Research of stochastic differential geometry

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1. The relationship between a stochastic differential equation and the stochastic flow of diffeomorphisms defined by it has been studied intensively since 1975. In particular, in case where the stochastic differential equation is based on Brownian motions, the problem was solved at the beginning of 1980. In this research, we studied the case where the stochastic differetntial equation is defined based on point processes or Levy processes with jumps, and showed that the solution defines a stochastic flow of the semigroup of smooth maps. Further we gave a sufficienct condition that it becomes a stochastic flow of diffeomorphisms.2. Asymptotic behaviors of solutions of differential equations with random coefficients are important objects of the research in the systems theory in engineering and the theory of population genetics in mathematical biology. Further they are also interesting as a mathematical theory, since these can be regarded as an application of the law of the large numbers a … More nd the central limit theorems to stochastic differential equations and stochastic partial differential equations. In this research, we studied the problems in the following cases. (1) The limits of stochastic differential equations or difference equations are represented by a Brownian flow. (2) Limit theorems for solutions of partial differential equations with random coefficients. (3) Fluctuation theorems and central limit theorems for stochastic ordinary differential equations and partial differential equations.3. Malliavin calculus was proposed by a French mathematician Malliavin in order to get the smoothness of the solutions of hypoelliptic partial differential equations and has been developed very rapidly. In this research, we showed that the Malliavin calculus cn be applied to a time-dependent system of hypoelliptic differential operator and that the calculus is also applicable to the study of the spectrum of a certian differential operator.4. We studied the boundary value problems of a degenerate elliptic operator and the problems of the Silov boundary from the point of the views of probability theory. Less