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

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper we establish a large deviations principle for the invariant measure of the non-Gaussian stochastic partial differential equation (SPDE) ∂tvε=ℒvε+f(x,vε)+εσ(x,vε) {Mathematical expression}. Here ℒ is a strongly-elliptic second-order operator with constant coefficients, ℒh:=DHxx-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 Ck(S1)-valued Markov process for each 0≦κ<1/2, where Cκ(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 Cκ(S1) is a globally exponentially stable critical point of the unperturbed equation ∂tυ0 = ℒυ0 +f(x,υ0), that υε has a unique stationary distribution vK, υ on (Cκ(S1), ℬ(CK(S1))) when the perturbation parameter ε is small enough. Some further calculations show that as ε tends to zero, vK, υ tends to vK,0, the point mass centered on the zero element of Cκ(S1). The main goal of this paper is to show that in fact vK, υ is governed by a large deviations principle (LDP). Our starting point in establishing the LDP for vK, υ is the LDP for the process υε, which has been shown in an earlier paper. Our methods of deriving the LDP for vK, υ based on the LDP for υε are slightly non-standard compared to the corresponding proofs for finite-dimensional stochastic differential equations, since the state space Cκ(S1) is inherently infinite-dimensional.

Original languageEnglish (US)
Pages (from-to)393-421
Number of pages29
JournalProbability Theory and Related Fields
Volume92
Issue number3
DOIs
StatePublished - Sep 1 1992
Externally publishedYes

Keywords

  • Mathematics Subject Classifications (1985): 60F10, 60H15

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Fingerprint Dive into the research topics of 'Large deviations for the invariant measure of a reaction-diffusion equation with non-Gaussian perturbations'. Together they form a unique fingerprint.

Cite this