L’objectif de cet article est de construire un mouvement brownien à valeurs dans une variété riemannienne conjointement avec un processus à valeurs ensembles , de telle sorte qu’au moins pour tout temps d’arrêt assez petit dans la filtration engendrée par , la loi de conditionnée par est la mesure riemannienne conditionnée sur . 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.
The purpose of this paper is to construct a Brownian motion taking values in a Riemannian manifold , together with a compact set-valued process such that, at least for small enough -stopping time and conditioned by , the law of is the normalized Lebesgue measure on . 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 play an important role. Several examples with moving intervals, discs, annuli, symmetric convex sets are investigated.
@article{JEP_2024__11__473_0, author = {Marc Arnaudon and Kol\'eh\`e Coulibaly-Pasquier and Laurent Miclo}, title = {Couplings of {Brownian} motions with set-valued dual processes on {Riemannian} manifolds}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {473--522}, publisher = {\'Ecole polytechnique}, volume = {11}, year = {2024}, doi = {10.5802/jep.258}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.258/} }
TY - JOUR AU - Marc Arnaudon AU - Koléhè Coulibaly-Pasquier AU - Laurent Miclo TI - Couplings of Brownian motions with set-valued dual processes on Riemannian manifolds JO - Journal de l’École polytechnique — Mathématiques PY - 2024 SP - 473 EP - 522 VL - 11 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.258/ DO - 10.5802/jep.258 LA - en ID - JEP_2024__11__473_0 ER -
%0 Journal Article %A Marc Arnaudon %A Koléhè Coulibaly-Pasquier %A Laurent Miclo %T Couplings of Brownian motions with set-valued dual processes on Riemannian manifolds %J Journal de l’École polytechnique — Mathématiques %D 2024 %P 473-522 %V 11 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.258/ %R 10.5802/jep.258 %G en %F JEP_2024__11__473_0
Marc Arnaudon; Koléhè Coulibaly-Pasquier; Laurent Miclo. Couplings of Brownian motions with set-valued dual processes on Riemannian manifolds. Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 473-522. doi : 10.5802/jep.258. https://jep.centre-mersenne.org/articles/10.5802/jep.258/
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