Concentration of the Brownian bridge on Cartan-Hadamard manifolds with pinched negative sectional curvature
Annales de l'Institut Fourier, Volume 55 (2005) no. 3, p. 891-930

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.

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.

DOI : https://doi.org/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
@article{AIF_2005__55_3_891_0,
     author = {Arnaudon, Marc and Simon, Thomas},
     title = {Concentration of the Brownian bridge on Cartan-Hadamard manifolds with pinched negative sectional curvature},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {3},
     year = {2005},
     pages = {891-930},
     doi = {10.5802/aif.2117},
     zbl = {1075.58019},
     mrnumber = {2149406},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2005__55_3_891_0}
}
Arnaudon, Marc; Simon, Thomas. Concentration of the Brownian bridge on Cartan-Hadamard manifolds with pinched negative sectional curvature. Annales de l'Institut Fourier, Volume 55 (2005) no. 3, pp. 891-930. doi : 10.5802/aif.2117. http://www.numdam.org/item/AIF_2005__55_3_891_0/

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