Moderate deviations for a stochastic Burgers equation

Rachid Belfadli,Mohamed Mellouk,Lahcen Boulanba

doi:10.15559/19-vmsta134

Rachid Belfadli, Mohamed Mellouk + Show 1 more

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https://doi.org/10.15559/19-vmsta134

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Abstract

A moderate deviations principle for the law of a stochastic Burgers equation is proved via the weak convergence approach. In addition, some useful estimates toward a central limit theorem are established.

Highlights

We consider the following stochastic Burgers equation with multiplicative space-time white noise, indexed by ε > 0, given by ∂ uε (t, x) = uε(t, x) + 1 ∂ uε(t, x) 2 ∂t√ + εσ 2 ∂x uε(t, x) W (t, x),(t, x) ∈ [0, T ] × [0, 1], (1)with Dirichlet’s boundary conditions uε(t, 0) = uε(t, 1) = 0 for t ∈ [0, T ], and the initial condition uε(0, x) = u0(x) for x ∈ [0, 1]
A related natural important question is to study moderate deviations results which deals with probabilities of deviations of
We will precise below the main difference between moderate and large deviations principles in the context of stochastic Burgers equation, and for a deeper description and detail about these two kinds of deviations principles and their relationship, we refer the reader to [6]

Summary

Introduction

We consider the following stochastic Burgers equation with multiplicative space-time white noise, indexed by ε > 0, given by. In the present paper, we mainly use the weak convergence approach to establish moderate deviations for stochastic Burgers’ equations while in the previous works ([8, 23, 14]) the authors studied the large deviations principle for this equation. The most likely advantage in using the weak convergence approach is that it allows one to avoid establishing technical exponential-type probability estimates usually needed in the classical studies of large deviations principle, and reduces the proofs to demonstrating qualitative properties like existence, uniqueness and tightness of certain analogues of the original processes. For ρ 1 and t ∈ [0, T ], the usual norms on Lρ([0, 1]) and Ht := L2([0, t] × [0, 1]) are respectively denoted by · ρ and · Ht

Preliminaries

Framework and the main result

Sufficient conditions for the general Laplace principle

Proof of the main result

Toward a central limit theorem