Comprehensive Study of Probability Theory

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We organized seven project teams in the first year and five ones in the second year and each team studied its own project. The belows are the main research results.1. The research of the stochastic analysis : We analyzed the measures on the Wiener spaces. As results, we obtain the followings. (1) A probabilistic interpretation of the asymptotic behaviors of the fundamental solutions of a heat equation as t 0. (2) The structure of the Sobolev space in infinite demensional spaces. (3) The construction of the differential geometry on the infinite dimensional spaced such as De Rham decomposition theorem.2. The research of Markov process : We studied the following problems for Marlkov processes with jumps. (1) The estimate of the fundamental solution. (2) The homogenization theorems. (3) Criteria for the recurrence (transience).3. The research of the limit theorems : We studied the probability distributions related to limit theorems. In particular, we showed that the distribution of a self similar process with independent increment is L-distribution and an arbitrary L-distributio is realized by the above stochastic process.4. The research of stochastic processes in random environments. We studied the asymptotic behaviors as t of random walks and diffusions in random environment.5. The research of Gaussian measures and Gaussian processes. As a result, it turned out that the capacity of the channel with feedbacks is not greater than the twice of the capacity of the channel without feedback. 6. The research of probability theory and spectrum. We studied the following objects.(1) The spectrum of the Schrodinger operators in a magnetic field. (2) The spectrum of the Laplacian in Sierpinsky's carpet and gaskets.7. The research of the ergodic theorems. We studied the structures of Anozov flow, holocycle flow and other dynamical systems. Also, we studied the relation between the local limit theroems related to continued fraction expansions and prime number theorem.