Global well-posedness of the two-dimensional stochastic nonlinear wave equation on an unbounded domain

TL;DR

This paper proves the global well-posedness of a two-dimensional stochastic nonlinear wave equation with cubic nonlinearity on an unbounded domain, using renormalisation techniques for initial data in a specific Sobolev space.

Contribution

It establishes the first global well-posedness result for this class of stochastic wave equations on unbounded domains with space-time white noise.

Findings

Global well-posedness for initial data in ^s, s > 4/5

Successful renormalisation of the nonlinearity

Extension of deterministic results to stochastic setting

Abstract

We study the two-dimensional wave equation with cubic nonlinearity posed on $\mathbb R^2$, with space-time white noise forcing. After a suitable renormalisation of the nonlinearity, we prove global well-posedness for this equation for initial data in $\mathcal H^s$, $s>\frac45$.