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