RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 TMF: Year: Volume: Issue: Page: Find

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

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

Full text: PDF file (587 kB)
First page: PDF file
References: PDF file   HTML file

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

Bibliographic databases:

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
\Bibitem{Sak17} \by V.~Zh.~Sakbaev \paper Averaging of random walks and shift-invariant measures on a~Hilbert space \jour TMF \yr 2017 \vol 191 \issue 3 \pages 473--502 \mathnet{http://mi.mathnet.ru/tmf9153} \crossref{https://doi.org/10.4213/tmf9153} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3662473} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2017TMP...191..886S} \elib{http://elibrary.ru/item.asp?id=29255339} \transl \jour Theoret. and Math. Phys. \yr 2017 \vol 191 \issue 3 \pages 886--909 \crossref{https://doi.org/10.1134/S0040577917060083} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000404743900008} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85021658827} 

• http://mi.mathnet.ru/eng/tmf9153
• https://doi.org/10.4213/tmf9153
• http://mi.mathnet.ru/eng/tmf/v191/i3/p473

 SHARE:

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
2. Remizov I.D., “Feynman and Quasi-Feynman Formulas For Evolution Equations”, Dokl. Math., 96:2 (2017), 433–437
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
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
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
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
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
•  Number of views: This page: 295 References: 41 First page: 26