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Coupling and Strong Feller for Jump Processes on Banach Spaces 2 ∗ 1 0 2 Feng-Yu Wanga),c) and Jian Wangb) p e a)School of Mathematical Sciences, Beijing Normal University, Beijing 100875,China S b)School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007,China. 2 1 c)Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, UK ] Email: [email protected]; [email protected];[email protected] R P September 13, 2012 . h t a m [ Abstract 2 v By using lower bound conditions of the Lévy measure w.r.t. a nice reference mea- 5 sure, the coupling and strong Feller properties are investigated for the Markov semi- 9 group associated with a class of linear SDEs driven by (non-cylindrical) Lévy processes 7 3 on a Banach space. Unlike in the finite-dimensional case where these properties have 1. alsobeenconfirmedforLévy processeswithoutdrift,intheinfinite-dimensionalsetting 1 the appearance of a drift term is essential to ensure the quasi-invariance of the process 1 by shifting the initial data. Gradient estimates and exponential convergence are also 1 : investigated. The main results are illustrated by specific models on the Wiener space v and separable Hilbert spaces. i X r AMS subject Classification: 60J75, 60J45. a Keywords: Coupling, Strong Feller, Lévy process, Wiener space. 1 Introduction In recent years, the coupling property, the strong Feller property, and gradient estimates have been intensively investigated for linear stochastic differential equations driven by Lévy ∗ Supported in part by NNSFC(11131003, 11126350 and 11201073), SRFDP, the Fundamental Research Funds for the Central Universities and the Programme of Excellent Young Talents in Universities of Fujian (JA10058 and JA11051). 1 processes on Rd, see e.g. [14, 22, 21, 17, 3, 19, 18, 9, 8] and references within. In these references the shift-invariance of the Lebesgue measure plays an essential role. When the state space is infinite-dimensional so that the Lebesgue measure is no longer available, we needareference measurewhichisquasi-invariant under areasonableclassofshifttransforms. Typical examples of the reference measure include the Wiener measure on the continuous path space and the Gaussian measure on a Hilbert space, see Section 5 for details. The purpose of this paper is to investigate regularity properties of linear SDEs driven by Lévy processes on a Banach space equipped with such a nice reference measure. To ensure the quasi-invariance of the solution, a strong enough linear drift term will be needed. On the other hand, concerning (semi-)linear SDEs on Hilbert spaces, when the noise is a cylindrical α-stable process, many regularity results derived in finite dimensions can be extended to the infinite-dimensional setting (see [15, 13, 25]); and when the noise has a non- trivial Gaussian part, the regularity properties can be derived by using the drift part and the Gaussian part (see e.g. [26, 6, 7, 16]). But there seems to be no results concerning the strong Feller and coupling properties for SDEs driven by purely jump non-cylindrical Lévy processes. In this paper we intend to investigate these properties for linear SDEs driven by non-cylindrical Lévy noise on Banach spaces. Let (B, B) be a Banach space and let µ be a probability measure on B having full k · k support. Let B′ be the dual space of B with , the duality between B and B′. Let (H, H) be another Banach space which is densely ahn·d·icontinuously embedded into B such thkat·kfor any h H, µ is quasi-invariant under the shift x x+h; that is, there exists a non-negative ∈ 7→ measurable function ϕ on B such that h (1.1) µ(dz h) = ϕ (z)µ(dz). h − Let L be a Lévy process on B with Lévy measure ν. Recall that a σ-finite measure ν on B t is called a Lévy measure if ν( 0 ) = 0 and the mapping from B to R given by ′ { } B a exp cos x,a 1 ν(dx) ′ ∋ 7→ h i− B (cid:20)Z (cid:21) (cid:0) (cid:1) is the characteristic function of a random variable on B. Note that since cos is an even function, one may replace ν by the symmetric measure ν+ν as in [2], where ν (A) = ν( A) ∗ ∗ for any A B. When B is a Hilbert space, ν is a Lévy measure if and only if ν( 0 )−= 0 ∈ { } and (1 x 2)ν(dx) < ; while in general, ν is a Lévy measure provided ν( 0 ) = 0 and B B ∧k k ∞ { } B(1 x B)ν(dx) < (see [1, 2]). LR∧etkσk: B B be a∞bounded linear operator and let (A,D(A)) be a linear operator on B R → generating a C semigroup (T ) . Consider the following linear SDE on B: 0 s s 0 ≥ (1.2) dX = AX dt+σdL . t t t For any x B, the solution with initial data x is ∈ t (1.3) Xx = T x+ T σdL , t 0. t t t−s s ≥ Z0 2 See [4, 12, 1, 2] for the detailed construction of this solution. Let B (B) be the class of all b bounded measurable functions on B. We aim to investigate the coupling property and the strong Feller property for the associated Markov semigroup P f(x) := Ef(Xx), t 0,x B,f B (B). t t ≥ ∈ ∈ b Recall that the solution has successful coupling if and only if (cf. [10, 5]) lim P (x, ) P (y, ) = 0, x,y B, t t var t k · − · k ∈ →∞ where P (x,dy) is the transition kernel of P and is the total variation norm. Let ρ t t var 0 k·k be a non-trivial non-negative measurable function on B such that (1.4) ν(dz) ρ (z)µ(dz) =: ν (dz) 0 0 ≥ holds. Thus, the Lévy process considered here is essentially different from the cylindrical α-stable process used in [15, 13]. Indeed, for B being a Hilbert space with ONB e , the i i 1 Lévy measure (if exists) for a cylindrical Lévy process is supported on Re {an}d≥hence, i 1 i ∪≥ is singular w.r.t. e.g. a non-trivial Gaussian probability measure µ. Assume (A) Ker(σ) = 0 and T B σH holds for any s > 0. s { } ⊂ Obviously, (A) implies that for any s > 0, the operator σ 1T : B H is well defined. − s → Theorem 1.1. Assume (A). Suppose that ν in (1.4) is infinite; i.e. ν (B) = . 0 0 ∞ (1) If for any h H ∈ (1.5) sup ϕ ( +εh) < , µ-a.e., εh · ∞ ε (0,1) ∈ then for any f B (B) and t > 0, P f is directionally continuous; i.e. lim P f(x+ b t ε 0 t εy) = P f(x) ho∈lds for any x,y B. → t ∈ (2) If for any s > 0 (1.6) sup ϕσ−1Tsy(·+σ−1Tsy) < ∞, µ-a.e., y B 1 k k ≤ then P is strong Feller for t > 0; i.e. P B (B) C (B). t t b b ⊂ A simple example for ν0(B) = to hold is as follows. Let z z B have a strictly ∞ → k k positive distribution density function ρ under the probability measure µ, for instance it is the case when µ is the Wiener measure (see Subsection 5.1 below). Let r (0, ], and let 0 ∈ ∞ α (0,2) when B is a Hilbert space and α (0,1) otherwise. Then ∈ ∈ ν (dz) := 1(0,r0)(kzkB) µ(dz) 0 ρ( z B) z 1B+α k k k k 3 is a Lévy measure on B with ν (B) = . This measure is an infinite-dimensional version of 0 ∞ the α-stable jump measure. Modifying arguments from [22, Theorem 3.1] and [18, Theorem 1.1] where the coupling property has been investigated in the finite-dimension setting, we have the following two assertions on the coupling property with estimates on the convergence rate. For r > 0 and z B, let B(z,r) = y B : z y B < r be the open ball at z with ∈ { ∈ k − k } radius r. Theorem 1.2. Assume (A). Suppose that ν in (1.4) is finite; i.e. ν (B) < , σ is invertible 0 0 ∞ with σ−1 B < , and Ts B c holds for some constant c > 0 and all s > 0. k k ∞ k k ≤ (i) If there exist z B and r > 0 such that 0 0 ∈ (1.7) δ (ε) := sup ϕσ−1Tsx(z)2ρ0(z −σ−1Tsx)2 µ(dz) < , ε > 0, 1 ρ (z) ∞ s≥ε,kxkB≤1ZB(z0,r0) 0 then there exists a constant C > 0 such that δ (ε) (1.8) Pt(x, ) Pt(x+y, ) var C(1+ y B) inf ε+ 1 , t > 0, x,y B k · − · k ≤ k k ε∈(0,1)(cid:18) r t (cid:19) ∈ holds. (ii) If there exist z B and r > 0 such that 0 0 ∈ (1.9) δ (ε) := sup ϕσ−1Tsx(z)2 ∨1 µ(dz) < , ε > 0, 2 ρ (z) ∞ s≥ε,kxkB≤1ZB(z0,r0) 0 then there exist two constants C > 0 such that for all x,y B and t > 0, ∈ δ (ε) (1.10) Pt(x, ) Pt(y, ) var C(1+ x y B) inf ε+ 2 . k · − · k ≤ k − k ε∈(0,1)(cid:18) r t (cid:19) Using ρ 1 in place of ρ , one may replace (1.7) by 0 0 ∧ δ˜ (ε) := sup ϕσ−1Tsx(z)2 µ(dz) < , ε > 0. 1 1 ρ (z) ∞ s≥ε,kxkB≤1ZB(z0,r0) ∧ 0 If inf ρ (z) > 0, then this condition and (1.9) are equivalent. But in general (1.7) z B(x0,r0) 0 ∈ and (1.9) are incomparable. Next, it is easy to see that the convergence rate implied by (1.8) or (1.10) is in general slower than 1 . Our next result shows that if ϕ and ρ are regular √t 0 enough, the convergence could be exponentially fast. Theorem 1.3. Assume (A). Suppose that ν in (1.4) is finite with λ := ν (B) (0, ), 0 0 0 ∈ ∞ Ts B ce−λs and k k ≤ (1.11) ρ0(z) ρ0(z +h) +ρ0(z) ϕh(z) 1 µ(dz) c h H, h H 1 | − | | − | ≤ k k k k ≤ B Z (cid:16) (cid:17) 4 holds for some constants c,λ > 0 and all s 0. If ≥ 1 t (1.12) stu≥p1 1−e−λ0t Z0 e−λ0r(cid:16)kzskuBp≤1ssu≥pr kσ−1TszkH(cid:17)dr < ∞, then there exists a constant C > 0 such that (1.13) Pt(x, ) Pt(y, ) var C(1+ x y B)e−λλ00+λλt , x,y B,t 0. k · − · k ≤ k − k ∈ ≥ Following the line of [23, Section 3], one may also naturally investigate gradient estimates and derivative formula for P . It is not difficult to present a formal result under a condition t similar to [23, (3.1)], for instance: Pexrisotpsoasitnioonn-n1e.g4a.tivAessfuumncetitohnatg{hon∈BHs:uscuhpst∈h[a0,t1]νkσ(−1gTs>hk0H <) =∞} is, dρengseisinboBun.dIefdthaenrde 0 0 { } ∞ Lipschitz continuous in H, and k·k µ( ϕ 1 ) q(t) := sup 1+ | h − | ∞e tν0(1 exp[ rg])dr − − − (1.14) khkH∈(0,1](cid:26)(cid:16) khkH (cid:17)Z0 µ g g( h) + | − ·− | ∞re tν0(1 exp[ rg])dr < , t > 0, − − − (cid:0) khkH (cid:1) Z0 (cid:27) ∞ then there exists a constant C > 0 such that 1 1 P f(x) : = limsup P f(x+εy) P f(x) y t t t |∇ | ε| − | ε 0 ↓ t C1 f q(t) σ−1Tsy Hds, f Bb(B),t > 0,x,y B. ≤ k k∞ k k ∈ ∈ Z0 Suppose moreover that Ts B ce−λs for some constants c,λ > 0 and all s 0. Then k k ≤ ≥ Pt(x, ) Pt(y, ) var C2(1+ x y B)e−λt, x,y B,t 0 k · − · k ≤ k − k ∈ ≥ holds for some constant C > 0. 2 Unfortunately, in the moment we do not have any non-trivial example in infinite dimensions to illustrate condition (1.14). Indeed, it seems that in infinite dimensions the uniform norm of the gradient of P t Pt := sup yPtf(x) : y B 1,x B, f 1 k∇ k∞ {|∇ | k k ≤ ∈ k k∞ ≤ } is most likely infinite for any t > 0. The intuition is that comparing with a cylindrical noise given in [13, Assumption 2.2], which is strong enough along single directions so that the noise might not take values in B, our non-cylindrical Lévy process seems too weak to imply a bounded gradient estimate of P . Nevertheless, we are able to estimate the uniform t 5 gradient of a modified version of P (cf. Proposition 4.1 below), which implies the desired t exponential convergence in (1.13). We remark that the derivative formula and gradient estimate are investigated in [20, 27] for SDEs on Rd driven by Lévy noises, where in [20] the process may contain a diffusion part but extensions of the main results to infinite dimensions are not yet available, while in [27] the main result was also extended to a class of semi-linear SPDEs driven by cylindrical α-stable processes. Both papers are quite different from the present one, where we aim to describe regularity properties of the semigroup merely using the Lévy measure of the noise. We will prove Theorems 1.1 (also Proposition 1.4), 1.2 and 1.3 in the following three sec- tions respectively. In Section 5 we present two specific examples, with µ the Wiener measure on a Brownian path space and the Gaussian measure on an Hilbert space respectively, to illustrated these results. 2 Proofs of Theorem 1.1 and Proposition 1.4 The key technique of the study is the coupling by change of measure. For readers’ conve- nience, let us briefly recall the main idea of the argument. To investigate e.g. the continuity of P f along y B, for any x B we construct a family of processes Xε and the t ε [0,1) associated proba∈bility densities∈R such that { ·} ∈ ε ε [0,1) { } ∈ (1) Xε = x+εy, Xε = X0, ε [0,1), t > 0; 0 t t ∈ (2) Under the probability R P, the process Xε is associated to the transition semigroup ε (P ) ; · s s 0 ≥ (3) lim R = R = 1 holds in L1(P). ε 0 ε 0 → Then, for any bounded measurable function f and t > 0, limP f(x+εy) = limE R f(Xε) = limE R f(X0) = E R f(X0) = P f(x). t ε t ε t 0 t t ε 0 ε 0 ε 0 → → → (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) To realize this idea in the present setting, the following Lemma 2.1 will play a crucial role. For fixed t > 0, let Λ be the distribution of L := (L ) which is a probability measure s s [0,t] ∈ on the paths pace W = w : [0,t] B is right-continuous having left limits t → (cid:8) (cid:9) equipped with the Skorokhod metric. For any w W , let t ∈ w(dz,ds) := δ , (∆ws,s) s∈[0,Xt],∆ws6=0 6 which records jumps of the path w, where ∆w = w w . Let s s s − − w(g) = g(z,s)w(dz,ds) = g(∆w ,s), g L1(w). s ∈ ZB×[0,t] s∈[0,Xt],∆ws6=0 A function g on B will be also regarded as a function on B [0,t] by letting g(z,s) = g(z) for (z,s) B [0,t]. × ∈ × Moreover, write L = L1+L0, where L1 and L0 are two independent Lévy processes with Lévy measure ν ν and ν respectively, and L0 does not have a Gaussian term. Let Λ1 and 0 0 − Λ0 be the distributions of L1 and L0 respectively. We have Λ = Λ1 Λ0. ∗ Repeating the proof of [23, Lemma 2.1] where B = Rd, we have the following result. Lemma 2.1. For any h L1(W B [0,t];Λ0 ν ds), t 0 ∈ × × × × h(w,z,s)Λ0(dw)ν (dz)ds 0 (2.1) ZWt×B×[0,t] = Λ0(dw) h(w z1 ,z,s)w(dz,ds). [s,t] − ZWt ZB [0,t] × To prove Theorem 1.1, we also need the following two more lemmas. Lemma 2.2. Let y B such that σ 1T y H for any s > 0, and let g be a non-negative − s measurable function ∈on B such that ν (g) :=∈ gdν < and w(g) > 0 for Λ0-a.e. w. Let 0 B 0 ∞ Φ (w,z,s) = ϕεσ−1Tsy(z)(Rρ0g)(z −εσ−1Tsy), ε 0. ε w(g)+g(z εσ 1T y) ≥ − s − If (1.5) holds for any h H, then Φ is uniformly integrable w.r.t. Λ0 µ ds on ε ε [0,1) W B [0,t]. ∈ { } ∈ × × t × × Proof. Since ϕ 1, applying (2.1) to h(w,z,s) = g(z) we obtain 0 ≡ w(g) Φ (w,z,s)Λ0(dw)µ(dz)ds 0 ZWt B [0,t] × × g(z) = Λ0(dw)ν (dz)ds 0 (2.2) ZWt B [0,t] w(g)+g(z) × × g(z) = Λ0(dw) w(dz,ds) w(g) ZWt ZB [0,t] × = 1. Next, by (1.1) and the integral transform z z εσ 1T y, for any F B (W B [0,t]) − s b t 7→ − ∈ × × 7 we have F(w,z +εσ 1T y,s)Φ (w,z,s)Λ0(dw)µ(dz)ds − s 0 ZWt B [0,t] × × F(w,z+εσ 1T y,s)(ρ g)(z) (2.3) = − s 0 Λ0(dw)µ(dz)ds w(g)+g(z) ZWt B [0,t] × × = F(w,z,s)Φ (w,z,s)Λ0(dw)µ(dz)ds. ε ZWt B [0,t] × × Letting F = 1 and combining this with (2.2), we conclude that Φ are probability ε ε [0,1) densities w.r.t. Λ0 µ ds. Moreover, applying (2.3) to F(w,z,s){=}1 ∈ for R > 0 and × × {Φε>R} letting (ρ g)(z) η(w,z,s) = sup w(g)0+g(z)ϕεσ−1Tsy(z +εσ−1Tsy) ε (0,1) ∈ which is finite Λ0 µ ds-a.e., we obtain × × sup (Φ 1 )(w,z,s)Λ0(dw)µ(dz)ds ε Φε>R ε∈(0,1)ZWt×B×[0,t] { } (Φ 1 )(w,z,s)Λ0(dw)µ(dz)ds 0 η>R ≤ ZWt B [0,t] { } × × which goes to zero as R by the dominated convergence theorem. → ∞ Lemma 2.3. Let E be a topology space and C (E) be the class of all bounded continuous b functions on B. Let µ be a finite measure on the Borel σ-field B such that C (E) is dense 0 b in L1(µ ). Let f be a sequence of uniformly integrable functions w.r.t. µ such that 0 n n 1 0 { } ≥ lim (Ff )dµ = (Ff )dµ n 0 0 0 n →∞ZE ZE holds for some f L1(µ ) and all F C (E). Then it holds also for any F B (E). 0 0 b b ∈ ∈ ∈ Proof. Let ε(R) = sup µ ( f f 1 ) which goes to zero as R . For any F B (E), let F n≥1 0C|(En −) su0c|h{t|fhna−tf0|>FR} F and µ ( F F→)∞ 1. Then ∈ b { m}m≥1 ⊂ b k mk∞ ≤ k k∞ 0 | m − | ≤ m F(f f )dµ F(f f )1 dµ + F ε(R) n − 0 0 ≤ n − 0 {|fn−f0|≤R} 0 k k∞ (cid:12)ZE (cid:12) (cid:12)ZE (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) R (cid:12) (cid:12) (cid:12) (cid:12) F (f f )1 dµ + F ε(R)+ (cid:12) (cid:12) ≤ (cid:12) m n − 0 {|fn−f0|≤R} (cid:12)0 k k∞ m (cid:12)ZE (cid:12) (cid:12) (cid:12) R (cid:12) (cid:12) F (f f )dµ +2 F ε(R)+ . (cid:12) m n 0 0 (cid:12) ≤ − k k∞ m (cid:12)ZE (cid:12) (cid:12) (cid:12) By first letting n then m (cid:12) and finally R (cid:12) , we complete the proof. (cid:12) (cid:12) → ∞ → ∞ → ∞ 8 Proof of Theorem 1.1. (1) Let f B (B) and x,y B be fixed. For any ε > 0, let b ∈ ∈ t F (w) = f T (x+εy)+ T σdw , ε t t s s − (cid:18) Z0 (cid:19) t where T σdw is the Itô stochastic integral which is Λ-a.e. well-defined. Let e.g. g = 0 t s s 1 . We ha−ve ν (g) < and, since ν (B) = and g > 0, w(g) > 0 for Λ0-a.e. w. Then, ρ0 1 R 0 ∞ 0 ∞ by∨(1.3) and Lemma 2.1 for F (w1 +w0 +(z +εσ 1T y)1 )g(z) h(w0,z,s) = 0 − s [s,t] , w0(g)+g(z) we obtain P f(x+εy) t = EF (L1 +L0) ε F (w1 +w0)g(z) = Λ1(dw1)Λ0(dw0) ε w0(dz,ds) w0(g) ZWt2 ZB×[0,t] F (w1 +w0 +εσ 1T y1 )g(z) = Λ1(dw1)Λ0(dw0) 0 − s [s,t] w0(dz,ds) w0(g) ZWt2 ZB×[0,t] F (w1 +w0 +(z +εσ 1T y)1 )g(z) = Λ1(dw1)Λ0(dw0) 0 − s [s,t] ν (dz)ds w0(g)+g(z) 0 ZWt2 ZB×[0,t] F (w1 +w0 +(z +εσ 1T y)1 )(ρ g)(z) = Λ1(dw1)Λ0(dw0) 0 − s [s,t] 0 µ(dz)ds. w0(g)+g(z) ZWt2 ZB×[0,t] Since εσ 1T y H so that (1.1) implies − s ∈ µ(dz −εσ−1Tsy) = ϕεσ−1Tsy(z)µ(dz), by using the integral transform z z εσ 1T y and noting that Λ = Λ1 Λ0, we obtain − s 7→ − ∗ P f(x+εy) t F (w+z1 )(ρ g)(z εσ 1T y) 0 [s,t] 0 − s (2.4) = Λ(dw) w0(g)+g(z εσ−1T y) ϕεσ−1Tsy(z)µ(dz)ds ZWt ZB×[0,t] − − s = Λ1(dw1) F (w1 +w0 +z1 )Φ (w0,z,s)Λ0(dw0)µ(dz)ds. 0 [s,t] ε ZWt ZWt B [0,t] × × Therefore, it suffices to show that lim (FΦ )(w,z,s)Λ0(dw)µ(dz)ds = (FΦ )(w,z,s)Λ0(dw)µ(dz)ds ε 0 ε→0ZWt B [0,t] ZWt B [0,t] × × × × holds for any F B (W B [0,t]). According to (2.3), this holds provided F C (W b t b t B [0,t]). Since∈the Borel×σ-fi×eld on the Polish space W B [0,t] is induced b∈y bounde×d t × × × 9 continuous functions, C (W B [0,t]) is dense in L1(Λ0 µ ds). Thus, the desired b t × × × × assertion follows from Lemmas 2.2 and 2.3. (2) For any sequence y B converging to 0 as n , define n { } ⊂ → ∞ Ψ (w,z,s) = ϕσ−1Tsyn(z)(ρ0g)(z −σ−1Tsyn), n 1. n w(g)+g(z σ 1T y ) ≥ − s n − Using σ 1T y to replace εh in the proof of Lemma 2.2, we see that (1.6) implies that − s n Ψ is uniformly integrable w.r.t. Λ0 µ ds on W B [0,t]. Therefore, using n n 1 t σ{ 1T}y≥ to replace εh in the proof of (1), w×e ob×tain lim ×P f(×x +y ) = P f(x) for any − s n n t n t f B (B),t > 0 and x B. →∞ b ∈ ∈ tPoroporfoovfePforropyositiHons1u.c4h. tShinatce {σh−1∈TsHy :Hsups1∈[0f,o1]rksσ−1T[s0h,k1H]. <Sin∞ce}tihsedebnosuenidnedBn,eistssuofffiρc0egs ∈ k k ≤ ∈ implies ν (g) < and ν ( g > 0 ) = implies w(g) > 0,Λ0-a.e., (2.4) holds true. By (2.4) 0 0 ∞ { } ∞ and (1.14) we have P f(x+εy) P f(x) t t | − | ε (2.5) f k k∞ Φε(w,z,s) Φ0(w,z,s) Λ0(dw)µ(dz)ds, ε > 0. ≤ ε | − | ZWt B [0,t] × × Since ρ0g is bounded and Lipschitz continuous in H, there exists a constant c1 > 0 such k·k that Φ (w,z,s) Φ (w,z,s) ε 0 | − | |ϕεσ−1Tsy(z)−1| + (ρ0g)(z −εσ−1Tsy) (ρ0g)(z) (2.6) ≤ w(g) w(g)+g(z εσ 1T y) − w(g)+g(z) (cid:12) − − s (cid:12) |ϕεσ−1Tsy(z)−1|+c(cid:12)(cid:12)1kεσ−1TsykH + c1|g(z −εσ−1Tsy)−g((cid:12)(cid:12)z)|. (cid:12) (cid:12) ≤ w(g) w(g)2 Moreover, according to [23, Lemma 2.2] with B in place of Rd, for any θ > 0, we have Λ0(dw) 1 = ∞rθ 1e tν0(1 e−rg)dr. − − − w(g)θ Γ(θ) ZWt Z0 Combining this with (2.5) and (2.6) and letting ε 0, we obtain the desired gradient → estimate. According to the proof of Theorem 1.3 in Section 4 with P1 replaced by P , this t t along with the assumption on T implies the second assertion. t 3 Proof of Theorem 1.2 By the triangle inequality for , it suffices to prove both assertions for small enough var k · k y B. k k 10