Research of stochastic differential geometry

1. 自1975年以来，人们对随机微分方程与其所定义的微分同态随机流之间的关系进行了深入的研究。特别是在随机微分方程是基于布朗运动的情况下，这个问题在1980年初就解决了。本文研究了基于点过程或具有跳跃的Levy过程定义随机微分方程的情况，并证明了其解定义了光滑映射半群的随机流。进一步给出了它成为微分同态随机流的充分条件。随机系数微分方程解的渐近行为是工程系统理论和数学生物学群体遗传学理论研究的重要对象。此外，它们作为一种数学理论也很有趣，因为它们可以被视为大数定律的应用，以及随机微分方程和随机偏微分方程的中心极限定理。在本研究中，我们研究了以下案例中的问题。(1)随机微分方程或差分方程的极限用布朗流表示。(2)随机系数偏微分方程解的极限定理。(3)随机常微分方程和偏微分方程的涨落定理和中心极限定理。Malliavin微积分是由法国数学家Malliavin为了求解准椭圆型偏微分方程解的光滑性而提出的，得到了非常迅速的发展。在本研究中，我们证明了Malliavin演算可以应用于时间相关的准椭圆型微分算子系统，并且该演算也可以应用于某一类微分算子谱的研究。从概率论的观点出发，研究了简并椭圆算子的边值问题和Silov边界问题。少

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