Nonlinear stochastic wave Equation driven by rough noise

TL;DR

This paper establishes the existence and uniqueness of strong solutions for a one-dimensional nonlinear stochastic wave equation driven by rough Gaussian noise with fractional spatial properties.

Contribution

It provides the first rigorous proof of strong solution existence and uniqueness for this class of nonlinear stochastic wave equations with fractional noise.

Findings

Proved strong solution existence for Hurst parameter H in (1/4, 1/2)

Established uniqueness of solutions under given conditions

Extended stochastic wave equation theory to rough noise settings

Abstract

In this paper, we obtain the existence and uniqueness of the strong solution to one spatial dimension stochastic wave equation $\frac{\partial^2 u(t,x)}{\partial t^2}=\frac{\partial^2 u(t,x)}{\partial x^2}+\sigma(t,x,u(t,x))\dot{W}(t,x)$ assuming $\sigma(t,x,0)=0$, where $\dot W$ is a mean zero Gaussian noise which is white in time and fractional in space with Hurst parameter $H\in(1/4, 1/2)$.