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

Abstract

In this paper we establish a large deviations principle for the invariant measure of the non-Gaussian stochastic partial differential equation (SPDE) ∂ t v e =ℒv e +f(x,v e )+eσ(x,v e ) $$\ddot W_{tx} $$ . Here ℒ is a strongly-elliptic second-order operator with constant coefficients, ℒh:=DH xx-αh, and the space variablex takes values on the unit circleS 1. The functionsf 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 wherem andM are some finite positive constants. The perturbationW is a Brownian sheet. It is well-known that under some simple assumptions, the solutionv 2 is aC k (S 1)-valued Markov process for each 0≦κ<1/2, whereC κ(S 1) is the Banach space of real-valued continuous functions onS 1 which are Holder-continuous of exponent κ. We prove, under some further natural assumptions onf and σ which imply that the zero element ofC κ(S 1) is a globally exponentially stable critical point of the unperturbed equation ∂ t υ0 = ℒυ0 +f(x,υ0), that υe has a unique stationary distributionv K, υ on (C κ(S 1), ℬ(C K (S 1))) when the perturbation parameter e is small enough. Some further calculations show that as e tends to zero,v K, υ tends tov K,0, the point mass centered on the zero element ofC κ(S 1). The main goal of this paper is to show that in factv K, υ is governed by a large deviations principle (LDP). Our starting point in establishing the LDP forv K, υ is the LDP for the process υe, which has been shown in an earlier paper. Our methods of deriving the LDP forv K, υ based on the LDP for υe are slightly non-standard compared to the corresponding proofs for finite-dimensional stochastic differential equations, since the state spaceC κ(S 1) is inherently infinite-dimensional.