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Teoriya Veroyatnostei i ee Primeneniya, 2022, Volume 67, Issue 2, Pages 247–263
DOI: https://doi.org/10.4213/tvp5428
(Mi tvp5428)
 

Distribution density of the first exit point of a two-dimensional diffusion process from a circle neighborhood of its initial point: the inhomogeneous case

B. P. Harlamov

Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg
References:
Abstract: A two-dimensional diffusion process is considered. The distribution of the first exit point of such a process from an arbitrary domain of its values is determined, as a function of the initial point of the process, by an elliptic second-order differential equation, and corresponds to the solution of the Dirichlet problem for this equation (the case of nonconstant coefficients). We examine the distribution density of the first exit point of the process from the small circular neighborhood of its initial point and study its relation to the Dirichlet problem. In terms of this asymptotics, we prove two theorems, which provide sufficient conditions and necessary conditions for the distribution of the first exit point, as a function of the initial point of the process, to satisfy a certain second-order elliptic differential equation corresponding to the standard Wiener process with drift and break. The removable second-order terms of the expansion in powers of the radius of the small circular neighborhood of the initial point of the process are identified. In terms of the removable terms, these two theorems are combined as a single theorem giving a necessary and sufficient condition for correspondence to this Wiener process.
Keywords: Green function, Dirichlet problem, Poisson kernel, integral equation, iteration.
Received: 08.08.2020
Accepted: 05.11.2020
English version:
Theory of Probability and its Applications, 2022, Volume 67, Issue 2, Pages 194–207
DOI: https://doi.org/10.1137/S0040585X97T990873
Bibliographic databases:
Document Type: Article
Language: Russian
Citation: B. P. Harlamov, “Distribution density of the first exit point of a two-dimensional diffusion process from a circle neighborhood of its initial point: the inhomogeneous case”, Teor. Veroyatnost. i Primenen., 67:2 (2022), 247–263; Theory Probab. Appl., 67:2 (2022), 194–207
Citation in format AMSBIB
\Bibitem{Har22}
\by B.~P.~Harlamov
\paper Distribution density of the first exit point of a~two-dimensional diffusion process from a~circle neighborhood of its initial point: the inhomogeneous case
\jour Teor. Veroyatnost. i Primenen.
\yr 2022
\vol 67
\issue 2
\pages 247--263
\mathnet{http://mi.mathnet.ru/tvp5428}
\crossref{https://doi.org/10.4213/tvp5428}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4466429}
\zmath{https://zbmath.org/?q=an:7573880}
\transl
\jour Theory Probab. Appl.
\yr 2022
\vol 67
\issue 2
\pages 194--207
\crossref{https://doi.org/10.1137/S0040585X97T990873}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85165255966}
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  • https://www.mathnet.ru/eng/tvp5428
  • https://doi.org/10.4213/tvp5428
  • https://www.mathnet.ru/eng/tvp/v67/i2/p247
  • Citing articles in Google Scholar: Russian citations, English citations
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    Теория вероятностей и ее применения Theory of Probability and its Applications
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