GEOMETRIC AND SYMMETRY PROPERTIES OF A NONDEGENERATE DIFFUSION PROCESS

A diffusion process with smooth and nondegenerate elliptic infinitesim al generator on a manifold M induces a Riemannian metric g on M. This paper discusses in detail different symmetry properties of such a diff usion 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 determi nistic systems and symmetries of diffusion processes are delineated. T he incidence of a stochastic framework on the properties of an origina l deterministic system are then illustrated in different examples. The construction of a diffusion process with given symmetries is also add ressed and resulting geometric problems are raised.