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Stochastic CGL equations without linear dispersion in any space dimension

Sergei Kuksin 1, Vahagn Nersesyan 2

(2012-05-04)

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.

1:	Centre de Mathématiques Laurent Schwartz (CMLS-EcolePolytechnique)
	CNRS : UMR7640 – Polytechnique - X
2:	Laboratoire de Mathématiques de Versailles (LM-Versailles)
	CNRS : UMR8100 – Université de Versailles Saint-Quentin-en-Yvelines

Subject

:

Mathematics/Analysis of PDEs

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From: Vahagn Nersesyan <>

Submitted on: Friday, 4 May 2012 13:13:03

Updated on: Friday, 4 May 2012 13:18:41