Where Did The Brownian Particle Go?

August, 1999
Report Number: 
564
Authors: 
Robin Pemantle and Yuval Peres and Jim Pitman and Marc Yor
Citation: 
Electronic Journal of Probability, Vol. 6 (2001) Paper no. 10, pages 1-22
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 $A \subseteq S^{d-1}$, 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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