Where did the Brownian particle go?

Details Where did the Brownian particle go? Where did the Brownian particle go?

2004-04-05

arxiv.org/abs/math/0404097v1

Elec. J. Prob vol. 6, paper 10 (2000)

Mathematics

Probability

26 pages

Scientific paper

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>=3, both the range and the terminal point of W can be recovered with probability 1 from mu. In particular, for d>=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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