Concentration of the Brownian bridge on Cartan-Hadamard manifolds with pinched negative sectional curvature

Marc Arnaudon[1]; Thomas Simon

  • [1] Université de Poitiers, département de Mathématiques, Téléport 2, BP 30179, Boulevard Marie et Pierre Curie, 86962 Futuroscope-Chasseneuil Cedex (France), Université d'Évry-Val d'Essonne, Equipe d'Analyse et Probabilités, Boulevard François Mitterrand, 91025 Evry Cedex (France)

Annales de l’institut Fourier (2005)

  • Volume: 55, Issue: 3, page 891-930
  • ISSN: 0373-0956

Abstract

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We study the rate of concentration of a Brownian bridge in time one around the corresponding geodesical segment on a Cartan-Hadamard manifold with pinched negative sectional curvature, when the distance between the two extremities tends to infinity. This improves on previous results by A. Eberle, and one of us . Along the way, we derive a new asymptotic estimate for the logarithmic derivative of the heat kernel on such manifolds, in bounded time and with one space parameter tending to infinity, which can be viewed as a counterpart to Bismut's asymptotic formula in small time.

How to cite

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Arnaudon, Marc, and Simon, Thomas. "Concentration of the Brownian bridge on Cartan-Hadamard manifolds with pinched negative sectional curvature." Annales de l’institut Fourier 55.3 (2005): 891-930. <http://eudml.org/doc/116211>.

@article{Arnaudon2005,
abstract = {We study the rate of concentration of a Brownian bridge in time one around the corresponding geodesical segment on a Cartan-Hadamard manifold with pinched negative sectional curvature, when the distance between the two extremities tends to infinity. This improves on previous results by A. Eberle, and one of us . Along the way, we derive a new asymptotic estimate for the logarithmic derivative of the heat kernel on such manifolds, in bounded time and with one space parameter tending to infinity, which can be viewed as a counterpart to Bismut's asymptotic formula in small time.},
affiliation = {Université de Poitiers, département de Mathématiques, Téléport 2, BP 30179, Boulevard Marie et Pierre Curie, 86962 Futuroscope-Chasseneuil Cedex (France), Université d'Évry-Val d'Essonne, Equipe d'Analyse et Probabilités, Boulevard François Mitterrand, 91025 Evry Cedex (France)},
author = {Arnaudon, Marc, Simon, Thomas},
journal = {Annales de l’institut Fourier},
keywords = {Brownian bridge; Cartan-Hadamard manifold; comparison theorems; Cox-Ingersoll-Ross process; heat kernel; large deviations; rank-one noncompact symmetric space},
language = {eng},
number = {3},
pages = {891-930},
publisher = {Association des Annales de l'Institut Fourier},
title = {Concentration of the Brownian bridge on Cartan-Hadamard manifolds with pinched negative sectional curvature},
url = {http://eudml.org/doc/116211},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Arnaudon, Marc
AU - Simon, Thomas
TI - Concentration of the Brownian bridge on Cartan-Hadamard manifolds with pinched negative sectional curvature
JO - Annales de l’institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 3
SP - 891
EP - 930
AB - We study the rate of concentration of a Brownian bridge in time one around the corresponding geodesical segment on a Cartan-Hadamard manifold with pinched negative sectional curvature, when the distance between the two extremities tends to infinity. This improves on previous results by A. Eberle, and one of us . Along the way, we derive a new asymptotic estimate for the logarithmic derivative of the heat kernel on such manifolds, in bounded time and with one space parameter tending to infinity, which can be viewed as a counterpart to Bismut's asymptotic formula in small time.
LA - eng
KW - Brownian bridge; Cartan-Hadamard manifold; comparison theorems; Cox-Ingersoll-Ross process; heat kernel; large deviations; rank-one noncompact symmetric space
UR - http://eudml.org/doc/116211
ER -

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