On nonlinear Schrödinger equations with random potentials: existence and probabilistic properties

Abstract

In this paper we are concerned with nonlinear Schrödinger equations with random potentials. Our class includes continuum and discrete potentials. Conditions on the potential $V_{\omega }$ are found for existence of solutions almost sure $\omega$. We study probabilistic properties like central limit theorem and law of larger numbers for the obtained solutions by independent ensembles. We also give estimates on the expected value for the $L^{\infty }$-norm of the solution showing how it depends on the size of the potential.

15 pages