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

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 $L^2[0,1]$ $$Y(\omega )=1/2{\int }_{-\infty }^0{T}_{\nu }(-s)\frac{\partial (Y(\theta (s,\omega ){)}^2}{\partial x}ds+{\int }_{-\infty }^0{T}_{\nu }(-s)d{W}_s(\omega ),$$ where $\theta$ is the group of $P$-preserving ergodic transformation on the canonical probability pace $(\Omega ,\mathcal{F},P)$ such that $\theta (t,\omega )(s)=W(t+s)-W(t)$.