Title: Invariant measures for a stochastic nonlinear Schrodinger equation

Authors: Jong Uhn Kim

Issue: Volume 55 (2006), Issue 2, 687-718

Abstract: We will prove the existence of an invariant measure for a nonlinear Schroedinger equation with random noise in $\mathbb{R}^n$. The existence of solutions of the Cauchy problem in $H^1$- and $L^2$-settings was established by de Bouard and Debussche (see Anne de Bouard and Arnaud Debussche, \textit{A stochastic nonlinear Schroedinger equation with multiplicative noise}, Comm. Math. Phys. \textbf{205} (1999), 161-181; Anne de Bouard and Arnaud Debussche, \textit{The stochastic nonlinear Schroedinger equation in $H^1$}, Stochastic Anal. Appl. \textbf{21} (2003), 97--126). Here we discuss only the defocusing equation with a zero-order dissipation. The proof for the existence of an invariant measure is based on various energy estimates and the approximation scheme to construct solutions, which are also crucial for the existence of global solutions to the Cauchy problem.