Explicit stochastic analysis of Brownian motion and point measures on Riemannian manifolds

 

Jean-Jacques Prat   Nicolas Privault
Université de La Rochelle   Université de la Rochelle
Avenue Marillac   Avenue Marillac
17042 La Rochelle Cedex   17042 La Rochelle Cedex 1
France   France

 

Abstract:

The gradient and divergence operators of stochastic analysis on Riemannian manifolds are expressed using the gradient and divergence of the flat Brownian motion. By this method we obtain the almost-sure version of several useful identities that are usually stated under expectations. The manifold-valued Brownian motion and random point measures on manifolds are treated successively in the same framework, and stochastic analysis of the Brownian motion on a Riemannian manifold turns out to be closely related to classical stochastic calculus for jump processes. In the setting of point measures we introduce a damped gradient that was lacking in the multidimensional case.

Key words: Stochastic calculus of variations, Brownian motion, random measures, Riemannian manifolds.
Mathematics Subject Classification (1991): 60H07, 60H25, 58-99, 58C20, 58G32, 58G99.

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