Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Markov chains approximation of jump–diffusion stochastic master equations

Clément Pellegrini

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Abstract

Quantum trajectories are solutions of stochastic differential equations obtained when describing the random phenomena associated to quantum continuous measurement of open quantum system. These equations, also called Belavkin equations or Stochastic Master equations, are usually of two different types: diffusive and of Poisson-type. In this article, we consider more advanced models in which jump–diffusion equations appear. These equations are obtained as a continuous time limit of martingale problems associated to classical Markov chains which describe quantum trajectories in a discrete time model. The results of this article goes much beyond those of [Ann. Probab. 36 (2008) 2332–2353] and [Existence, uniqueness and approximation for stochastic Schrödinger equation: The Poisson case (2007)]. The probabilistic techniques, used here, are completely different in order to merge these two radically different situations: diffusion and Poisson-type quantum trajectories.

Résumé

Les trajectoires quantiques sont des solutions d’équations différentielles stochastiques décrivant des phénomènes aléatoires associés aux prinicpes de mesure (quantique) des systèmes quantiques ouverts. Ces équations, également appelées équations de Belavkin ou équations maîtresses stochastiques sont habituellement de deux types: soit diffusif soit de type saut. Dans cet article, nous considérons des modèles plus avancés où des équations de type saut-diffusion apparaissent. Ces équations sont obtenues comme solutions de problèmes de martingales. Ces problèmes de martingales sont obtenus comme limites continus (en temps) à partir de chaînes de Markov classiques décrivant des trajectoires quantiques pour des modèles à temps discret. Les résultats de cet article généralisent ceux de [Ann. Probab. 36 (2008) 2332–2353] et [Existence, uniqueness and approximation for stochastic Schrödinger equation: The Poisson case (2007)]. Ici, les techniques probabilistes utilisés sont complétement différentes afin de pouvoir mixer les deux types d’évolutions: diffusives et poissoniennes.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 46, Number 4 (2010), 924-948.

Dates
First available in Project Euclid: 4 November 2010

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1288878330

Digital Object Identifier
doi:10.1214/09-AIHP330

Mathematical Reviews number (MathSciNet)
MR2744878

Zentralblatt MATH identifier
1211.60020

Subjects
Primary: 60F99: None of the above, but in this section 60G99: None of the above, but in this section 60H10: Stochastic ordinary differential equations [See also 34F05]

Keywords
Stochastic master equations Quantum trajectory Jump–diffusion stochastic differential equation Stochastic convergence Markov generators Martingale problem

Citation

Pellegrini, Clément. Markov chains approximation of jump–diffusion stochastic master equations. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 4, 924--948. doi:10.1214/09-AIHP330. https://projecteuclid.org/euclid.aihp/1288878330


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