Large Deviation Principle for a Stochastic Heat Equation with Spatially Correlated Noise 1

In this paper we prove a large deviation principle (ldp) for a perturbed stochastic heat equation defined on [0, T ] × [0, 1] d. This equation is driven by a Gaussian noise, white in time and correlated in space. Firstly, we show the Hölder continuity for the solution of the stochastic heat equation. Secondly, we check that our Gaussian process satisfies a ldp and some requirements on the skeleton of the solution. Finally, we prove the called Freidlin-Wentzell inequality. In order to obtain all these results we need precise estimates of the fundamental solution of this equation.


Introduction
Consider the following perturbed d-dimensional spatial stochastic heat equation on the compact set [0, 1] with ε > 0, L = ∂ ∂t − ∆ where ∆ is the Laplacian on IR d and ∂([0, 1] d ) is the boundary of [0, 1] d .We consider null initial conditions and the compact set [0, 1] d instead of [0, ζ] d , ζ > 0, for the sake of simplicity.Assume that the coefficients satisfy the following assumptions: (C) the functions α and β are Lipschitz.
The noise F = {F (ϕ), ϕ ∈ D(IR d+1 )} is an L 2 (Ω, F, P )-valued Gaussian process with mean zero and covariance functional given by J(ϕ, ψ) = and f : IR d → IR + is a continuous symmetric function on IR d − {0} such that there exists a non-negative tempered measure λ on IR d whose Fourier transform is f .The functional J in (1.2) is said to be a covariance functional if all these asumptions are satisfied.Then, in addition, where F is the Fourier transform and z is the conjugate complex of z.In (1.2) we could also work with a non-negative and non-negative definite tempered measure, therefore symmetric, instead of the function f but, in this case, all the notation over the sets appearing in the integrals is becoming tedious.In this paper, moreover, we assume the following hypothesis on the measure λ: for some η ∈ (0, 1].For instance, the function f (x) = x −κ , κ ∈ (0, d), satisfies (H η ).As in Dalang [4] (see also Dalang and Frangos [5]) the Gaussian process F can be extended to a worthy martingale measure, in the sense given by Walsh [23], Then, following the approach of Walsh [23], one can give a rigorous meaning to (1.1) by means of a weak formulation.Assumptions (C) and (H 1 ) will ensure the existence and uniqueness of a jointly measurable adapted process {u ε (t, x), (t, x) G(t − s, x, y)α(u ε (s, y))F (ds, dy) where ε > 0 and G(t, x, y) denotes the fundamental solution of the heat equation on [0, 1] d : G(0, x, y) = δ(x − y).
The stochastic integral in (1.3) is defined with respect to the F t -martingale measure M .Denote D d T = [0, T ] × [0, 1] d .If d > 1, the evolution equation can not be driven by the Brownian sheet because G does not belong to L 2 (D d T ) and we need to work with a smoother noise.The study of existence and uniqueness of solution to this sort of equation (1.3) on IR d has been analyzed by Dalang in [4].Many other authors have also studied existence and uniqueness of solution to d-dimensional spatial stochastic equations, in particular, wave and heat equations (see, for instance, Dalang and Frangos [5], Karszeswka and Zabszyk [12], Millet and Morien [15], Millet and Sanz-Solé [16], Peszat and Zabszyk [17], [18]).Assuming the above-mentioned hypothesis, (C) and (H η ) for some η ∈ [0, 1], we will also check that the trajectories of the process are (γ 1 , γ 2 )-Hölder continuous with respect to the parameters t and x, satisfaying γ 1 ∈ (0, 1−η 2 ) and γ 2 ∈ (0, 1 − η).The Hölder continuity for the stochastic heat equation on IR d has been studied by Sanz-Solé and Sarrà [20], [21].On the other hand, the wave case has been dealt in [15], [16] and [20].Under (C) and (H η ) for some η ∈ (0, 1), we will prove the most important result of this paper: the existence of a large deviation principle (ldp) for the law of the solution u ε to (1.3) on C γ,γ (D d T ), with γ ∈ (0, 1−η  4 ).This means that we check the existence of a lower semi-continuous function I : called rate function, such that {I ≤ a} is compact for any a ∈ [0, ∞), and where, for a given subset The proof of this goal is based on a classical result given by Azencott in [1] (see also Priouret [19]), that allow us to go beyond a ldp from εF to u ε .Azencott's method is the following: , be two families of random variables.Suppose the following requirements: 1. {X ε 1 , ε > 0} obeys a ldp with the rate function 2. There exists a function K : {I 1 < ∞} → E 2 such that, for every a < ∞, the function is continuous.

3.
For every R, ρ, a > 0, there exist θ > 0 and ε 0 > 0 such that, for h ∈ E 1 satisfying I 1 (h) ≤ a and ε ≤ ε 0 , we have Then, the family {X ε 2 , ε > 0} obeys a ldp with the rate function Consequently, we will need to check that our initial Gaussian process satisfies a ldp, the existence of a function K and, finally, to prove (1.4).We will follow the approach of Freidlin and Wentzell [10] for diffusion process (see also Dembo and Zeitouni [6]).Another remarkable article is Chenal and Millet [3] where they prove the existence of a ldp for a one-dimensional stochastic heat equation.Sowers has also studied this last equation but using a different method.For more information about the study of onedimensional stochastic heat equations, we refer to Chenal and Millet [3].Finally, Chenal [2] has checked a ldp for a stochastic wave equation on IR 2 .The paper is organized as follows.In Section 2 we establish a ldp for our Gaussian process (first point of Theorem 1.1).In Section 3 we state some properties on u ε (t, x), mainly the Hölder continuity.Section 4 contains the proofs of some requirements on the skeleton of u ε (second point of Theorem 1.1).Section 5 is devoted to the proof of called Freidlin-Wentzell's inequality (third point of Theorem 1.1).All the arguments of this paper need precise estimates of the fundamental solution G which will be enunciated and proved in an appendix.Moreover, this appendix also contains an exponential inequality used in Section 5.
In this paper we fix T > 0 and all constants will be denoted independently of its value.We finish this introduction by giving some basic notations.We write, for functions φ, φ : Then, we define the topology of (γ 1 , γ 2 )-Hölder convergence on Finally, for θ > 0, a set A ⊂ IR n , n ≥ 1 and φ : A → IR, we introduce the following notation, 2 Large deviation principle for the Gaussian process Let E be the space of measurable functions ϕ : IR d → IR such that endowed with the inner product Let H be the completation of (E, , E ).For T > 0, let This space is a real separable Hilbert space such that, if where F is the noise introduced in Section 1.
where R(x) is the rectangle [0, x] (here [0, x] is a product of rectangles).We have Γ((t, x), (t , x )) = ϕ t,x , ϕ t ,x H T with One can easily check the following three conditions: (i) Γ is symmetric.
Then, W F is a Gaussian process with covariance function Γ.The trajectories of W F belong to C γ ,γ (D d T ; IR), this means that the law of the process W F is ν.The space H is the set of functions h such that, for h ∈ H T and for any (t, x) with the following scalar product h, k H is a Hilbert space isomorphic to H T .Classical results on Gaussian processes (see, for instance, Theorem 3.4.12 in [7]) show that, for γ ∈ [0, 1  2 ), the family {εW F , ε > 0} satisfies a large deviation principle on C γ ,γ (D d T ; IR) with rate function Remark that, if

Properties on the solution
In this section we analyze the existence and uniqueness of solution (1.3) and the Hölder continuity with respect to the parameters.
Proposition 3.1 Assume (C) and (H 1 ).Then (1.3) has a unique solution.Moreover, for any Proof: Define the following Picard's approximations for n ≥ 1, where G(t − s, x, y) is the fundamental solution to (1.1) described in the Appendix.The proof of this result is almost the same as Proposition 2.4 in [13] but adding the dependence on ε.
Burkholder's and Hölder's inequalities, (3.1) and (6.1.7)imply Using similar arguments together (6.1.8),one can obtain In order to finish the proof of (3.2) we notice that A 3 and A 4 can be dealt in a similar but easier way by means of (6.1.9)and (6.1.10),respectively.We now examine (3.3).For 0 with Then, the same steps as before but using (6.1.19) and (6.1.20)conclude the proof of this theorem.

Continuity of the skeleton
For any h ∈ H, we consider the solution to the deterministic evolution equation for all (t, x) ∈ D d T , where •, • H T is defined in (2.1).Remark.We specify the formulation used in (4.1).Due to the kernel G, Then, in fact, (4.1) can rewrite as follows In this section, we show that the map h → S h is continuous on C γ 1 ,γ 2 (D d T ) for some (γ 1 , γ 2 ) which will be given later.In order to obtain this goal we will need the next proposition, we omit the proof of this result because it is similar to Proposition 3.1 and Theorem 3.2.
ii) Assume (H η ) for some η ∈ (0, 1).For any For s ∈ [0, T ], y ∈ [0, 1] d , we will use the following discretization in time and space We give some obvious properties: The kernel is discretized as follows and we also use the following notation S h n (s, y) = S h(s n , y n ).The proof of the next theorem is the most important aim of this section.
, where { Ĩ ≤ a} is endowed with the topology of uniform convergence.
Proof: In order to prove the continuity of S h we decompose d γ 1 ,γ 2 (S h, S g) into two parts: sup and sup We first examine (4.8).According to (1.5), if we want to study (4.8), we need to deal with the following two functions, for t ∈ [0, T ], We start the proof studying ψ(T ).For (s, x) ∈ D d T , we have The usual argument based on Hölder's inequality together with (C) and the estimates (6.1.5)and (6.1.6)imply that, for t ∈ [0, T ] and h, g ∈ { Ĩ ≤ a}, Then, Gronwall's lemma yields The rest of the study of ψ(T ) consists in dealing with A i , for i = 1, 2, 3.
We now analyze ψ γ 1 (T ).Fix x ∈ [0, 1] d , assume t, t ∈ [0, T ], t < t.Then, from the Lipschitz property of β, it follows with and B 7 are the same terms as B 2 and B 3 , changing h and h for g and g, respectively.

The Freidlin-Wentzell inequality
In this section we will prove the inequality (1.4) of Theorem 1.1.
Step 1.Using the stopping time and by applying a usual localization procedure (see, for instance, Proposition 3.9 in [14]) we can assume that the coefficients α and β are bounded.
Step 2. Given h ∈ H such that Ĩ( h) ≤ a and ε ≤ ε 0 , An extension of Girsanov's Theorem (see, for instance, Section IV.5 in [2]) allows us to reduce the proof of (5.2) to establishing the following (5.3) Step 3. We will now observe that (5.3) is equivalent to where In order to check this equivalence we proceed as in Theorem 4.2.We will give the basic ideas of the proof.Schwarz's inequality, the Lipschitz property on α and β, (6.1.5),(6.1.6)and Gronwall's Lemma yield It only remains to study the Hölder property, that means to deal with Then, Schwarz's inequality, the Lipschitz property, (5.5), (6.1.7)-(6.1.10),(6.1.19) and (6.1.20)imply the equivalence between (5.3) and (5.4).
Step 4.Here the stochastic integral K ε, h will be discretized as follows We will also consider the sets It is immediate to check that In order to conclude the proof of this proposition we need to prove the next two facts: 1) There exists n 0 such that, for n ≥ n 0 , 2) For any n > 0, we can choose δ > 0 such that, considering then B ε n (δ) = ∅.

Study of the kernel
Here we will enunciate and prove all the results on the kernel G used in this paper.Recall that G denotes the fundamental solution of the heat equation The details about the construction of the fundamental solution G can be found in Chapter 1 of [11], more specifically in Section 4. This fundamental solution G is non-negative and can be decomposed into different terms as follows: where H is the heat kernel on IR d This decomposition is given in [9].Finally, it is well-known that where F is the Fourier transform.The proof of following bounds of G, D t G and D x G can be found in Theorem 1.1 of [8].
Lemma 6.1.1There exists a positive constant C such that In the sequel we give new results on G.
Lemma 6.1.3Assume (H η ) for some η ∈ (0, 1).There exists a positive constant C such that, for any Remark.Recall that the fundamental solution G of (6.1.1)satisfies Proof of Lemma 6.1.3:Observe that (6.1.2) implies In order to check (6.1.7)we only need to bound the right-hand side of (6.1.11).The terms H i can be studied using the same arguments as H.
Let L be the Lipschitz constant of the function R. Clearly where Θ = and f is defined in Section 1.Now we analyze the term with H. First of all, from •, • H is a scalar product, we have with Since t < t, using basic tools of mathematical calculus and (6.1.3),we obtain .
For γ 1 ∈ (0, 1−η 2 ) and s < t , we have and then, where Γ is the Gamma function.On the other hand, (6.1.3)again yields and similar computations to the study of A 1 show that for all γ 1 ∈ (0, 1−η 2 ).
Then, (6.1.7)follows from (6.1.12)-(6 . As before the study of B 1 is obvious and we can deal with B 3 by means of B 2 .Then, we will only need to work with B 2 .From (6.1.3),by integrating we obtain