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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