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February, 1979 An Inverse Balayage problem for Brownian Motion
A. F. Karr, A. O. Pittenger
Ann. Probab. 7(1): 186-191 (February, 1979). DOI: 10.1214/aop/1176995164


Let $B$ be a standard $n$-dimensional Brownian motion, let $A$ be compact and let $\nu$ be a probability measure on $\partial A$. We treat the following inverse exit problem: describe the set $M(\nu)$ of all probability measures $\mu$ on $A$ such that $P^\mu\{B(T)\in \cdot\} = \nu(\cdot)$, where $T$ is the time of first exit from $A$. Elements of $M(\nu)$ are characterized in terms of integrals of harmonic functions with respect to them. For $n = 1$, extreme points of $M(\nu)$ are computed in closed form; for $n \geqslant 2$, extreme points of $M(\nu)$ are characterized. Geophysical and potential-theoretic aspects of the problem are discussed.


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A. F. Karr. A. O. Pittenger. "An Inverse Balayage problem for Brownian Motion." Ann. Probab. 7 (1) 186 - 191, February, 1979.


Published: February, 1979
First available in Project Euclid: 19 April 2007

zbMATH: 0392.60059
MathSciNet: MR515829
Digital Object Identifier: 10.1214/aop/1176995164

Primary: 60J65
Secondary: 31B20 , 60G40 , 60J45

Keywords: balayage , Brownian motion , exit distribution , extreme point , Harmonic function , inverse balayage problem , inverse exit problem , inverse problem of potential theory , stopping time

Rights: Copyright © 1979 Institute of Mathematical Statistics

Vol.7 • No. 1 • February, 1979
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