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.

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: 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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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://archive.numdam.org/articles/10.5802/aif.2117/

[1] J. P. Anker; P. Ostellari The heat kernel on non compact symmetric spaces (Lie groups and symmetric spaces) (2003), pp. 27-46 | Zbl

[2] M. Berger A Panoramic View of Riemannian Geometry, Springer-Verlag, 2003 | MR | Zbl

[3] J. M. Bismut Large Deviations and the Malliavin Calculus, Birkhäuser, 1984 | MR | Zbl

[4] M. R. Bridson; A. Haefliger Metric Spaces of Non-Positive Curvature, Springer-Verlag, Berlin, 1999 | MR | Zbl

[5] J. Cheeger; D. G. Ebin Comparison Theorems in Riemannian Geometry, North-Holland, 1975 | MR | Zbl

[6] A. Dembo; O. Zeitouni Large Deviations Techniques and Applications, Jones and Barlett Publishers, Boston, 1993 | MR | Zbl

[7] A. Eberle 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] M. Emery Stochastic Calculus in Manifolds, Springer-Verlag, Berlin, 1989 | MR | Zbl

[9] W. Feller Diffusion processes in one dimension, Trans. Amer. Math. Soc, Volume 77 (1954), pp. 1-31 | DOI | MR | Zbl

[10] S. Giulini; W. Woess The Martin compactification of the Cartesian product of two hyperbolic spaces, J. Reine Angew. Math., Volume 444 (1993), pp. 17-28 | MR | Zbl

[11] S. Helgason Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, 1978 | MR | Zbl

[12] S. Helgason Groups and Geometric Analysis, Academic Press, 1984 | MR | Zbl

[13] E. P. Hsu Stochastic Analysis on Manifolds, Amer. Math. Society, Providence, RI, 2002 | Zbl

[14] T. H. Koornwinder; R. A. Askey et al. Jacobi functions and analysis on non compact semisimple Lie groups (1984), pp. 1-85 | Zbl

[15] N. Lohoue; T. Rychener Die Resolvente von Δ auf symmetrischen Raümen von nichtkompakten Typ, Comment. Math. Helvet., Volume 57 (1982), pp. 445-468 | DOI | MR | Zbl

[16] G. Lorang; B. Roynette 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] W. Magnus; F. Oberhettinger; R. P. Soni Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-Verlag, New York, 1966 | MR | Zbl

[18] P. Mandl Analytical Treatment of One-Dimensional Markov Processes, Academia, Prague, and Springer-Verlag, New-York, 1968 | MR | Zbl

[19] J. R. Norris Path integral formulae for heat kernels and their derivatives, Probab. Theory Related Fields, Volume 94 (1993), pp. 525-541 | DOI | MR | Zbl

[20] D. Revuz; M. Yor Continuous Martingales and Brownian Motion, Springer-Verlag, Berlin, 1999 | MR | Zbl

[21] T. Simon Concentration of the Brownian bridge on the hyperbolic plane, Ann. Probab., Volume 30 (2002) no. 4, pp. 1977-1989 | DOI | MR | Zbl

[22] A. Thalmaier On the Differentiation of Heat Semigroups and Poisson Integrals, Stoch. Stoch. Rep., Volume 61 (1997), pp. 297-321 | MR | Zbl

[23] A. Thalmaier; F.Y. Wang Gradient estimates for harmonic functions on regular domains in Riemannian manifolds, J. Funct. Anal., Volume 155 (1998), pp. 109-124 | DOI | MR | Zbl

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