Infinite dimensional stochastic analysis and geometry

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We studied Markov processes by using the stochastic analysis. We are mainly interested in the behavior of the semigroup associated with a Markov process. First we considered diffusion processes on a Riemannian manifold. We give a generator as a sum of the Laplace-Beltrami operator and a vector field. We discussed the uniqueness of the semigroup associated with the generator. The condition was given in terms of the vector field.Next we consider the conditions for which the semigroup reserves a convex set. We consider this problem in the framework of general Banach space. We give some necessary and sufficient conditions in terms of generator. In Hilbert space setting, we formulate this problem by using Dirichlet forms.In addition, we considered the rate of convergence of the semigroup under the condition of logarithmic Sobolev inequality. We also discussed the dual ultracontractive property of the semigroup.