Stochastic CGL equations without linear dispersion in any space dimension

Abstract

We consider the stochastic CGL equation $$\dot{u}-\nu \Delta u+(i+a)|u{|}^{2}u=\eta (t,x),\text{dim}x=n,$$ where $\nu >0$ and $a\ge 0$, in a cube (or in a smooth bounded domain) with Dirichlet boundary condition. The force $\eta$ is white in time, regular in $x$ and non-degenerate. We study this equation in the space of continuous complex functions $u(x)$, and prove that for any $n$ it defines there a unique mixing Markov process. So for a large class of functionals $f(u(\cdot ))$ and for any solution $u(t,x)$, the averaged observable $\E f(u(t,\cdot))$ converges to a quantity, independent from the initial data $u(0,x)$, and equal to the integral of $f(u)$ against the unique stationary measure of the equation.