Brownian motion with respect to time-changing riemannian metrics, applications to Ricci flow
Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 2, p. 515-538

We generalize brownian motion on a riemannian manifold to the case of a family of metrics which depends on time. Such questions are natural for equations like the heat equation with respect to time dependent laplacians (inhomogeneous diffusions). In this paper we are in particular interested in the Ricci flow which provides an intrinsic family of time dependent metrics. We give a notion of parallel transport along this brownian motion, and establish a generalization of the Dohrn-Guerra or damped parallel transport, Bismut integration by part formulas, and gradient estimate formulas. One of our main results is a characterization of the Ricci flow in terms of the damped parallel transport. At the end of the paper we give a canonical definition of the damped parallel transport in terms of stochastic flows, and derive an intrinsic martingale which may provide information about singularities of the flow.

Nous généralisons la notion de mouvement brownien sur une variété au cas du mouvement brownien dépendant d'une famille de métriques. Cette généralisation est naturelle quand on s'intéresse aux équations de la chaleur avec un laplacien qui dépend du temps, ou de manière générale dans le cadre de diffusions in-homogènes. Dans cet article, nous nous sommes particulièrement intéressés au flot de Ricci, flot géométrique fournissant une famille intrinsèque de métriques. Nous donnons une notion de transport parallèle le long d'un tel processus, puis nous généralisons celle du transport parallèle déformé, et donnons une formule d'intégration par parties à la Bismut dont nous tirons des formules de contrôle de norme de gradients de solutions d'équation de la chaleur in-homogène. Un des résultats principaux de cet article est une caractérisation probabiliste du flot de Ricci, en terme du transport parallèle déformé. Dans les dernières sections, nous donnons une définition canonique du transport parallèle déformé en utilisant le flot stochastique, et nous en dérivons une martingale intrinsèque, qui pourrait donner des informations sur les singularités du flot.

@article{AIHPB_2011__47_2_515_0,
     author = {Coulibaly-Pasquier, Kol\'eh\`e A.},
     title = {Brownian motion with respect to time-changing riemannian metrics, applications to Ricci flow},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {2},
     year = {2011},
     pages = {515-538},
     doi = {10.1214/10-AIHP364},
     zbl = {1222.58030},
     mrnumber = {2814421},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2011__47_2_515_0}
}
Coulibaly-Pasquier, Koléhè A. Brownian motion with respect to time-changing riemannian metrics, applications to Ricci flow. Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 2, pp. 515-538. doi : 10.1214/10-AIHP364. http://www.numdam.org/item/AIHPB_2011__47_2_515_0/

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