Stochastic CGL equations without linear dispersion in any space dimension

TL;DR

This paper proves the existence and uniqueness of a mixing Markov process for the stochastic CGL equation without linear dispersion in any spatial dimension, demonstrating convergence to a stationary measure regardless of initial conditions.

Contribution

It establishes the well-posedness and ergodic properties of the stochastic CGL equation in arbitrary dimensions, extending previous results to a broader class of equations.

Findings

Unique mixing Markov process for the stochastic CGL equation in any dimension.

Convergence of time-averaged observables to a stationary measure.

Applicability to a broad class of functionals and initial conditions.

Abstract

We consider the stochastic CGL equation $$ \dot u- \nu\Delta u+(i+a) |u|^2u =\eta(t,x),\;\;\; \text {dim} \,x=n, $$ 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.