On the long-time behavior of Hilbert space diffusion

Stochastic differential equations in Hilbert space as random nonlinear modified Schrodinger equations have achieved great attention in recent years; of particular interest is the long-time behavior of their solutions. In this note we discuss the long-time behavior of the solutions of the stochastic differential equation describing the time evolution of a free quantum particle subject to spontaneous collapses in space. We explain why the problem is subtle and report on a recent rigorous result, which asserts that any initial state converges almost surely to a Gaussian state having a fixed spread both in position and momentum. Copyright (C) EPLA, 2008