Pathwise stationary solutions of stochastic Burgers equations with $L^2[0,1]$-noise and stochastic Burgers integral equations on infinite horizon

TL;DR

This paper establishes the existence and uniqueness of stationary solutions for stochastic Burgers equations with $L^2[0,1]$-noise, including integral equations on an infinite horizon, using random dynamical systems and ergodic transformations.

Contribution

It introduces a novel approach to prove stationary solutions for stochastic Burgers equations with large viscosity and $L^2$-noise, including integral equations on an infinite horizon.

Findings

Existence and uniqueness of stationary solutions in $L^2[0,1]$.

Representation of solutions via integral equations on infinite horizon.

Application of ergodic transformations to analyze solution properties.

Abstract

In this paper, we show the existence and uniqueness of the stationary solution $u(t,\omega)$ and stationary point $Y(\omega)$ of the differentiable random dynamical system $U:R\times L^2[0,1]\times \Omega\to L^2[0,1]$ generated by the stochastic Burgers equation with $L^2[0,1]$-noise and large viscosity, especially, $u(t,\omega)=U(t,Y(\omega),\omega)=Y(\theta(t,\omega))$, and $Y(\omega) \in H^1[0,1]$ is the unique solution of the following equation in $L^2[0,1]$ $$ Y(\omega)={1/2}\int_{-\infty}^0T_\nu(-s)\frac{\partial (Y(\theta(s,\omega))^2}{\partial x}ds +\int_{-\infty}^0T_\nu(-s)dW_s(\omega), $$ where $\theta$ is the group of $P$-preserving ergodic transformation on the canonical probability pace $(\Omega, {\cal F}, P)$ such that $\theta(t,\omega)(s)=W(t+s)-W(t)$.