演講者：李向東教授

單　位：Academy of Mathematics and System Science, Chinese Academy of Sciences

日　期：2013年12月9日（星期一）　AM 10:00

地　點：國立高雄大學理學院408室

講　題：Malliavin calculus on geometric Langevin processes and kinetic Fokker-Planck equation

摘　要：

In 1974, P. Malliavin introduced the Stochastic Calculus of Variations and gave a probabilsitic proof of the Hörmander hypoellipticity theorem. Since then the Malliavin calculus has been a central topic in stochastic analysis. It has played a crucial role in J.-M. Bismut's probabilistic proof of the Atiyah-Singer index theorem of the Dirac operator and in P. L. Lions's work on Greeks formula for complete financial markets.

In recent years, J.-M. Bismut has introduced a family of hypoelliptic Laplacians on the cotangent bundle over compact Riemannian manifolds. The hypoelliptic Laplacian is the infinitesimal generator of the Langevin process on the cotangent bundle, which is an interpolation between the geodesic flow and the Brownian motion on the underlying manifold. The purpose of this paper is to use the Malliavin calculus to study the generic geometric Langevin processes in connection with the kinetic Fokker-Planck equation on the cotangent bundle over compact Riemannian manifolds. We will prove an integration by parts formula and the Greeks formula with respect to the law of the geometric Langevin processes on the cotangent bundle, which extend previous results due to Bismut and P. L. Lions et al.