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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