An Overshoot Approach to Recurrence and Transience of Markov Processes
aa r X i v : . [ m a t h . P R ] J u l An Overshoot Approach toRecurrence and Transience of Markov Processes
Bj¨orn B¨ottcherJuly, 2010
Fakult¨at Mathematik und Naturwissenschaften, Institut f¨ur mathematische Stochastik,01062 Dresden, Germany, bjoern.boettcher at tu-dresden.de
Abstract
We develop criteria for recurrence and transience of one-dimensional Markov processeswhich have jumps and oscillate between + ∞ and −∞ . The conditions are based on a Markovchain which only consists of jumps (overshoots) of the process into complementary parts ofthe state space.In particular we show that a stable-like process with generator − ( − ∆) α ( x ) / such that α ( x ) = α for x < − R and α ( x ) = β for x > R for some R > α, β ∈ (0 ,
2) is transientif and only if α + β <
2, otherwise it is recurrent.As a special case this yields a new proof for the recurrence, point recurrence and tran-sience of symmetric α -stable processes. Keywords:
Markov processes with jumps, recurrence, transience, stable-like processes
The recurrence and transience of Markov process has been studied by various authors and varioustechniques, there is the potential theoretic approach (see Getoor [7] for a unification of thecriteria) and the Markov chain approach by Meyn and Tweedie [10]. In particular for Fellerprocesses there have been several attempts to classify their behavior based on the generator orthe associated Dirichlet form, see Chapter 6 of Jacob [8] and the references given therein.In one dimension a transient process either drifts to infinity (i.e. lim t →∞ X t = + ∞ or = −∞ )or it may be oscillating: lim sup t →∞ X t = + ∞ and lim inf t →∞ X t = −∞ . An oscillating process may be recurrent, transient or neither of those (cf. Sections 2 and 4 forthe definitions). Even for such a simple process as the stable-like process (a Markov processwith generator − ( − ∆) α ( x ) / and symbol | ξ | α ( x ) , respectively; see Bass [2] for a construction) isthe recurrence and transience behavior in general unknown. Besides symmetric α -stable L´evyprocesses the only processes of this type treated in the literature are processes where α ( · ) isperiodic [6] or related processes where the generator is a symmetric Dirichlet form [17, 18]. Theinitial motivation for this paper was to treat the non-symmetric case. But in the following wedevelop a more general framework.In Section 2 we introduce a “local” notion of recurrence and transience for which we willgive sufficient conditions in Section 3. Afterwards in Section 4 the local notions are linked tothe (global) recurrence and transience of the processes. In particular conditions which imply therecurrence-transience dichotomy are given. Furthermore we give a result which allows to comparethe behavior of Markov processes which coincide outside some compact ball.The paper closes with an application to stable and stable-like processes. We consider time homogeneous strong Markov processes (Ω , F , F t , X t , θ t , P x ) with c`adl`ag pathson R d ( d ∈ N ), where the filtration ( F t ) t ≥ satisfies the usual conditions. Note that ( θ t ) t ≥ is the1amily of shift operators on Ω, i.e. X s ( θ t ( ω )) = X t + s ( ω ) for ω ∈ Ω.To simplify notation we denote such a process by ( X t ) t ≥ . The state space R d will be equippedwith the Borel- σ -algebra B ( R d ) and sets will be elements of B ( R d ) if not stated otherwise. For aset A the first entrance time is defined, with the convention inf ∅ = ∞ , by τ A := inf { t ≥ | X t ∈ A } . Note that τ A is a stopping time for any A ∈ B ( R d ), since the process is right continuous andadapted, hence progressive. Furthermore for any stopping time σ also τ A,σ := inf { t ≥ σ | X t ∈ A } is a stopping time since { τ A,σ ≤ t } = [ s ∈ Q ∩ [0 ,t ] { X s ∈ A } ∩ { σ ≤ s } ∈ F t (compare [5], Chapter 2, Prop. 1.5).Now we define a pointwise (local) notion of recurrence and transience. Definition 2.1.
Let ( X t ) t ≥ be R d -valued process and b ∈ R d . With respect to ( X t ) t ≥ the point b is called • recurrent if P b ( ∀ T > ∃ t > T : X t = b ) = 1 , • left limit recurrent if P b ( ∀ T > ∃ t > T : X t − = b ) = 1 , • locally recurrent if P b (lim inf t →∞ | X t − b | = 0) = 1 , • locally transient if P b (lim inf t →∞ | X t − b | = 0) < , • transient if P b (lim inf t →∞ | X t − b | = ∞ ) = 1 . Remark 2.2.
The notion of local is meant in a spatial sense, as opposed to a temporal sense. Onewould get the latter by transferring the definition of (deterministic) locally recurrent functions(e.g. [4]) to processes.Note that only for left limit recurrence we need that the paths have left limits, the rightcontinuity is not necessary for these definitions. The reason of introducing left limit recurrenceat all, is that our method will not allow to prove recurrence for points but at most left limitrecurrence. Nevertheless we have the following Lemma to conclude recurrence for a point.
Lemma 2.3.
Let ( X t ) t ≥ be quasi left continuous, i.e. for every increasing sequence of stoppingtimes σ n with limit σ : X σ n n →∞ −−−−→ X σ a.s. on { σ < ∞} . Then the following implication holds: b is left limit recurrent ⇒ b is recurrent. roof. Define σ := k ∈ N and for n ∈ N σ n := inf { t ≥ σ n − (cid:12)(cid:12) | X t − b | < n } and σ := lim n →∞ σ n . Clearly ( σ n ) n ∈ N is increasing. Thus σ is well defined and P b ( σ < ∞ ) = 1 , since b is left limit recurrent. Note that σ n might be constant for n large, but in this case theprocess is already in b. In general by the quasi left continuity P b ( X σ = lim n →∞ X σ n = b ) = 1holds. Since k was arbitrary this yields that b is recurrent.Further simple consequences of Definition 2.1 are that (left limit) recurrence implies localrecurrence and that we have the dichotomy b is either locally recurrent or locally transient . (2.1)A process ( X t ) t ≥ is point recurrent if and only if all b ∈ R d are recurrent. The other commonnotions for recurrence and transience of processes do not have such a simple relation to the abovelocal notions. Details will be given in Section 4. In this section we treat for simplicity the case d = 1, see Remark 3.5 for the extension to higherdimensions. Let ( X t ) t ≥ be a process on R satisfyinglim sup t →∞ X t = ∞ and lim inf t →∞ X t = −∞ a.s.. (3.1)Further assume that there exists some b ∈ R such that for the stopping times τ b := inf { t ≥ | X t ≤ b } and σ b := inf { t ≥ | X t ≥ b } the process satisfies P x ( X τ b = b ) = 0 for all x > b, P x ( X σ b = b ) = 0 for all x < b, (3.2)i.e. the process almost surely enters ( −∞ , b ] and [ b, ∞ ) not by hitting b . The distributions of X τ b and X σ b are called overshoot distributions . Remark 3.1.
Note that assumption (3.2) is not equivalent to assuming that the process isnon-creeping. For example consider a compound Poisson process on R with jump distribution δ − + δ . The process started in 0 is non-creeping but hits b = 1 with probability one.Now define σ := 0 and for each n ∈ N set τ n := inf { t ≥ σ n − | X t < b } ,σ n := inf { t ≥ τ n | X t > b } . Note that σ is always the the first time of passing b from below. Contrary τ is for the processstarted in x > b the first time of passing b from above, but τ = 0 for x < b .These stopping times have the following properties. Proposition 3.2.
Let x = b , then i) P x ( τ n < ∞ ) = 1 and P x ( σ n < ∞ ) = 1 for all n ∈ N , { X τ n < b } ⊂ { σ n > τ n } , iii) P x ( X τ n < b ) = 1 implies P x ( X σ n > b ) = 1 , iv) P x ( X σ n > b, X τ n < b, ∀ n ∈ N ) = 1 , v) P x ( σ n − < τ n < σ n , ∀ n ∈ N ) = 1 .Proof. i) By (3.1) the process will pass b infinitely often almost surely, i.e. τ n and σ n are finitealmost surely.ii) Let ω ∈ { X τ n < b } . Then by the right continuity there exists an ε ω > X τ n + ε ω ( ω ) < b , since ( X t ) t ≥ is c`adl`ag. Thus σ n ( ω ) ≥ τ n ( ω ) + ε ω , i.e. σ n ( ω ) > τ n ( ω ) . iii) First note that P x ( X τ n < b ) = 1 implies by ii) that P x ( σ n > τ n ) = 1 , and τ n is a finitestopping time by i). By the right continuity { X σ = b } contains all paths which enter( b, ∞ ) continuously from b and { X σ b = b } contains all paths which enter [ b, ∞ ) at b. Thus { X σ = b } ⊂ { X σ b = b } , i.e. P y ( X σ = b ) ≤ P y ( X σ b = b ) = 0 , which implies P y ( X σ > b ) = 1 . Now for y < b the strong Markov property (note: σ n = σ ◦ θ τ n ) yields P x ( X σ n > b | X τ n = y ) = P y ( X σ > b ) = 1 . Then P x ( X σ n > b ) = Z ( −∞ ,b ] P x ( X σ n > b | X τ n = y ) P x ( X τ n ∈ dy )= Z ( −∞ ,b ] P x ( X τ n ∈ dy ) = 1 . iv) Analogously to ii) and iii) one gets:ii*) { X σ n > b } ⊂ { τ n +1 > σ n } , iii*) P x ( X σ n > b ) = 1 implies P x ( X τ n +1 < b ) = 1 , and further P x ( X τ < b ) = 1holds. Thus repeated applications of iii) resp. iii*) yield P x ( X τ n < b ) = P x ( X σ n > b ) = 1 for each n ∈ N . Thus P x ( X τ n < b, X σ n > b, ∀ n ∈ N ) = 1as a countable intersection of sets of measure one.v) This is a consequence of ii), ii*) and iv).Now define for x > b on the set { σ n − < τ n < σ n , ∀ n ∈ N } , which has probability one byProposition 3.2 v), the sequence ( Y n ) n ≥ by Y n := X σ n and note that by the strong Markov property for B ∈ B ( R ) P ( Y n ∈ B | Y n − = x ) = P ( X σ n ∈ B | X σ n − = x ) = P ( X σ ∈ B | X = x ) = P ( Y ∈ B | Y = x ) , i.e. ( Y n ) n ≥ is a Markov chain on ( b, ∞ ). This Markov chain captures only the first set of countablymany overshoots passing b from ( −∞ , b ) of the process ( X t ) t ≥ , since the times ( σ n ) n ≥ arestrictly increasing but possibly bounded.Nevertheless this Markov chain can be used to determine the local recurrence/transiencebehavior of ( X t ) t ≥ by the following theorem. 4 heorem 3.3. Let ( X t ) t ≥ and ( Y n ) n ≥ be as defined above. i) If P x (lim n →∞ Y n = ∞ ) = 1 for all x > b and there exists r, R > and c < such that sup y ∈ [ b − r,b + r ] y = b P y ( X σ > b + R ) < c (3.3) then b is locally transient. ii) If P x (lim inf n →∞ Y n = b ) = 1 for all x > b then b is locally recurrent. iii) If P x (lim n →∞ Y n = b ) = 1 for all x > b and there exists r ′ , R ′ > and c < such that sup y ≥ b + r ′ P y ( X σ < b + R ′ ) < c (3.4) then b is left limit recurrent. Remark 3.4.
Roughly speaking, condition (3.3) ensures that the overshoots represent the wholeprocess, whereas condition (3.4) ensures that the limit b is reached in finite time. The followingtwo examples show these conditions cannot be removed.1. Let ( N t ) t ≥ be a Poisson process and ( ˜ X n ) n ≥ be a Markov chain with transition distribu-tion P ( ˜ X ∈ dy | ˜ X = x ) = δ x ( dy ) for | x | > ,δ − | x | x ( dy ) for 0 < | x | ≤ ,δ ( dy ) for x = 0 . The Markov chain is in fact deterministic and, when started in 0, the chain moves as0 , , − , − , , , − , − , . . . . Now the chain subordinated by the Poisson process is a c´adl´ag time homogeneous strongMarkov process satisfying (3.1) and (3.2) for b = 0 . Furthermore 0 is locally recurrent andthus not locally transient. The associated chain of overshoots is deterministic, especiallyfor x ∈ (0 ,
1] : Y = x, Y = 1 x + 2 and for n ∈ N Y n = 1 x + 2 n, i.e. lim n →∞ Y n = ∞ and ∀ R, r > y ∈ [ − r,r ] ,y =0 P y ( X σ > R ) ≥ sup y ∈ (0 ,r ] P y ( Y > R ) = 1 .
2. Changing the transition distribution to P ( ˜ X ∈ dy | ˜ X = x ) = δ − x ( dy ) for | x | > ,δ | x | x ( dy ) for 0 < | x | ≤ ,δ ( dy ) for x = 0 . yields that the chain started in 0 moves as0 , , , − , − , , , − , − , . . . . Thus for the chain subordinated by the Poisson process 0 is locally recurrent but not leftlimit recurrent (in finite time). For the associated jump chain for x > Y = x, Y = 1 x + 1 and in general Y n = 1 x + 2 n − , i.e. lim n →∞ Y n = 0 and ∀ R, r > y ≥ r P y ( X σ < R ) = sup y ≥ r P y ( Y < R ) = 1 . roof of Theorem 3.3. i) By (3.1) ( X t ) t ≥ does not explode in finite time. This and ∞ =lim n →∞ Y n = lim n →∞ X σ n a.s. imply that σ n → ∞ almost surely. Let r, R and c be as in(3.3). Now fix ε > . Then there exists a
N > ∀ n ≥ N : P x ( X σ n > R + b ) ≥ − ε, since lim n →∞ X σ n = ∞ . Let n ≥ N and define ν n as the time of the first visit to B := [ b − r, b + r ] \{ b } after time σ n , i.e. ν n := inf { t ≥ σ n | X t ∈ B } and σ k be the time of the first jump into ( b, ∞ ) after ν n , i.e. k := inf { l ∈ N | σ l > ν n } . Now suppose b is locally recurrent. An overshoot hits b with probability zero, thus thelocal recurrence of b implies that that P x ( ν n < ∞ ) = 1 and P x ( X ν n ∈ B ) = 1 . Thus P x ( k < ∞ ) = 1 and σ k = σ ◦ θ ν n , where θ ν n is the shift operator corresponding to ν n . Thenthe strong Markov property yields1 − ε ≤ P x ( X σ k > R + b )= Z B P x ( X σ k > R + b | X ν n = y ) P x ( X ν n ∈ dy )= Z B P y ( X σ > R + b ) P x ( X ν n ∈ dy ) ≤ sup y ∈ B P y ( X σ > R + b ) < c < , which is a contradiction. Thus b is locally transient.ii) Let lim inf n →∞ Y n = b almost surely. If σ n → ∞ a.s. the statement is obvious. In general let ε > , T > η T := inf { t ≥ T | X t ∈ [1 , ∞ ) } . By (3.1) for all y ∈ R we have P y ( η T < ∞ ) = 1 . Thus for x > P x ( ∃ t > T : | X t − b | < ε ) ≥ P x ( ∃ t > | X t + η T − b | < ε )= Z [1 , ∞ ) P x ( ∃ t > | X t + η T − b | < ε (cid:12)(cid:12) X η T = y ) P x ( X η T ∈ dy )= Z [1 , ∞ ) P y ( ∃ t > | X t − b | < ε ) P x ( X η T ∈ dy ) ≥ Z [1 , ∞ ) P y ( ∃ n ∈ N : | Y n − b | < ε ) P x ( X η T ∈ dy ) = 1 . Since T and ε where arbitrary this implies that b is locally recurrent.iii) If ( σ n ) n ∈ N is a.s. bounded then b is reached as the left limit at least once and the sameargument as in part ii ) implies that b is left limit recurrent.Otherwise set σ ∞ := lim n →∞ σ n and let r ′ , R ′ and c be as in (3.4). Further let ε > N > n ≥ N P x ( X σ n < b + R ′ ) ≥ − ε, such an N exists since lim n →∞ Y n = b a.s.. Now let n ≥ N and define ν n as the time of thefirst visit to ( b + r ′ , ∞ ) after time σ n , i.e. ν n := inf { t ≥ σ n | X t ≥ b + r ′ } k := inf { l ∈ N | σ l > ν n } . Note that σ ∞ = ∞ with positive probability but in general not almost surely. Thus only on { σ k > ν n } the stopping time σ k is the time of the first jump into ( b, ∞ ) after ν n , i.e. on thisset σ k = σ ◦ θ ν n holds. Now { σ k >ν n } is F ν n measurable and the strong Markov propertyby conditioning on F ν n (the σ -algebra associated with ν n ) yields1 − ε ≤ P x ( X σ k < b + R ′ )= P x ( X σ k < b + R ′ , σ k ≤ ν n ) + P x ( X σ k < b + R ′ , σ k > ν n ) ≤ P x ( σ k ≤ ν n ) + E x ( E x ( { X σ ( θ νn ) ν n } | F ν n ))= P x ( σ k ≤ ν n ) + E x ( { σ k >ν n } P X νn ( X σ < b + R ′ )) < P x ( σ k ≤ ν n ) + c P ( σ k > ν n ) ≤ ˜ c < , which is a contradiction, since ε was arbitrary. Thus ( σ n ) n ∈ N is bounded. Remark 3.5.
In order to use this approach for d > −∞ , b ] and [ b, ∞ )by parts of the state space separated by a ( d − b ∈ R d is analogous to the one dimensional case and part iii) requires again areformulation of (3.4) in terms of the hyperplane.But note that for d > In this section we will link local recurrence and local transience to the notion of recurrenceand transience for processes, as used by Meyn and Tweedie e.g. in [10] (our presentation ispartly motivated by [15]). Note that all results of this section would also hold if we weaken ourassumption on the processes from c`adl`ag to only right continuous .By λ we denote the Lebesgue measure. Definition 4.1.
A process ( X t ) t ≥ on R d is called • λ -irreducible if λ ( A ) > ⇒ E x (cid:18)Z ∞ A ( X t ) dt (cid:19) > for all x, • recurrent with respect to λ if λ ( A ) > ⇒ E x (cid:18)Z ∞ A ( X t ) dt (cid:19) = ∞ for all x, • Harris recurrent with respect to λ if λ ( A ) > ⇒ P x (cid:18)Z ∞ A ( X t ) dt = ∞ (cid:19) = 1 for all x, • transient if there exists a countable cover of R d with sets A j such that for each j there isa finite constant M j > such that: E x (cid:18)Z ∞ A j ( X t ) dt (cid:19) < M j , a T -model if for some probability measure µ on [0 , ∞ ) there exists a kernel T ( x, A ) with T ( x, R d ) > for all x such that the function x T ( x, A ) is lower semi-continuous for all A ∈ B ( R d ) and Z ∞ E x ( A ( X t )) µ ( dt ) ≥ T ( x, A ) holds for all x , A ∈ B ( R d ) . We start with the recurrence-transience-dichotomy for λ -irreducible T -models. Theorem 4.2.
Let ( X t ) t ≥ be a λ -irreducible T -model, then it is either Harris recurrent ortransient.Proof. (Compare with the proof of Prop. 3.1 in [15].) A λ -irreducible process is by Thm. 2.3 in[16] either recurrent or transient. In the case of recurrence the reference measure is the so calledmaximal irreducible measure, but this yields in our case especially recurrence with respect to λ. Now suppose the process is recurrent with respect to λ then for all x ∈ R d and all ε > E x (cid:18)Z ∞ B ε ( x ) ( X t ) dt (cid:19) = ∞ where B ε ( x ) = { y ∈ R d | | x − y | < ε } i.e. each x ∈ R d is topological recurrent (cf. Sec. 4 [16])). Thus by Thm. 4.2 in [16] the wholestate space R d is a maximal Harris set, that means there exists a measure φ on B ( R d ) such that( X t ) t ≥ is Harris recurrent with respect to φ. Now φ = µR (cf. proof of Thm. in 2.4 [9] and theproof of Prop. 3.1 in [15]) for some non trivial measure µ and a kernel R which satisfies λ ( A ) > ⇒ R ( x, A ) > ∀ x ∈ R d . Thus ( X t ) t ≥ is Harris recurrent with respect to λ. Now we can state the main theorem of this section which links the local notions introducedin Section 2 to the stability of the process.
Theorem 4.3.
Let ( X t ) t ≥ on R d be a λ -irreducible T -model, then i) ∃ b which is locally recurrent ⇔ ( X t ) t ≥ is Harris recurrent. ii) ∃ b which is locally transient ⇔ ( X t ) t ≥ is transient.Proof. By Theorem 4.2 the process is either Harris recurrent or transient. Thus it is enough toprove the equivalence in i) since also local recurrence and local transience are complementary.As in the previous proof, a point x is called topologically recurrent if E x ( R ∞ A ( X t ) dt ) = ∞ for all neighborhoods A of x . Note that for a λ -irreducible process each point is reachable, i.e.for every x and every neighborhood A we have P x ( τ A < ∞ ) >
0. Thus Thm. 4.1 in [16] yieldsfor a λ -irreducible T-model: ∃ b topologically recurrent ⇔ ( X t ) t ≥ is recurrent . Now assume b is locally recurrent. For any neighborhood A of b we find a open ball withcenter b and radius ε > B ε ( b ) ⊂ A . The local recurrence implies that the process hits B ε ( b ) with probability one, also after arbitrary large times, i.e. for all R > P b ( ∃ t > R : X t ∈ B ε ( b )) = 1 . Furthermore since X t is right continuous the average time spent in B ε ( b ) after hitting B ε ( b ) ispositive, i.e. 0 < inf y ∈ B ε ( b ) E y ( τ R d \ B ε ( b ) ) . Thus we get E b (cid:18)Z ∞ A ( X t ) dt (cid:19) ≥ E b (cid:18)Z ∞ B ε ( X t ) dt (cid:19) ≥ ∞ , b is topological recurrent. Therefore ( X t ) t ≥ is recurrent. By the dichotomy we get that infact ( X t ) t ≥ is Harris recurrent, since it is not transient.On the other Hand, let ( X t ) t ≥ be Harris recurrent. Thus P x (cid:18)Z ∞ A ( X t ) dt = ∞ (cid:19) = 1 for all x and all A with λ ( A ) > B ε ( b ) for any ε > b is locallyrecurrent.We further recall the following theorem, which provides some way to check that ( X t ) t ≥ is aT-model. Theorem 4.4 (Thm. 5.1 and Thm. 7.1 in [16]) . i) ( X t ) t ≥ is a T-model, if every compact set C is petite, i.e. there exists a probability measure µ on [0 , ∞ ) and a non-trivial measure ν on R d such that Z ∞ E x ( A ( X t )) µ ( dt ) ≥ ν ( A ) for all x ∈ C and all A. ii) Let ( X t ) t ≥ be λ -irreducible and x E x ( f ( X t )) be continuous for all continuous and boundedfunctions f , then ( X t ) t ≥ is a T-model. Part ii) in particular shows that every λ -irreducible C b -Feller process is a T-model, and notethat [14] gives necessary and sufficient conditions for a C ∞ -Feller process to be also C b -Feller.Useful for applications is the following theorem which gives sufficient criteria for a process tobe a λ -irreducible T-model. Theorem 4.5.
Let ( X t ) t ≥ be a process on R d and denote its transition probabilities by P t ( x, A ) := P x ( X t ∈ A ) . Then i) ( X t ) t ≥ is λ -irreducible if λ ( A ) > ⇒ P t ( x, A ) > for all t > , x ∈ R d , (4.1)ii) ( X t ) t ≥ is a λ -irreducible T-model if (4.1) holds and there exits a compact set K ⊂ [0 , ∞ ] and a non trivial measure ν such that for all compact sets C ⊂ R d inf t ∈ K inf x ∈ C P t ( x, A ) ≥ ν ( A ) for all A ∈ B ( R d ) . (4.2) Further, a special case of ii) : iii) ( X t ) t ≥ is a λ -irreducible T-model if the transition probability P t ( x, . ) is the sum of a, possiblytrivial, discrete measure and a measure which has a (sub-)probability density ˜ p t ( x, y ) withrespect to λ such that ˜ p t ( x, y ) > for all x, y ∈ R d , t > , (4.3)inf t ∈ [1 , inf x ∈ C ˜ p t ( x, y ) > for all y ∈ R d and all compact sets C. (4.4) Proof.
Assume (4.1) holds and let A be such that λ ( A ) >
0. Then P x ( τ A < ∞ ) ≥ P t ( x, A ) > t > X t ) t ≥ is φ -irreducible with φ ( . ) := λ ( . ) Z [0 , ∞ ) e − t P t ( x, . ) dt. A with λ ( A ) > Z [0 , ∞ ) e − t P t ( x, A ) dt > φ is equivalent to λ , i.e. ( X t ) t ≥ is λ -irreducible.If further (4.2) holds then Theorem 4.4 part i) with µ ( dt ) = e − t dt implies that ( X t ) t ≥ is aT-model.For part iii) note that (4.3) implies that (4.1) holds and (4.4) implies that (4.2) holds with ν being a subprobability measure with density inf t ∈ [1 , inf x ∈ C ˜ p t ( x, . ) e − .We give a further characterization of recurrence and transience in this context, which showsthat it is in fact enough to know the behavior of the process outside some compact set. Theorem 4.6.
Let ( X t ) t ≥ be λ -irreducible T-model, R be some positive constant and B R (0) denote the closed ball centered at 0 with radius R , then i) ∀ x : P x (cid:16) τ BR (0) < ∞ (cid:17) = 1 ⇐⇒ ( X t ) t ≥ is Harris recurrent. ii) ∃ x : P x (cid:16) τ BR (0) < ∞ (cid:17) < ⇐⇒ ( X t ) t ≥ is transient.Proof. Given a λ -irreducible T-model then by Thm. 5.1 in [16] every compact set is petite. ThusThm. 3.3 in [9] implies “ ⇒ ” of i).For ii) “ ⇒ ” note that λ ( B R (0)) >
0. Thus ( X t ) t ≥ cannot be Harris recurrent and thedichotomy implies that it is transient.Harris recurrence and transience are complementary and so are the left hand sides of i) andii). Thus the “ ⇐ ” directions hold.In fact the Theorem 4.6 shows that processes which coincide outside a ball have the samerecurrence and transience behavior, respectively. Corollary 4.7.
Let ( X t ) t ≥ and ( Y t ) t ≥ be λ -irreducible T-models. If there exists an R > suchthat τ X BR (0) d = τ Y BR (0) for all X = Y = x ∈ R d \ B R (0) then ( X t ) t ≥ and ( Y t ) t ≥ have the same recurrence/transience behavior.Here τ X and τ Y are the entrance times corresponding to X t and Y t , respectively and d = denotesequality in distribution.Proof. In the setting of Theorem 4.6 we find P x (cid:16) τ BR (0) = 0 (cid:17) = 1 for all x ∈ B R (0) . This shows that Theorem 4.6 ii) might only hold for some x ∈ R d \ B R (0), i.e. only the distributionsof τ B R (0) for x ∈ R d \ B R (0) need to be checked. Thus, if these distributions coincide for twoprocesses, Theorem 4.6 yields the same behavior. α -stable and stable-like Processes Let ( X t ) t ≥ be a real valued symmetric α -stable process, i.e. it is a L´evy process with character-istic exponent | ξ | α with α ∈ (0 , X t ) t ≥ sampled at integer times ( X n ) n ∈ N is a symmetric randomwalk and (3.1) holds. Define σ b and τ b as in Section 3, i.e. τ b := inf { t ≥ | X t ≤ b } and σ b := inf { t ≥ | X t ≥ b } .
10n 1958 Ray [12] showed that for b > P ( X σ b ∈ dy ) = sin( απ ) π y (cid:18) by − b (cid:19) α [ b, ∞ ) ( y ) dy and in particular for 0 < α < P ( X σ b = b ) = 0 . The translation invariance of ( X t ) t ≥ yields for all b P x ( X σ b ∈ dy ) = P ( X σ b − x + x ∈ dy ) = sin( απ ) π y − x (cid:18) b − xy − b (cid:19) α [ b, ∞ ) ( y ) dy for x < b (5.1)and the symmetry yields P x ( X τ b ∈ dy ) = P − x ( − X σ − b ∈ dy ) = sin( απ ) π x − y (cid:18) x − bb − y (cid:19) α ( −∞ ,b ] ( y ) dy for x > b. (5.2)In particular (3.2) is satisfied.Note that by the translation invariance for any b for x < P x ( X σ < r ) = P x + b ( X σ b < r + b ) , for x > P x ( X τ < r ) = P x + b ( X τ b < r + b ) . Thus we will for simplicity only consider the case b = 0 in the sequel and define the upwards-overshoot density u and the downwards-overshoot density v for α ∈ (0 ,
2) byfor x < u α ( x, y ) := sin( απ ) π y − x (cid:18) − xy (cid:19) α [0 , ∞ ) ( y )for x > v α ( x, y ) := sin( απ ) π x − y (cid:18) − xy (cid:19) α ( −∞ , ( y )We will write X ∼ f for a random variable X with density f . Lemma 5.1.
Let α , β ∈ (0 , and U ∼ u α ( − , · ) and V ∼ v β (1 , · ) be independent. Then i) the overshoot densities satisfy for y ∈ R for x < u α ( x, y ) = − x u α ( − , − yx ) and for x > v β ( x, y ) = 1 x v β (1 , yx ) , ii) for arbitrary probability densities f on [0 , ∞ ) and g on ( −∞ , , and random variables F ∼ f , G ∼ g independent of V and U respectively, it holds that (for s ∈ R ) P ( F V ≤ s ) = Z s −∞ Z ∞−∞ f ( x ) v β ( x, y ) dx dy and P ( − GU ≤ s ) = Z s −∞ Z ∞−∞ g ( x ) u α ( x, y ) dx dy, iii) for r ∈ R E ( U r ) = sin (cid:0) απ (cid:1) sin (cid:16) ( α − r ) π (cid:17) for α − < r < α , ∞ otherwise , nd E (( − V U ) r ) = sin (cid:0) απ (cid:1) sin (cid:16) βπ (cid:17) sin (cid:16) ( α − r ) π (cid:17) sin (cid:16) ( β − r ) π (cid:17) for α ∨ β − < r < α ∧ β , ∞ otherwise, iv) for α + β = 2 there exists a moment of a downwards-overshoot followed by an upwards-overshoot which is less than 1, i.e. α + β < ∃ r < E ( − V U ) r < ,α + β > ∃ r > E ( − V U ) r < , and for α + β = 2 there is a symmetry: ∀ s : P ( − V U ≤ s ) = P (( − V U ) − ≤ s ) . Proof. i) For x < − x u α (cid:16) − , − yx (cid:17) = sin( απ ) π − x (cid:0) − yx + 1 (cid:1) (cid:18) − yx (cid:19) α [0 , ∞ ) (cid:16) − yx (cid:17) = sin( απ ) π y − x (cid:18) − xy (cid:19) α [0 , ∞ ) ( y )= u α ( x, y )holds and analogously for x > x v β (cid:16) , yx (cid:17) = sin( βπ ) π x (cid:0) − yx (cid:1) (cid:18) − yx (cid:19) β ( −∞ , (cid:16) yx (cid:17) = sin( βπ ) π x − y (cid:18) − xy (cid:19) β ( −∞ , ( y )= v β ( x, y ) . ii) Using i) yields for s ∈ R with substitution ˜ yx = y Z s −∞ Z ∞−∞ f ( x ) v β ( x, y ) dx dy = Z ∞−∞ Z s −∞ f ( x ) 1 x v β (1 , yx ) dy dx = Z ∞−∞ Z ∞−∞ ( −∞ ,s ] (˜ yx ) f ( x ) 1 x v β (1 , ˜ y ) x d ˜ y dx = P ( F V ≤ s )and with substitution − ˜ yx = y Z s −∞ Z ∞−∞ g ( x ) u α ( x, y ) dx dy = Z ∞−∞ Z s −∞ g ( x ) (cid:18) − x (cid:19) u α ( − , − yx ) dy dx = Z ∞−∞ Z ∞−∞ ( −∞ ,s ] ( − ˜ yx ) g ( x ) (cid:18) − x (cid:19) u α ( − , ˜ y ) ( − x ) d ˜ y dx = P ( − GU ≤ s ) . iii) Note that Z ∞ ( y + 1) − y − s dy = B (1 − s, s ) = Γ(1 − s )Γ( s )Γ(1) = π sin ( sπ ) for all 0 < s < B ( · , · ) is the Beta function and the last equality holds by the reflection formula forthe Gamma function (e.g. 6.1.17 in [1]). Thus E ( U r ) = Z ∞ y r u α ( − , y ) dy = sin (cid:0) απ (cid:1) π Z ∞ ( y + 1) − y − α + r dy = sin (cid:0) απ (cid:1) sin (cid:16) ( α − r ) π (cid:17) for all r such that α − < r < α . Further for r ≥ α and y ≥ y + 1) − y − α + r ≥ y − α + r − and this is not integrable on [1 , ∞ ) thus for r ≥ α the moment is ∞ . Similarly for r ≤ α − y ≤
1: ( y + 1) − y − α + r ≥ y − α + r and this is not integrable on (0 ,
1] thus for r ≤ α − ∞ .Furthermore for y > v β (1 , − y ) = sin( βπ ) π y + 1 y − β [0 , ∞ ) ( y ) = u β ( − , y )and thus E (( − V ) r ) = Z ∞−∞ ( − y ) r v β (1 , y ) dy = Z ∞−∞ ˜ y r u β ( − , ˜ y ) d ˜ y and the independence of V, U yields E (( − V U ) r ) = sin (cid:0) απ (cid:1) sin (cid:16) βπ (cid:17) sin (cid:16) ( α − r ) π (cid:17) sin (cid:16) ( β − r ) π (cid:17) for r in (cid:16) α ∨ β − , α ∧ β (cid:17) .iv) For r ⋆ = α + β − E (( − V U ) r ⋆ ) = sin (cid:0) απ (cid:1) sin (cid:16) βπ (cid:17) sin (cid:16) α − β π + π (cid:17) sin (cid:16) β − α π + π (cid:17) = sin (cid:0) απ (cid:1) sin (cid:16) βπ (cid:17) cos (cid:16) α − β π (cid:17) = 1 − (cid:16) α + β π (cid:17) (cid:16) α − β π (cid:17) where we used first the translation and symmetry of sin and cos. In the last step formula4.3.31 [1] was used for for the numerator and 4.3.25 [1] for the denominator.Thus the r ⋆ -moment is less than one for α + β = 2 . Note that r ⋆ is negative for α + β < α + β >
2. Finally P ( − U V ≤ s ) = Z Z ( −∞ ,s ] ( − ˜ x ˜ y ) v β (1 , ˜ x ) u α ( − , ˜ y ) d ˜ y d ˜ x = Z Z ( −∞ ,s ] (cid:18) − xy (cid:19) v β (1 , − x ) u α ( − , y ) 1 x y dy dx = Z Z ( −∞ ,s ] (cid:18) − xy (cid:19) sin( απ ) π sin( βπ ) π
11 + x x β − y ( − y ) α x y dy dx = Z Z ( −∞ ,s ] (cid:18) − xy (cid:19) sin( απ ) π sin( βπ ) π x + 1 x β − y − − y ) α − dy dx = Z Z ( −∞ ,s ] (cid:18) − xy (cid:19) v β (1 , y ) u α ( − , x ) x β + α − ( − y ) α + β − dy dx = P ( − ( U V ) − ≤ s ) 13here we used in the second line the substitution ˜ x = − x and ˜ y = − y and for the last stepthe assumption α + β = 2. Theorem 5.2.
Let ( X t ) t ≥ be a c`adl`ag time homogeneous strong Markov process on R such that (3.1) holds and such that there exist b ∈ R , α, β ∈ (0 , such that lim t → E x (cid:18) e iX t ξ − t (cid:19) = ( −| ξ | β for x > b, −| ξ | α for x < b. Then i) b is left limit recurrent if α + β > , ii) b is recurrent if α + β ≥ , iii) b is transient if α + β < .Proof. If b = 0 consider ( X t − b ) t ≥ for which the properties at 0 correspond to those of ( X t ) t ≥ at b . Thus, without loss of generality, we may assume b = 0.Let ( Y n ) n ≥ be the overshoot Markov chain corresponding to ( X t ) t ≥ as defined in Section3. Then for x > s ∈ RP x ( Y n ≤ s ) = Z s −∞ Z ∞−∞ Z ∞−∞ v β ( y, v ) u α ( v, u ) P x ( Y n − ∈ dy ) dv du and by Lemma 5.1 ii) Y n d = Y n − Y i =1 ( − U i V i )where U i ∼ u α ( − , · ), V i ∼ v β (1 , · ) and ( U i ) i =1 ,...,n − , ( V i ) i =1 ,...,n − , Y are independent. Inparticular for r ∈ R E x ( Y rn ) = E x ( Y r ) ( E (( − U V ) r )) n − holds and furthermore using the definition of Y and Lemma 5.1 ii) for ˜ V ∼ v β ( x, · ) independentof U E x ( Y r ) = E ( − ˜ V r ) E ( U r )and E ( − ˜ V r ) = − Z R ˜ v r v β ( x, ˜ v ) d ˜ v = − Z ∞−∞ v r x v β (cid:16) , vx (cid:17) dv = − x r Z R v r v β (1 , v ) dv = x r E ( − V r ) . To prove i), let α + β > r >
0, cf. Lemma 5.1 iv), such that E (( − U V ) r ) < . Then E x ( Y r ) < ∞ and for all ε > ∞ X n =1 P x ( Y n ≥ ε ) ≤ ∞ X n =1 E x ( Y rn ) ε r = E x ( Y r ) ε r ∞ X n =1 E (( − U V ) r ) n − = E x Y r ε r − E (( − U V ) r ) < ∞ holds. Thus the Borel-Cantelli Lemma implies that Y n n →∞ −−−−→ q ∈ ( α ∨ β − , , then 0 < E (( − U V ) q ) < ∞ by Lemma 5.1 iii). With R ′ := (2 E (( − U V ) q )) q we getsup y ≥ P y ( X σ < R ′ ) = sup y ≥ P y ( Y < R ′ ) = sup y ≥ P y ( Y q > R ′ q ) ≤ sup y ≥ y | q | E (( − U V ) q ) R ′ q = 12 , i.e. (3.4) holds. Thus 0 is left limit recurrent by Theorem 3.3 iii).Analogously to prove iii), let α + β < r <
0, cf. Lemma 5.1 iv), such that E (( − U V ) r ) < . E x ( Y r ) < ∞ and for all ε > ∞ X n =1 P x (cid:18) Y n ≥ ε (cid:19) ≤ ∞ X n =1 E x ( Y −| r | n ) ε | r | = E x ( Y r ) ε | r | ∞ X n =1 E (( − U V ) r ) n − = E x Y r ε | r | − E (( − U V ) r ) < ∞ holds. Thus the Borel-Cantelli Lemma implies / Y n n →∞ −−−−→ Y n n →∞ −−−−→ ∞ almost surely. Now let q ∈ (cid:16) , α ∧ β (cid:17) , then 0 < E (( − U V ) q ) < ∞ and R := (2 E (( − U V ) q )) q yieldssup y ∈ (0 , P y ( X σ > R ) = sup y ∈ (0 , P y ( Y > R ) ≤ sup y ∈ (0 , E y ( Y q ) R q = sup y ∈ (0 , y q E (( − U V ) q ) R q = 12 . Moreover, for y < P y ( X σ > R ) = Z ∞ R u α ( y, z ) dz = Z ∞ R − y u α ( − , − zy ) dz = Z ∞− Ry u α ( − , ˜ z ) d ˜ z holds and thus sup y ∈ [ − , P y ( X σ > R ) = Z ∞ R u α ( − , ˜ z ) d ˜ z < R > u α is a probability density with u α ( − , x ) > x >
0. Therefore (3.3) holds and 0 is by Theorem 3.3 i) locally transient.Finally let α + β = 2 and note thatlog Y n d = log Y + n − X i =1 log( − U i V i )holds. By Lemma 5.1 iv) for any r ∈ RP (log( − U V ) ≤ r ) = P (log(( − U V ) − ) ≤ r ) = P (log( − U V ) ≥ − r )and thus log Y n has the same distribution as a symmetric random walk with initial distributiongiven by log Y . Hence lim sup n →∞ log( Y n ) = + ∞ and lim inf n →∞ log( Y n ) = −∞ holds and therefore lim sup n →∞ Y n = + ∞ and lim inf n →∞ Y n = 0 . Now Theorem 3.3 ii) implies that 0 is locally recurrent.
Remark 5.3.
Note that we assumed the existence of the process in Theorem 5.2. The proof ofthe existence of such a process (and that it is a λ -irreducible T-model) is part of ongoing researchand will be postponed to a forthcoming paper. This seems reasonable to us, since the existenceof the process is related to the question of solving SDEs with discontinuous coefficients and thesolution theory for such equations requires tools which go beyond the scope of the present paper.The the next result for symmetric α -stable L´evy processes is well known (e.g. [13]). We justpresent it with a new proof. Corollary 5.4.
Let ( X t ) t ≥ be a symmetric α -stable L´evy process with stability index α ∈ (0 , ,then ( X t ) t ≥ is i) point recurrent if α > , Harris recurrent if α ≥ , iii) transient if α < .Proof. Just apply Theorem 5.2 for α = β and note that b can be chosen arbitrary. Further notethat the process is clearly a λ -irreducible T-model, since it is a C b -Feller process with positivetransition density. Thus Theorem 4.3 yields the recurrence-transience dichotomy.Furthermore Lemma 2.3 is applicable since the process is a Hunt process, i.e. in particular it isquasi-left continuous (e.g. Thm. I.9.4 in [3]).The results of Section 2 show that two λ -irreducible C b -Feller processes have the same recur-rence (transience) behavior if they have the same generator outside an arbitrary ball. In particularwe get the following Corollary for stable-like processes. Corollary 5.5.
Assume the process in Theorem 5.2 exists and is a λ -irreducible T-model. Let ( X t ) t ≥ be a stable-like process on R with symbol | ξ | α ( x ) and suppose there exists α, β ∈ (0 , such that for some arbitrary R > α ( x ) = α for x < − R,α ( x ) = β for x > R, then ( X t ) t ≥ is • Harris recurrent if and only if α + β ≥ , • transient if and only if α + β < .Proof. X t is λ -irreducible since it has a transition density with respect to the Lebesgue measure(cf. [11]) and a T-model, since it is a C b -Feller process by Prop. 6.2 in [2].The process coincides on R \ B R (0) with the process of Theorem 5.2 and therefore by Corollary4.7 both processes have the same recurrence/transience behavior. Thus Theorem 5.2 implies theresult. Acknowledgement:
The paper was initiated during a visit to Zagreb funded by DAAD.The author is grateful for discussions on the topic with Ren´e Schilling and Zoran Vondracek.