Spatial ergodicity of stochastic wave equations in dimensions 1, 2 and 3

David Nualart,Guangqu Zheng

doi:10.1214/20-ecp361

Abstract

In this note, we study a large class of stochastic wave equations with spatial dimension less than or equal to $3$. Via a soft application of Malliavin calculus, we establish that their random field solutions are spatially ergodic.

Highlights

In this article, we fix d ∈ {1, 2, 3} and consider the stochastic wave equation ∂2u ∂t2 = ∆u + σ(u)W, (1.1) on R+ ×Rd with initial conditions u(0, x) and
We study a large class of stochastic wave equations with spatial dimension less than or equal to 3
Via a soft application of Malliavin calculus, we establish that their random field solutions are spatially ergodic

Summary

Introduction

Laplacian in the space variables and Wis a centered Gaussian noise with covariance. E[W (t, x)W (s, y)] = δ0(t − s)γ(x − y). Spatial ergodicity of stochastic wave equations where the stochastic integral is defined in the sense of Dalang-Walsh and G(t − s, x − y) denotes the fundamental solution to the corresponding deterministic wave equation, i.e. with σt denoting the surface measure on ∂Bt := {x ∈ R3 : |x| = t}; see Example 6 and Theorem 13 in Dalang’s paper [3]. It is natural to investigate the corresponding second-order fluctuations They have been established in several cases briefly recalled below: ECP 25 (2020), paper 80. Such an inequality fails to work when d = 3, as the fundamental solution G(t, ) is a measure for d = 3 (see (1.5)).

Preliminaries

Malliavin calculus