Large deviations for stochastic heat equation with rough dependence in space

Yaozhong Hu,Tusheng Zhang,David Nualart

doi:10.3150/16-bej880

Abstract

In this paper, we establish a large deviation principle for the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance of a fractional Brownian motion with Hurst parameter $H\in (\frac{1}{4},\frac{1}{2})$ in the space variable.

Highlights

In this paper, we consider the one-dimensional stochastic partial differential equations ∂ uε ∂t = κ 2 ∂ 2 uε ∂x2 √ + εσ uεW, t ∈ [0, T ], x ∈ R, (1.1)where W is a zero-mean Gaussian process with covariance given byE W (s, x)W (t, y) = 1 |x|2H + |y|2H − |x − y|2H 2 (s ∧ t)
A section of preliminaries contains the definition of stochastic integral, the notion of solution for equation (1.1) and the functional spaces introduced in [13]
Let H be the space of predictable processes g defined on [0, T ] × R such that almost surely g

Summary

Introduction

We consider the one-dimensional stochastic partial differential equations. The existence and uniqueness of a mild solution whose trajectories belong to a suitable space of trajectories is proved by using techniques inspired by the works of Gyöngy [11] and Gyöngy and Nualart [12]. The purpose of this paper is to establish a large deviation principle for the laws of the solutions uε to equation (1.1). We use the weak convergence approach to large deviations based on the Laplace principle, developed by P. A section of preliminaries contains the definition of stochastic integral, the notion of solution for equation (1.1) and the functional spaces introduced in [13].

Preliminaries

A criteria for large deviations

Skeleton equations

Large deviation principle