Large deviations for the invariant measure of a reaction-diffusion equation with non-Gaussian perturbations

In this paper we establish a large deviations principle for the invariant measure of the non-Gaussian stochastic partial differential equation (SPDE) ∂_tv^ε=ℒv^ε+f(x,v^ε)+εσ(x,v^ε) {Mathematical expression}. Here ℒ is a strongly-elliptic second-order operator with constant coefficients, ℒh:=DH_xx-αh, and the space variable x takes values on the unit circle S¹. The functions f and σ are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic PDE, and in particular we require that 0<m≦σ≦M where m and M are some finite positive constants. The perturbation W is a Brownian sheet. It is well-known that under some simple assumptions, the solution v² is a C^k(S¹)-valued Markov process for each 0≦κ<1/2, where C^κ(S¹) is the Banach space of real-valued continuous functions on S¹ which are Hölder-continuous of exponent κ. We prove, under some further natural assumptions on f and σ which imply that the zero element of C^κ(S¹) is a globally exponentially stable critical point of the unperturbed equation ∂_tυ⁰ = ℒυ⁰ +f(x,υ⁰), that υ^ε has a unique stationary distribution v^{K, υ} on (C^κ(S¹), ℬ(C^K(S¹))) when the perturbation parameter ε is small enough. Some further calculations show that as ε tends to zero, v^{K, υ} tends to v^K,0, the point mass centered on the zero element of C^κ(S¹). The main goal of this paper is to show that in fact v^{K, υ} is governed by a large deviations principle (LDP). Our starting point in establishing the LDP for v^{K, υ} is the LDP for the process υ^ε, which has been shown in an earlier paper. Our methods of deriving the LDP for v^{K, υ} based on the LDP for υ^ε are slightly non-standard compared to the corresponding proofs for finite-dimensional stochastic differential equations, since the state space C^κ(S¹) is inherently infinite-dimensional.