On the long time behavior of Hilbert space diffusion

TL;DR

This paper investigates the long-term behavior of solutions to a stochastic differential equation modeling a quantum particle with spontaneous collapses, showing convergence to a Gaussian state with fixed spreads in position and momentum.

Contribution

It provides a rigorous analysis of the asymptotic behavior of Hilbert space diffusion models for quantum particles with spontaneous collapses, highlighting convergence to Gaussian states.

Findings

Solutions converge almost surely to Gaussian states

The Gaussian states have fixed spreads in position and momentum

The problem involves subtle mathematical considerations

Abstract

Stochastic differential equations in Hilbert space as random nonlinear modified Schroedinger 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.