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TMF, 2017, Volume 191, Number 3, Pages 473–502 (Mi tmf9153)  

This article is cited in 11 scientific papers (total in 11 papers)

Averaging of random walks and shift-invariant measures on a Hilbert space

V. Zh. Sakbaev

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: We study random walks in a Hilbert space $H$ and representations using them of solutions of the Cauchy problem for differential equations whose initial conditions are numerical functions on $H$. We construct a finitely additive analogue of the Lebesgue measure: a nonnegative finitely additive measure $\lambda$ that is defined on a minimal subset ring of an infinite-dimensional Hilbert space $H$ containing all infinite-dimensional rectangles with absolutely converging products of the side lengths and is invariant under shifts and rotations in $H$. We define the Hilbert space $\mathcal H$ of equivalence classes of complex-valued functions on $H$ that are square integrable with respect to a shift-invariant measure $\lambda$. Using averaging of the shift operator in $\mathcal H$ over random vectors in $H$ with a distribution given by a one-parameter semigroup (with respect to convolution) of Gaussian measures on $H$, we define a one-parameter semigroup of contracting self-adjoint transformations on $\mathcal H$, whose generator is called the diffusion operator. We obtain a representation of solutions of the Cauchy problem for the Schrödinger equation whose Hamiltonian is the diffusion operator.

Keywords: invariant measure on Hilbert space, finitely additive measure, random walk, Schrödinger equation, Cauchy problem.

Funding Agency Grant Number
Russian Science Foundation 14-11-00687
This research was supported by a grant from the Russian Science Foundation (Project No. 14-11-00687).


DOI: https://doi.org/10.4213/tmf9153

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English version:
Theoretical and Mathematical Physics, 2017, 191:3, 886–909

Bibliographic databases:

Received: 25.01.2016
Revised: 28.04.2016

Citation: V. Zh. Sakbaev, “Averaging of random walks and shift-invariant measures on a Hilbert space”, TMF, 191:3 (2017), 473–502; Theoret. and Math. Phys., 191:3 (2017), 886–909

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Orlov Yu.N., Sakbaev V.Zh., Smolyanov O.G., “Feynman Formulas For Nonlinear Evolution Equations”, Dokl. Math., 96:3 (2017), 574–577  crossref  mathscinet  zmath  isi  scopus
    2. Remizov I.D., “Feynman and Quasi-Feynman Formulas For Evolution Equations”, Dokl. Math., 96:2 (2017), 433–437  crossref  mathscinet  zmath  isi  scopus
    3. I. D. Remizov, “Approximations to the solution of Cauchy problem for a linear evolution equation via the space shift operator (second-order equation example)”, Appl. Math. Comput., 328 (2018), 243–246  crossref  mathscinet  isi  scopus
    4. V. Zh. Sakbaev, “Averaging of random flows of linear and nonlinear maps”, European Conference - Workshop Nonlinear Maps and Applications, Journal of Physics Conference Series, 990, IOP Publishing Ltd, 2018, UNSP 012012  crossref  isi  scopus
    5. I. D. Remizov, M. F. Starodubtseva, “Quasi-Feynman Formulas providing Solutions of Multidimensional Schrödinger Equations with Unbounded Potential”, Math. Notes, 104:5 (2018), 767–772  mathnet  crossref  crossref  isi  elib
    6. V. Zh. Sakbaev, D. V. Zavadsky, “Shift-invariant measures on infinite-dimensional spaces: integrable functions and random walks”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 160, no. 2, Izd-vo Kazanskogo un-ta, Kazan, 2018, 384–391  mathnet
    7. I. D. Remizov, “Explicit formula for evolution semigroup for diffusion in Hilbert space”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 21:4 (2018), 1850025  crossref  mathscinet  isi
    8. E. O. Kiktenko, “Asimmetriya lokalno dostupnoi i lokalno peredavaemoi informatsii v termalnom dvukhkubitnom sostoyanii”, Kvantovaya veroyatnost, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 151, VINITI RAN, M., 2018, 45–61  mathnet  mathscinet
    9. V. Zh. Sakbaev, “Polugruppy preobrazovanii prostranstva funktsii, kvadratichno integriruemykh po translyatsionno invariantnoi mere na banakhovom prostranstve”, Kvantovaya veroyatnost, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 151, VINITI RAN, M., 2018, 73–90  mathnet  mathscinet
    10. D. V. Zavadsky, V. Zh. Sakbaev, “Diffusion on a Hilbert Space Equipped with a Shift- and Rotation-Invariant Measure”, Proc. Steklov Inst. Math., 306 (2019), 102–119  mathnet  crossref  crossref  mathscinet  isi
    11. Yu. N. Orlov, V. Zh. Sakbaev, O. G. Smolyanov, “Feynman Formulas and the Law of Large Numbers for Random One-Parameter Semigroups”, Proc. Steklov Inst. Math., 306 (2019), 196–211  mathnet  crossref  crossref  mathscinet  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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