Decay of the stochastic linear Schrödinger equation in $d\ge 3$ with small multiplicative noise

Abstract

We give decay estimates of the solution to the linear Schrödinger equation in dimension $d\ge 3$ with a small noise which is white in time and colored in space. As a consequence, we also obtain certain asymptotic behaviour of the solution. The proof relies on the bootstrapping argument used by Journé-Soffer-Sogge for decay of deterministic Schrödinger operators.

15 pages