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October, 1987 Conditional Brownian Motion in Rapidly Exhaustible Domains
Neil Falkner
Ann. Probab. 15(4): 1501-1514 (October, 1987). DOI: 10.1214/aop/1176991989


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.


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Neil Falkner. "Conditional Brownian Motion in Rapidly Exhaustible Domains." Ann. Probab. 15 (4) 1501 - 1514, October, 1987.


Published: October, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0627.60068
MathSciNet: MR905344
Digital Object Identifier: 10.1214/aop/1176991989

Primary: 60J45
Secondary: 60J65

Keywords: $h$-path Brownian motion , conditional gauge

Rights: Copyright © 1987 Institute of Mathematical Statistics


Vol.15 • No. 4 • October, 1987
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