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Where did the Brownian particle go? (English) Zbl 0977.60071

Summary: 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 \(d \geq 3\), both the range and the terminal point of \(W\) can be recovered with probability 1 from \(\mu\). In particular, for \(d \geq 3\) 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 \(\mu\).

MSC:

60J65 Brownian motion
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