A large deviation principle for nonlinear stochastic wave equation driven by rough noise

TL;DR

This paper establishes a large deviation principle for a one-dimensional nonlinear stochastic wave equation driven by rough noise with fractional spatial component, using a variational framework and a modified weak convergence approach.

Contribution

It extends large deviation principles to nonlinear stochastic wave equations influenced by fractional noise, employing a novel variational and weak convergence methodology.

Findings

Proves a large deviation principle for the specified stochastic wave equation.

Adapts the variational framework and weak convergence criterion to fractional noise.

Provides a rigorous mathematical foundation for analyzing rare events in such systems.

Abstract

This paper is devoted to investigating Freidlin-Wentzell's large deviation principle for one (spatial) dimensional nonlinear stochastic wave equation $\frac{\partial^2 u^{\e}(t,x)}{\partial t^2}=\frac{\partial^2 u^{\e}(t,x)}{\partial x^2}+\sqrt{\e}\sigma(t, x, u^{\e}(t,x))\dot{W}(t,x)$, where $\dot{W}$ is white in time and fractional in space with Hurst parameter $H\in(\frac 14,\frac 12)$. The variational framework and the modified weak convergence criterion proposed by Matoussi et al. \cite{MSZ} are adopted here.