GEOMETRY OF STOCHASTIC DIFFERENTIAL EQUATIONS

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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)