Hitting a boundary point by diffusions in the closed half space
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.
scientific article; zbMATH DE number 3984261
scientific article