Brownian motion with respect to time-changing riemannian metrics, applications to Ricci flow

Koléhè A. Coulibaly-Pasquier

Annales de l'I.H.P. Probabilités et statistiques (2011)

  • Volume: 47, Issue: 2, page 515-538
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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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 47.2 (2011): 515-538. <http://eudml.org/doc/242349>.

@article{Coulibaly2011,
abstract = {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.},
author = {Coulibaly-Pasquier, Koléhè A.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Brownian motion; damped parallel transport; Ricci flow; Bismut formula; gradient estimates; heat equation},
language = {eng},
number = {2},
pages = {515-538},
publisher = {Gauthier-Villars},
title = {Brownian motion with respect to time-changing riemannian metrics, applications to Ricci flow},
url = {http://eudml.org/doc/242349},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Coulibaly-Pasquier, Koléhè A.
TI - Brownian motion with respect to time-changing riemannian metrics, applications to Ricci flow
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2011
PB - Gauthier-Villars
VL - 47
IS - 2
SP - 515
EP - 538
AB - 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.
LA - eng
KW - Brownian motion; damped parallel transport; Ricci flow; Bismut formula; gradient estimates; heat equation
UR - http://eudml.org/doc/242349
ER -

References

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