Probability Theory
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.

Given an n-dimensional compact manifold M, endowed with a family of Riemannian metrics g(t), a Brownian motion depending on the deformation of the manifold (via the family g(t) 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 (M,g(t)) and a characterization of the Ricci flow in terms of the damped parallel transport.

Soit M une variété compacte de dimension n et g(t) une famille de métriques sur M, nous allons définir un g(t)-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 g(t)). 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 (M,g(t)) et une caractérisation du flot de Ricci en terme de transport parallèle déformé.

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DOI: 10.1016/j.crma.2008.05.004
Arnaudon, Marc 1; Coulibaly, Kolehe Abdoulaye 1; Thalmaier, Anton 2

1 Laboratoire de Mathématiques et Applications (UMR6086), Université de Poitiers, Téléport 2, BP 30179, 86962 Futuroscope Chasseneuil cedex, France
2 Unité de recherche en mathématiques, Université du Luxembourg, 162A, avenue de la Faïencerie, L-1511 Luxembourg, Grand-Duché de Luxembourg
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Arnaudon, Marc; Coulibaly, Kolehe Abdoulaye; Thalmaier, Anton. 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. http://www.numdam.org/articles/10.1016/j.crma.2008.05.004/

[1] Arnaudon, M.; Driver, B.K.; Thalmaier, A. Gradient estimates for positive harmonic functions by stochastic analysis, Stochastic Process. Appl., Volume 117 (2007) no. 2, pp. 202-220

[2] Chow, B.; Knopf, D. The Ricci Flow: An Introduction, Mathematical Surveys and Monographs, vol. 110, American Mathematical Society, Providence, RI, 2004

[3] Elworthy, K.D.; Li, X.-M. Formulae for the derivatives of heat semigroups, J. Funct. Anal., Volume 125 (1994) no. 1, pp. 252-286

[4] Elworthy, K.D.; Yor, M. Conditional expectations for derivatives of certain stochastic flows, Séminaire de Probabilités, XXVII, Lecture Notes in Math., vol. 1557, Springer, Berlin, 1993, pp. 159-172

[5] Émery, M. Stochastic Calculus in Manifolds, Springer-Verlag, Berlin, 1989 (With an appendix by P.-A. Meyer)

[6] Hsu, E.P. Stochastic Analysis on Manifolds, Graduate Studies in Mathematics, vol. 38, American Mathematical Society, Providence, RI, 2002

[7] Huisken, G. Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., Volume 20 (1984) no. 1, pp. 237-266

[8] Malliavin, P. Stochastic Analysis, Grundlehren der Mathematischen Wissenschaften, vol. 313, Springer, Berlin, 1997

[9] Thalmaier, A.; Wang, Feng-Yu Gradient estimates for harmonic functions on regular domains in Riemannian manifolds, J. Funct. Anal., Volume 155 (1998) no. 1, pp. 109-124

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