The Annals of Applied Probability

Basic properties of nonlinear stochastic Schrödinger equations driven by Brownian motions

Carlos M. Mora and Rolando Rebolledo

Full-text: Open access

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.

Article information

Source
Ann. Appl. Probab., Volume 18, Number 2 (2008), 591-619.

Dates
First available in Project Euclid: 20 March 2008

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1206018198

Digital Object Identifier
doi:10.1214/105051607000000311

Mathematical Reviews number (MathSciNet)
MR2399706

Zentralblatt MATH identifier
1145.60036

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.) 37L40: Invariant measures 81S25: Quantum stochastic calculus 81P15: Quantum measurement theory

Keywords
Nonlinear stochastic Schrödinger equations regular invariant measures existence and uniqueness of solutions quantum mechanics stochastic evolution equations

Citation

Mora, Carlos M.; Rebolledo, Rolando. Basic properties of nonlinear stochastic Schrödinger equations driven by Brownian motions. Ann. Appl. Probab. 18 (2008), no. 2, 591--619. doi:10.1214/105051607000000311. https://projecteuclid.org/euclid.aoap/1206018198


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