aa r X i v : . [ m a t h . P R ] F e b ASYMPTOTICS OF STOCHASTIC BURGERS EQUATION WITHJUMPS
SHULAN HU , RAN WANG ∗ Abstract.
For one-dimensional stochastic Burgers equation driven by Brownian motionand Poisson process, we study the ψ -uniformly exponential ergodicity with ψ ( x ) = 1 + k x k , the moderate deviation principle and the large deviation principle for the occupationmeasures. Introduction
As is well-known, Burgers equation was first studied to understand the turbulent fluidflow, see Burgers (1974). Since then the Burgers equation perturbed by different randomnoises have been considered, see monographs Da Prato and Zabczyk (2014), Peszat and Zabczyk(2007) and recent articles Dong and Xu (2007), Wu and Xie (2012), Dong et al. (2014) andreferences therein.The ergodicity of the stochastic Burgers equation driven by Brownian motion and Poissonprocess was proved in Dong (2008) in the sense that the system converges to a uniqueinvariant measure under the weak topology, but the convergence speed is not addressed.In this paper, we prove that the system converges to the invariant measure exponentiallyfaster under a topology stronger than total variation by constructing a Lyapunov functionin the same way as in Dong et al. (2019). The moderate deviation principle (MDP) for theoccupation measure is also obtained.The large deviation principle (LDP) for the occupation measure is one of the strongestergodicity results for the long time behavior of Markov processes. It has been one of theclassical research topics in probability since the pioneering work of Donsker and Varadhan(1975-1984). It gives an estimate on the probability that the occupation measures are devi-ated from the invariant measure, refer to Deuschel and Stroock (1989) for an introductionto large deviation theory of Markov processes. Wu (2001) gave the hyper-exponential recur-rence criterion of the LDP of occupation measures for strong Feller and irreducible Markovprocesses. Based on this criterion, the large deviations of the occupation measures for thestochastic Burgers equation and stochastic Navier-Stokes equation driven by Brownian mo-tion are proved in Gourcy (2007a) and Gourcy (2007b). There are some other papers aboutthe applications of Wu’s criterion, see Jak˘si`c et al. (2015), Nersesyan (2018) for some dissi-pative SPDEs. Wang et al. (2019) proposed a framework for verifying the hyper-exponentialrecurrence condition, which contains a family of strong dissipative SPDEs. In that frame-work, the strong dissipation produces to a stronger-norm moment estimate for the systemafter a fixed time uniformly over the initial values, which implies the hyper-exponential
Mathematics Subject Classification.
Key words and phrases.
Stochastic Burgers equation; Exponential ergodicity; Large deviation principle;Poisson processes. ecurrence condition. See Wang and Xu (2018) for an application to stochastic reaction-diffusion equation driven by the subordinate Brownian motion.However, the framework in Wang et al. (2019) is no longer available for the stochasticBurgers equation, which does not have the strong dissipation. In this paper, we check thehyper-exponential recurrence condition by using an exponential martingale argument. Dueto the present of the jumps, the proof here is more complicated than that for the Brownianmotion case in Gourcy (2007a).The paper is organized as follows. The framework is given in Section 2. Section 3is devoted to proving the ψ -uniformly exponential ergodicity and the moderate deviationprinciple. In Section 4, we prove the large deviation principle.2. The framework
Let H := L (0 ,
1) with the Dirichlet boundary condition and with vanishing mean values.Then H is a real separable Hilbert space with inner product h x, y i := Z x ( ξ ) y ( ξ ) dξ, ∀ x, y ∈ H . Denote k x k H := ( h x, x i H ) . Let ∆ x = x ′′ be the second order differential operator on H .Then − ∆ is a positive self-adjoint operator on H . Let α k = π k and e k ( ξ ) := √ kπξ ),for any k ∈ N ∗ = { , , · · · } . Then { e k } k ∈ N ∗ forms an orthogonal basis of H and − ∆ e k = α k e k for any k ∈ N ∗ .Let V be the domain of the fractional operator ( − ∆) , i.e., V := ( X k ∈ N ∗ α k a k e k ; ( a k ) k ∈ N ⊂ R with X k ∈ N ∗ a k < + ∞ ) , with the inner product h x, y i V := X k ∈ N ∗ α k h x, e k i H · h y, e k i H , and with the norm k x k V := h x, x i V . Clearly, V is densely and compactly embedded in H .Let (Ω , F , {F t } t ≥ , P ) be a completed filtered probability space, and N ( dt, du ) the Poissonmeasure with finite intensity measure n ( du ) on a given measurable space ( U , B ( U )). Then e N ( dt, du ) := N ( dt, du ) − n ( du ) dt is the compensated martingale measure. Let W be the cylindrical Wiener process, whichis independent with N ( dt, du ), e.g., W := P k ∈ N ∗ W k e k , where { W k } k ∈ N ∗ are a sequence ofindependent standard one-dimensional Brownian motions independent with N ( dt, du ).Consider the following stochastic Burgers equation in the Hilbert space H : (cid:26) dX t = ∆ X t dt + B ( X t ) dt + QdW t + R U f ( X t − , u ) e N ( dt, du ) ,X (0) = x ∈ H (2.1)Here B ( x ) := B ( x, x ) is a bilinear operator, which is defined by B ( x, y ) := xy ′ for x ∈ H , y ∈ V , and Q ∈ L ( H ) (the space of all Hilbert-Schmidt operators from H to H ) is givenby Qx = X k ∈ N ∗ β k h x, e k i e k , x ∈ H , ith k Q k HS := p tr( Q ∗ Q ) = pP k ∈ N ∗ | β k | < ∞ .Assume that the coefficient f satisfies the following conditions:(H.1) f ( · , · ) : H × U → H is measurable;(H.2) R U k f (0 , u ) k H n ( du ) < ∞ ;(H.3) R U k f ( x, u ) − f ( y, u ) k H n ( du ) ≤ K k x − y k H , ∀ x, y ∈ H ;(H.4) f ( · , u ) ∈ C b ( H ) , ∀ u ∈ U .Let D ([0 , + ∞ ); H ) be the space of all c`adl`ag functions from [0 , + ∞ ) to H equipped withthe Skorokhod topology. Denote by S ( t ) = e ∆ t . Definition 2.1.
The process X = { X t } t ≥ is called a mild solution of (2.1), if for any x ∈ H , X ∈ D ([0 , + ∞ ); H ) satisfying that for any t > Z t (cid:2) k X ( s ) k H + k B ( X ( s )) k H (cid:3) ds < ∞ , and X t = S ( t ) x + Z t S ( t − s ) B ( X s ) ds + Z t S ( t − s ) QdW s + Z t Z U S ( t − s ) f ( X s − , u ) e N ( ds, du ) , P − a.s. For all ϕ ∈ B b ( H ) (the space of all bounded measurable functions on H ), define P t ϕ ( x ) := E x [ ϕ ( X t )] for all t ≥ , x ∈ H , where E x denotes the expectation with respect to (w.r.t. for short) the law of stochasticprocess X with initial value X = x . For any t > P t is said to be strong Feller if P t ϕ ∈ C b ( H ) for any ϕ ∈ B b ( H ), where C b ( H ) is the space of all bounded continuousfunctions on H . P t is irreducible in H if P t O ( x ) > x ∈ H and any non-empty opensubset O of H .Recall the following properties about the solution to Eq.(2.1). Theorem 2.2 (Dong and Xu (2007), Dong (2008)) . Under (H.1)-(H.4), the following state-ments hold:(i) For every x ∈ H and ω ∈ Ω a.s., Eq. (2.1) admits a unique mild solution X = { X t } t ≥ ∈ D ([0 , ∞ ); H ) ∩ L ((0 , ∞ ); V ) , which is a Markov process.(ii) X = { X t } t ≥ is strong Feller and irreducible in H , and it admits a unique invariantprobability measure µ . Define the occupation measure L t by(2.2) L t (Γ) := 1 t Z t δ X s (Γ) ds, where Γ is a Borel measurable set in H , δ · is the Dirac measure. Then L t is in M ( H ), thespace of probability measures on H . On M ( H ), let σ ( M ( H ) , B b ( H )) be the τ -topology ofconvergence against measurable and bounded functions, which is much stronger than theusual weak convergence topology σ ( M ( H ) , C b ( H )). . ψ -uniformly exponential ergodicity and moderate deviation principle Let M b ( H ) be the space of signed σ -additive measures of bounded variation on H equippedwith the Borel σ -field B ( H ). On M b ( H ), we consider the τ -topology σ ( M b ( H ) , B b ( H )).Given a measurable function ψ : H → R + , define B ψ := { g : H → R ; | g ( x ) | ≤ ψ ( x ) } . For a function b ( t ) : R + → (0 , + ∞ ), define(3.1) M t := 1 b ( t ) √ t Z t ( δ X s − µ ) ds. where b ( t ) satisfies(3.2) lim t →∞ b ( t ) = + ∞ , lim t →∞ b ( t ) √ t = 0 . Let P ν be the probability measure of the system X with initial measure ν .(H.5) Assume that there exists a constant M > M := sup x ∈ H Z U k f ( x, u ) k H n ( du ) < + ∞ . Theorem 3.1.
Assume (H.1)-(H.5) hold. Then the following statements hold for ψ ( x ) =1 + k x k H . (1) The invariant measure µ satisfies that µ ( ψ ) < ∞ and the Markov semigroup { P t } t ≥ is ψ -uniformly exponentially ergodic, i.e., there exist some constants C, γ > satis-fying that for sup g ∈B ψ | P t g ( x ) − µ ( g ) | ≤ Ce − γt ψ ( x ) , x ∈ H , t ≥ . (2) For any initial measure ν verifying ν ( ψ ) < + ∞ , the measure P ν ( M t ∈ · ) satisfies thelarge deviation principle w.r.t. the τ -topology with speed b ( t ) and the rate function (3.4) I ( ν ) := sup (cid:26)Z ϕdν − σ ( ϕ ); ϕ ∈ B b ( H ) (cid:27) , ∀ ν ∈ M b ( H ) , where (3.5) σ ( ϕ ) := lim t →∞ t E µ (cid:18)Z t ( ϕ ( X s ) − µ ( ϕ )) ds (cid:19) exists in R for every ϕ ∈ B ψ . More precisely, the following three properties hold: (a1) for any r ≥ , { β ∈ M b ( H ); I ( β ) ≤ r } is compact in ( M b ( H ) , τ ) ; (a2) ( the upper bound ) for any closed set E in ( M b ( H ) , τ ) , lim sup t →∞ b ( t ) log P β ( M t ∈ E ) ≤ − inf β ∈E I ( β );(a3) ( the lower bound ) for any open set D in ( M b ( H ) , τ ) , lim inf t →∞ b ( t ) log P β ( M t ∈ D ) ≥ − inf β ∈D I ( β ) . ecall that a measurable function h : H → R belongs to the extended domain D e ( L ) ofthe generator L of { P t } t ≥ , if there is a measurable function g : H → R satisfying that forall t > R t | g | ( X s ) ds < + ∞ , P x -a.s., and(3.6) h ( X t ) − h ( X ) − Z t g ( X s ) ds, is a c`adl`ag P x -local martingale for all x ∈ H . In that case, we write g := L h. Proof of Theorem 3.1.
According to (Down et al. , 1995, Theorem 5.2c) and (Wu, 2001, The-orem 2.4), it is sufficient to prove that there exist some continuous function 1 ≤ ψ ∈ D e ( L ),compact subset K ⊂ H and constants ε, C > − L ψψ ≥ ε K c − C K . (3.7)Here, we construct the Lyapunov function ψ in the same way as in Dong et al. (2019). Since1 + k x k H is comparable with (1 + k x k H ) , we will take(3.8) ψ ( x ) = (1 + k x k H ) instead of 1 + k x k H . First observe that(3.9) ∇ ψ ( x ) = x (1 + k x k H ) , and(3.10) Hess ψ ( x ) = − x × x (1 + k x k H ) + I H (1 + k x k H ) , here I H stands for the identity operator. Then, we have(3.11) sup x ∈ H k Hess ψ ( x ) k ≤ , sup x ∈ H k∇ ψ ( x ) k ≤ , here k Hess ψ ( x ) k and k∇ ψ ( x ) k denote their operator norms. Moreover, we have(3.12) h ∆ x, ∇ ψ ( x ) i = h ∆ x, x i (1 + k x k H ) = − k x k V (1 + k x k H ) , ∀ x ∈ V , and(3.13) h B ( x ) , ∇ ψ ( x ) i = h B ( x ) , x i (1 + k x k H ) = 0 , ∀ x ∈ V . By Taylor’s expansion, for any x ∈ H , u ∈ U , there exists constant θ ∈ (0 ,
1) satisfyingthat ψ (cid:0) x + f ( x, u ) (cid:1) − ψ ( x ) − h∇ ψ ( x ) , f ( x, u ) i = 12 (cid:10) Hess ψ (cid:0) x + θf ( x, u ) (cid:1) f ( x, u ) , f ( x, u ) (cid:11) . (3.14)By Itˆo’s formula, we have dψ ( X t ) = h ∆ X t , ∇ ψ ( X t ) i dt + h B ( X t ) , ∇ ψ ( X t ) i dt + h∇ ψ ( X t ) , QdW t i + 12 tr( Q ∗ Hess ψ ( X t ) Q ) dt Z U (cid:0) ψ ( X t − + f ( X t − , u )) − ψ ( X t − ) (cid:1) e N ( dt, du )+ Z U (cid:0) ψ ( X t − + f ( X t − , u )) − ψ ( X t − ) − h∇ ψ ( X t − ) , f ( X t − , u ) i (cid:1) n ( du ) dt. (3.15)Then, by (H.5), (3.9)-(3.14), we know that L ψ ( x ) = h ∆ x, ∇ ψ ( x ) i + h B ( x ) , ∇ ψ ( x ) i + 12 tr( Q ∗ Hess ψ ( x ) Q )+ Z U ( ψ ( x + f ( x, u )) − ψ ( x ) − h∇ ψ ( x ) , f ( x, u ) i ) n ( du ) ≤ − k x k V (1 + k x k H ) + 12 k Q k + 12 Z U k f ( x, u ) k H n ( du ) ≤ − k x k V (1 + k x k H ) + 1(1 + k x k H ) + 12 k Q k + M ≤ − (1 + k x k V ) + c , (3.16)where in the last inequality the Poincar´e inequality k x k V ≥ π k x k H is used, c := 1 + ( k Q k + M ).Let K := { x ∈ H ; k x k V ≤ c } . Then K is a compact set in H . For any x ∈ K , we have(1 + k x k V ) − c (1 + k x k H ) ≥ − c ;(3.17)for any x / ∈ K , we have (1 + k x k V ) − c (1 + k x k H ) ≥ (1 + k x k V ) − k x k V (1 + k x k H ) ≥ . (3.18)Putting (3.16)-(3.18) together, we obtain that − L ψ ( x ) ψ ( x ) ≥ (1 + k x k V ) − c (1 + k x k H ) ≥
12 1 K c − c K , (3.19)which implies (3.7). The proof is complete. (cid:3) Large deviation principle (H.6) Assume that there exists a constant a > x ∈ H Z U k f ( x, u ) k H exp ( a k f ( x, u ) k H ) n ( du ) < + ∞ . For any λ > , L >
0, let(4.2) M λ ,L := (cid:26) ν ∈ M ( H ); Z e λ k x k H ν ( dx ) ≤ L (cid:27) . Theorem 4.1.
Assume (H.1)-(H.4) and (H.6) hold. Then the family P ν ( L t ∈ · ) as t → + ∞ satisfies the LDP with respect to the τ -topology, with the speed t and the rate function J ,uniformly for any initial measure ν in M λ ,L . More precisely, the following three propertieshold: a1) for any a ≥ , { β ∈ M ( H ); J ( β ) ≤ a } is compact in ( M ( H ) , τ ) ; (a2) (the upper bound) for any closed set E in ( M ( E ) , τ ) , lim sup t →∞ t log sup ν ∈M λ ,L P ν ( L t ∈ E ) ≤ − inf β ∈E J ( β );(a3) (the lower bound) for any open set D in ( M ( E ) , τ ) , lim inf t →∞ t log inf ν ∈M λ ,L P ν ( L t ∈ D ) ≥ − inf β ∈D J ( β ) . Remark . Assumptions (H.1)-(H.4) are standard conditions for the existence and unique-ness of the solution for Eq.(2.1), see Dong and Xu (2007). Condition (H.6) ((H.5) resp.)guarantees for the exponential (square resp.) integrability of the solution. The similar condi-tions are often used in the study of the large deviation theory for small Poisson noise pertur-bations of SPDEs, e.g., see (R¨ockner and Zhang, 2007, Section 4), (Budhiraja et al. , 2013,Condition 3.1) and (Yang, Zhai and Zhang, 2015, Condition 3.1). Inspiring by (Budhiraja et al. ,2013, Section 4.1), we give the following example of the Poisson random measure N and f satisfying (H.1)-(H.4) and (H.6):Let { N ( t ) } t ≥ be a Poisson process with the rate 1, { A j } j ∈ N independent and identi-cally distributed random variables, with a common distribution function F , which are alsoindependent of { N ( t ) } t ≥ . Then N ([0 , t ] ⊗ B ) = N ( t ) X j =1 B ( A j ) , t ≥ , B ∈ B ( R + ) , is a Poisson random measure on the space R + × R + . The intensity measure of { N ( t ) } t ≥ isgiven by ν ( A ⊗ B ) = ρ ( A ) · F ( B ) , A, B ∈ B ( R + ) . Here ρ ( · ) denotes the Lebesgue measure.Assume that there exists a > Z ∞ u e a u F ( du ) < ∞ . For any function G ∈ C b ( H ), the function f ( x, u ) := G ( x ) u, x ∈ H , u ∈ R + satisfies all the conditions required in Theorem 4.1. Remark . The rate function J can be expressed by the entropy of Donsker-Varadhan,see Donsker and Varadhan (1975-1984), (Deuschel and Stroock, 1989, Chapter V) or (Wu,2001, Section 2.2). Under the Feller assumption: P t ( C b ( H )) ⊂ C b ( H ) , ∀ t ≥ , we know that (for instance see Lemma B.7 in Wu (2000)) J ( ν ) = sup (cid:26) − Z L ϕϕ dν ; 1 ≤ ϕ ∈ D e ( L ) (cid:27) , ν ∈ M ( H ) . (4.3) emark . For every ϕ : H → R measurable and bounded, as ν R H ϕdν is continuousw.r.t. the τ -topology, then by the contraction principle (Deuschel and Stroock (1989)), P ν (cid:18) t Z t ϕ ( X s ) ds ∈ · (cid:19) satisfies the LDP on R uniformly over ν in M λ ,L , with the rate function given by J ϕ ( r ) := inf (cid:26) J ( β ) < + ∞ ; β ∈ M ( H ) and Z ϕdβ = r (cid:27) , ∀ r ∈ R . The proof of Theorem 4.1.
By Theorem 2.2, we know that P t is strong Feller and irreduciblein H for any t >
0. According to (Wu, 2001, Theorem 2.1), to prove Theorem 4.1, it issufficient to prove that for any λ >
0, there exists a compact set K in H (4.4) sup ν ∈M λ ,L E ν (cid:2) e λτ K (cid:3) < ∞ , and sup x ∈K E x h e λτ (1) K i < ∞ , where(4.5) τ K := inf { t ≥ X t ∈ K} , τ (1) K := inf { t ≥ X t ∈ K} . The basic ingredient for the proof of (4.4) is to show the exponential decay of the tailsof the stopping times τ K and τ (1) K for a suitable choice of compact set K ⊂ H . It can beproved by using arguments in (Gourcy, 2007a, Lemma 6.1) or (Wang et al. , 2019, Lemma3.8), combining with the critical exponential estimate in Proposition 4.6 below.The proof is complete. (cid:3) The following result is similar to Lemma 4.1 in R¨ockner and Zhang (2007).
Lemma 4.5.
For any g ∈ C b ( H ) , M gt := exp (cid:18) g ( X t ) − g ( X ) − Z t h ( X s ) ds (cid:19) is an F t -local martingale, where h ( x ) = h ∆ x, ∇ g ( x ) i + h B ( x ) , ∇ g ( x ) i + 12 k Q ∗ ∇ g ( x ) k H + 12 tr ( Q ∗ Hess g ( x ) Q )(4.6) + Z U (cid:16) exp (cid:2) g ( x + f ( x, u )) − g ( x ) (cid:3) − − (cid:10) ∇ g ( x ) , f ( x, u ) (cid:11)(cid:17) n ( du ) . Proof.
We follow the argument in (R¨ockner and Zhang, 2007, Lemma 4.1). Applying Itˆo’sformula first to exp( g ( X t )) and then to exp (cid:16) g ( X t ) − g ( X ) − R t h ( X s ) ds (cid:17) proves the lemma. (cid:3) Proposition 4.6.
Assume that (H.1)-(H.4) and (H.6) hold. For any λ ∈ (0 , a ] , θ ∈ (0 , ,there exist constants c ( θ ) , c ( θ ) , c ( λ ) such that for any T > , E x (cid:20) exp (cid:18) θλ Z T k X t k V dt (cid:19)(cid:21) ≤ c ( θ ) + c ( θ ) e c ( λ ) T e λ k x k H . (4.7) emark . For the stochastic Burgers equation driven by Brownian motion (i.e., f ≡ λ ∈ (0 , π k Q k / k Q k is the norm of Q as an operator in H , E x (cid:20) exp (cid:18) λ Z T k X t k V dt (cid:19)(cid:21) ≤ e λ t r ( Q ) T e λ k x k H . (4.8)This is difficult to prove in the jump case. Here, we replace R T k X t k V dt by R T k X t k V dt andadd an extra parameter θ ∈ (0 ,
1) in (4.7), which is also enough to show the exponentialdecay of the tails of the stopping times τ K and τ (1) K for a suitable choice of K . The proof of Proposition 4.6.
For any λ ∈ (0 , a ], let Z λ := Z T λ k X t k V (1 + λ k X t k H ) dt. Since k x k V ≥ π k x k H , we have λ k X t k V (1 + λ k X t k H ) ≥ (cid:0) λ k X t k V (cid:1) − . (4.9)Thus, to prove (4.7), it is enough to prove that for any λ ∈ (0 , a ], there exist constants c ( θ ) , c ( θ ) , c ( λ ) such that E x [exp ( θZ λ )] ≤ c ( θ ) + c ( θ ) e c ( λ ) T e λ k x k H . (4.10)Let ψ λ ( x ) := (1 + λ k x k H ) , a generalization of ψ given in (3.8). Then ψ λ has the similar estimates (3.9)-(3.13) with ψ up to some constants.Let G ( x ) := e ψ λ ( x ) . Note that Hess G ( x ) = G ( x ) Hess ψ λ ( x ) + G ( x ) ∇ ψ λ ( x ) × ψ λ ( x ) . It implies that(4.11) k Hess G ( x ) k HS ≤ λ G ( x ) , ∀ x ∈ V . By Taylor’s expansion, there exist constants θ ∈ (0 , , θ ∈ (0 , θ ) satisfying that (cid:12)(cid:12) exp (cid:2) ψ λ ( x + f ( x, u )) − ψ λ ( x ) (cid:3) − − (cid:10) ∇ ψ λ ( x ) , f ( x, u ) (cid:11)(cid:12)(cid:12) = (cid:12)(cid:12) e − ψ λ ( x ) (cid:2) G ( x + f ( x, u )) − G ( x ) − h∇ G ( x ) , f ( x, u ) i (cid:3)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) e − ψ λ ( x ) (cid:10) Hess ( G )( x + θ f ( x, u )) f ( x, u ) , f ( x, u ) (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) ≤ λ (cid:0) ψ λ ( x + θ f ( x, u )) − ψ λ ( x ) (cid:1) k f ( x, u ) k H = λ (cid:0) h∇ ψ λ ( x + θ f ( x, u )) , θ f ( x, u ) i (cid:1) k f ( x, u ) k H ≤ λ (cid:0) λ k f ( x, u ) k H (cid:1) k f ( x, u ) k H . (4.12) pplying Lemma 4.5 with the above choice of ψ λ , we know that M ψ λ ( t ) := exp (cid:18) ψ λ ( X t ) − ψ λ ( x ) − Z t h ( X s ) ds (cid:19) is an F t -local martingale, where h ( x ) := h ∆ x, ∇ ψ λ ( x ) i + h B ( x ) , ∇ ψ λ ( x ) i + 12 k Q ∗ ∇ ψ λ ( x ) k H + 12 tr( Q ∗ Hess ψ λ ( x ) Q )+ Z U (exp [ ψ λ ( x + f ( x, u )) − ψ λ ( x )] − − h∇ ψ λ ( x ) , f ( x, u ) i ) n ( du ) ≤ − λ k x k V (1 + λ k x k H ) + λ k Q k + λ Z U exp ( λ k f ( x, u ) k H ) k f ( x, u ) k H n ( du ) . (4.13)By (H.6), we know that for any fixed λ ∈ (0 , a ],(4.14) M λ := sup x ∈ H Z U k f ( x, u ) k H exp ( λ k f ( x, u ) k H ) n ( du ) < ∞ . Hence, by (4.13) and (4.14), we have that for any r ≥ P ( Z λ > r ) ≤ P ψ λ ( X T ) + Z T λ k X s k V (1 + λ k X s k H ) ds > r ! = P ψ λ ( X T ) − ψ λ ( x ) − Z T h ( X s ) ds + ψ λ ( x ) + Z T h ( X s ) ds + Z T λ k X s k V (1 + λ k X s k H ) ds > r ! ≤ P (cid:18) ψ λ ( X T ) − ψ λ ( x ) − Z T h ( X s ) ds > r − ψ λ ( x ) − T λ (cid:18) M λ k Q k (cid:19)(cid:19) ≤ E h M ψ λ T i exp (cid:18) − r + ψ λ ( x ) + T λ (cid:18) M λ k Q k (cid:19)(cid:19) ≤ exp (cid:18) − r + ψ λ ( x ) + T λ (cid:18) M λ k Q k (cid:19)(cid:19) , (4.15)where in the last inequality we have used the fact that M gt is a non-negative local martingale.For any θ ∈ (0 ,
1) and λ ≤ a , by (4.15), we have E [exp ( θZ λ )]= θ + θ Z ∞ e θr P x ( Z λ > r ) dr ≤ θ + θ Z ∞ e θr exp (cid:18) − r + ψ λ ( x ) + T λ (cid:18) M λ k Q k (cid:19)(cid:19) dr = θ + exp (cid:18) ψ λ ( x ) + T λ (cid:18) M λ k Q k (cid:19)(cid:19) θ − θ . (4.16)This implies (4.7). The proof is complete. (cid:3) cknowledgments : We sincerely thank the referee for helpful comments and remarks. S.Hu is supported by the National Social Science (17BTJ034); R. Wang is supported by theNSFC(11871382), the Chinese State Scholarship Fund Award by the CSC and the YouthTalent Training Program by Wuhan University. ReferencesA. Budhiraja, J. Chen, P. Dupuis (2013):
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E-mail address : hu [email protected] ∗ Corresponding author, School of Mathematics and Statistics, Wuhan University,Wuhan, 430072, China.
E-mail address : [email protected]@whu.edu.cn