Couplings of Brownian motions with set-valued dual processes on Riemannian manifolds

Arnaudon, Marc; Coulibaly-Pasquier, Koléhè; Miclo, Laurent

Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 473-522

The purpose of this paper is to construct a Brownian motion ${\left(X_t\right)}_{t\ge 0}$ taking values in a Riemannian manifold $M$, together with a compact set-valued process ${\left(D_t\right)}_{t\ge 0}$ such that, at least for small enough ${ℱ}^D$-stopping time $\tau >0$ and conditioned by ${ℱ}_{\tau }^D$, the law of $X_{\tau }$ is the normalized Lebesgue measure on $D_{\tau }$. 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 $D_t$ play an important role. Several examples with moving intervals, discs, annuli, symmetric convex sets are investigated.

L’objectif de cet article est de construire un mouvement brownien ${\left(X_t\right)}_{t\ge 0}$ à valeurs dans une variété riemannienne $M$ conjointement avec un processus à valeurs ensembles ${\left(D_t\right)}_{t\ge 0}$, de telle sorte qu’au moins pour tout temps d’arrêt $\tau >0$ assez petit dans la filtration ${ℱ}^D$ engendrée par ${\left(D_t\right)}_{t\ge 0}$, la loi de $X_{\tau }$ conditionnée par ${ℱ}_{\tau }^D$ est la mesure riemannienne conditionnée sur $D_{\tau }$. Ce résultat d’entrelacement est une généralisation du théorème de Pitman. Nous commençons par construire des processus entrelacés réguliers par le biais du théorème de Stokes. Puis en utilisant différentes procédures de limites, nous construisons des processus entrelacés synchrones, libres et miroirs. Les temps locaux du mouvement brownien sur le squelette (morphologique) ou sur la frontière jouent des rôles importants. Nous étudions plusieurs exemples consistant en des intervalles, des disques, des anneaux et des ensembles convexes symétriques.

Classification : 60J60, 60J65, 60H10, 58J65, 53E10, 60J55, 35K93
Keywords: Brownian motions on Riemannian manifolds, intertwining relations, set-valued dual processes, couplings of primal and dual processes, stochastic mean curvature evolutions, boundary and skeleton local times, generalized Pitman theorem
Mots-clés : Mouvements browniens sur des variétés riemanniennes, relations d’entrelacement, processus duaux à valeurs ensembles, couplage de processus primaux et duaux, évolutions par courbure moyenne stochastiques, temps locaux sur le squelette et sur la frontière, généralisation du théorème de Pitman

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