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Let $\mathbb S^1$ be the unit circle and $\Delta$ be the Laplace-Beltrami operator on $\mathbb S^1$ which is an infinitesimal generator of the correspondent Markov semigroup $P_t$. Is the explicit distribution of this Markov process known?

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You should check blms.oxfordjournals.org/content/28/6/643. –  Jon Dec 5 '11 at 15:16

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up vote 3 down vote accepted

Brownian motion on the circle can be written as $X_t = e^{i B_t}$ where $B_t$ is a standard one-dimensional Brownian motion.

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Thank a lot, Nate –  Ilya Dec 9 '11 at 15:44

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