The Annals of Probability

Connecting Brownian Paths

Burgess Davis and Thomas S. Salisbury

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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.

Article information

Ann. Probab., Volume 16, Number 4 (1988), 1428-1457.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60J65: Brownian motion [See also 58J65]

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


Davis, Burgess; Salisbury, Thomas S. Connecting Brownian Paths. Ann. Probab. 16 (1988), no. 4, 1428--1457. doi:10.1214/aop/1176991577.

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