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

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

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

## Access Full Article

top## Abstract

top## How to cite

topCoulibaly-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

top- [1] M. Arnaudon and A. Thalmaier. Complete lifts of connections and stochastic Jacobi fields. J. Math. Pures Appl. (9) 77 (1998) 283–315. Zbl0916.58045MR1618537
- [2] M. Arnaudon and A. Thalmaier. Stability of stochastic differential equations in manifolds. In Séminaire de Probabilités, XXXII 188–214. Lecture Notes in Math. 1686. Springer, Berlin, 1998. Zbl0918.60040MR1655151
- [3] M. Arnaudon and A. Thalmaier. Horizontal martingales in vector bundles. In Séminaire de Probabilités, XXXVI 419–456. Lecture Notes in Math. 1801. Springer, Berlin, 2003. Zbl1046.58013MR1971603
- [4] M. Arnaudon, R. O. Bauer and A. Thalmaier. A probabilistic approach to the Yang–Mills heat equation. J. Math. Pures Appl. (9) 81 (2002) 143–166. Zbl1042.58021MR1994607
- [5] B. Chow and D. Knopf. The Ricci Flow: An Introduction. Mathematical Surveys and Monographs 110. Amer. Math. Soc., Providence, RI, 2004. Zbl1086.53085MR2061425
- [6] M. Cranston. Gradient estimates on manifolds using coupling. J. Funct. Anal. 99 (1991) 110–124. Zbl0770.58038MR1120916
- [7] D. M. DeTurck. Deforming metrics in the direction of their Ricci tensors. J. Differential Geom. 18 (1983) 157–162. Zbl0517.53044MR697987
- [8] K. D. Elworthy and X.-M. Li. Formulae for the derivatives of heat semigroups. J. Funct. Anal. 125 (1994) 252–286. Zbl0813.60049MR1297021
- [9] K. D. Elworthy and M. Yor. Conditional expectations for derivatives of certain stochastic flows. In Séminaire de Probabilités, XXVII 159–172. Lecture Notes in Math. 1557. Springer, Berlin, 1993. Zbl0795.60046MR1308561
- [10] K. D. Elworthy, Y. Le Jan and X.-M. Li. On the Geometry of Diffusion Operators and Stochastic Flows. Lecture Notes in Math. 1720. Springer, Berlin, 1999. Zbl0942.58004MR1735806
- [11] M. Emery. Une topologie sur l’espace des semimartingales. In Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78) 260–280. Lecture Notes in Math. 721. Springer, Berlin, 1979. Zbl0406.60057MR544800
- [12] M. Émery. Stochastic Calculus in Manifolds: With an Appendix by P.-A. Meyer. Springer, Berlin, 1989. Zbl0697.60060MR1030543
- [13] M. Émery. On two transfer principles in stochastic differential geometry. In Séminaire de Probabilités, XXIV, 1988/89 407–441. Lecture Notes in Math. 1426. Springer, Berlin, 1990. Zbl0704.60066MR1071558
- [14] R. S. Hamilton. Three-manifolds with positive Ricci curvature. J. Differential Geom. 17 (1982) 255–306. Zbl0504.53034MR664497
- [15] E. P. Hsu. Stochastic Analysis on Manifolds Graduate Studies in Mathematics 38. Amer. Math. Soc., Providence, RI, 2002. Zbl0994.58019MR1882015
- [16] J. Jost. Harmonic Mappings between Riemannian Manifolds. Proceedings of the Centre for Mathematical Analysis, Australian National University 4. Australian National University Centre for Mathematical Analysis, Canberra, 1984. Zbl0542.58001MR756629
- [17] J. Jost. Riemannian Geometry and Geometric Analysis, 4th edition. Springer, Berlin, 2005. Zbl0828.53002MR2165400
- [18] W. S. Kendall. Nonnegative Ricci curvature and the Brownian coupling property. Stochastics 19 (1986) 111–129. Zbl0584.58045MR864339
- [19] S. Kobayashi and K. Nomizu. Foundations of Differential Geometry. Vol. I. Wiley Classics Library. Wiley, New York, 1996. Reprint of the 1963 original, Wiley. Zbl0119.37502MR1393940
- [20] H. Kunita. Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics 24. Cambridge Univ. Press, Cambridge, 1990. Zbl0743.60052MR1070361
- [21] J. M. Lee. Riemannian Manifolds: An Introduction to Curvature. Graduate Texts in Mathematics 176. Springer, New York, 1997. Zbl0905.53001MR1468735
- [22] D. W. Stroock and S. R. Srinivasa Varadhan. Multidimensional Diffusion Processes. Classics in Mathematics. Springer, Berlin, 2006. Reprint of the 1997 edition. Zbl1103.60005MR2190038
- [23] A. Thalmaier and F.-Y. Wang. Gradient estimates for harmonic functions on regular domains in Riemannian manifolds. J. Funct. Anal. 155 (1998) 109–124. Zbl0914.58042MR1622800
- [24] P. Topping. Lectures on the Ricci Flow. London Mathematical Society Lecture Note Series 325. Cambridge Univ. Press, Cambridge, 2006. Zbl1105.58013MR2265040

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.