The Limiting Distribution of Brownian Motion in a Bounded Region with Instantaneous Return

Abstract

A point executes Brownian motion in a bounded, connected, and open three dimensional region $D$. When it reaches the boundary $\Gamma$, at point $\alpha$, it is instantaneously returned to $D$ according to probability measure $\mu(\alpha)$ (we write $\mu(\alpha, A)$ for the measure of set $A$), and the Brownian motion is resumed. This is a Markov process and, subject to certain regularity conditions on $\Gamma$ and $\mu(\alpha)$, we derive the limiting distribution of the process. Processes of this sort have been considered by Feller [1]; he has obtained the transition probabilities of such processes. He is concerned more generally with Markov processes with continuous sample functions on a linear interval; the return may be instantaneous or after a random period of time. Let $p^0(t, \xi, A)$ be the probability that the point is in set $A$ of $D$ at time $t$ when it is initially at point $\xi$ of $D$, with the additional restriction that no boundary contacts have been made. It is known that \begin{equation*}\tag{1}p^0(t, \xi, A) = \int_A u(t, \xi, x)dx,\end{equation*} where $dx$ is the volume element about $x$ and $u$ is the solution of the equation $$\frac{1}{2}\Delta u = u_t,$$ subject to the conditions $$u(t, \xi, \alpha) = 0,\quad\alpha\varepsilon\Gamma,\quad\lim_{t \rightarrow 0} \int_C u(t, \xi, x)dx = 1,$$ where $C$ is any sphere of non-zero radius with center $\xi$ which is entirely within $D$. We may write explicitly $$u(t,\xi,x) = \sum^\infty_{k = 1} v_k(\xi)v_k(x)e^{-\lambda_kt},$$ where $\lambda_k$ is the $k$th eigenvalue and $v_k(x)$ the corresponding eigenfunction of the equation $\Delta u + 2\lambda u = 0$ subject to the boundary condition $u = 0$ on $\Gamma$. If $K(\xi, x)$ is the Green's function of $\Delta u = 0$ in $D$, then ([2], and [3], page 273) \begin{equation*}\tag{2}K(\xi, x) = \frac{1}{2} \int^\infty_0 u(t, \xi, x) dt.\end{equation*} Let $\phi(t, \xi, \alpha)dt d\alpha$ be the probability the point is absorbed at surface element $d\alpha$ of $\Gamma$ between $t$ and $t + dt$ when it is initially at point $\xi$ of $D$. Then $\phi$ is half the interior normal derivative of $u$ at point $\alpha$ of $\Gamma$ ([3], page 273). When the point is initially at $\xi$ the probability of ultimate absorption in set $S$ of $\Gamma$ is given by \begin{equation*}\tag{3}\pi(\xi,S) = \int_S \int^\infty_0 \phi(t, \xi, \alpha) dt d\alpha.\end{equation*} We may define a discrete parameter Markov process with $\Gamma$ as state space by taking as transition probability \begin{equation*}\tag{4}\pi(\alpha, S) = \int_D \pi(\xi, S)\mu(\alpha, d\xi)\end{equation*} This Markov process has a limiting distribution $\pi$ which satisfies the equation \begin{equation*}\tag{5}\pi(S) = \int_\Gamma \pi(\alpha, S)\pi(d\alpha).\end{equation*} We define a measure of sets of $D$ by $$\lambda(A) = \int_\Gamma \mu(\alpha, A)\pi(d\alpha).$$ We may now write the density function for the limiting distribution. If $M(\xi)$ is the mean time of reaching the boundary when the point is initially at $\xi$, $$M(\xi) = \int_\Gamma \int^\infty_0 t_\phi(t, \xi, \alpha) dt d\alpha.$$ then the density function of the limiting distribution is \begin{equation*}\tag{6}\frac{2 \int_D K(\xi, x)\lambda(d\xi)}{\int_D M(\xi)\lambda(d\xi)}.\end{equation*} If we are given a probability measure $\lambda$ in $D$ and the return is always according to $\lambda$, then it is clear that the limiting density of this process is also given by (6). If $\lambda$ concentrates at a single point $\xi$ we may drop the integrals in (6), and in particular we get $$M(\xi) = 2 \int_D K(\xi, x)dx.$$ We note that (6) is essentially the steady distribution of temperature in the following problem: $D$ is a homogeneous heat conducting body whose boundary is kept at temperature 0 and in which there is a constant source of heat distributed according to $\lambda$. Regarding the regularity conditions, we shall assume that $\Gamma$ is made up of finitely many surfaces, each with a continuously turning tangent plane and that $D$ has a Green's function ([4], page 262). We will assume there is a closed set $B$ in $D$ such that $$\inf_{\alpha \varepsilon \Gamma} \mu(\alpha, B) = \gamma > 0.$$