Geometric and symmetry properties of a nondegenerate diffusion process (Q1917193)

The case of diffusion process with (time-independent) infinitesimal generator \(\mathcal L\) on a manifold \(M\) of dimension \(n\), when \(\mathcal L\) is smooth (or analytic) and nondegenerate elliptic is studied. Systematically and in detail different symmetric properties of such a diffusion by geometric methods are discussed. Partial differential equations associated with the generator are also studied. Moreover, the author has a try at delineate the relationships between symmetries of deterministic systems and symmetries of diffusion processes. Finally, several applications with an emphasis on how the choice of a diffusion process to represent a dynamic system can affect the symmetries of the original system are discussed. At the same time, specific discussions on gradient vector fields and on practical filtering are given. In these last cases the geometric problems raised by the construction of a diffusion process with given symmetries are outlined.