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

This article is cited in 7 scientific papers (total in 7 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. Zavadskii, “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  mathscinet  isi
    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
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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