Probability and PDEs

Stochastic CGL equations without linear dispersion in any space dimension

speaker: Sergei Kuksin (Ecole Polytecnique)

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 \(\mathbb E f(u(t,\cdot))\) converges to a quantity, independent from the initial data, and equal to the integral of \(f(u)\) against the unique stationary measure of the equation. This is a joint work with Vahagn Nersesyan.

timetable:
Tue 21 May, 11:30 - 12:20, Aula Dini
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