Large deviations for stochastic heat equations with memory driven by Levy-type noise

For a heat equation with memory driven by a L\'evy-type noise we establish the existence of a unique solution. The main part of the article focuses on the Freidlin-Wentzell large deviation principle of the solutions of heat equation with memory driven by a L\'evy-type noise. For this purpose, we exploit the recently introduced weak convergence approach.


Introduction
In this work we consider a non-linear heat equation with memory driven by a Lévytype noise. Heat equations with memory have been considered for a long time and their study has recently been published in the monograph [2]. In order to correct the non-physical property of instantaneous propagation for the heat equation, Gurtin and Pipkin introduced in [15] a modified Fourier's law, which resulted in a heat equation with memory. More precisely, let upt, xq denote the temperature at time t at position x in a bounded domainŌ. By following the theory developed in [10], [15] and [18], the temperature upt, xq and the density ept, xq of the internal energy and the heat flux ϕpt, xq are related by ept, xq " e 0`b0 upt, xq for t P Ê`, x PŌ, ϕpt, xq "´c 0 ∇upt, xq`ż t 8 γprq∇upt`r, xq dr for t P Ê`, x PŌ.
Here, the constant e 0 denotes the internal energy at equilibrium and the constants b 0 ą 0 and c 0 ą 0 are the heat capacity and thermal conduction. The heat flux relaxation is described by the function γ : p´8, 0s Ñ Ê`. The energy balance for the system has the form B t ept, xq "´divϕpt, xq`r F pt, upt, xqq, where r F is the nonlinear heat supply which might describe temperature-dependent radiative phenomena. After rescaling the constants and generalizing r F , we arrive at B t upt, xq " ∆upt, xq`F`t, upt, xq˘`div B`t, upt, xq˘`ż This equation models the heat flow in a rigid, isotropic, homogeneous heat conductor with linear memory and is considered for example in [12], [13] and [15]. It is well known that many physical phenomena are better described by taking into account some kind of uncertainty, for instance some randomness or random environment. For this purpose, we assume in this work that the heat supplyF in the derivation above contains a stochastic term representing an environmental noise. In order to accommodate a general non-Gaussian environmental noise with possibly discontinuous trajectories, we model the noise by a Lévy-tpe stochastic process. Repeating the above derivation with r F containing such random noise, we arrive at a stochastically perturbed version of Equation (1.1); see Equation (2.1) in the next section. The main aim of our work is to establish the existence and uniqueness theorem and to investigate the Freidlin-Wentzell large deviation principle of the solutions of the stochastically perturbed equation.
For the case of a Gaussian environmental noise, there exists a great amount of literature. For instance, results on the well-posedness of the resulting equation were obtained in [4], [8] and [17]. Long time behaviors of the stochatically perturbed equation for a Gaussian environmental noise were studied in [4], [5], [8] and [9]. The Freidlin-Wentzell large deviation principle in this situation had been studied in [17] under certain restriction on the the nonlinearity.
In our case of a Lévy-type environmental noise, we will obtain the well-posedness of the stochastically perturbed equation by the classical cutting-off method. Due to the appearance of jumps in our setting, the Freidlin-Wentzell large deviation principle are distinctively different to the Gaussian case in [17]. We will use the weak convergence approach introduced in [6] and [7] for the case of Poisson random measures. This approach is a powerful tool to establish the Freidlin-Wentzell large deviation principle for various finite and infinite dimensional stochastic dynamical systems with irregular coefficients driven by a non-Gaussian Lévy noise, see for example [3], [6], [11], [19], [20] and [21]. The main point of our approach is to prove the tightness of some controlled stochastic dynamical systems. For this purpose, we exploit estimates of the controlled stochastic dynamical systems which significantly differ from those in the Gaussian setting.
The organization of this paper is as follows. In Section 2, we introduce the assumptions and establish the well-posedness of the stochastically perturbed equation. Section 3 is devoted to establishing the Freidlin-Wentzell's large deviation principle.

Existence of a solution
Let pΩ, F, :" tF t u tPr0,T s , P q be a filtered probability space with an adapted, standard cylindrical Brownian motion W on a Hilbert space H. Let r N be a compensated time homogeneous Poisson random measure on a Polish space X with intensity ν and assume r N to be independent of W . If S is a separable metric space we denote the space of equivalence classes of random variables V c : Ω Ñ S by L 0 pΩ, Sq and the Borel σ-algebra by BpSq.
Consider the following stochastic heat equation on L 2 :" L 2 pOq for a bounded domain O Ď Ê d for t P r0, T s: where the initial condition is given by u 0 P L 2 and the square Bochner integrable function ̺ : p´8, 0s Ñ L 2 . The past dependence is described by the function γ P L 1 pÊ´, Ê`q. The non-linear drift is described by the operator F : r0, T sˆL 2 Ñ L 2 .
The diffusion coefficients are described by the operators G 1 : r0, T sˆL 2 Ñ L 2 and G 2 : r0, T sˆL 2ˆX Ñ L 2 , where L 2 denotes the space of Hilbert-Schmidt operators from H to L 2 . We begin with introducing some notations. For p ě 1, we denote by L p pOq the usual L p -space over O with the standard norm }¨} p . For m P AE, let H m 0 pOq be the usual m-order Sobolev space over O with Dirichlet boundary conditions, and denote its norm and the dual space by }¨} 2,m and H´mpOq, respectively. For simplicity, we will write L p :" L p pOq, H m 0 :" H m 0 pOq, H´m :" H´mpOq. Sobolev's embedding theorem (see e.g. [1]) guarantees for any q ě 2 and q˚:" q{pq´1q that H d 0 ãÑ L q and L q˚ã Ñ H´d. It is well known that the Laplacian ∆ establishes an isomorphism between H 1 0 and H´1. Since H 1 0 coincides with the domain of the operator p´∆q 1{2 , we will use the following equivalent norm in H 1 0 : Notice that there exists a constant λ 1 ą 0 such that λ 1 }u} 2 2 ď }∇u} 2 2 for all u P H 1 0 . For q ě 2 we define V q :" H 1 0 X L q and Vq :" H´1`L q˚. By identifying L 2 with itself by the Riesz representation, we obtain an evolution triple That is, for any v P V q and w " w 1`w2 P H´1`L q˚w e have xv, wy Vq ,Vq " xv, w 1 y H 1 0 ,H´1`x v, w 2 y L q ,L q˚. For the simplicity of notation, when no confusion may arise, we will use the unified notation x¨,¨y to denote the above dual relations between different spaces.
Denote the Lebesgue measures on r0, T s and r0, 8q by Leb T and Leb 8 , respectively and define ν T :" Leb T b ν. We introduce the function space for all Γ P Bpr0, T sˆXq with ν T pΓq ă 8 and for some δ ą 0 .
Throughout this work we will assume the following assumptions: H1: there exist constants c 1 , c 2 ą 0 and h 1 P L 1 pr0, T s, Ê`q such that we have for all v 1 , v 2 P L 2 and t P r0, T s: H2: there exist constants c 3 , c 4 , c 5 ą 0, q ě 2 and h 2 , h 3 P L 1 pr0, T s, Ê`q such that we have for all v 1 , v 2 P L 2 and t P r0, T s: for any x P L q , y, z P H 1 0 , the mapping η Þ Ñ xx,F pt, y`ηzqy L q ,L q˚i s continuous on r0, 1s. (2.5) H3: there exist c 6 ą 0, h 4 P L 1 pr0, T s, Ê`q such that we have for all v 1 , v 2 P L 2 and t P r0, T s: (2.7) H4: there exist h 5 , h 6 P L 2 x P X and t P r0, T s: Our main theorem in this section guarantees the existence of a unique solution of Equation (2.1).
Step 1. As ν is σ-finite on the Polish space X, there exist measurable subsets K m Ò X As in the proof of [17,Th.3.2] it follows that there exists an -adapted stochastic process U m,n P L 0`Ω , Dpr0, T s, L 2 q X L 2 pr0, T s, H 1 0 q˘satisfying (2.10) for each m, n P AE.
In a first step we show that there exists T 0 ą 0 and a constant C ą 0 such that for all m, n P AE we have: For this purpose, we apply Itô's formula to obtain and Assumption (2.7) guarantees that By applying (2.13) -(2.15) to equation (2.12) we obtain By applying Burkholder's inequality and Young's inequality, we obtain that for every r ą 0 and κ 1 P p0, 1q there exists a constant c κ 1 ą 0 such that where we used Assumption (2.7) in the last equality. Similarly, we conclude from Burkholder's inequality, Young's inequality and Assumption (2.9), that for each r ą 0 and κ 2 P p0, 1q there exists a constant c κ 2 ą 0 such that with h 6 psq :" ş X h 2 6 ps, xq νpdxq for all s P r0, T s. Another application of Assumption (2.9) implies for each r P r0, T s that By choosing κ 1 " κ 2 :" 1 4 and applying (2.17) -(2.19) to inequality (2.16) we obtain for each r P r0, T s that Inequality (2.20) implies for each r P r0, T s: By choosing T 0 P r0, T s such that 2δ 2 T 0 " 1{2, we conclude from Gronwall's inequality that for a constant C ą 0. Applying this to inequality (2.20) completes the proof of (2.11).
Step 2. For m P AE and n 1 , n 2 P AE define Γ m n 1 ,n 2 ptq :" U m,n 1 ptq´U m,n 2 ptq. By similar arguments as in Step 1 and by Gronwall's lemma we obtain Hence, there exists a adapted process U m P L 0`Ω , Dpr0, T s, and, due to (2.11), satisfying Taking the limit in (2.10) as n Ñ 8 shows that U m is the unique solution of the equation Step 3. For m, n P AE with n ą m define Γ n,m ptq :" U n ptq´U m ptq. Applying Itô's formula and similar arguments as in Step 1 result in It follows from (2.6) by Burkholder's inequality and Young's inequality that for each In the same way it follows from (2.8) that for each κ 2 , κ 3 P p0, 1q there exist constants and from (2.9) that Another application of (2.8) and (2.9) imply By choosing κ 1 " κ 2 " κ 3 " 1{6 and recalling 2δ 2 T 0 " 1 2 , we obtain by applying (2.24) -(2.28) to the inequality (2.23) that α n,m pT 0 q`β n,m pT 0 q " 0.
Hence, there exists an -adapted process U P L 0`Ω , Dpr0, T 0 s, L 2 q X L 2 pr0, T 0 s, Taking limits in (2.22) shows that that U is the unique solution of 2.1 on the interval r0, T 0 s.
Step 4. By repeating the above arguments we obtain the existence of a unique solution of (2.1) on the interval rT 0 , 2T 0 s which finally leads to the completion of the proof by further iterations.

Large Deviation Principle
Recall that H is a separable Hilbert space with an orthonormal basis tu i u iPAE and assume that X is a locally compact Polish space with a σ-finite measure ν defined on BpXq.
Let S be a locally compact Polish space. The space of all Borel measures on S is denoted by M pSq and the set of all µ P M pSq with µpKq ă 8 for each compact set K Ď S is denoted by M F C pSq. We endow M F C pSq with the weakest topology such that for each f P C c pSq the mapping µ P M F C pSq Ñ ş S f psqµpdsq is continuous. This topology is metrizable such that M F C pSq is a Polish space, see [7] for more details.
We introduce the functions W : Ω Ñ C`r0, T s; H˘, W pα, βqptq " Define for each t P r0, T s the σ-algebra (c) W and N are independent.
For each f P L 2 pr0, T s, Hq, we introduce the quantity and we define for each m P AE the space Equiped with the weak topology, S m 1 is a compact subset of L 2 pr0, T s, Hq. We will throughout consider S m 1 endowed with this topology. By defining the function ℓ : r0, 8q Ñ r0, 8q, ℓpxq " x log x´x`1 we introduce for each measurable function g : r0, T sˆX Ñ r0, 8q the quantity Q 2 pgq :" ż r0,T sˆX ℓ`gps, xq˘ds νpdxq.
A function g P S m 2 can be identified with a measureĝ P M F C pr0, T sˆXq, defined bŷ gpAq " ż A gps, xq ds νpdxq for all A P Bpr0, T sˆXq. This identification induces a topology on S m 2 under which S m 2 is a compact space, see the Appendix of [6]. Throughout, we use this topology on S m 2 .
Proof. Define the space C :" Cpr0, T s; HqˆM F C pr0, T sˆXq. Using the correspondence (3.1), we define a function G 0 : C Ñ D such that G 0ˆż0 f psq ds,ĝ˙" upf, gq for all f P S m 1 , g P S m 2 , m P AE, where upf, gq is the unique solution of (3.3). Theorem 2.1 implies that for each ε ą 0 there exists a mapping G ε : C Ñ D such that where U ε is the solution of (3.2) (and D " denotes equality in distribution).
Define for each m P AE a space of stochastic processes on Ω by S m 1 :" tϕ : r0, T sˆΩ Ñ H : -predictable and ϕp¨, ωq P S m 1 for P -a.a. ω P Ωu.
Let pK n q nPAE be a sequence of compact sets K n Ď X with K n Õ X. For each n P AE, for all pt, ωq P r0, T sˆΩ ) , and let R b " Ť 8 n"1 R b,n . Define for each m P AE a space of stochastic process on Ω by S m 2 :" tψ P R b : ψp¨,¨, ωq P S m 2 for P -a.a. ω P Ωu.
According to Theorem 2.4 in [6] and Theorem 4.2 in [7], our claim is established once we have proved: (C1) if pf n q nPAE Ď S m 1 converges to f P S m 1 and pg n q nPAE Ď S m 2 converges to g P S m 2 for some m P AE, then G 0´ż0 f n psq ds,ĝ n¯Ñ G 0´ż0 f psq ds,ĝ¯in D.
In the sequel, we will prove Condition (C2). The proof of Condition (C1) follows analogously.
Proof. The proof can be accomplished by following [6, p.543  for i " 5, 6. Using this inequality, the proof can be accomplished as the proof of (2.11).
For each ε ě 0 let Y ε be the unique solution of the SPDE and let Z ε be the unique solution of the random PDE Z ε ptq " ż t 0 ∆Z ε psq ds`ż t 0 ż X G 2 ps, V ε psq, xqpψ ε ps, xq´1q νpdxq ds for all t P r0, T s.
It follows from (2.7) by Burkholder's inequality and Young's inequality that for each κ 1 P p0, 1q there exists a constant c κ 1 ą 0 such that E´sup tPr0,T sˇI Analogously, one obtains from (2.9) that for each κ 2 P p0, 1q there exists a constant c κ 2 ą 0 such that  "ż0 e p¨´sq∆ f psq ds, f P A M,m * is relatively compact in Cpr0, T s, L 2 q. From Lemma 3.3 we conclude that which completes the proof of part (b). Part (c). The very definition of J ε yields According to Lemma 4.3 in [14], the estimates (3.6) and (3.7) imply that tJ ε , ε P p0, 1qu is tight in Cpr0, T s, H´dq X L 2 pr0, T s, L 2 q, which completes the proof of part (c).
(c) Define for each k P AE the stochastic processes Note, that it follows from (3.8) and V ε " Y ε`Zε`Jε that V 1 k has the same distribution as V ε k . The convergence (3.9) implies In particular, it follows by (3.13), (3.14) and Lemma 3.3 that Step 2: We will prove that S 1 solves equation (3.3) for f " ϕ 1 and g " ψ 1 , that is w, G 2`s , S 1 psq, z˘`ψ 1 ps, zq´1˘E νpdzq ds, For this purpose note that the very definition of S 1 k yields xw,  [17, p.5234], that there is a subsequence such that the left hand side of (3.23) converges P 1 -a.s. to xw, S 1 ptqy´xw, u 0 ỳ ż t 0 @ w, ∆S 1 psq`div Bps, S 1 psqq`F ps, S 1 psqq`ż 0 8 γprq∆S 1 ps`rq dr D ds ż t 0 xw, G 1 ps, S 1 psqqϕ 1 psqy ds.
Step 3. In the last step we establish that L k :" V 1 Recall that V 1 k has the same distribution as V ε k and S 1 is a solution of (3.3) according to Step 2. Thus, once we will have established (3.25), it shows that Condition (C2) is satisfied.
As the definition of T 1 only depends on γ, we can repeat the above procedure for the interval rT 1 , 2T 1 s, which after further iterations finally leads to the proof of (3.25).