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Where Did the Brownian Particle Go?

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Where Did the Brownian Particle Go?
2.	Creator	Author's name, affiliation, country	Robin Pemantle; Ohio State University
2.	Creator	Author's name, affiliation, country	Yuval Peres; University of California, Berkeley
2.	Creator	Author's name, affiliation, country	Jim Pitman; University of California, Berkeley
2.	Creator	Author's name, affiliation, country	Marc Yor; Université Pierre et Marie Curie
3.	Subject	Discipline(s)	Mathematics
3.	Subject	Keyword(s)	Brownian motion, conditional distribution of a path given its occupation measure, radial projection.
3.	Subject	Subject classification	60J65.
4.	Description	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 $mu$.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2001-01-10
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/83
10.	Identifier	Digital Object Identifier	10.1214/EJP.v6-83
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 6
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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