Arnaudon, Marc, Miclo, Laurent and Coulibaly-Pasquier, Abdoulaye Koléhè (2024) Couplings of Brownian motions with set-valued dual processes on Riemannian manifolds. Journal de l'École polytechnique — Mathématiques, vol. 11. pp. 473-522.

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Abstract

The purpose of this paper is to construct a Brownian motion (Xt)t⩾0 taking values in a Riemannian manifold M , together with a compact set-valued process (Dt)t⩾0 such that, at least for small enough F D -stopping time τ > 0 and conditioned by F Dτ , the law of Xτ is the normalized Lebesgue measure on Dτ . This intertwining result is a generalization of Pitman’s theorem. We first construct regular intertwined processes related to Stokes’ theorem.
Then using several limiting procedures we construct synchronous intertwined, free intertwined,mirror intertwined processes. The local times of the Brownian motion on the (morphological) skeleton or the boundary of each Dt play an important role. Several examples with moving intervals, discs, annuli, symmetric convex sets are investigated.