no. 3
Concentration of the Brownian bridge on Cartan-Hadamard manifolds with pinched negative sectional curvature
[Concentration du pont brownien dans les variétés de Cartan-Hadamard à courbure négative pincée]
Arnaudon, Marc1 ; Simon, Thomas
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, Tome 55 (2005) no. 3, pp. 891-930.

Dans une variété de Cartan-Hadamard à courbure négative pincée, nous déterminons la concentration d’un pont brownien en temps 1 autour du segment géodésique correspondant, lorsque la distance entre les extrémités tend vers l’infini. Notre résultat améliore et généralise ceux de A. Eberle (2002) et T. Simon (2002). Nous établissons pour cela une nouvelle estimée de la convergence de la dérivée logarithmique du noyau de la chaleur en temps borné lorsque la distance entre les deux points tend vers l’infini, qui peut être vue comme un analogue de la formule de Bismut asymptotique en temps petit.

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.

DOI : 10.5802/aif.2117
Classification : 58J65, 60F10, 60H30
Keywords: Brownian bridge, Cartan-Hadamard manifold, comparison theorems, Cox-Ingersoll-Ross process, heat kernel, large deviations, rank-one noncompact symmetric space
Mot clés : pont brownien, variété de Cartan-Hadamard, théorèmes de comparaison, processus de Cox-Ingersoll-Ross, noyau de la chaleur, grandes déviations, espace symétrique non compact de rang un

Arnaudon, Marc 1 ; Simon, Thomas 

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) 
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     title = {Concentration of the {Brownian} bridge on {Cartan-Hadamard} manifolds with pinched negative sectional curvature},
     journal = {Annales de l'Institut Fourier},
     pages = {891--930},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Arnaudon, Marc; Simon, Thomas. Concentration of the Brownian bridge on Cartan-Hadamard manifolds with pinched negative sectional curvature. Annales de l'Institut Fourier, Tome 55 (2005) no. 3, pp. 891-930. doi : 10.5802/aif.2117. https://aif.centre-mersenne.org/articles/10.5802/aif.2117/

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