Distribution density of the first exit point of a~two-dimensional diffusion process from a~circle neighborhood of its initial point: the inhomogeneous case
Teoriâ veroâtnostej i ee primeneniâ, Tome 67 (2022) no. 2, pp. 247-263
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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, integral equation, iteration.
Mots-clés : Poisson kernel
Mots-clés : Poisson kernel
@article{TVP_2022_67_2_a2,
author = {B. P. Harlamov},
title = {Distribution density of the first exit point of a~two-dimensional diffusion process from a~circle neighborhood of its initial point: the inhomogeneous case},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {247--263},
publisher = {mathdoc},
volume = {67},
number = {2},
year = {2022},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_2022_67_2_a2/}
}
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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. Teoriâ veroâtnostej i ee primeneniâ, Tome 67 (2022) no. 2, pp. 247-263. http://geodesic.mathdoc.fr/item/TVP_2022_67_2_a2/