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