Electronic Journal of Probability

Where Did the Brownian Particle Go?

Robin Pemantle, Yuval Peres, Jim Pitman, and Marc Yor

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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 $mu$.

Article information

Electron. J. Probab., Volume 6 (2001), paper no. 10, 22 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

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Pemantle, Robin; Peres, Yuval; Pitman, Jim; Yor, Marc. Where Did the Brownian Particle Go?. Electron. J. Probab. 6 (2001), paper no. 10, 22 pp. doi:10.1214/EJP.v6-83. https://projecteuclid.org/euclid.ejp/1461097640

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