Given an n-dimensional compact manifold M, endowed with a family of Riemannian metrics , a Brownian motion depending on the deformation of the manifold (via the family of metrics) is defined. This tool enables a probabilistic view of certain geometric flows (e.g. Ricci flow, mean curvature flow). In particular, we give a martingale representation formula for a non-linear PDE over M, as well as a Bismut type formula for a geometric quantity which evolves under this flow. As application we present a gradient control formula for the heat equation over and a characterization of the Ricci flow in terms of the damped parallel transport.
Soit M une variété compacte de dimension n et une famille de métriques sur M, nous allons définir un -mouvement brownien, qui sera l'analogue d'un mouvement brownien sur une variété mais tenant compte de la déformation (c'est-à-dire de la famille de métriques ). Cet outil nous donnera une vision probabiliste de différents flots géométriques (e.g. flot de Ricci, flot de courbure moyenne). Nous donnerons aussi des formules de représentation en terme des martingales de solutions d'EDP non-linéaires sur M, ainsi que des formules du type Bismut pour des quantités géométriques évoluant le long d'un tel flot. Pour finir, nous donnerons comme application une formule de contrôle du gradient d'une solution de l'équation de la chaleur sur et une caractérisation du flot de Ricci en terme de transport parallèle déformé.
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Marc Arnaudon 1; Kolehe Abdoulaye Coulibaly 1; Anton Thalmaier 2
@article{CRMATH_2008__346_13-14_773_0, author = {Marc Arnaudon and Kolehe Abdoulaye Coulibaly and Anton Thalmaier}, title = {Brownian motion with respect to a metric depending on time; definition, existence and applications to {Ricci} flow}, journal = {Comptes Rendus. Math\'ematique}, pages = {773--778}, publisher = {Elsevier}, volume = {346}, number = {13-14}, year = {2008}, doi = {10.1016/j.crma.2008.05.004}, language = {en}, }
TY - JOUR AU - Marc Arnaudon AU - Kolehe Abdoulaye Coulibaly AU - Anton Thalmaier TI - Brownian motion with respect to a metric depending on time; definition, existence and applications to Ricci flow JO - Comptes Rendus. Mathématique PY - 2008 SP - 773 EP - 778 VL - 346 IS - 13-14 PB - Elsevier DO - 10.1016/j.crma.2008.05.004 LA - en ID - CRMATH_2008__346_13-14_773_0 ER -
%0 Journal Article %A Marc Arnaudon %A Kolehe Abdoulaye Coulibaly %A Anton Thalmaier %T Brownian motion with respect to a metric depending on time; definition, existence and applications to Ricci flow %J Comptes Rendus. Mathématique %D 2008 %P 773-778 %V 346 %N 13-14 %I Elsevier %R 10.1016/j.crma.2008.05.004 %G en %F CRMATH_2008__346_13-14_773_0
Marc Arnaudon; Kolehe Abdoulaye Coulibaly; Anton Thalmaier. Brownian motion with respect to a metric depending on time; definition, existence and applications to Ricci flow. Comptes Rendus. Mathématique, Volume 346 (2008) no. 13-14, pp. 773-778. doi : 10.1016/j.crma.2008.05.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.05.004/
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