Propagation of Singularities for the Stochastic Wave Equation

TL;DR

This paper investigates how singularities in solutions to a one-dimensional stochastic wave equation propagate over time, focusing on the effects of Gaussian noise that is white in time and colored in space.

Contribution

It introduces a novel approach combining a law of the iterated logarithm with Gaussian process methods to analyze singularity propagation in stochastic wave equations.

Findings

Singularities propagate along characteristic lines.

The behavior of solutions is significantly influenced by the spatial correlation of noise.

The methods can be extended to other stochastic partial differential equations.

Abstract

We study the existence and propagation of singularities of the solution to a one-dimensional linear stochastic wave equation driven by an additive Gaussian noise that is white in time and colored in space. Our approach is based on a simultaneous law of the iterated logarithm and general methods for Gaussian processes.