Geometric and Symmetry Properties of a Nondegenerate Diffusion Process

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.