Exact asymptotics of the stochastic wave equation with time-independent noise

TL;DR

This paper derives precise asymptotic behaviors for the moments of solutions to the stochastic wave equation in low dimensions driven by time-independent Gaussian noise, including critical cases.

Contribution

It provides the first exact asymptotic formulas for moments of the stochastic wave equation with time-independent noise, covering all dimensions up to three.

Findings

Exact asymptotics for the p-th moment when time or p is large

Transition time for second moment in the critical case (d=3, white noise)

Asymptotic behavior characterized for noise with Riesz kernel covariance

Abstract

In this article, we study the stochastic wave equation in all dimensions $d\leq 3$, driven by a Gaussian noise $\dot{W}$ which does not depend on time. We assume that either the noise is white, or the covariance function of the noise satisfies a scaling property similar to the Riesz kernel. The solution is interpreted in the Skorohod sense using Malliavin calculus. We obtain the exact asymptotic behaviour of the $p$-th moment of the solution either when the time is large or when $p$ is large. For the critical case, that is the case when $d=3$ and the noise is white, we obtain the exact transition time for the second moment to be finite.