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Teor. Veroyatnost. i Primenen., 2019, Volume 64, Issue 2, Pages 308–327 (Mi tvp5263)  

Generators of quantum one-dimensional diffusions

A. S. Kholevo

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: Quantum dynamical semigroups represent a noncommutative analogue of (sub)Markov semigroups in classical probability: while the latter are semigroups of maps in functional spaces, the former are semigroups of maps in operator algebras having certain properties of positivity and normalization. In this paper we describe quantum dynamical semigroups, which are the noncommutative analogues of classical diffusions on $\mathbf{R}$ and $\mathbf{R}_{+}$, and demonstrate various properties of the semigroup and its generator depending on the boundary condition. We also give a proof of a result describing the domain of the generator of “noncommutative diffusion on $\mathbf{R}_{+}$ with extinction at 0” and give an explicit example of the trace-class operator in this domain, which does not belong to the domain of closure of the initial operator.

Keywords: quantum dynamical semigroup, quantum diffusion, generator, quantum Markovian master equations, minimal solution.

Funding Agency Grant Number
Russian Academy of Sciences - Federal Agency for Scientific Organizations PRAS-18-01
This work was supported by the Program of the Presidium of the Russian Academy of Sciences no. 01 “Fundamental Mathematics and Its Applications” (grant PRAS-18-01).


DOI: https://doi.org/10.4213/tvp5263

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English version:
Theory of Probability and its Applications, 2019, 64:2, 249–263

Bibliographic databases:

Received: 23.10.2018
Revised: 26.11.2018
Accepted:25.10.2018

Citation: A. S. Kholevo, “Generators of quantum one-dimensional diffusions”, Teor. Veroyatnost. i Primenen., 64:2 (2019), 308–327; Theory Probab. Appl., 64:2 (2019), 249–263

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