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.
Publié le : 1979-02-14
Classification:
Brownian motion,
exit distribution,
balayage,
inverse exit problem,
inverse balayage problem,
inverse problem of potential theory,
harmonic function,
extreme point,
stopping time,
60J65,
60J45,
60G40,
31B20
@article{1176995164,
author = {Karr, A. F. and Pittenger, A. O.},
title = {An Inverse Balayage problem for Brownian Motion},
journal = {Ann. Probab.},
volume = {7},
number = {6},
year = {1979},
pages = { 186-191},
language = {en},
url = {http://dml.mathdoc.fr/item/1176995164}
}
Karr, A. F.; Pittenger, A. O. An Inverse Balayage problem for Brownian Motion. Ann. Probab., Tome 7 (1979) no. 6, pp. 186-191. http://gdmltest.u-ga.fr/item/1176995164/