Hitting a boundary point by diffusions in the closed half space
type: Item
instance of: scholarly article
title: Hitting a boundary point by diffusions in the closed half space
zbMATH Open document ID: 0608.60073
DOI: 10.1016/0047-259X(86)90024-2
author: B. George
published in: Journal of Multivariate Analysis
publication date: 1986-01-01T00:00:00Z
review text: For \(d\geq 2\) let \(G=\{x=(x_ 1,...,x_ d):x_ 1>0\}\), \(\partial G=\{x\in {\mathbb{R}}^ d:x_ 1=0\}\), and consider a diffusion \((X_ t,P_ x)\) in \(G\cup \partial G\) with certain Wentzell-type boundary conditions on \(\partial G\). For \(z\in \partial G\) let \(\partial B(z,\delta)=\{x:| x-z| =\delta \}\) and \(\tau_{\delta}=\inf \{t: X_ t=\partial B(z,\delta)\}\). The main result of the paper is that for all x such that \(0<| x-z| \leq \delta\) \[ \lim_{c\downarrow 0}P_ x(\tau_{\delta}<\tau_ c)=1. \] This is proved under two different sets of conditions for the generators of X. In the first case it is assumed that the second order terms in the boundary condition vanish. Under this assumption it is seen that the problem can be reduced to the case of normal reflection.
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 3984261
zbMATH Keywords: hitting a point
zbMATH Keywords: Wentzell-type boundary conditions
zbMATH Keywords: reflection
reviewed by: Paavo Salminen
MaRDI profile type: Publication
cites work: Criteria for recurrence and existence of invariant measures for multidimensional diffusions
cites work: Recurrence and ergodicity of diffusions
cites work: Q4089589
cites work: Q5585820
cites work: On the existence of solutions of stochastic differential equations with boundary conditions
cites work: On the uniqueness of solutions of stochastic differential equations with boundary conditions
cites work: Diffusion processes with boundary conditions
cites work: Q3862204
cites work: On stochastic differential equations for multi-dimensional diffusion processes with boundary conditions. II
full work available at URL: 0047-259x(86)90024-2
OpenAlex ID: W2011694656
Recommended article: Hitting half-spaces by Bessel-Brownian diffusions
Recommended article: Hitting half-spaces or spheres by Ornstein–Uhlenbeck type diffusions
Recommended article: On hitting single points by a multidimensional diffusion
Recommended article: On boundary crossing probabilities for diffusion processes
Recommended article: Q4028967
Recommended article: Crossing probabilities for diffusion processes with piecewise continuous boundaries
Recommended article: Boundary traces of shift-invariant diffusions in half-plane
Recommended article: Boundary behavior of diffusion approximations to Markov jump processes
Recommended article: Q4955447
description: scientific article; zbMATH DE number 3984261