Basic properties of nonlinear stochastic Schrödinger equations driven by Brownian motions

Abstract

The paper is devoted to the study of nonlinear stochastic Schrödinger equations driven by standard cylindrical Brownian motions (NSSEs) arising from the unraveling of quantum master equations. Under the Born--Markov approximations, this class of stochastic evolutions equations on Hilbert spaces provides characterizations of both continuous quantum measurement processes and the evolution of quantum systems. First, we deal with the existence and uniqueness of regular solutions to NSSEs. Second, we provide two general criteria for the existence of regular invariant measures for NSSEs. We apply our results to a forced and damped quantum oscillator.

Published in at http://dx.doi.org/10.1214/105051607000000311 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Quantum Trajectories in Random Environment: the Statistical Model for a Heat Bath
• Basic properties of a mean field laser equation
• The Gisin-Percival Stochastic Schrödinger Equation from Standard Quantum Filtering Theory
• The Estimation Lie Algebra Associated with Quantum Filters