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JOURNALS // Teoriya Veroyatnostei i ee Primeneniya // Archive

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

This article is cited in 5 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.

Received: 06.02.2017
Accepted: 20.02.2017

DOI: 10.4213/tvp5121


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

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