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Teor. Veroyatnost. i Primenen., 2017, Volume 62, Issue 3, Pages 446–467 (Mi tvp5121)  

This article is cited in 4 scientific papers (total in 4 papers)

Initial-boundary value problems in a bounded domain: probabilistic representations of solutions and limit theorems. II

I. A. Ibragimovab, N. V. Smorodinaba, M. M. Faddeeva

a Saint Petersburg State University
b St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: The paper puts forward a new method of construction of a probabilistic representation of solutions to initial-boundary value problems for a number of evolution equations (in particular, for the Schrödinger equation) in a bounded subdomain of $\mathbb R^2$ with smooth boundary. Our method is based on the construction of a special extension of the initial function from the domain to the entire plane. For problems with Neumann boundary condition, this method produces a new approach to the construction of a Wiener process “reflected from the boundary,” which was first introduced by A. V. Skorokhod.

Keywords: initial-boundary value problems, evolution equations, Schrödinger equation, limit theorems, Skorokhod problem, Feynman integral, Feynman measure.

Funding Agency Grant Number
Russian Foundation for Basic Research 16-01-00258
15-01-01453
16-01-00443
Russian Academy of Sciences - Federal Agency for Scientific Organizations
The first author was supported by the “Contemporary Problems of Theoretical Mathematics” program of the Division of Mathematical Sciences of the Russian Academy of Sciences and by the Russian Foundation for Basic Research (grant 16-01-00258). The second author was supported by the “Contemporary Problems of Theoretical Mathematics” program of the Division of Mathematical Sciences of the Russian Academy of Sciences and by the Russian Foundation for Basic Research (grant 15-01-01453). The third author was supported by the Russian Foundation for Basic Research (grant 16-01-00443).


DOI: https://doi.org/10.4213/tvp5121

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English version:
Theory of Probability and its Applications, 2018, 62:3, 356–372

Bibliographic databases:

Received: 06.02.2017
Accepted:20.02.2017

Citation: I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev, “Initial-boundary value problems in a bounded domain: probabilistic representations of solutions and limit theorems. II”, Teor. Veroyatnost. i Primenen., 62:3 (2017), 446–467; Theory Probab. Appl., 62:3 (2018), 356–372

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. P. N. Ievlev, “Veroyatnostnye predstavleniya dlya reshenii nachalno-kraevykh zadach dlya uravneniya Shrëdingera v $d$-mernom share”, Veroyatnost i statistika. 27, Zap. nauchn. sem. POMI, 474, POMI, SPb., 2018, 149–170  mathnet
    2. Ibragimov I.A., Smorodina N.V., Faddeev M.M., “A Construction of Reflecting Levy Processes”, Dokl. Math., 99:1 (2019), 71–74  crossref  isi
    3. I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev, “Reflecting Lévy processes and associated families of linear operators”, Theory Probab. Appl., 64:3 (2019), 335–354  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. P. N. Ievlev, “Brounovskoe dvizhenie s otrazheniem v $d$-mernom share”, Veroyatnost i statistika. 28, Zap. nauchn. sem. POMI, 486, POMI, SPb., 2019, 158–177  mathnet
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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