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CTRW modeling of quantum measurement and fractional equations of quantum stochastic filtering and control
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Kolokoltsov, Vassili N. (2022) CTRW modeling of quantum measurement and fractional equations of quantum stochastic filtering and control. Fractional Calculus and Applied Analysis, 25 (1). pp. 128-165. doi:10.1007/s13540-021-00002-2 ISSN 1314-2224.
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WRAP-CTRW-modeling-quantum-stochastic-filtering-control-2022.pdf - Published Version - Requires a PDF viewer. Available under License Creative Commons Attribution 4.0. Download (518Kb) | Preview |
Official URL: https://doi.org/10.1007/s13540-021-00002-2
Abstract
Initially developed in the framework of quantum stochastic calculus, the main equations of quantum stochastic filtering were later on derived as the limits of Markov models of discrete measurements under appropriate scaling. In many branches of modern physics it became popular to extend random walk modeling to the continuous time random walk (CTRW) modeling, where the time between discrete events is taken to be non-exponential. In the present paper we apply the CTRW modeling to the continuous quantum measurements yielding the new fractional in time evolution equations of quantum filtering and thus new fractional equations of quantum mechanics of open systems. The related quantum control problems and games turn out to be described by the fractional Hamilton-Jacobi-Bellman (HJB) equations on Riemannian manifolds. By-passing we provide a full derivation of the standard quantum filtering equations, in a modified way as compared with existing texts, which (i) provides explicit rates of convergence (that are not available via the tightness of martingales approach developed previously) and (ii) allows for the direct applications of the basic results of CTRWs to deduce the final fractional filtering equations.
Item Type: | Journal Article | ||||||
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Subjects: | Q Science > QA Mathematics | ||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | ||||||
SWORD Depositor: | Library Publications Router | ||||||
Library of Congress Subject Headings (LCSH): | Fractional differential equations, Differential equations, Partial, Quantum theory, Stochastic control theory, Stochastic partial differential equations, Stochastic analysis | ||||||
Journal or Publication Title: | Fractional Calculus and Applied Analysis | ||||||
Publisher: | Springer International Publishing | ||||||
ISSN: | 1314-2224 | ||||||
Official Date: | 7 February 2022 | ||||||
Dates: |
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Volume: | 25 | ||||||
Number: | 1 | ||||||
Page Range: | pp. 128-165 | ||||||
DOI: | 10.1007/s13540-021-00002-2 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Re-use Statement: | |||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||
Date of first compliant deposit: | 15 July 2022 | ||||||
Date of first compliant Open Access: | 15 July 2022 | ||||||
RIOXX Funder/Project Grant: |
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