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ccsd-00002886,version2-20Sep2004 CONCENTRATION OF THE BROWNIAN BRIDGE ON
 

Summary: ccsd-00002886,version2-20Sep2004
CONCENTRATION OF THE BROWNIAN BRIDGE ON
CARTAN-HADAMARD MANIFOLDS WITH PINCHED
NEGATIVE SECTIONAL CURVATURE
MARC ARNAUDON AND THOMAS SIMON
Abstract. We study the rate of concentration of a Brownian bridge in time
one around the corresponding geodesical segment on a Cartan-Hadamard man-
ifold with pinched negative sectional curvature, when the distance between
the two extremities tends to infinity. This improves on previous results by
A. Eberle [7], and one of us [21]. 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 [3].
Contents
1. Introduction 1
2. The case of real hyperbolic spaces 4
2.1. Asymptotics of first-passage times for CIR-type diffusions 4
2.2. Three further estimates 10
2.3. End of the proof 12
3. The case of rank-one noncompact symmetric spaces 14

  

Source: Arnaudon, Marc - Département de mathématiques, Université de Poitiers
Ecole Polytechnique, Centre de mathématiques

 

Collections: Mathematics