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October, 1988 Connecting Brownian Paths
Burgess Davis, Thomas S. Salisbury
Ann. Probab. 16(4): 1428-1457 (October, 1988). DOI: 10.1214/aop/1176991577


We study two processes obtained as follows: Take two independent $d$-dimensional Brownian motions started at points $x, y$, respectively. For the first process, let $d \geq 3$ and condition on $X_t = Y_t$ for some $t$ (a set of probability 0). Run $X$ out to the point of intersection and then run $Y$ in reversed time from this point back to $y$. For the second process, let $d \geq 5$ and perform the same construction, this time conditioning on $X_s = Y_t$ for some $s, t$. The first process is shown to be Doob's conditioned (to go from $x$ to $y$) Brownian motion $Z$, and the second has distribution absolutely continuous with respect to that of $Z$, the Radon-Nikodym density being a constant times the time $Z$ takes to travel from $x$ to $y$. Similar results (including extensions to the critical dimensions $d = 2$ and $d = 4$) are obtained by conditioning the motions to hit before they leave domains. We use the asymptotics of the probability of "near misses" and results on the weak convergence of $h$-transforms.


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Burgess Davis. Thomas S. Salisbury. "Connecting Brownian Paths." Ann. Probab. 16 (4) 1428 - 1457, October, 1988.


Published: October, 1988
First available in Project Euclid: 19 April 2007

zbMATH: 0658.60111
MathSciNet: MR958196
Digital Object Identifier: 10.1214/aop/1176991577

Primary: 60J65

Keywords: $h$-transforms , bi-Brownian motion , Conditioned Brownian motion , path intersections , Wiener sausage

Rights: Copyright © 1988 Institute of Mathematical Statistics


Vol.16 • No. 4 • October, 1988
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