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

Summary In this paper we establish a large deviations principle for the invariant measure of the non-Gaussian stochastic partial differential equation (SPDE) _tv=v+f(x,v)+(x,v). Here is a strongly-elliptic second-order operator with constant coefficients, h:=DH_xx-h, and the space variablex takes values on the unit circleS¹. The functionsf and are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic PDE, and in particular we require that 0<mM wherem andM are some finite positive constants. The perturbationW is a Brownian sheet. It is well-known that under some simple assumptions, the solutionv² is aC^k(S¹)-valued Markov process for each 0<1/2, whereC(S¹) is the Banach space of real-valued continuous functions onS¹ which are Hölder-continuous of exponent . We prove, under some further natural assumptions onf and which imply that the zero element ofC(S¹) is a globally exponentially stable critical point of the unperturbed equation _t⁰ = ⁰ +f(x,⁰), that has a unique stationary distributionv^K, on (C(S¹), (C^K(S¹))) when the perturbation parameter is small enough. Some further calculations show that as tends to zero,v^K, tends tov^K,0, the point mass centered on the zero element ofC(S¹). The main goal of this paper is to show that in factv^K, is governed by a large deviations principle (LDP). Our starting point in establishing the LDP forv^K, is the LDP for the process , which has been shown in an earlier paper. Our methods of deriving the LDP forv^K, based on the LDP for are slightly non-standard compared to the corresponding proofs for finite-dimensional stochastic differential equations, since the state spaceC(S¹) is inherently infinite-dimensional.This work was performed while the author was with the Department of Mathematics, University of Maryland, College Park, MD 20742, USA