T A ∧ τ B ( x , r ) ); T A > τ B ( x , r ) ] ≥ ζ K P u ( T A < τ B ( x , r ) ) − E u [ f − ( X τ B ( x , r ) )]= ζ K P u ( T A < τ B ( x , r ) ) − E u (cid:20) ˆ τ B ( x , r ) ˆ B ( x , r ) c f − ( z ) n ( X t , z − X t ) dz dt (cid:21) ≥ c ζ K | A || B ( x , r ) | − c r α ℓ ( r ) sup v ∈ B ( x , r ) ˆ B ( x , r ) c f − ( z ) n ( v, z − v ) dz. Using (3.4) we obtain | A || B ( x, λs ) | ≤≤ c r α ℓ ( r ) sup v ∈ B ( x , r ) ˆ B ( x , r ) c f − ( z ) n ( v, z − v ) dz ! | B ( x , r ) | c ζ K | B ( x, λs ) | = K c ζ K | B ( x , r ) || B ( x, λs ) | = , which implies | A ′ || B ( x, λs ) | ≤ . Let C ⊂ B ( x, λs ) \ A ′ be a compact subset such that | C || B ( x, λs ) | ≥ . (3.5)Let H = f + B c s/ . Assume that E x [ H ( X τ λs )] > ηK. (3.6) Then for any y ∈ B ( x, λs ) we have f ( y ) = E y f ( X τ s ) = E y f + ( X τ s ) − E y f − ( X τ s )= E y f + ( X τ s ) − E y [ f − ( X τ s ); X τ s B ( x , r )] ≥ E y [ f + ( X τ s ); X τ s B s/ ] − E y [ f − ( X τ s ); X τ s B ( x , r )] . Applying Proposition 2.9 to H it follows f ( y ) ≥ C − E x [ f + ( X τ λs ); X τ λs B s/ ] − c r α ℓ ( r ) sup v ∈ B ( x , r ) ˆ B ( x , r ) c f − ( z ) n ( v, z − v ) dz. Combining the last display with the assumption (3.6) and the definition of ζ in (3.3)gives f ( y ) ≥ C − ηK − K = ζ K (cid:16) − K ζK (cid:17) ≥ ζ K for all y ∈ B ( x, λs ) , which is a contradiction to (3.5). Therefore E x [ H ( X τ λs )] ≥ ηK .Let M = sup v ∈ B s/ f ( v ). Then K = f ( x ) = E x [ f ( X T C ); T C < τ s ] + E x [ f ( X τ s ); τ s < T C , X τ s ∈ B s/ ]+ E x [ f ( X τ s ); τ s < T C , X τ s B s/ ] ≤ ζ K P x ( T C < τ s ) + M (1 − P x ( T C < τ s )) + ηK and thus MK ≥ − η − ζ P x ( T C < τ s )1 − P x ( T C < τ s ) . From the last display we conclude that M ≥ K (1 + 2 β ) with β = c − c ) + ζ > x ′ ∈ B ( x, s ) so that f ( x ′ ) ≥ K (1 + β ).Using this procedure we obtain sequences ( x n ) and ( s n ) such that x n +1 ∈ B ( x n , s n )and K n := f ( x n ) ≥ (1 + β ) n − K . Thus ∞ X n =1 | x n +1 − x i | ≤ ∞ X n =1 s i ≤ c (cid:0) K K (cid:1) /d r, for some constant c > K > K c d , then ( x n ) is a sequence in B ( x , r ) such thatlim n → + ∞ f ( x n ) ≥ lim n → + ∞ (1 + β ) n − K = ∞ . ONDEGENERATE JUMP PROCESSES 19
This is a contradiction with the boundedness of f and so K ≤ c d K . Thussup v ∈ B ( x ,r ) f ( v ) ≤ c d K = c d r α ℓ ( r ) sup v ∈ B ( x , r ) ˆ B ( x , r ) c f − ( z ) n ( v, z − v ) dz ! . Now, let x, y ∈ B ( x , r ). Then f ( x ) ≤ c d r α ℓ ( r ) sup v ∈ B ( x , r ) ˆ B ( x , r ) c f − ( z ) n ( v, z − v ) dz ! ≤ c d f ( y ) + c d r α ℓ ( r ) sup v ∈ B ( x , r ) ˆ B ( x , r ) c f − ( z ) n ( v, z − v ) dz. The proof is complete. (cid:3) Regularity estimates
In this section we prove a general tool that allows to deduce regularity estimatesfrom the version of the Harnack equality given in Theorem 1.2. This approach isdeveloped in [Kas], see also Theorem 3 in [DK].
Theorem 4.1.
Let m : R d × (cid:0) R d \ { } (cid:1) → [0 , ∞ ) be a measurable function such that sup x ∈ R d ´ R d ( | h | ∧ m ( x, h ) dh is finite. Assume there is a function γ : (0 , ∞ ) → (0 , ∞ ) such that for all x, h ∈ R d , h = 0 k (cid:16) h | h | (cid:17) γ ( | h | ) ≤ m ( x, h ) ≤ γ ( | h | ) , (4.1) where k : S d − → [0 , ∞ ) is a measurable bounded symmetric function such thatthere is δ > and a non-empty open set I ⊂ S d − with k ( ξ ) ≥ δ for every ξ ∈ I .Furthermore, assume that lim sup R →∞ R σ ˆ B (0 ,R ) c γ ( | u | ) du ≤ , lim inf r → r σ ˆ B (0 ,r ) c γ ( | u | ) du ≥ , (4.2) with < σ ≤ σ . Let L be a non-local operator defined by L f ( x ) = ˆ R d \{ } ( f ( x + h ) − f ( x ) − h∇ f ( x ) , h i {| h |≤ } ) m ( x, h ) dh (4.3) for f ∈ C b ( R d ) .Assume that harmonic functions with respect to L satisfy a Harnack inequality, i.e. there exist constants c , c ≥ such that for every x ∈ R d , r ∈ (0 , ) and for everybounded function f : R d → R which is non-negative in B ( x , r ) and harmonic in B ( x , r ) the following Harnack inequality holds for all x, y ∈ B ( x , r ) f ( x ) ≤ c f ( y ) + c M ( x , r ) sup v ∈ B ( x , r ) ˆ B ( x , r ) c f − ( z ) m ( v, z − v ) dz , (4.4) where M ( x , r ) = ( ´ B ( x , r ) c m ( x , z − x ) dz ) − .Then there exist β ∈ (0 , , c ≥ such that for every x ∈ R d , every R ∈ (0 , ,every function f : R d → R which is harmonic in B ( x , R ) and every ρ ∈ (0 , R/ x,y ∈ B ( x ,ρ ) | f ( x ) − f ( y ) | ≤ c k f k ∞ ( ρ/R ) β . (4.5) Remark:
Conditions (4.1), (4.2), (4.3) do not imply in general that L satisfies aHarnack inequality, see the discussion of Example 2.Let us illustrate this result by giving two examples. Example 3: m ( x, h ) = | h | − d − α , i.e. k ≡ γ ( t ) = t − d − α , σ = σ = α . Then L = c ( α )∆ α/ . The Harnack inequality (4.4) then becomes f ( x ) ≤ c f ( y ) + c r α ˆ B ( x , r ) c f − ( z ) | z − x | − d − α dz , (4.6)and the theorem can be applied. Note that the function f in (4.6) might be negativeoutside of B ( x , r ). Example 4: m ( x, h ) ≍ | h | − d − α , i.e. k ≡ γ ( t ) = t − d − α , σ = σ = α , cf. [BL02].The Harnack inequality can be formulated as in (4.6). Proof of Theorem 1.4.
We apply Theorem 4.1. Let k = k as in (1.4) and I = B as in (1.5). Set m ( x, h ) = n ( x, h ), γ ( t ) = j ( t ), σ = σ and σ = α − ε where ε ∈ (0 , α − σ ) is arbitrary. Then the first condition in (4.2) follows from (J3). Thesecond condition follows from r σ ˆ ∞ r s d − j ( s ) ds = r α − ε ˆ ∞ r s − − α ℓ ( s ) ds ∼ (1 /α ) r − ε ℓ ( r ) → + ∞ for r → , where we use Proposition 2.2 (ii). It remains to check that there is a constant c > x ∈ R d and every r ∈ (0 , ) r α ℓ ( r ) ≤ cM ( x , r ) , i.e. ˆ B ( x , r ) c m ( x , z − x ) dz ≤ c ℓ ( r ) r α . ONDEGENERATE JUMP PROCESSES 21
This condition follows from ˆ B ( x , r ) c m ( x , z − x ) dz ≤ ˆ B ( x , r ) c j ( | z − x | ) dz ≤ c ℓ (4 r )(4 r ) α ≤ c ℓ ( r ) r α , (4.7)where we use Proposition 2.2 (ii) again. (cid:3) Proof of Theorem 4.1.
For x ∈ R d and r ∈ (0 ,
1) let ν xr denote the measure on B ( x , r ) c defined by ν xr ( A ) = (cid:16) ˆ A γ ( | z − x | ) dz (cid:17)(cid:16) ˆ B ( x ,r ) c γ ( z − x ) dz (cid:17) − for every Borel set A ⊂ B ( x , r ) c . With some positive constant c ≥ k we obtain for every bounded function f : R d → R M ( x , r ) sup x ∈ B ( x ,r/ ˆ B ( x ,r ) c f − ( z ) m ( x, z − x ) dz ≤ c (cid:16) ˆ B ( x ,r ) c γ ( | y − x | ) dy (cid:17) − sup x ∈ B ( x ,r/ ˆ B ( x ,r ) c f − ( z ) γ ( | z − x | ) dz . This observation together with the main assumption of the theorem ensures thatthere exist constants c , c ≥ x ∈ R d , r ∈ (0 ,
1) andevery bounded function f : R d → R which is non-negative in B ( x , r ) and harmonicin B ( x , r ) the following estimate holdssup B ( x ,r/ f ≤ c inf B ( x ,r/ f + c sup x ∈ B ( x ,r/ ˆ B ( x ,r ) c f − ( z ) ν xr ( dz ) . (4.8)We aim to apply Lemma 11 from [DK]. Note that it is not important for theapplication of [DK, Lemma 11] whether harmonicity is defined with respect to anoperator L or some Dirichlet form. Assumption (4.2) implies that there are c ≥ R > R > R , r ∈ (0 ,
1) and x ∈ B ( x , r/ ˆ B ( x ,R ) c γ ( | z − x | ) dz ≤ c R − σ (4.9)Moreover, there is c ≥ (cid:16) ˆ B ( x ,r ) c γ ( | z − x | ) dz (cid:17) − ≤ c r σ . (4.10) Estimates (4.9) and (4.10) imply: ∃ c ≥ ∀ r ∈ (0 , ∃ j ≥ ∀ j ≥ j ∀ x ∈ B ( x , r ) : ν xr (cid:0) B ( x , j r ) c (cid:1) ≤ c (2 j r ) − σ r σ ≤ c − σj . Recall that we assumed σ ≤ σ . Note that 2 − σ < c /j → j → ∞ . Wefinally provedsup We explain the geometric arguments behind the proof of Proposition 2.9Given η ∈ S d − and ρ > V ( η, ρ ) ⊂ R d as follows. Set S ( η, ρ ) = (cid:0) B ( η, ρ ) ∪ B ( − η, ρ ) (cid:1) ∩ S d − and V ( η, ρ ) = { x ∈ R d | x = 0 , x | x | ∈ S ( η, ρ ) } . From now on, we keep η ∈ S d − and ρ > V instead of V ( η, ρ ).Choose ϑ ∈ (0 , π ] so that ρ = 2(1 − cos ϑ ).Using a simple geometric argument one can establish the following fact:Let λ ∈ (0 , sin ϑ ), x ∈ R d , r ∈ (0 , u ∈ B λr ( x ) and z ∈ B ( x , r ) c . Assume z ∈ u + V . Set e x = x − r ξ ∈ ∂B ( x , r ) where ξ ∈ { + η, − η } is chosen so that h z − u , ξ i > 0, see Figure 2. Then the choice of λ implies(1) B (˜ x , λr ) ⊂ T u ∈ B ( x , λr ) ( u + V ) .Moreover, there is ˜ z ∈ ∂B ( x , r ) such that(2) B (˜ z , λr ) ⊂ T v ∈ B (˜ x , λr ) ( v + V ) ,(3) z ∈ T w ∈ B (˜ z , λr ) ( w + V ) ,(4) | z − ˜ z | < | z − x | and thus | z − w | < | z − u | for all u ∈ B ( x , λr ) , w ∈ B (˜ z , λr ) . ONDEGENERATE JUMP PROCESSES 23 b b b z b b B (˜ z , λr ) B ( x , λr ) B (˜ x , λr ) B ( x , r ) u Figure 2. The choice of ˜ x and ˜ z .These conditions assure that the Markov jump process under consideration has astrictly positive probability to jump from a neighborhood of x via neighborhoodsof ˜ x and ˜ z to z . One could avoid the introduction of ˜ z and let the process jumpdirectly from the neighborhood of ˜ x to z but this would result in a slightly strongerassumption than (J2). References [BGT87] N. H. Bingham, C. M. Goldie, and J. L. Teugels. Regular variation . Cambridge UniversityPress, Cambridge, 1987.[BK05a] R. F. Bass and M. Kassmann. Harnack inequalities for non-local operators of variableorder. Trans. Amer. Math. Soc. , 357:837–850, 2005.[BK05b] R. F. Bass and M. Kassmann. H¨older continuity of harmonic functions with respect tooperators of variable orders. Comm. Partial Differential Equations , 30:1249–1259, 2005.[BL02] R. F. Bass and D. Levin. Harnack inequalities for jump processes. Potential Anal. , 17:375–388, 2002.[BS05] K. Bogdan and P. Sztonyk. Harnack’s inequality for stable L´evy processes. PotentialAnal. , 22(2):133–150, 2005.[BSS02] K. Bogdan, A. St´os, and P. Sztonyk. Potential theory for L´evy stable processes. Bull.Polish Acad. Sci. Math. , 50(3):361–372, 2002.[CS09] L. A. Caffarelli and L. Silvestre. Regularity theory for fully nonlinear integro-differentialequations. Communications on Pure and Applied Mathematics , 62(5):597–638, 2009. [DK] B. Dyda and M. Kassmann. Comparability and regularity estimates for symmetric non-local dirichlet forms. Preprint.[Kas] M. Kassmann. Harnack’s inequality and H¨older regularity estimates for nonlocal opera-tors. Preprint.[KS07] P. Kim and R. Song. Potential theory of truncated stable processes. Math. Z. , 256:139–173, 2007.[Mim10] A. Mimica. Harnack inequalities for some L´evy processes. Potential Anal. , 32:275–303,2010.[Pic96] J. Picard. On the existence of smooth densities for jump processes. Probab. Theory Relat.Fields , 105:481–511, 1996.[Sil06] L. Silvestre. H¨older estimates for solutions of integro-differential equations like the frac-tional Laplace. Indiana Univ. Math. J. , 55(3):1155–1174, 2006.[SV04] R. Song and Z. Vondraˇcek. Harnack inequalities for some classes of Markov processes. Math. Z. , 246:177–202, 2004. Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, Postfach 100131, D-33501 Biele-feld, Germany E-mail address : [email protected] Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, Postfach 100131, D-33501 Biele-feld, Germany E-mail address ::