On the geometry of diffusion operators and stochastic flows / K. D. Elworthy, Y. Le Jan, Xue-Mei Li

The authors study the concepts of second-order semi-elliptic operators, stochastic differential equations, stochastic flows and Gaussian vector fields with related metric linear connections on tangent bundles and subbundles of tangent bundles. Given a semi-elliptic differential operator on a manifold M, a representation in such called Hörmander form (a sum of square vector fields) gets an extension to an operator on differential forms. In the same way, representing a diffusion process as the one point motion of a stochastic flow determines a semi-group acting on differential forms. Given a regularity condition there is an associated linear connection and adjoint “semi-connection” in term of which the operators can be simply described.

Contents: Chapter 1: the construction of linear connections of vector bundles as push forwards of connections on trivial bundles; Chapter 2: the infinitesimal generator given in Hörmander form and its associated stochastic differential equations; Chapter 3: the diffusion coefficient of a stochastic differential equation has a kernel and filter out the “redundant noise” (from the point of view of the one point motion); Chapter 4: applications to analysis in spaces of paths (integration by parts, logarithmic Sobolev inequality); Chapter 5: applications to stability properties of stochastic flows (bounds for moment exponents, moment stability); technical appendices are contained in Chapter 6. (Zentralblatt)