Studies on Probability theory.

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We have concerned with the theory of stochastic processes and related fields. For the purpose of promoting the activity in these fields, we held workshops and symposia on various topics during the period between 1987, April and 1989, March. In the meetings supported by the project, we were interested in the studies of the topics described below: Dirichlet space and Markov process, Malliavin calculus and applications, Gaussian process, stationary process and time series, Langivin equation, limit theorem for stochastic processes, stochastic process on fractals, stochastic process in random media, fluctuation of spectra in random media, percolation, Ising model, phase transition and critical phenomena, ergodic theory, analysis and geometry on loop space, stochastic control, large deviation and applications, applications of non-standard analysis to probability theory etc. Like many other branches of mathematics, probability theory has grown and developed with inspiration from other areas of science. This is especially clear in the theory of stochastic processes, where ideas from mathematical physics and engineering, for example, have exerted a noteworthy influence. In a series of the workshops organized by Uchiyama and Funaki, a serious attempt has been made to probabilistic methods in mathematical physics. In particular, we were devoted to the sutdy of the following topics: time dependent Ginzburg-Landau model and phase transitions, the laplacian in regions with many obstacles, hydrodynamic limit of various models. Stochastic analysis is a new branch of probability theory and is becoming more and more important in close connection with partial differential equations, geometry, ergodic theory and mathematical physics. For purpose of promoting the activity in stochastic analysis we have also organized a series of workshops. In these meetings, notable results related to various topics in stochastic analysis were announced.