GEOMETRY OF STOCHASTIC DIFFERENTIAL EQUATIONS

1997年9月，包括5名外国研究人员在内的我们在九州大学聚会，讨论了无限维空间的随机微分方程、随机分析和几何方面的各种问题。同样在1998年9月，我们与3名外国与会者讨论了随机偏微分方程及其相关分析。我们得到了以下结果。1)流形上的随机微分方程使我们能够定义流形上的连接。随机流动的许多性质都可以通过连接来解释（k.d.l elworthy）2)我们研究了一个不具有Lipschits连续性的粗糙路径常微分方程。它使我们能够应用由布朗运动驱动的随机微分方程。（T.Lyons）3)环群几何。作为一种典型的无限维群，我们研究了与布朗运动密切相关的环群。我们讨论了在环路群上构造几何的可能性，并在环路群上定义了一个连接。我们将已知的随机偏微分方程的L(2)理论推广到L(p)理论，得到了关于随机偏微分方程解的光滑性的更清晰的结果。利用预测积分（Skorohod积分）和Malliavin微积分，得到了一类随机偏微分方程（Nualart）基本解的新结果。

In September 1997, including 5 foreign investigators, we gathered together at Kyushu Univ.and discussed various problems concerning stochastic differential equations and stochastic analysis and geometry of infinite dimensional spaces.Also in September 1998, with 3 foreign participants, we discussed stochastic partial differential equations and related analysis. We obtained the following results.1)Stochastic differential equations on a manifold enable us to define a connect ion on the manifolds. Many proeprties of stochastoic flows can be interpreted through the connection (K.D.Elworthy)2)We studied an ordinary differential equation with rough path without the Lipschits continuity. It enabled us to apply the stochastic differential equations driven by a Brownian motion. (T.Lyons)3)Geometry of the loop group. As a typical infinite dimensional group, we studied the loop group which is closely related to the Brownian motion. We discussed the possibility of canstructiong the geometry on the loop group, defining a connection on it. (P.Malliavin)4)We extend the known L(2) theory of stochastic partial differential equations to that of L(p) theory and obtained a sharper result on the smoothmess of the soluti on. (Krylov)5)By using anticipating integral (Skorohod integral) and Malliavin calculus, we obtained a new result on the fundamental solution of a stochastic partial differential equation (Nualart)