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On the Lévy–Khinchin Formula in Noncommutative Probability Theory

Александр Семенович Холево

An important role in modern statistical quantum measurement theory is played by measures that assume values in a noncommutative algebra of transformations. This paper investigates convolution semigroups of such measures arising in connection with measurement processes that proceed continuously in time. The principal result is a noncommutative generalization of the Lévy–Khinchin formula, which describes the structure of the convolution semigroups in terms of their Fourier transforms.

Theory of Probability and Its ApplicationsDec 1, 19943 citations

A Quantum Decomposition of Lévy Processes

Yuh-Jia Lee, Hsin-Hung Shih

Stochastic Analysis: Classical and QuantumOct 1, 20053 citations

Markov structure of quantum Lévy processes

Uwe Franz, René Schott

Springer eBooksJan 1, 19990 citations

Quantum quasi-Markov processes, L-dynamics, and noncommutative Girsanov transformation

V. P. Belavkin

We review the basic concepts of quantum stochastics using the universal Itô *-algebra approach.The main notions and results of classical and quantum stochastics are reformulated in this unifying approach.The thermal quantum Lévy process with commuting increments is defined in terms of the modular *-Itô algebra.The quasi-Markov quantum stochastic processes over increasing W*-algebras generated by a quantum Lévy process is characterized in terms of their quantum stochastic germs.The corresponding quantum stochastic master equation on the increasing predual L-spaces is derived as a noncommutative and non-Markov generalization of the Zakai equation driven by the dual quantum Lévy process.This is done by a noncommutative analog of the Girsanov transformation which we introduce here in full generality.We review the basic concepts of quantum stochastics using the universal Itô *-algebra approach.The main notions and results of classical and quantum stochastics are reformulated in this unifying approach.The thermal quantum Lévy process with commuting increments is defined in terms of the modular *-Itô algebra.The quasi-Markov quantum stochastic processes over increasing W*-algebras generated by a quantum Lévy process is characterized in terms of their quantum stochastic germs.The corresponding quantum stochastic master equation on the increasing predual L-spaces is derived as a noncommutative and non-Markov generalization of the Zakai equation driven by the dual quantum Lévy process.This is done by a noncommutative analog of the Girsanov transformation which we introduce here in full generality.

Communications on Stochastic AnalysisDec 1, 20100 citations

Quasi-distributions for arbitrary non-commuting operators

J. Ben-Benjamin, Leon W. Cohen

arXiv:2003.05509Physics Letters AMar 9, 20205 citations

Anyon wave equations and the noncommutative plane

P. A. Horváthy, Mikhail S. Plyushchay

The “Jackiw–Nair” nonrelativistic limit of the relativistic anyon equations provides us with infinite-component wave equations of the Dirac–Majorana–Lévy-Leblond type for the “exotic” particle, associated with the two-fold central extension of the planar Galilei group. An infinite dimensional representation of the Galilei group is found. The velocity operator is studied, and the observable coordinates describing a noncommutative plane are identified.

arXiv:hep-th/0404137Physics Letters BJun 22, 2004110 citations

Noncommutative Lévy Processes for Generalized (Particularly Anyon) Statistics

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