Asymptotics of the solutions of the stochastic lattice wave equation

TL;DR

This paper studies the long-term behavior of solutions to a stochastic lattice wave equation with energy- and momentum-conserving noise, deriving asymptotic limits and describing their properties.

Contribution

It establishes the asymptotic distribution of solutions, deriving a time-inhomogeneous Ornstein-Uhlenbeck equation for the limit wave function under various initial data conditions.

Findings

Limit wave function satisfies a time-inhomogeneous Ornstein-Uhlenbeck equation.

Weak limit for square integrable initial data is deterministic.

Results hold for both square integrable and statistically homogeneous initial data.

Abstract

We consider the long time limit theorems for the solutions of a discrete wave equation with a weak stochastic forcing. The multiplicative noise conserves the energy and the momentum. We obtain a time-inhomogeneous Ornstein-Uhlenbeck equation for the limit wave function that holds both for square integrable and statistically homogeneous initial data. The limit is understood in the point-wise sense in the former case, and in the weak sense in the latter. On the other hand, the weak limit for square integrable initial data is deterministic.