On the long-time behavior of Hilbert space diffusionA. Bassi1, 2 and D. Dürr2
1 Department of Theoretical Physics, University of Trieste - Strada Costiera 11, 34014 Trieste, Italy, EU
2 Mathematisches Institut der L.M.U. - Theresienstr. 39, 80333 München, Germany, EU
received 28 May 2008; accepted in final form 20 August 2008; published October 2008
published online 16 September 2008
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.
03.65.Ta - Foundations of quantum mechanics; measurement theory.
02.50.Ey - Stochastic processes.
02.30.Jr - Partial differential equations.
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