[Concentration du pont brownien dans les variétés de Cartan-Hadamard à courbure négative pincée]
Dans une variété de Cartan-Hadamard à courbure négative pincée, nous déterminons la concentration d’un pont brownien en temps 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.
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
@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},
pages = {891--930},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {55},
number = {3},
year = {2005},
doi = {10.5802/aif.2117},
mrnumber = {2149406},
zbl = {1075.58019},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.2117/}
}
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 SP - 891 EP - 930 VL - 55 IS - 3 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2117/ DO - 10.5802/aif.2117 LA - en ID - AIF_2005__55_3_891_0 ER -
%0 Journal Article %A Arnaudon, Marc %A Simon, Thomas %T Concentration of the Brownian bridge on Cartan-Hadamard manifolds with pinched negative sectional curvature %J Annales de l'Institut Fourier %D 2005 %P 891-930 %V 55 %N 3 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2117/ %R 10.5802/aif.2117 %G en %F 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, Tome 55 (2005) no. 3, pp. 891-930. doi : 10.5802/aif.2117. http://www.numdam.org/articles/10.5802/aif.2117/
[1] The heat kernel on non compact symmetric spaces (Lie groups and symmetric spaces) (2003), pp. 27-46 | Zbl
[2] A Panoramic View of Riemannian Geometry, Springer-Verlag, 2003 | MR | Zbl
[3] Large Deviations and the Malliavin Calculus, Birkhäuser, 1984 | MR | Zbl
[4] Metric Spaces of Non-Positive Curvature, Springer-Verlag, Berlin, 1999 | MR | Zbl
[5] Comparison Theorems in Riemannian Geometry, North-Holland, 1975 | MR | Zbl
[6] Large Deviations Techniques and Applications, Jones and Barlett Publishers, Boston, 1993 | MR | Zbl
[7] Absence of spectral gaps on a class of loop spaces, J. Math. Pures Appl, Volume 81 (2002) no. 9, pp. 915-955 | MR | Zbl
[8] Stochastic Calculus in Manifolds, Springer-Verlag, Berlin, 1989 | MR | Zbl
[9] Diffusion processes in one dimension, Trans. Amer. Math. Soc, Volume 77 (1954), pp. 1-31 | DOI | MR | Zbl
[10] The Martin compactification of the Cartesian product of two hyperbolic spaces, J. Reine Angew. Math., Volume 444 (1993), pp. 17-28 | MR | Zbl
[11] Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, 1978 | MR | Zbl
[12] Groups and Geometric Analysis, Academic Press, 1984 | MR | Zbl
[13] Stochastic Analysis on Manifolds, Amer. Math. Society, Providence, RI, 2002 | Zbl
[14] Jacobi functions and analysis on non compact semisimple Lie groups (1984), pp. 1-85 | Zbl
[15] Die Resolvente von auf symmetrischen Raümen von nichtkompakten Typ, Comment. Math. Helvet., Volume 57 (1982), pp. 445-468 | DOI | MR | Zbl
[16] Étude d'une fonctionnelle liée au pont de Bessel, Ann. Inst. H. Poincaré Probab. Statist., Volume 32 (1996) no. 1, pp. 107-133 | Numdam | MR | Zbl
[17] Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-Verlag, New York, 1966 | MR | Zbl
[18] Analytical Treatment of One-Dimensional Markov Processes, Academia, Prague, and Springer-Verlag, New-York, 1968 | MR | Zbl
[19] Path integral formulae for heat kernels and their derivatives, Probab. Theory Related Fields, Volume 94 (1993), pp. 525-541 | DOI | MR | Zbl
[20] Continuous Martingales and Brownian Motion, Springer-Verlag, Berlin, 1999 | MR | Zbl
[21] Concentration of the Brownian bridge on the hyperbolic plane, Ann. Probab., Volume 30 (2002) no. 4, pp. 1977-1989 | DOI | MR | Zbl
[22] On the Differentiation of Heat Semigroups and Poisson Integrals, Stoch. Stoch. Rep., Volume 61 (1997), pp. 297-321 | MR | Zbl
[23] Gradient estimates for harmonic functions on regular domains in Riemannian manifolds, J. Funct. Anal., Volume 155 (1998), pp. 109-124 | DOI | MR | Zbl
Cité par Sources :
