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2001 Where Did the Brownian Particle Go?
Robin Pemantle, Yuval Peres, Jim Pitman, Marc Yor
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Electron. J. Probab. 6: 1-22 (2001). DOI: 10.1214/EJP.v6-83

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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Robin Pemantle. Yuval Peres. Jim Pitman. Marc Yor. "Where Did the Brownian Particle Go?." Electron. J. Probab. 6 1 - 22, 2001. https://doi.org/10.1214/EJP.v6-83

Information

Accepted: 10 January 2001; Published: 2001
First available in Project Euclid: 19 April 2016

zbMATH: 0977.60071
MathSciNet: MR1831805
Digital Object Identifier: 10.1214/EJP.v6-83

Subjects:
Primary: 60J65

Keywords: Brownian motion , conditional distribution of a path given its occupation measure , radial projection

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