## The Annals of Probability

### Conditional Brownian Motion in Rapidly Exhaustible Domains

Neil Falkner

#### Abstract

Let $D$ be a domain in $\mathbb{R}^d$ and let $\Delta_1$ be the set of minimal points of the Martin boundary of $D$. For $x \in D$ and $z \in \Delta_1$, let $(X_t)$ under the law $P^{x; z}$ be Brownian motion in $D$, starting at $x$ and conditioned to converge to $z$. Let $\tau$ be the lifetime of $(X_t)$, so $X_{\tau-} = z P^{x; z}$ a.s. Let $q \in L^p(D)$ for some $p > d/2$. Under the assumption that $D$ is what we call rapidly exhaustible, which is essentially a very weak boundary smoothness condition, we show that if the quantity $E^{x;z}\big\{\exp\big\lbrack\int^\tau_0 q(X_s) ds\big\rbrack\big\}$ is finite for one $x \in D$ and one $z \in \Delta_1$, then this quantity is bounded on $D \times \Delta_1$. This result may be viewed as saying, in a fairly strong sense, that the amount of time $(X_t)$ spends in each part of $D$ does not depend very much on the minimal Martin boundary point $z$ to which $(X_t)$ is conditioned to converge.

#### Article information

Source
Ann. Probab., Volume 15, Number 4 (1987), 1501-1514.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176991989

Digital Object Identifier
doi:10.1214/aop/1176991989

Mathematical Reviews number (MathSciNet)
MR905344

Zentralblatt MATH identifier
0627.60068

JSTOR