## 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^{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 S^{1}. 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 v^{2} is a C^{k}(S^{1})-valued Markov process for each 0≦κ<1/2, where C^{κ}(S^{1}) is the Banach space of real-valued continuous functions on S^{1} 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}(S^{1}) is a globally exponentially stable critical point of the unperturbed equation ∂_{t}υ^{0} = ℒυ^{0} +f(x,υ^{0}), that υ^{ε} has a unique stationary distribution v^{K, υ} on (C^{κ}(S^{1}), ℬ(C^{K}(S^{1}))) 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}(S^{1}). 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}(S^{1}) is inherently infinite-dimensional.

Original language | English (US) |
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Pages (from-to) | 393-421 |

Number of pages | 29 |

Journal | Probability Theory and Related Fields |

Volume | 92 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 1992 |

Externally published | Yes |

## Keywords

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

## ASJC Scopus subject areas

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty