On the long time behavior of Hilbert space diffusion

Stochastic differential equations in Hilbert space as random nonlinear modified Schrödinger 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. Hilbert space valued stochastic differential equations appear all over in quantum physics and their meaning ranges from fundamental to effective descriptions of quantum systems. They can on the one hand be seen as basic equations in collapse models, where the aim is to find a unified description of microscopic quantum phenomena and macroscopic classical ones; this is achieved by modifying the Schrödinger equation, adding stochastic nonlinear terms which model the spontaneous collapse of the wave function [1–10]. This fundamental meaning of Hilbert space diffusions have been originated in a discrete version, the so called GRW model [1], which relies on jump processes rather than a continuous diffusion process. One can view the diffusion process as continuum limit of the GRW jump process [2]; whether they can be empirically distinguished is unclear. Mathematically, diffusion processes are analytically easier to handle than jump processes: that is the reason why we consider diffusions. The same type of equations occur in the theory of continuous measurement [11–15]; in this context, they describe the effect of continuous measurements on the evolution of quantum systems. Synonymous to measurement is decoherence, and therefore the very same effective equations appear also in decoherence theory [16, 17]. The common class of stochastic differential equations used in Quantum Mechanics has the following structure: