Harnack inequalities for SDEs driven by subordinate Brownian motions

J. Math. Anal. Appl. 417 (2014) 970–978

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Harnack inequalities for SDEs driven by subordinate Brownian motions ✩ Chang-Song Deng a,b a b

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China TU Dresden, Institut für Mathematische Stochastik, 01062 Dresden, Germany

a r t i c l e

i n f o

Article history: Received 28 November 2013 Available online 1 April 2014 Submitted by U. Stadtmueller Keywords: Harnack inequality Gradient estimate Subordinate Brownian motion Time-change Stochastic differential equation

a b s t r a c t By using regularization approximations of the underlying subordinator and a gradient estimate approach, the dimension-independent Harnack inequalities are established for the inhomogeneous semigroup associated with a class of SDEs with Lévy noise containing a subordinate Brownian motion. Our estimates in Harnack type inequalities improve the corresponding ones in the recent paper by Wang and Wang (2014) [10]. © 2014 Elsevier Inc. All rights reserved.

1. Introduction The dimension-independent Harnack inequality in the sense of [6] has a lot of applications, such as the study of functional inequalities, heat kernel estimates, transportation-cost inequalities and so on. Recently, this inequality has also been intensively investigated for various SDEs and SPDEs driven by Brownian noise (see e.g. [1,7] and references therein), which can be naturally regarded as a continuous Markov model. However, many physical and economic systems should be described by discontinuous Markov processes, which attract much attention in recent years due to their importance both in theory and in applications. Thus, it is quite meaningful to establish Harnack inequality for SDEs driven by jump processes (see [8,10, 9,14,13] for recent development in this direction). The central aim of this work is to investigate Harnack inequalities for SDEs driven by subordinate Brownian motions (cf. [15,10,9,14]). Let us first recall some basic notations for subordinate Brownian motions. A subordinator (St )t0 is a real-valued increasing Lévy process with S0 = 0 and Ee−rSt = e−tB(r) ,

r, t  0,

✩ Supported in part by SRFDP (20130141120036), the International Postdoctoral Exchange Fellowship Program (2013) and the Fundamental Research Funds for the Central Universities. E-mail address: [email protected].

http://dx.doi.org/10.1016/j.jmaa.2014.03.082 0022-247X/© 2014 Elsevier Inc. All rights reserved.

C.-S. Deng / J. Math. Anal. Appl. 417 (2014) 970–978

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where B is a Bernstein function. We refer to [5] for more details on subordinators and Bernstein functions. Let (Wt )t0 be a standard Brownian motion on Rd and (St )t0 be a subordinator independent of (Wt )t0 . The process (WSt )t0 is called a subordinate Brownian motion, which is a rotationally invariant Lévy process with symbol B(| · |2 /2). Since (WSt )t0 is obtained by replacing the time of Brownian motion by an independent subordinator, this enables us to study subordinate Brownian motion by non-random time-changed Brownian motion. This point is crucial for Zhang’s regularization approximation argument in [15]. Let σ : [0, ∞) → Rd ⊗ Rd be measurable and locally bounded, and b : [0, ∞) × Rd → Rd measurable, locally bounded and continuous in the second variable. Consider the following inhomogeneous stochastic equation on Rd : t Xs,t (x) = x +

  br Xs,r (x) dr +

s

t σr dWSr + Vt − Vs ,

0  s  t,

(1.1)

s

where W = (Wt )t0 , S = (St )t0 and V = (Vt )t0 are independent stochastic processes such that (i) W is a standard d-dimensional Brownian motion with W0 = 0; (ii) S is a subordinator; (iii) V is a locally bounded measurable process on Rd with V0 = 0. As in [10], we will assume the following conditions on b and σ: (H1) There exists a locally bounded measurable function K : [0, ∞) → R such that 

 bt (x) − bt (y), x − y  Kt |x − y|2 ,

x, y ∈ Rd , t  0.

(H2) For every t  0, σt is invertible and there exists a measurable function λ : [0, ∞) → (0, ∞) such that σt−1   λt for all t  0. Then the existence, uniqueness and non-explosion of the solution to (1.1) can be guaranteed by (H1). Consider the inhomogeneous Markov semigroup (Ps,t )0st on Bb (Rd ):   Ps,t f (x) := Ef Xs,t (x) ,

  0  s  t, f ∈ Bb Rd , x ∈ Rd .

Let t K(s, t) =

Kr dr,

0  s  t.

s

When S is the interesting α-stable subordinator, σ does not depend on time t and   sup ∇x bt (x) < ∞, t0,x∈Rd

with the help of regularization approximations of the time-change, Zhang established in [15] the Bismut derivative formula for P0,t . Following Zhang’s regularization approximation argument together with the method of coupling and Girsanov transformation, Harnack and log-Harnack inequalities for P0,t were derived in [10] (see also [14] for the anisotropic case). We find that a minor modification of the coupling constructed in [10] can lead to better estimates (see Theorem 1.1 below). To see this, let’s take only the log-Harnack

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inequality for example. If we assume that t → λt is increasing as in [10], then one of the main results in [10] states that

 |x − y|2 λ2r , x, y ∈ Rd inf E r −2K(0,u) P0,t log f (y)  log P0,t f (x) + (1.2) 2 r∈(0,t] e dS u 0 holds for all f ∈ Bb (Rd ) with f  1. Therefore, our estimate in Theorem 1.1(1) below improves (1.2) since by the monotonicity of λ  t

−1 −2K(0,u) λ−2 u e

= inf r

dSu

r∈(0,t]

0

0

1 −2K(0,u) dS λ−2 u e u

 inf r r∈(0,t]

0

λ2r e−2K(0,u) dSu

.

Theorem 1.1. Assume that both (H1) and (H2) hold. (1) For any 0  s  t,  t −1 |x − y|2 −2K(s,u) E Ps,t log f (y)  log Ps,t f (x) + λ−2 dSu u e 2 s

holds for all x, y ∈ Rd and f ∈ Bb (Rd ) with f  1. (2) For any 0  s  t and p > 1,

  p   Ps,t f (y)  Ps,t f p (x) E exp

p|x − y|2 t −2 2(p − 1)2 s λu e−2K(s,u) dSu

p−1

holds for all x, y ∈ Rd and non-negative f ∈ Bb (Rd ). To prove Theorem 1.1, however, we shall use a gradient estimate approach rather than simply repeat the coupling argument as in [10]. Harnack inequality was first obtained in [6] via the gradient estimate (see also [4,11,12] for further development). When the underlying semigroup is not regular enough to satisfy the gradient estimate, the coupling method turned out to be very efficient to establish Harnack type inequalities (cf. [1,7,3]). As far as we know, these two techniques are the main tools to establish Wang’s Harnack type inequalities. Since our present state space is Rd and the coefficients are quite regular, it seems feasible (and also natural) for us to derive Harnack inequality via the gradient estimate approach. This can be regarded as the motivation of our paper. Now we move on to consider the gradient estimates for Ps,t . For a function f on Rd and x ∈ Rd , let |∇f |(x) = lim sup y→x

|f (y) − f (x)| . |y − x|

In other words, |∇f |(x) is the local Lipschitz constant of f at point x. As pointed out in [10], according to [2, Proposition 2.3], the log-Harnack inequality implies an L2 -gradient estimate. Therefore, the following result is a direct consequence of Theorem 1.1(1). Corollary 1.2. Assume that both (H1) and (H2) hold. Then for any 0  s  t and f ∈ Bb (Rd ), 



2

 t

2 

|∇Ps,t f | (x)  Ps,t f (x) − Ps,t f (x) 2

E

−1 −2K(s,u) λ−2 u e

dSu

,

s

In the next section, we will prove Theorem 1.1 via a gradient estimate approach.

x ∈ Rd .

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2. Proof of Theorem 1.1 by using gradient estimate We proceed as in [15,10,14] to use regularization approximations of time-change. Let  be an arbitrary sample of S. Then  is an increasing and càdlàg function from [0, ∞) to [0, ∞) with 0 = 0. Consider the following regularization of :

εt

1 := ε

t+ε 1 s ds + εt = εs+t ds + εt, t

ε > 0, t  0.

(2.1)

0

Clearly, t → εt is absolutely continuous and for each t  0 one has εt ↓ t as ε ↓ 0. Let v be a measurable and ε ,v (x) solve the following locally bounded function from [0, ∞) to Rd vanishing at starting point 0. Let Xs,t SDE on Rd : ε ,v Xs,t (x)

t =x+



ε ,v br Xs,r (x)



t σr dWεr + vt − vs ,

dr +

s

0  s  t,

(2.2)

s

which is indeed driven by the Brownian motion with absolutely continuous time-change. This will be crucial for our study. Let  ε ,v  ε ,v Ps,t f (x) = Ef Xs,t (x) ,

  0  s  t, ε > 0, f ∈ Bb Rd , x ∈ Rd .

In this section, we shall first establish gradient estimate for SDEs with standard Brownian noise (see ε ,v will be derived (see PropoLemma 2.1 below), and then as applications Harnack type inequalities for Ps,t sition 2.2 below). Finally, one can apply the approximation technique in [10, proof of Theorem 1.1] to get the desired Harnack and log-Harnack inequalities for Ps,t in Theorem 1.1. 2.1. Gradient estimates for SDEs under absolutely continuous time-change We first transform SDE (2.2) into an equation driven by standard Brownian motions on Rd , and so that the classical technique can be used. Fix any ε > 0. Since t → t is non-decreasing, it is easy to see from (2.1) that t → εt is strictly increasing. Denote by γ ε the inverse function of ε , i.e. εγtε = t for t  ε0 and γεεt = t for t  0. Then t → γtε is also absolutely continuous and strictly increasing for t ∈ [ε0 , ∞). Let us now define ε

ε

 ,v ε Ys,t (x) = Xγsε,v ,γtε (x) − vγt + vγsε ,

ε0  s  t, x ∈ Rd .

By (2.2) and the change of variables, we know that for ε0  s  t ε

γt

ε

 ,v Ys,t (x) = x +

ε

 ε  br Xγsε,v ,r (x) dr +

γsε

t =x+

γt

σr dWεr γsε

˜bε ,v r

 ε ,v  Ys,r (x) dr +

s

t σγrε dWr , s

where ˜bε ,v (·) := bγ ε (· + vγ ε − vγ ε )γ˙ ε , r r r r s

s  r  t.

(2.3)

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ε

 ,v Now one can associate to (Ys,t (x))ε0 st the (time dependent) second order elliptic partial differential ε  ,v operators (Lt )tε0 defined by

Lt

ε

,v

f (x) =

d d  ∂ 2 f (x)   ε ,v  ∂f (x) 1  ˜b (x) σγtε σγTtε ij + , t i ∂x 2 i,j=1 ∂xi ∂xj i i=1

t  ε0 , x ∈ Rd ,

 ,v where f is a smooth function on Rd . Let Ps,t be the corresponding inhomogeneous Markov semigroup, i.e. ε

 ε ,v  ε ,v = Ef Ys,t (x) , Ps,t

  ε0  s  t, f ∈ Bb Rd .

It is not hard to verify that Lt

ε

,v 

 ε 1 ϕ(f ) = ϕ (f )Lt ,v f + ϕ (f )|σγtε ∇f |2 2

(2.4)

holds for any C 1 function ϕ : R → R and smooth function f on Rd . Note that by (H1) 

 ε ε x − y, ˜br ,v (x) − ˜br ,v (y)  Kγrε γ˙ rε |x − y|2 ,

r  ε0 , x, y ∈ Rd .

(2.5)

 ε    ε ε ,v Ps,t f (x) = Ef Yε ,,vεt (x) + vt − vs = Pε ,,vεt f (· + vt − vs ) (x)

(2.6)

Moreover, we have

s

s

for all 0  s  t, f ∈ Bb (Rd ) and x ∈ Rd . Lemma 2.1. For any f ∈ Cb1 (Rd ) and ε > 0,   ε ε ε ,v  ε ,v ∇Ps,t f  eK(γs ,γt ) Ps,t |∇f |,

ε0  s  t.

ε

 ,v Proof. Let (Ys,t (y))ε0 st solve Eq. (2.3) with x replaced by y. It follows from (2.5) that

2  ε ,v ε ,v Ys,t (y) − Ys,t (x) t = |y − x| + 2 2



 ε ,v   ε ,v  ε ε ε ,v ε ,v Ys,r (y) − Ys,r (x), ˜br ,v Ys,r (y) − ˜br ,v Ys,r (x) dr

s

t  |y − x| + 2 2

 ε ,v 2 ε ,v Kγrε γ˙ rε Ys,r (y) − Ys,r (x) dr.

s

This together with Gronwall’s inequality yields that  t  2  ε ,v ε  ,v 2 ε Ys,t (y) − Ys,t (x)  |y − x| exp 2 Kγ ε γ˙ r dr = |y − x|2 e2K(γsε ,γtε ) . r s

Combining this with Jensen’s inequality and Fatou’s lemma, we obtain

C.-S. Deng / J. Math. Anal. Appl. 417 (2014) 970–978 ε

975

ε

 ,v  ,v   |Ef (Ys,t (y)) − Ef (Ys,t (x))| ε ,v  ∇Ps,t f (x) = lim sup |y − x| y→x

 ε ,v ε ,v ε ,v ε ,v (y)) − f (Ys,t (x))| |Ys,t (y) − Ys,t (x)| |f (Ys,t ·  lim sup E ε ,v ε ,v |y − x| y→x |Ys,t (y) − Ys,t (x)| ε ε

  ,v  ,v |f (Ys,t (y)) − f (Ys,t (x))| ε ε  eK(γs ,γt ) E lim sup ε ,v ε ,v y→x |Ys,t (y) − Ys,t (x)|   ε ε ε  ,v = eK(γs ,γt ) E|∇f | Ys,t (x)  ,v = eK(γs ,γt ) Ps,t |∇f |(x). ε

ε

ε

2

2.2. Φ-Harnack inequality for SDEs under absolutely continuous time-change ε

 ,v This subsection is devoted to establish the so-called Φ-Harnack inequality for Ps,t :

 ε ,v   ε ,v  Φ Ps,t f (y)  Ps,t {Φ ◦ f }(x) eΨ (x,y) ,

  0  f ∈ Bb Rd , x, y ∈ Rd ,

(2.7)

where Φ and Ψ are non-negative functions on Rd and Rd ×Rd , respectively. Clearly, when Φ(r) = rp for some p > 1 this inequality reduces to the Harnack inequality with power p, while if Φ(r) = er then it becomes the log-Harnack inequality introduced in [4]. In order to derive the Φ-Harnack inequality (2.7), we need the following assumption: (H3) Φ ∈ C([0, ∞)), Φ is second-order differentiable with Φ > 0 on (0, ∞), and Φ(r)Φ (r) > 0. r>0 [Φ (r)]2

c := inf Note that (H3) implies that Φ|(0,∞) > 0 and

  2 Φ (r) Φ (r) c , Φ(r) Φ(r)

r > 0.

(2.8)

Typical examples for Φ satisfying (H3) contain Φ(r) = rp (p > 1) with c = (p − 1)/p and Φ(r) = er with c = 1. Moreover, one can give lots of other examples, e.g. the mixing of Φ(r) = rp (p > 1) and Φ(r) = er : Φ(r) = αrp + βer (α, β > 0). Proposition 2.2. Assume (H1)–(H3). For any 0  s  t and ε > 0,   ε ,v   ε ,v  Φ Ps,t f (y)  Ps,t {Φ ◦ f }(x) exp

2c

t s

|x − y|2



−2K(s,u) dε λ−2 u e u

holds for all non-negative f ∈ Bb (Rd ) and x, y ∈ Rd . Proof. We shall adopt the idea used in [4, proof of Theorem 2.1]. First, by a standard approximation argument, we may and do assume that f ∈ C 2 (Rd ), inf f > 0 and f is constant outside a compact set. Fix arbitrarily ε0  s  t. It follows from Itô’s formula and (2.4) that  ε ,v  ε ,v  dΦ Pr,t f Ys,r (x)   ε ,v  ε ,v     ε ,v  ε ,v  ε = ∇Φ Pr,t f Ys,r (x) , σγrε dWr + Lr ,v Φ Pr,t f Ys,r (x) dr

C.-S. Deng / J. Math. Anal. Appl. 417 (2014) 970–978

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  ε ,v   ε ,v  ε ,v  ε − Φ Pr,t f · Lr ,v Pr,t f Ys,r (x) dr    ε ,v  ε ,v   1   ε ,v    ε ,v 2  ε ,v = ∇Φ Pr,t f Ys,r (x) , σγrε dWr + Φ Pr,t f · σγrε ∇Pr,t f Ys,r (x) dr. 2 This implies that for any r ∈ [s, t]  ε ,v  ε ,v   ε ,v  EΦ Pr∧τ f Ys,r∧τn (x) − Φ Ps,t f (x) n ,t 1 = E 2

r∧τ  n



  ε ,v    ε ,v 2  ε ,v Φ Pu,t Ys,u (x) du, f · σγuε ∇Pu,t f

s

where  ε ,v    τn := inf r  s; Ys,r (x)  n ,

n ∈ N.

Noting that the solution of (2.3) is non-explosive, one has τn ↑ ∞ as n ↑ ∞. Consequently, by the dominated convergence theorem, Fubini’s theorem and the monotone convergence theorem (note that (H3) implies Φ |(0,∞) > 0), we know that  ε ,v    ε ,v  ε ,v Ps,r f (x) Φ Pr,t f (x) − Φ Ps,t  ε ,v  ε ,v   ε ,v  f Ys,r∧τn (x) − Φ Ps,t f (x) = lim EΦ Pr∧τ n ,t n→∞

1 = lim E 2 n→∞

r∧τ  n



  ε ,v    ε ,v 2  ε ,v Φ Pu,t f · σγuε ∇Pu,t f Ys,u (x) du

s

=

1 2

r

    ε ,v   ε ,v 2  ε ,v Φ Pu,t f · σγuε ∇Pu,t f (x) du Ps,u

(2.9)

s

holds for all r ∈ [s, t]. Let xr = (x − y)hr + y,

r ∈ [s, t],

where h ∈ AC([s, t]; R) (the set of all absolutely continuous functions from [s, t] to R) with hs = 0 and ht = 1. Recall that (H3) yields that  ε ,v  Φ Pr,t f  Φ(inf f ) > 0. Now, it follows from (2.9), (H2), Lemma 2.1 and (2.8) that   ε ,v  d ε ,v log Ps,r Φ Pr,t f (xr ) dr 1 d ε ,v   ε ,v  = ε ,v Ps,r Φ Pr,t f (xr ) ε ,v  Ps,r {Φ(Pr,t f )}(xr ) dr =

ε 1  ε ,v    ,v 2 Ps,r {Φ (Pr,t f )

 ,v 2  ,v  ,v · |σγrε ∇Pr,t f | } + hr x − y, ∇Ps,r {Φ(Pr,t f )} (xr ) ε ,v ε ,v   Ps,r {Φ(Pr,t f )} ε

ε

ε

C.-S. Deng / J. Math. Anal. Appl. 417 (2014) 970–978



=

1 2λ2γ ε

977

ε ε  ,v  ,v 2  ,v  ,v ε ,v ε ,v Ps,r {Φ (Pr,t f )|∇Pr,t f | } − |hr ||x − y|eK(γs ,γr ) Ps,r {Φ (Pr,t f )∇Pr,t f} ε

ε

ε

r

  ε ,v ε ,v Ps,r f ) 2λ12 ε Φ(Pr,t γr

ε

(xr )

ε ,v ε ,v Ps,r {Φ(Pr,t f )} ε

 ,v Φ (Pr,t f) ε ,v 2 |∇Pr,t f| ε  Φ(P ,v f ) r,t

− |hr ||x − y|eK(γs ,γr ) ε

ε



ε

 ,v f) Φ (Pr,t ε ,v |∇Pr,t f| ε  Φ(P ,v f ) r,t

ε ,v ε ,v {Φ(Pr,t f )} Ps,r    ε ε ε  ,v ε ,v Ps,r f ) 2λc2 ε ξ 2 − |hr ||x − y|eK(γs ,γr ) ξ Φ(Pr,t γr  (xr ) ε ,v ε ,v {Φ(Pr,t f )} Ps,r

−

(xr )

|x − y|2   2 2 2K(γsε ,γrε ) hr λγrε e , 2c

 ,v  ,v  ,v where ξ := |∇Pr,t f |Φ (Pr,t f )/Φ(Pr,t f ). Taking integral for both sides over [s, t] w.r.t. dr, we arrive at ε

ε

log

ε

t ε ,v {Φ ◦ f }(x) Ps,t |x − y|2   2 2 2K(γsε ,γrε ) hr λγrε e  − dr. ε ,v 2c Φ(Ps,t f (y)) s

Optimizing the right hand side in h ∈ AC([s, t]; R) with hs = 0 and ht = 1, i.e. taking r hr =

s t s

−2K(γs ,γu ) λ−2 du ε e γu ε

ε

−2K(γs ,γu ) du λ−2 ε e γu ε

ε

r ∈ [s, t],

,

we obtain  ,v {Φ ◦ f }(x) Ps,t |x − y|2 log  − . ε t ε)  ,v −2K(γsε ,γu Φ(Ps,t f (y)) 2c λ−2 du γε e ε

s

u

This implies that  ε ,v   ε ,v  Φ Ps,t f (y)  Ps,t {Φ ◦ f }(x) exp

 2c

t s



|x − y|2 −2K(γs ,γu ) du λ−2 ε e γu ε

ε

.

Noting (2.6), and replacing first f by f (· + vγtε − vγsε ) and then s and t by εs and εt respectively, we get the desired inequality. 2 2.3. Proof of Theorem 1.1 Proof of Theorem 1.1. Taking Φ(r) = er with c = 1 and Φ(r) = rp with c = (p − 1)/p in Proposition 2.2, we get respectively the log-Harnack inequality ε

ε

 ,v  ,v Ps,t log f (y)  log Ps,t f (x) +

2

t

|x − y|2

λ−2 e−2K(s,u) s u

dεu

,

  1  f ∈ Bb Rd , x, y ∈ Rd

and the Harnack inequality with power p 

p ε ,v Ps,t f (y)

 ε ,v p   Ps,t f (x) exp



p|x − y|2

2(p − 1)

t

λ−2 e−2K(s,u) s u

dεu

 ,

  0  f ∈ Bb Rd , x, y ∈ Rd .

Finally, it remains to apply the approximation argument in [10, proof of Theorem 2.1] to finish the proof. 2

978

C.-S. Deng / J. Math. Anal. Appl. 417 (2014) 970–978

Acknowledgments This paper was revised when the author was a Humboldt fellow at TU Dresden. He is grateful for the financial support through the Alexander von Humboldt Foundation (Postdoctoral Fellowship CHN/1150506) and would like to thank Professor René L. Schilling for providing him with nice working environment. Thanks are also given to the referee for careful reading and a list of very useful suggestions and corrections. References [1] M. Arnaudon, A. Thalmaier, F.-Y. Wang, Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below, Bull. Sci. Math. 130 (2006) 223–233. [2] M. Arnaudon, A. Thalmaier, F.-Y. Wang, Equivalent Harnack and gradient inequalities for pointwise curvature lower bound, Bull. Sci. Math. (2014), http://dx.doi.org/10.1016/j.bulsci.2013.11.001, in press. [3] C.-S. Deng, Harnack inequality on configuration spaces: the coupling approach and a unified treatment, Stochastic Process. Appl. 124 (2014) 220–234. [4] M. Röckner, F.-Y. Wang, Log-Harnack inequality for stochastic differential equations in Hilbert spaces and its consequences, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 (2010) 27–37. [5] R.L. Schilling, R. Song, Z. Vondraček, Bernstein Functions—Theory and Applications, Studies in Mathematics, vol. 37, De Gruyter, Berlin, 2010. [6] F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Related Fields 109 (1997) 417–424. [7] F.-Y. Wang, Harnack inequality and applications for stochastic generalized porous media equations, Ann. Probab. 35 (2007) 1333–1350. [8] F.-Y. Wang, Derivative formula and Harnack inequality for SDEs driven by Lévy processes, Stoch. Anal. Appl. 32 (2014) 30–49. [9] F.-Y. Wang, Integration by parts formula and applications for SDEs with Lévy noise, arXiv:1308.5799. [10] F.-Y. Wang, J. Wang, Harnack inequalities for stochastic equations driven by Lévy noise, J. Math. Anal. Appl. 410 (2014) 513–523. [11] F.-Y. Wang, J.-L. Wu, L. Xu, Log-Harnack inequality for stochastic Burgers equations and applications, J. Math. Anal. Appl. 384 (2011) 151–159. [12] F.-Y. Wang, T. Zhang, Log-Harnack inequality for mild solutions of SPDEs with multiplicative noise, Stochastic Process. Appl. 124 (2014) 1261–1274. [13] J. Wang, Harnack inequalities for Ornstein–Uhlenbeck processes driven by Lévy processes, Statist. Probab. Lett. 81 (2011) 1436–1444. [14] L. Wang, X. Zhang, Harnack inequalities for SDEs driven by cylindrical α-stable processes, arXiv:1310.3927v2. [15] X. Zhang, Derivative formula and gradient estimates for SDEs driven by α-stable processes, Stochastic Process. Appl. 123 (2013) 1213–1228.