On shift Harnack inequalities for subordinate semigroups and moment estimates for Lévy processes

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1 Available online a ScienceDirec Sochasic Processes and heir Applicaions 15 (15) On shif Harnack inequaliies for subordinae semigroups and momen esimaes for Lévy processes Chang-Song Deng a,b,, René L. Schilling b a School of Mahemaics and Saisics, Wuhan Universiy, Wuhan 437, China b TU Dresden, Fachrichung Mahemaik, Insiu für Mahemaische Sochasik, 16 Dresden, Germany Received December 14; received in revised form 4 April 15; acceped 16 May 15 Available online 7 May 15 Absrac We show ha shif Harnack ype inequaliies (in he sense of Wang (14)) are preserved under Bochner s subordinaion. The proofs are based on wo ypes of momen esimaes for subordinaors. As a by-produc we esablish momen esimaes for general Lévy processes. c 15 Elsevier B.V. All righs reserved. MSC: 6J75; 47G; 6G51 Keywords: Shif Harnack inequaliy; Subordinaion; Subordinae semigroup; Lévy process 1. Inroducion and moivaion Subordinaion in he sense of Bochner is a mehod o generae new ( subordinae ) sochasic processes from a given process by a random ime change wih an independen one-dimensional increasing Lévy process ( subordinaor ). A corresponding noion exiss a he level of semigroups. If he original process is a Lévy process, so is he subordinae process. For insance, any symmeric α-sable Lévy process can be regarded as a subordinae Brownian moion, cf. [11]. This provides us wih anoher approach o invesigae jump processes via he corresponding Corresponding auhor a: TU Dresden, Fachrichung Mahemaik, Insiu für Mahemaische Sochasik, 16 Dresden, Germany. addresses: dengcs@whu.edu.cn (C.-S. Deng), rene.schilling@u-dresden.de (R.L. Schilling). hp://dx.doi.org/1.116/j.spa / c 15 Elsevier B.V. All righs reserved.

2 385 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) resuls for diffusion processes. See [5] for he dimension-free Harnack inequaliy for subordinae semigroups, [1] for subordinae funcional inequaliies and [4] for he quasi-invariance propery under subordinaion. In his paper, we will esablish shif Harnack inequaliies, which were inroduced in [16], for subordinae semigroups. Le (S ) be a subordinaor. Being a one-sided Lévy process, i is uniquely deermined by is Laplace ransform which is of he form E e us = e φ(u), u >,. The characerisic exponen φ : (, ) (, ) is a Bernsein funcion having he following Lévy Khinchine represenaion φ(u) = bu + 1 e ux ν(dx), u >. (1.1) (, ) The drif parameer b and he Lévy measure ν his is a Radon measure on (, ) saisfying (, )(x 1) ν(dx) < uniquely characerize he Bernsein funcion. The corresponding ransiion probabiliies µ := P(S ) form a vaguely coninuous convoluion semigroup of probabiliy measures on [, ), i.e. one has µ +s = µ µ s for all, s and µ ends weakly o he Dirac measure concenraed a as. If (X ) is a Markov process wih ransiion semigroup (P ), hen he subordinae process is given by he random ime-change X φ := X S. The process (X φ ) is again a Markov process, and i is no hard o see ha he subordinae semigroup is given by he Bochner inegral P φ f := P s f µ (ds),, f bounded, measurable. (1.) [, ) The formula (1.) makes sense for any Markov semigroup (P ) on any Banach space E and defines again a Markov semigroup. We refer o [11] for deails, in paricular for a funcional calculus for he generaor of (P φ ). If (S ) is an α-sable subordinaor ( < α < 1), he dimension-free Harnack ype inequaliies in he sense of [13] were esablished in [5], see [14] for more deails on such Harnack inequaliies. For example, if (P ) saisfies he log-harnack inequaliy P log f (x) log P f (y) + Φ(, x, y), x, y E, >, 1 f B b (E) for some funcion Φ : (, ) E E [, ), hen a similar inequaliy holds for he subordinae semigroup (P φ ) ; ha is, he log-harnack inequaliy is preserved under subordinaion. For he sabiliy of he power-harnack inequaliy, we need an addiional condiion on α: if he following power-harnack inequaliy holds P f (x) p P f p (y) exp [Φ(, p, x, y)], x, y E, >, f B b (E), where p > 1 and Φ(, p, x, y) : (, ) [, ) is a measurable funcion such ha for some κ > Φ(, p, x, y) = O( κ ) as, hen (P φ ) saisfies also a power-harnack inequaliy provided ha we have α (κ/(1 + κ), 1), see [5, Theorem 1.1]. We sress ha he resuls of [5] hold for any subordinaor whose Bernsein funcion saisfies φ(u) Cu α for large values of u wih some consan

3 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) C > and α (, 1) (as before, α (κ/(1 + κ), 1) is needed for he power-harnack inequaliy), see [17, Proof of Corollary.]. Recenly, new ypes of Harnack inequaliies, called shif Harnack inequaliies, have been proposed in [16]: A Markov semigroup (P ) saisfies he shif log-harnack inequaliy if P log f (x) log P [ f ( + e)](x) + Ψ(, e), >, 1 f B b (E), (1.3) and he shif power-harnack inequaliy wih power p > 1 if P f (x) p P [ f p ( + e)](x) exp[φ(, p, e)], >, f B b (E); (1.4) here, x, e E and Ψ(, e), Φ(, p, e) : (, ) [, ) are measurable funcions. These new Harnack ype inequaliies can be applied o hea kernel esimaes and quasi-invariance properies of he underlying ransiion probabiliy under shifs, see [16,14] for deails. The power-harnack and log-harnack inequaliies were esablished in [17] for a class of sochasic differenial equaions driven by addiive noise conaining a subordinae Brownian moion as addiive componen, see also [3] for an improvemen and [18] for furher developmens. The shif power-harnack inequaliy was also derived in [15] for his model by using Malliavin s calculus and finie-jump approximaions. In paricular, when he noise is an α-sable process wih α (1, ), an explici shif power-harnack inequaliy was presened, cf. [15, Corollary 1.4(3)]. However, shif log-harnack inequaliies for sochasic differenial equaions driven by jump processes have no ye been sudied. We wan o consider he sabiliy of shif Harnack inequaliies under subordinaion. In many specific cases, see e.g. Example.5 in Secion, Ψ(s, e) and Φ(s, p, e) are of he form C 1 s κ 1 + C s κ + C 3, wih consans C i, i = 1,, 3, depending only on e E and p > 1, and exponens κ 1, κ >. As i urns ou, his means ha we have o conrol ES κ, κ R\{}, for he shif log-harnack inequaliy and E e λsκ, λ > and κ R\{}, for he shif power-harnack inequaliy. Throughou he paper, we use he convenion ha 1 :=. Since momen esimaes for sochasic processes are ineresing on heir own, we sudy such (exponenial) momen esimaes firs for general Lévy processes, and hen for subordinaors. For real-valued Lévy processes wihou Brownian componen esimaes for he ph (p > ) momen were invesigaed in [8,7] via he Blumenhal Geoor index inroduced in []. While he focus of hese papers were shor-ime asympoics, we need esimaes also for 1 which require a differen se of indices for he underlying processes. Le us briefly indicae how he paper is organized. In Secion we esablish he shif Harnack ype inequaliies for he subordinae semigroup P φ from he corresponding inequaliies for P. Some pracical condiions are presened o ensure he sabiliy of he shif Harnack inequaliy under subordinaion; in Example.5 we illusrae our resuls using a class of sochasic differenial equaions. Secion 3 is devoed o momen esimaes of Lévy processes: Secion 3.1 conains, under various condiions, several concree (non-)exisence resuls and esimaes for momens, while Secion 3. provides he esimaes for ES κ and E e λsκ which were used in Secion. Finally, we show in he Appendix ha he index β of a Lévy symbol a he origin saisfies β [, ]. As usual, we indicae by subscrips ha a posiive consan C = C α,β,γ,... depends on he parameers α, β, γ,..... Shif Harnack inequaliies for subordinae semigroups In his secion, we use he momen esimaes for subordinaors from Secion 3 o esablish shif Harnack inequaliies for subordinae semigroups. Le (P ) be a Markov semigroup on a

4 3854 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) Banach space E and S = (S ) be a subordinaor whose characerisic (Laplace) exponen φ is he Bernsein funcion given by (1.1). Recall ha he subordinae semigroup (P φ ) is defined by (1.). Firs, we inroduce wo indices for subordinaors: φ(u) σ := sup α : lim u u α =, (.1) φ(u) φ(u) ρ := sup α : lim inf u u α > = inf α : lim inf u u α =. (.) Since φ(u) lim = lim φ (u) = b + lim xe ux ν(dx) = b + x ν(dx) (, ], (.3) u u u u (, ) (, ) i is clear ha σ [, 1]. Moreover, he following formula holds, see [4]: φ(u) σ = sup α : lim sup u u α < = sup α 1 : y α ν(dy) <. (.4) y 1 For any ϵ >, noing ha yields φ(u) u 1+ϵ b u ϵ + 1 u ϵ lim u φ(u) =, u1+ϵ (,1) x ν(dx) + 1 ν(x 1), u >, u1+ϵ one has ρ 1 + ϵ. Since ϵ > is arbirary, we conclude ha ρ [, 1]. Remark.1. We will frequenly use he condiion lim inf u φ(u)u ρ > for some ρ (c, ρ ], where c. This is clearly equivalen o eiher c < ρ < ρ or ρ = ρ > c and lim inf u φ(u)u ρ >. Assume ha P saisfies he following shif log-harnack inequaliy P log f (x) log P [ f ( + e)](x) + C 1(e) κ 1 + C (e) κ + C 3 (e), >, 1 f B b (E), (.5) for some x, e E; here κ 1 >, κ (, 1], and C i (e), i = 1,, 3, are consans depending only on e. We are going o show ha he subordinae semigroup P φ saisfies a similar shif log-harnack inequaliy. The following assumpions on he subordinaor will be imporan: (H1) κ 1 > and lim inf u φ(u)u ρ > for some ρ (, ρ ]. 1 (H) κ (, 1] saisfies y 1 yκ ν(dy) <. (H3) κ (, 1] saisfies inf θ [κ,1] y> yθ ν(dy) <. (H4) κ (, σ ), where σ (, σ ] saisfies lim sup u φ(u)u σ <. Noe ha by (.4), (H4) is sricly sronger han (H). 1 This is equivalen o eiher < ρ < ρ or ρ = ρ > and lim inf u φ(u)u ρ >, see Remark.1. In analogy o Remark.1, his is equivalen o eiher < σ < σ or σ = σ > and lim sup u φ(u)u σ <.

5 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) Theorem.. Suppose ha P saisfies (.5) for some x, e E. In each of he following cases he subordinae semigroup P φ saisfies also a shif log-harnack inequaliy P φ log f (x) log P φ [ f ( + e)](x) + Ψ(, e), >, 1 f B b (E). (.6) (a) Assume (H1) and (H). Then (.6) holds wih Ψ(, e) of he form C κ1,ρ C 1 (e) ( 1) κ 1/ρ + C (e)c κ ( 1) + C 3 (e). (b) Assume (H1) and (H3). Then (.6) holds wih Ψ(, e) of he form C κ1,ρ C 1 (e) ( 1) κ 1/ρ + C (e)c κ ( 1) κ + C 3 (e). (c) Assume (H1) and (H4). Then (.6) holds wih Ψ(, e) of he form C κ1,ρ C 1 (e) ( 1) κ 1/ρ + C (e)c κ,σ ( 1) κ/σ + C 3 (e). Proof. Because of (.5) he shif log-harnack inequaliy (1.3) for P holds wih Ψ(s, e) = C 1 (e)s κ 1 + C (e)s κ + C 3 (e). Noe ha (H1) implies φ(u) as u, hence excluding he compound Poisson subordinaor, so µ ({}) = P(S = ) =, >. By Jensen s inequaliy we find for all > P φ log f (x) = P s log f (x) µ (ds) (, ) log Ps [ f ( + e)](x) + Ψ(s, e) µ (ds) (, ) log P s [ f ( + e)](x) µ (ds) + Ψ(s, e) µ (ds) (, ) (, ) = log P φ [ f ( + e)](x) + Ψ(s, e) µ (ds). Therefore, P φ (, ) log f (x) log P φ [ f ( + e)](x) + C 1 (e)es κ 1 + C (e)es κ + C 3 (e) and he desired esimaes follow from he corresponding momen bounds in Secion 3. Now we urn o he shif power-harnack inequaliy for P φ. We will assume ha for some p > 1 and x, e E, P saisfies he following shif power-harnack inequaliy P f (x) p P [ f p ( + e)](x) exp H1 (p, e) κ 1 + H (p, e) κ + H 3 (p, e), >, f B b (E), (.7) where κ 1 >, κ (, 1] and H i (p, e), i = 1,, 3, are consans depending on p and e.

6 3856 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) Theorem.3. Assume ha P saisfies (.7) for some p > 1 and x, e E; furhermore, assume ha lim inf u φ(u)u ρ > for some ρ (κ 1 /(1 + κ 1 ), ρ ]. 3 If r H (p, e) exp y κ ν(dy) < p 1 y 1 holds for some r > 1, hen he subordinae semigroup P φ saisfies he shif power-harnack inequaliy (1.4) wih an exponen Φ(, p, e) given by C κ1,ρ,p,r,e κ 1 + C κ,p,r,e( 1) + H 3 (p, e). (.8) ρ (1 ρ)κ ( 1) 1 Proof. As in he proof of Theorem., we see µ ({}) = for any >. By (.7) and he Hölder inequaliy one has φ P f (x) p p = P s f (x) µ (ds) (, ) Ps [ f p ( + e)](x) 1 H1 p (p, e) exp (, ) ps κ + H (p, e) s κ + H 3(p, e) p µ (ds) 1 p p P s [ f p ( + e)](x) µ (ds) (, ) exp (, ) = P φ [ f p ( + e)](x) e H 3(p,e) P φ [ f p ( + e)](x) e H 3(p,e) E exp By Theorem 3.8(b) in Secion 3, E exp r H 1 (p, e) (r 1)(p 1) S κ 1 H1 (p, e) (p 1)s κ + H (p, e) 1 p 1 sκ + H 3(p, e) p 1 E exp r H 1 (p, e) (r 1)(p 1) S κ 1 (r 1)(p 1) r p 1 µ (ds) H1 (p, e) p 1 S κ 1 + H (p, e) p 1 Sκ (r 1)(p 1) r exp On he oher hand, i follows from Theorem 3.3(a) ha p 1 p 1 r H (p, e) E exp S κ r. p 1 Cκ1,ρ,p,r,e κ 1 ρ (1 ρ)κ 1 ( 1) p 1 r H (p, e) E exp S κ r exp C κ,p,r,e( 1). p 1 Combining he above esimaes finishes he proof. Remark.4. By Theorem 3.8(b), in (.8) C κ1,ρ,p,r,e can be explicily given as a funcion of p, r, e. In order o wrie C κ,p,r,e in (.8) explicily as a funcion of p, r, e, we need some addiional condiion. If (,1) y κ ν(dy) <, his can be easily realized by using. 3 This is equivalen o eiher κ1 /(1 + κ 1 ) < ρ < ρ or ρ = ρ > κ 1 /(1 + κ 1 ) and lim inf u φ(u)u ρ >, see Remark.1.

7 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) Theorem 3.3(b). If ν(y 1) =, hen we can use [9, Theorem 5.17]: for any λ >, κ (, 1] and > E e λsκ E e λ(1 S ) e λ E e λs = exp λ + bλ + e λy 1 ν(dy). (,1) Example.5. Consider he following sochasic differenial equaion on R d : dx = l (X ) d + Σ dw, (.9) where l : [, ) R d R d and Σ : [, ) R d R d are measurable and locally bounded, and (W ) is a sandard d-dimensional Brownian moion. We assume he following condiions on l and Σ: (A1) There exiss a locally bounded measurable funcion K : [, ) [, ) such ha l (x) l (y) K x y, x, y R d,. (A) For each, he marix Σ is inverible and here exiss a measurable funcion γ : [, ) (, ) such ha γ L Σ loc ([, )) and 1 γ for all. I is well known ha (A1) ensures ha (.9) has for each saring poin X = x R d a unique soluion (X x ) wih infinie life-ime. By P we denoe he associaed Markov semigroup, i.e. P f (x) = E f (X x ),, f B b(r d ), x R d. Assume ha for some κ 1 1 and κ (, 1] lim sup lim sup 1 κ 1 1 +κ γ r (r K r + 1) dr <, (.1) γ r (r K r + 1) dr <. (.11) Typical examples for K and γ saisfying (.1) and (.11) are γ s = 1 and K s = s θ 1 for θ. Then i is easy o see ha (.1) is fulfilled wih κ 1 1 and (.11) is saisfied wih κ [(θ + 1), 1]\{}. γ s = 1 {} (s) + s θ 1 (, ) (s) for θ ( 1/, ] and K s = 1 (log s). Then (.1) holds wih κ 1 1 θ and (.11) holds wih κ (1 + θ, 1]. We are going o show ha here exiss a consan C > such ha for all > and x, e R d he following shif log- and power-(p > 1)-Harnack inequaliies hold: 1 P log f (x) log P [ f ( + e)](x) + C e κ + κ + 1, 1 f B b (R d ), 1 P f (x) p P [ f p ( + e)](x) Cp e 1 exp p 1 κ + κ + 1, f B b (R d ). 1 In paricular, Theorems. and.3 can be applied. Alhough he proof of Example.5 relies on known argumens, see e.g. [16,14], we include he complee proof for he convenience of he reader.

8 3858 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) Proof of Example.5. Fix > and x, e R d. We adop he new coupling argumen from [16] (see also [14]) o consruc anoher process (Ys x) s also saring from x such ha Y x X x = e a he fixed ime. The process (Ys x) s is he soluion of he following equaion Clearly, dy x s = l s (X x s ) ds + Σ s dw s + e ds, Y x Y x r X x r = r = x. (.1) e ds = r e, r, (.13) and, in paricular, Y x X x = e. Rewrie (.1) as where Le dy x s = l s (Y x s ) ds + Σ s d W s, Y x = x, s W s := W s + M s := s Σr 1 l r (Xr x ) l r (Yr x ) + e dr, s. Σr 1 l r (Xr x ) l r (Yr x ) + e dw r, s. Since i follows from (A), (A1) and (.13) ha Σr 1 l r (X x r ) l r (Y x r ) + e he compensaor of he maringale M saisfies Se M = Σr 1 R := exp M 1 M. lr γr (Xr x ) l r (Yr x ) + e γ r K r X x r Yr x + e = e γ r (r K r + 1), r, l r (Xr x ) l r (Yr x ) + e e dr γr (r K r + 1) dr. (.14) Novikov s crierion shows ha ER = 1. By he Girsanov heorem, ( W s ) s is a d-dimensional Brownian moion under he probabiliy measure R P. To derive he shif log-harnack inequaliy for P, we firs noe ha (.14) implies log R = M 1 M = Σr 1 l r (Xr x ) l r (Yr x ) + e d W r + 1 M Σr 1 l r (Xr x ) l r (Yr x ) + e d W r + e γ r (r K r + 1) dr.

9 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) Since ER = 1, we find wih he Jensen inequaliy for any random variable F 1 R log F dp log F dp, hence, R log F dp log F dp + R log R dp. R Thus, we ge for any f B b (R d ) wih f 1 P log f (x) = E R P log f (Y x ) = E R log f (X x + e) log E f (X x + e) + E[R log R ] = log P [ f ( + e)](x) + E R P log R log P [ f ( + e)](x) + e γ r (r K r + 1) dr. (.15) On he oher hand, for any p > 1 and f B b (R d ) wih f, we deduce wih he Hölder inequaliy ha P f (x) p = ER P f (Y x ) p = E[R f (X x + e)] p E f p (X x + e) ER = P [ f p ( + e)](x) ER p p 1 p 1 p p 1 p 1. Because of (.14), i follows ha p p 1 p ER = E exp (p 1) M p p 1 M p (p 1) M p exp (p 1) e γr (r K r + 1) dr E exp p p 1 M p (p 1) M p e = exp (p 1) γr (r K r + 1) dr. In he las sep we have used he fac ha exp p p 1 M due o (.14) and Novikov s crierion. Therefore, P f (x) p P [ f p ( + e)](x) exp p e (p 1) p M (p 1) is a maringale; his is γr (r K r + 1) dr (.16) holds for all p > 1 and non-negaive f B b (R d ). Finally, i remains o observe ha (.1) and (.11) imply ha here exiss a consan C > such ha 1 1 γr (r K r + 1) dr C κ + κ for all >. Subsiuing his ino (.15) and (.16), respecively, we obain he desired shif log- and power- Harnack inequaliies.

10 386 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) Momen esimaes for Lévy processes 3.1. General Lévy processes A Lévy process X = (X ) is a d-dimensional sochasic process wih saionary and independen incremens and almos surely càdlàg (righ-coninuous wih finie lef limis) pahs X. As usual, we assume ha X =. Our sandard references are [9,6]. Since Lévy processes are (srong) Markov processes, hey are compleely characerized by he law of X, hence by he characerisic funcion of X. I is well known ha Ee iξ X = e ψ(ξ), >, ξ R d, where he characerisic exponen (Lévy symbol) ψ : R d C is given by he Lévy Khinchine formula ψ(ξ) = il ξ + 1 ξ Qξ + 1 e iξ y + iξ y1 (,1) ( y ) ν(dy), y where l R d is he drif coefficien, Q is a non-negaive semidefinie d d marix, and ν is he Lévy measure on R d \{} saisfying y (1 y ) ν(dy) <. The Lévy riple (l, Q, ν) uniquely deermines ψ, hence X. The infiniesimal generaor of X is given by L f = l f + 1 Q f + f (y + ) f y f 1(,1) ( y ) ν(dy), y f C b (Rd ). Firs, we consider E X κ for κ R\{}. Recall, cf. [9, Theorem 5.3], ha he Lévy process X has a κh (κ > ) momen, i.e. E X κ < for some (hence, all) >, if and only if y κ ν(dy) <. (3.1) y 1 To presen our resuls, we also need o inroduce four indices for Lévy processes. Define he Blumenhal Geoor index β of a Lévy process by ψ(ξ) β := inf α : lim ξ ξ α =. Clearly, β [, ]. For a Lévy process wihou diffusion par and dominaing drif, his coincides wih Blumenhal-and-Geoor s original definiion based on he Lévy measure ν, see for example [,1], i.e. β = inf α : y α ν(dy) <. < y <1 Moreover, i is easy o see ha β = inf α : lim sup ξ ψ(ξ) ψ(ξ) ξ α = = inf α : lim sup ξ ξ α <. (3.) In Secion we have inroduced he index σ for subordinaors using he characerisic Laplace exponen (Bernsein funcion) φ. A similar index exiss for a general Lévy process X i is he

11 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) counerpar a he origin of he classical Blumenhal Geoor index bu is definiion is based on he characerisic exponen ψ: ψ(ξ) β := sup α : lim ξ ξ α =. (3.3) I holds ha β [, ], see Appendix for an easy proof. As in (3.), one has ψ(ξ) β = sup ξ α = = sup α : lim sup ξ α : lim sup ξ ψ(ξ) ξ α Moreover, for a Lévy process wihou dominaing drif, i holds ha, cf. [1], β = sup α : y α ν(dy) <. Le δ := sup δ := inf y 1 α : lim inf ξ α : lim inf ξ Re ψ(ξ) ξ α > Re ψ(ξ) ξ α > = sup = inf α : lim inf ξ α : lim inf ξ Clearly, δ β and β δ. Noe ha δ [, ]. < Re ψ(ξ) ξ α =, Re ψ(ξ) ξ α =. Theorem 3.1. Le X be a Lévy process in R d wih Lévy riple (l, Q, ν) and characerisic exponen ψ. (a) Assume ha Q =, κ (, 1] and (3.1) holds. Then for any > E X κ l κ κ + y κ ν(dy) y 1 d κ/ + κ(3 κ) y ν(dy) 1 κ/ 1 + ν( y 1) κ/. < y <1 In paricular, E X κ C κ for any 1. (b) Assume ha Q =, κ (, 1] and y θ ν(dy) <. inf θ [κ,1] y Then for any > where E X κ inf θ [κ,1] κ/θ ˆl θ θ + y θ ν(dy), (3.4) y ˆl := l y ν(dy). (3.5) < y <1 In paricular, E X κ C κ κ for any 1..

12 386 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) (c) Assume ha κ (, β), where β (, β ] saisfies lim sup ξ ψ(ξ) ξ β <. 4 Then for any 1 E X κ C κ,β,d κ/β. If furhermore lim sup ξ ψ(ξ) ξ β <, 5 hen his esimae holds for all >. (d) Le κ (, d). If for some δ (, δ ], lim inf ξ Re ψ(ξ) ξ δ >, 6 hen for any (, 1] E X κ C κ,δ,d κ/δ. If furhermore lim inf ξ Re ψ(ξ) ξ δ >, 7 hen his esimae holds for all >. (e) If ψ is real-valued, hen for any κ d and > E X κ =. Remark 3.. (a) The echnique (Taylor s heorem) used in he proof of Theorem 3.1(a) (see also he proof of Theorem 3.3(a)) can be used o ge bounds on more general momens of Lévy processes, e.g. E X κ wih κ > 1. However, he cos we have o pay is ha he esimaes may be oo rough. (b) I is well known ha (cf. [9, Theorem 5.3]) he finieness of momens of a Lévy process depends only on he ail behaviour of he underlying Lévy measure ν, i.e. on he big jumps. Theorem 3.1(b) i is essenially conained in [8, Theorem.1], using a very differen mehod (Laplace ransform) requires also a condiion on he small jumps. This allows us o ge a clean formula for he -dependence in (3.4). (c) Theorem 3.1(b) is sharp for he Gamma subordinaor, see Example 3.5. (d) Le X be a symmeric α-sable Lévy process in R d wih < α <. Then ψ(ξ) = ξ α and we can choose in Theorem 3.1(c) β = β = β = α. For > i is well known ha E X κ is finie if, and only if, κ (, α) = (, β), in which case E X κ = κ/α E X 1 κ. This means ha Theorem 3.1(c) is sharp for symmeric α-sable Lévy processes. (e) If he Lévy process has no dominaing drif, hen lim sup ξ ψ(ξ) ξ β < implies (3.1) for κ (, β). The converse, however, is in general wrong. Thus he condiions in pars (a) and (c) of Theorem 3.1 are incomparable. (f) For a one-dimensional symmeric α-sable Lévy process we have δ = δ = δ = α. By he scaling propery, we have E X κ = C κ,α κ/α for κ (, 1). Thus, Theorem 3.1(d) is sharp for symmeric α-sable Lévy processes in R. Proof of Theorem 3.1. (a) Rewrie X as X = l + X,, where X = (X ) is he Lévy process in R d generaed by L f = f (y + ) f y f 1(,1) ( y ) ν(dy), f Cb (Rd ). Noing ha y (x + y) a x a + y a, x, y, a [, 1], (3.6) 4 This is equivalen o eiher < β < β or β = β > and lim sup ξ ψ(ξ) ξ β <. 5 This is equivalen o eiher β < β β or β = β (, β ] and lim sup ξ ψ(ξ) ξ β <. 6 This is equivalen o eiher < δ < δ or δ = δ > and lim inf ξ Re ψ(ξ) ξ δ >. 7 This is equivalen o eiher δ < δ δ or δ = δ (, δ ] and lim inf ξ Re ψ(ξ) ξ δ >.

13 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) i suffices o show ha for all > d κ/ E X κ y κ ν(dy) + κ(3 κ) y ν(dy) y 1 < y <1 and 1 + ν( y 1 ) 1 κ/ κ/. (3.7) Fix ϵ > and >. Le f (x) := ϵ + x κ/, x R d, τ n := inf s : X s > n, n N. (3.8) By he Dynkin formula we ge for any n N E ϵ + X κ/ τn ϵ κ/ = E = E + E [, τ n ) [, τ n ) y 1 [, τ n ) L f (X s ) ds f ( X s + y) f (X s ) ν(dy) ds < y <1 f ( X s + y) f (X s ) y f (X s ) ν(dy) ds. (3.9) We esimae he wo erms on he righ side separaely. For he firs expression we have f ( X s + y) f (X s ) ν(dy) ϵ κ/ ν( y 1) + y κ ν(dy). (3.1) y 1 y 1 For he second erm, we observe ha for any x R d f (x) x j x = κ(κ ) i By Taylor s heorem, κ( κ) ϵ + x κ/ xi x j + κ ϵ + x κ/ 1 1{i= j} ϵ + x κ/ x + κ ϵ + x κ/ 1 κ( κ)ϵ κ/ 1 + κϵ κ/ 1 = κ(3 κ)ϵ κ/ 1. f (X s + y) f (X s ) y f (X s ) = 1 d i, j=1 f x j x i X s + θ X s,y y y i y j 1 d κ(3 κ)ϵκ/ 1 i, j=1 d κ(3 κ)ϵκ/ 1 y, y i y j

14 3864 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) where θ X s,y [ 1, 1] depends on X s and y. Thus, we ge f ( X s + y) f (X s ) y f (X s ) ν(dy) < y <1 d κ(3 κ)ϵκ/ 1 < y <1 y ν(dy). Combining his wih (3.1) and (3.9), we arrive a ϵ E + X τn κ/ ϵ κ/ + ϵ κ/ ν( y 1) + y 1 y κ ν(dy) E [ τ n ] d + κ(3 κ)ϵκ/ 1 y ν(dy) E [ τ n ]. < y <1 Since τ n as n, we can le n and use he monoone convergence heorem o obain ϵ E X κ E + X κ/ ϵ κ/ + ϵ κ/ ν( y 1) + y κ ν(dy) y 1 + d κ(3 κ)ϵκ/ 1 y ν(dy) = + y κ ν(dy) y 1 d κ(3 κ) < y < ν( y 1) ϵ κ/ y ν(dy) < y <1 Since ϵ > is arbirary, we can opimize in ϵ >, i.e. le d κ(3 κ) < y <1 ν(dy) y ϵ, 1 + ν( y 1) ϵ κ/ 1. o ge (3.7). (b) Our assumpion < y <1 y ν(dy) < enails ha X has bounded variaion. Therefore, X = ˆl + X,, where (X ) is a drif-free Lévy process wih generaor Le L f = y f (y + ) f ν(dy), f C b (R d ). σ n := inf : X > n, n N. I follows from Dynkin s formula and (3.6) ha for any θ [κ, 1] and n N E X σn θ = E X s + y θ X s θ ν(dy) ds y θ ν(dy). [, σ n ) y y Since σ n as n, we can le n and use he monoone convergence heorem o ge E X θ y θ ν(dy). y

15 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) Using (3.6) again, we obain ha E X θ ˆl θ θ + E X θ ˆl θ θ + y θ ν(dy). y Togeher wih Jensen s inequaliy, his yields for any θ [κ, 1] and > E X κ E X θ κ/θ κ/θ ˆl θ θ + y θ ν(dy), y which implies he desired esimae. (c) Since < κ < β β, we have, see e.g. [1, III.18.3], x κ = c κ,d (1 cos(x ξ)) ξ κ d dξ, x R d, R d \{} where c κ,d := κκ 1 Γ κ+d π d/ Γ 1 κ. By Tonelli s heorem, we ge E X κ = c κ,d E = c κ,d = c κ,d R d \{} R d \{} (1 cos(x ξ)) ξ κ d dξ R d \{} 1 Re E e ix ξ ξ κ d dξ 1 Re e ψ(ξ) ξ κ d dξ. Since Re ψ, we have 1 Re e ψ(ξ) 1 e ψ(ξ) ψ(ξ), ξ R d \{}. If lim sup ξ ψ(ξ) ξ β <, hen ψ(ξ) C β ξ β, < ξ 1. (3.11) Thus, we find for all 1 E X κ c κ,d R d \{} ψ(ξ) ξ κ d dξ c κ,d ψ(ξ) ξ < ξ κ d dξ + c κ,d ξ κ d dξ 1/β ξ > 1/β c κ,d C β ξ β κ d dξ + c κ,d ξ κ d dξ < ξ 1/β ξ > 1/β = c κ,d C β κ/β ξ β κ d dξ + c κ,d κ/β ξ κ d dξ, (3.1) < ξ 1 which implies he firs esimae. If furhermore lim sup ξ ψ(ξ) ξ β <, hen (3.11) holds for all ξ R d, and so (3.1) holds rue for all >. This gives he second asserion. ξ >1

16 3866 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) (d) Recall ha e u x = c d R d u e ix ξ dξ, x R d, u >, u d+1 + ξ where c d := π (d+1)/ Γ d+1. Using Fourier ransforms, we have for all u > and x R d ξ d+κ e u ξ e ix ξ dξ = c R d κ,d c d x y κ u dy, R d u d+1 + y where c κ,d := κ π d/ Γ κ Γ d κ. This implies ha for all n N and > Rd ξ d+κ e n 1 ξ Re e ix ξ dξ = c R d κ,d c d X + y κ n 1 dy n d+1 + y = c κ,d c d X + n 1 y κ dy 1 + y. d+1 Taking expecaion and using Fubini s heorem, we ge ξ d+κ e n 1 ξ Re e ψ(ξ) dξ = c R d κ,d c d E R d R d R d X + n 1 y Combining his wih Faou s lemma and Tonelli s heorem, we obain E X κ = E X κ dy c d 1 + y d+1 = c d E R d c d lim inf = 1 c κ,d 1 c κ,d = 1 c κ,d n E lim inf n lim inf n lim inf n R d X + n 1 y X + n 1 y κ κ dy 1 + y d+1 dy 1 + y d+1 R d ξ d+κ e n 1 ξ Re e ψ(ξ) dξ R d ξ d+κ e n 1 ξ e Re ψ(ξ) dξ R d ξ d+κ e Re ψ(ξ) dξ, κ dy 1 + y. d+1

17 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) where we have used he monoone convergence heorem. If lim inf ξ Re ψ(ξ) ξ δ >, hen here exis consans C 1 = C 1 (δ) > and C = C (δ) such ha Re ψ(ξ) C 1 ξ δ, ξ C. (3.13) Thus, we find for all > E X κ 1 c κ,d ξ d+κ dξ + ξ d+κ e C1 ξ δ dξ ξ C R d = 1 c κ,d ξ d+κ dξ + κ/δ ξ d+κ e C1 ξ δ dξ. (3.14) ξ C R d This gives he firs asserion. If furhermore lim inf ξ Re ψ(ξ) ξ δ >, hen (3.13) holds wih C =, so ha he second asserion follows by using (3.14) wih C =. (e) Using 1 y r = 1 Γ (r) e uy u r 1 du, r >, y (3.15) and he Fourier ransform formula from he beginning of par (d), we ge by Tonelli s heorem ha E X κ = 1 Γ (κ) E e u X u κ 1 du = c d Γ (κ) E R d e ix ξ u dξ u κ 1 du. u d+1 + ξ Since ψ(ξ) R for all ξ R d, < e ψ(ξ) 1, and we can use Fubini s heorem for he inner inegrals and hen Tonelli s heorem for he wo ouer inegrals o ge E X κ = c d e ψ(ξ) u Γ (κ) dξ u κ 1 du R d u d+1 + ξ = c d u κ Γ (κ) du e ψ(ξ) dξ R d u d+1 + ξ = c d Γ (κ) R d c d d+1 Γ (κ) R d where he las equaliy follows from κ d. ξ u κ u d+1 + u du e ψ(ξ) dξ u κ d 1 du e ψ(ξ) dξ =, ξ Le us now consider E e λ X κ for λ > and κ R\{}. For κ (, 1] he funcion R d x e λ x κ R is submuliplicaive; hus, E e λ X κ < for some (hence, all) > if, and only if, e λ y κ ν(dy) <, (3.16) y 1 see [9, Theorem 5.3].

18 3868 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) Theorem 3.3. Le X be a Lévy process in R d wih Lévy riple (l, Q, ν) and characerisic exponen ψ. (a) If Q =, κ (, 1] and (3.16) holds, hen for any λ > here is some (non-explici) consan C κ,λ > such ha E e λ X κ e C κ,λ κ/, if < 1, e C κ,λ, if 1. (b) If Q =, κ (, 1], (3.16) holds and y κ ν(dy) <, (3.17) < y <1 hen for any λ, > E e λ X κ exp λ ˆl κ κ + M κ,λ, where ˆl is given by (3.5) and M κ,λ := e λ y κ 1 ν(dy). y (c) If ν and κ > 1, hen for any λ, > E e λ X κ =. (d) If ψ is real-valued, hen for any λ,, κ > E e λ X κ =. Remark 3.4. (a) Since ν is a Lévy measure, i is easy o see ha (3.16) and (3.17) imply M κ,λ <. (b) I is well known, see e.g. [9, Theorem 6.1(ii)], ha E e λ X log X =, λ > for any Lévy process wih Lévy measure having unbounded suppor supp ν. Since for any κ > 1 here exiss a consan C κ > such ha e λ x log x C κ e λ x κ, λ >, x R d, his implies Theorem 3.3(c) if supp ν is unbounded; Theorem 3.3(c), however, is valid for all non-zero ν. Proof of Theorem 3.3. (a) Le X be a Lévy process in R d wih riple (,, ν). I is enough o show ha E e λ X κ E exp λ ϵ + X κ/ exp λϵ κ/ 1 + ε 1 C 1 + C (3.18) for all ϵ (, 1] and >, where C 1 := d κ(λκ + 3 κ)eλ C := e λ y 1 < y <1 y ν(dy), e λ y κ ν(dy) ν ( y 1).

19 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) Fix ϵ (, 1] and >. Le g(x) := exp λ ϵ + x κ/, x R d, and define τ n by (3.8). As in he proof of Theorem 3.1(a) we can use a Taylor expansion o ge for s < τ n g( X s + y) g(x s ) ν(dy) C g(x s ), y 1 g( X s + y) g(x s ) y g(x s ) ν(dy) C 1 λϵ κ/ 1 g(x s ). < y <1 Now we use Dynkin s formula and Tonelli s heorem o obain E g(x )1 {<τn } e λϵ κ/ Eg(X τn ) e λϵκ/ C 1 λϵ κ/ 1 + C E g(x s ) ds [, τ n ) = C 1 λϵ κ/ 1 + C E g(x s )1 {s<τn } ds. From Gronwall s inequaliy we see E g(x )1 {<τn } exp λϵ κ/ + C 1 λϵ κ/ 1 + C for all n N. Finally, (3.18) follows as n. (b) As in he proof of Theorem 3.1(b), we use Dynkin s formula, (3.6) and Tonelli s heorem o obain ha for all n N E e λ X κ 1 {<σn } 1 E e λ X σn κ 1 = E E [, σ n ) [, σ n ) = M κ,λ E = M κ,λ E y e λ X s +y κ e λ X s κ ν(dy) ds e λ X s κ e λ y κ 1 y e λ X s κ ds [, σ n ) e λ X s κ 1 {s<σn } This, ogeher wih Gronwall s inequaliy, yields ha E e λ X κ 1 {<σn } e M κ,λ, n N. ds. ν(dy) ds I remains o le n and use (3.6) o ge he desired resul. (c) Since ν we may, wihou loss of generaliy, assume ha here exis some Borel se B 1 R wih eiher inf B 1 > or sup B 1 < and Borel ses B,..., B d R\{} such ha η := ν(λ) (, ), where Λ := B 1 B B d. The jump imes of jumps wih size in he se Λ define a Poisson process, say (N ), wih inensiy η. Noe ha X can be decomposed ino wo independen

20 387 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) Lévy processes X = X (1) + X (),, where X (1) is a compound Poisson process wih Lévy measure ν Λ, and X () is a Lévy process wih Lévy measure ν ν Λ ; moreover, X (1) and X () are independen processes. Se r := inf B 1 sup B 1 (, ). By he riangle inequaliy we find for any y R d, X (1) + y X (1) y r N y. Using Sirling s formula n! πn n+ 1 e n+ 1 1n πn n+1 e n+1, n N, we obain ha for any λ, > (1) λ X E e +y κ E e λ(r N y ) κ 1 {r N > y } = n:rn> y e η π e e λ(rn y )κ (η)n e η n:rn> y n! (ηe) n e λ(rn y )κ n n+1 =, where he divergence is caused by κ > 1. Combining his wih Tonelli s heorem, we ge E e λ X κ = E e λ X (1) +y κ P X () dy =. R d (d) This follows from Theorem 3.1(e) as we may choose n N such ha nκ d and E e λ X κ λn n! E X nκ. 3.. Subordinaors A subordinaor is an increasing Lévy process in R. Le S = (S ) be a subordinaor wih Bernsein funcion φ given by (1.1). Since S is a Lévy process, all resuls of Secion 3.1 hold wih X replaced by S. The following example shows ha he resul in Theorem 3.1(b) is sharp for Gamma subordinaors. Example 3.5. Le S = (S ) be he Gamma process wih parameers α, β > ; his is a subordinaor wih b =, ν(dy) = αy 1 e βy 1 (, ) (y) dy. I is known ha he disribuion of S a ime > is a Γ (α, β)-disribuion, i.e. P(S dx) = βα Γ (α) xα 1 e βx 1 (, ) (x) dx.

21 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) Le κ (, 1]. Then we have and G() := ES κ = H() := I is easy o check ha Γ (α + κ) β κ Γ (α) κ/θ inf y θ ν(dy) = 1 θ [κ,1] y> β κ inf [αγ θ [κ,1] (θ)]κ/θ. inf [αγ θ [κ,1] (θ)]κ/θ = αγ (κ), (α) κ, if is small enough, if is large enough. Since G() lim H() = lim Γ (α + κ) Γ (α)αγ (κ) = 1 Γ (κ) lim Γ (α + κ) Γ (α + 1) = 1, he upper bound in (3.4) is sharp for small. Moreover, by Sirling s formula one has Γ (x) π x x 1 e x, x, (3.19) G() lim H() = lim Γ (α + κ) Γ (α)(α) κ = 1 e κ This means ha (3.4) is also sharp as. lim 1 + κ α+κ 1 = 1. α We will need he following Blumenhal Geoor index for subordinaors φ(u) σ := inf α : lim u u α =. Comparing his index wih ρ defined in (.), i is easy o see ha ρ σ 1. I is well known, cf. [], ha σ inf α : y α ν(dy) < (,1) wih equaliy holding for drif-free subordinaors, i.e. b = in (1.1). I is also no hard o see, cf. (3.), ha φ(u) σ = inf α : lim sup u u α <. Noe ha we can exend Bernsein funcions analyically ono he righ complex half-plane {z C : Re z > } and coninuously ono he closed half-plane {z C : Re z }, see [11, Proposiion 3.6]. Then he Lévy symbol of S is given by ψ(ξ) = φ( iξ) for ξ R. Since e 1 e (1 x) 1 e x 1 x for all x and 1 e ix x for all x R, we have for

22 387 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) all ξ R\{} φ( iξ) b ξ + b ξ + (, ) e e 1 φ( ξ ). This implies ha for any α > and lim sup u lim sup u φ(u) u α φ(u) u α 1 e iξ x ν(dx) (1 [ ξ x]) ν(dx) (, ) < = lim sup ξ < = lim sup ξ φ( iξ) ξ α < φ( iξ) ξ α <. Thus, he following resul is a direc consequence of Theorem 3.1(c). Corollary 3.6. Assume ha κ (, σ ), where σ (, σ ] and lim sup u φ(u)u σ <. 8 Then ES κ C κ,σ ( 1) κ/σ, >. If furhermore lim sup u φ(u)u σ <, 9 hen ES κ C κ,σ κ/σ, >. Remark 3.7. As in Remark 3.(d), i is easy o see ha Corollary 3.6 is sharp for he α-sable subordinaor, < α < 1. Recall ha σ is defined by (.1). Le φ(u) φ(u) ρ := inf α : lim inf u u α > = sup α : lim inf u u α =. Because of (.3), i is easy o see ha σ ρ 1. The following resul is essenially due o [5]. For he sake of compleeness, we presen he argumen. The proof is based on he fac ha he funcions x x κ and x e λx κ, κ, λ, x >, are compleely monoone funcions, cf. [11, Chaper 1]. Theorem 3.8. (a) Le ρ > and κ >. If for some ρ (, ρ ], lim inf u φ(u)u ρ >, 1 hen for all > ES κ C κ,ρ ( 1) κ/ρ. 8 This is equivalen o eiher < σ < σ or σ = σ > and lim sup u φ(u)u σ <. 9 This is equivalen o eiher σ < σ σ or σ = σ (, σ ] and lim sup u φ(u)u σ <. 1 This is equivalen o eiher < ρ < ρ or ρ = ρ > and lim inf u φ(u)u ρ >.

23 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) If furhermore lim inf u φ(u)u ρ >, 11 hen for all > ES κ C κ,ρ κ/ρ. (b) Le ρ > and κ (, ρ /(1 ρ )). If lim inf u φ(u)u ρ >, 1 for some ρ (κ/(1 + κ), ρ ], hen for all λ, > ρ E e λs κ λ ρ (1 ρ)κ λ exp C κ,ρ λ + + κ/ρ κ/ρ. In paricular, for any λ > E e λs κ exp ( 1) C κ,ρ,λ κ ρ (1 ρ)κ, >. If furhermore lim inf u φ(u)u ρ >, 13 hen for all λ, > ρ E e λs κ λ ρ (1 ρ)κ λ exp C κ,ρ + κ/ρ κ/ρ. (c) Le σ < 1. If κ > σ /(1 σ ), hen for all λ, > E e λs κ =. Remark 3.9. Le S be an α-sable subordinaor ( < α < 1). Then we can choose in Theorem 3.8(a) ρ = ρ = ρ = α. Noe ha ES κ = C κ,α κ/α for all κ, >, see e.g. [9, (5.5)]. This means ha Theorem 3.8(a) is sharp for α-sable subordinaors. Proof of Theorem 3.8. (a) If lim inf u φ(u)u ρ >, hen here are consans C 1 = C 1 (ρ) > and C = C (ρ) such ha φ(u) C 1 u ρ, u C. (3.) Combining his wih (3.15), we obain ES κ = 1 Γ (κ) 1 Γ (κ) C = Cκ κγ (κ) + u κ 1 e φ(u) du u κ 1 du + 1 u κ 1 e C 1u ρ du Γ (κ) Γ κρ ργ (κ)(c 1 ) ρ κ, (3.1) which implies he firs asserion. If furhermore lim inf u φ(u)u ρ >, hen (3.) holds wih C =. Thus, he second esimae follows from (3.1) wih C =. 11 This is equivalen o eiher ρ < ρ ρ or ρ = ρ (, ρ ] and lim inf u φ(u)u ρ >. 1 This is equivalen o eiher κ/(1 + κ) < ρ < ρ or ρ = ρ > κ/(1 + κ) and lim inf u φ(u)u ρ >. 13 This is equivalen o eiher κ 1+κ ρ < ρ ρ or ρ = ρ (κ/(1 + κ), ρ ] and lim inf u φ(u)u ρ >.

24 3874 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) (b) I follows from (3.15) ha for x λ n 1 e λx κ = 1 + n! x nκ = 1 + where k(u) := = 1 + e ux k(u) du, λ n n! λ n n!γ (nκ) unκ 1, u >. 1 Γ (nκ) Now we can use Tonelli s heorem o obain E e λs κ = 1 + E e us k(u) du = 1 + If lim inf u φ(u)u ρ >, hen by (3.), E e λs κ 1 + = 1 + C k(u) du + λ n C nκ n! Γ (nκ)nκ + 1 ρ e C 1u ρ k(u) du λ n Γ u nκ 1 e ux du e φ(u) k(u) du. (3.) nκ ρ n! Γ (nκ) (C 1 ) nκ ρ Combining his wih he inequaliies π x x 1 e x Γ (x) π x x 1 e x+ 1 1x, x >, we arrive a n! π n n+ 1 e n, n N, E e λs κ π κ λc κ e κ+1 κ κ n + 1 πρ π κ + eρ/(1κ) πρ nn (1+κ)n κρ κ 1 n e ρ/(1nκ) n n λc κ e κ+1 κ κ n n (1+κ)n κρ κ 1 n Se G := λc κ eκ+1 κ κ ; because of n e κ+1 κ κ e κ+1 κ κ κ ρec 1 κ/ρ κ. ρec 1 κ/ρ n λ κ/ρ. n λ κ/ρ we have n n n!, n N, G n n (1+κ)n G n n n G n n! = e G 1. (3.3)

25 Se C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) ϵ := κ ρ + κ + 1 (, 1] and H := eκ+1 κ κ Using Jensen s inequaliy and (3.3), i holds ha H n n ϵn = (H) n ϵ ϵ 1 n n n (H) n ϵ 1 n n n 1 (H) 1 n ϵ ϵ n! (H) 1 ϵ ϵ = exp 1. κ ρec 1 ϵ κ/ρ λ >. κ/ρ Combining he above esimaes and using he following elemenary inequaliies we obain 1 + z(e x 1) y e xy+z(1+z)x y and E e λs κ = 1 π κ π κ 1 exp + 1 exp e G 1 + eρ/(1κ) e G π κ + 1 π κ + 1 π κ + 1 π κ πρ exp 1 ex + 1 ey e x+y, x, y, z, (H) 1 ϵ ϵ / eρ/(1κ) πρ exp (H) 1 ϵ ϵ / 1 G + (H) 1 exp 1ϵ e ρ/(1κ) e ρ/(1κ) ϵ πρ πρ G + ϵ(h) 1ϵ + ϵ eρ/(1κ) e ρ/(1κ) + 1 πρ πρ H. H ϵ Thus, he firs asserion follows. If furhermore lim inf u φ(u)u ρ >, hen we can ake C = and so G =. Hence, he desired asserion follows from he above esimae wih G =. (c) Pick σ (σ, κ/(1 + κ)). By he definiion of σ, here exis wo posiive consans C 3 = C 3 (σ ) and C 4 = C 4 (σ ) such ha φ(u) C 3 u σ, u C 4. Togeher wih (3.) his yields ha E e λs κ = = C 4 C 4 e φ(u) k(u) du e C 3uσ k(u) du λ n n! Γ (nκ) (C 3 ) nκ σ λ n n! Γ (nκ) (C 3 ) nκ σ λ n n! Γ (nκ) (C 3 ) nκ σ C 3 C σ 4 u nκ nκ Γ σ Γ nκ σ σ 1 e u du C3 C4 σ u nκ σ 1 e u du σ C3 C4 σ nκ nκ σ.

26 3876 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) By (3.19) and n! πn n+ 1 e n, n, we have for large n N λ n n! Γ (nκ) (C 3 ) nκ σ Γ nκ σ C3 C4 σ σ nκ λ n πκ 1 n (1+κ)n e κ 1 κ κ (C 3 ) σ κ λ n πκ 1 n (1+κ)n e κ 1 κ κ (C 3 ) σ κ σ 1 = n ( 1 κ+ σ κ )n λ π n e κ 1 κ κ nκ σ πσ n nκ n nκ κ nκ σ σ σ C3 C σ nκ 4 σ σ e nκ πσ n nκ n nκ κ σ σ e κ n σ, κ σ ec 3 nκ σ which, because of 1 κ + κ σ >, ends o infiniy as n. Acknowledgemens The auhors would like o hank an anonymous referee for her/his insighful commens and useful suggesions which helped o improve he presenaion of our manuscrip. The firsnamed auhor graefully acknowledges suppor hrough he Alexander-von-Humbold foundaion (Posdocoral Fellowship CHN/11556), he Naional Naural Science Foundaion of China (114144) and he Inernaional Posdocoral Exchange Fellowship Program (13). Appendix In his secion, we show ha he index β defined by (3.3) akes values in [, ]. Wihou loss of generaliy we may assume ha he Lévy measure ν ; oherwise, he asserion is rivial. Since 1 cos u (1 cos 1)u, u 1, we have for all ξ R d ha ψ(ξ) Re ψ(ξ) = 1 ξ Qξ + (1 cos ξ y ) ν(dy) y (1 cos ξ y ) ν(dy) y, ξ y 1 (1 cos 1) ξ y ν(dy). y, ξ y 1 Because ν, we know ha here exiss a uni vecor x R d such ha ν D := 1 D ν, where D := z R d \{} : arccos x z, π. z 8

27 C.-S. Deng, R.L. Schilling / Sochasic Processes and heir Applicaions 15 (15) Since ξ, y D saisfy we see arccos ξ y ξ y arccos ξ x + arccos x y π ξ y 8 + π 8 = π 4, ξ y ξ y 1, ξ, y D. Thus, we ge for all ξ D ha ψ(ξ) (1 cos 1) 1 cos 1 y D, ξ y 1 ξ y D, ξ y 1 ξ y ν(dy) y ν(dy), and, by Faou s lemma, ψ(ξ) lim inf ξ D, ξ ξ 1 cos 1 lim inf y 1 { ξ y 1} ν D (dy) ξ D, ξ R d 1 cos 1 y lim inf 1 { ξ y 1} ν D (dy) R d ξ D, ξ = 1 cos 1 y ν D (dy) (, ]. R d This shows ha β. References [1] C. Berg, G. Fors, Poenial Theory on Locally Compac Abelian Groups, in: Ergebnisse der Mahemaik und ihrer Grenzgebiee, II, Ser. Bd. 87, Springer, Berlin, [] R.M. Blumenhal, R.K. Geoor, Sample funcions of sochasic processes wih saionary independen increamens, J. Mah. Mech. 1 (1961) [3] C.-S. Deng, Harnack inequaliies for SDEs driven by subordinae Brownian moions, J. Mah. Anal. Appl. 417 (14) [4] C.-S. Deng, R.L. Schilling, On a Cameron Marin ype quasi-invariance heorem and applicaions o subordinae Brownian moion, Soch. Anal. Appl. in press, arxiv: [5] M. Gordina, M. Röckner, F.-Y. Wang, Dimension-independen Harnack inequaliies for subordinaed semigroups, Poenial Anal. 34 (11) [6] N. Jacob, Pseudo Differenial Operaors and Markov Processes (Volume I), Imperial College Press, London, 1. [7] H. Luschgy, G. Pagès, Momen esimaes for Lévy processes, Elecron. Comm. Probab. 13 (8) [8] P.W. Millar, Pah behavior of processes wih saionary independen incremens, Z. Wahrscheinlichkeisheor. Verwande Geb. 17 (1971) [9] K. Sao, Lévy Processes and Infiniely Divisible Disribuions, Cambridge Universiy Press, Cambridge, [1] R.L. Schilling, Growh and Hölder condiions for he sample pahs of Feller processes, Probab. Theory Relaed Fields 11 (1998) [11] R.L. Schilling, R. Song, Z. Vondraček, Bernsein Funcions. Theory and Applicaions, second ed., in: Sudies in Mahemaics, vol. 37, De Gruyer, Berlin, 1. [1] R.L. Schilling, J. Wang, Funcional inequaliies and subordinaion: sabiliy of Nash and Poincaré inequaliies, Mah. Z. 7 (1) [13] F.-Y. Wang, Logarihmic Sobolev inequaliies on noncompac Riemannian manifolds, Probab. Theory Relaed Fields 19 (1997) [14] F.-Y. Wang, Harnack Inequaliies for Sochasic Parial Differenial Equaions, Springer, New York, 13. [15] F.-Y. Wang, Inegraion by pars formula and applicaions for SDEs wih Lévy noise, arxiv:

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