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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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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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Check in the journal of applied functional analysis vol7, page 26 . The article is the random motion on the sphere generated by the Laplace Beltrami Operator. The authors are V.G.Papanicolaou and D. Kouloumpou. Is my phd work

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