## The Annals of Probability

### Geometric and Symmetry Properties of a Nondegenerate Diffusion Process

M. Cohen de Lara

#### Abstract

A diffusion process with smooth and nondegenerate elliptic infinitesimal generator on a manifold $M$ induces a Riemannian metric $g$ on $M$. This paper discusses in detail different symmetry properties of such a diffusion by geometric methods. Partial differential equations associated with the generator are studied likewise. With an eye to modelling and applications to filtering, relationships between symmetries of deterministic systems and symmetries of diffusion processes are delineated. The incidence of a stochastic framework on the properties of an original deterministic system are then illustrated in different examples. The construction of a diffusion process with given symmetries is also addressed and resulting geometric problems are raised.

#### Article information

Source
Ann. Probab., Volume 23, Number 4 (1995), 1557-1604.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176987794

Digital Object Identifier
doi:10.1214/aop/1176987794

Mathematical Reviews number (MathSciNet)
MR1379159

Zentralblatt MATH identifier
0859.60069

JSTOR
links.jstor.org

Subjects
Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 58G35

#### Citation

de Lara, M. Cohen. Geometric and Symmetry Properties of a Nondegenerate Diffusion Process. Ann. Probab. 23 (1995), no. 4, 1557--1604. doi:10.1214/aop/1176987794. https://projecteuclid.org/euclid.aop/1176987794