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 Electronic Journal of Probability > Vol. 6 (2001) > Paper 10 open journal systems 


Where Did the Brownian Particle Go?

Robin Pemantle, Ohio State University
Yuval Peres, University of California, Berkeley
Jim Pitman, University of California, Berkeley
Marc Yor, Université Pierre et Marie Curie


Abstract
Consider the radial projection onto the unit sphere of the path a $d$-dimensional Brownian motion $W$, started at the center of the sphere and run for unit time. Given the occupation measure $mu$ of this projected path, what can be said about the terminal point $W(1)$, or about the range of the original path? In any dimension, for each Borel set $A$ in $S^{d-1}$, the conditional probability that the projection of $W(1)$ is in $A$ given $mu(A)$ is just $mu(A)$. Nevertheless, in dimension $d ge 3$, both the range and the terminal point of $W$ can be recovered with probability 1 from $mu$. In particular, for $d ge 3$ the conditional law of the projection of $W(1)$ given $mu$ is not $mu$. In dimension 2 we conjecture that the projection of $W(1)$ cannot be recovered almost surely from $mu$, and show that the conditional law of the projection of $W(1)$ given $mu$ is not $mu$.


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Pages: 1-22

Published on: January 10, 2001
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Electronic Journal of Probability. ISSN: 1083-6489