Mathematical Analysis of Infinite Dimensional Stochastic Models

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Performing the reseach based on the project plan we obtained the following reseach results.1. Fleming-Viot processes play an important role in population genetics, for which we obtained two significant results.First, we considered the model with mutation and unbounded selectionas genetic factors. In this case it has not proved even the well-posedness of the diffusion processes, which we settled together with the uniqueness problem of the stationary distributions. This work was caried out jointly with S.N.Ethier (USA). Furthermore we solved the problem of diffusion approximation from discrete time Markov chain models.Second, we solved a reversibility problem for the Fleming-Viot processes with mutation and selection, that is to characterize the mutation operator for the process to have a reversible distribution. This work was done with Z.H.Li (China) and L.Yau (USA). (Shiga)2. We considered a suvival probability problem of random walker in temporarily and spatially varing random environ … More ment, and obtained a precise asymprotics of the suvival probability for small parameter rigion. To prove it we developed a detailed analysis of linear stochastic partial differential equations which are dual objects of the random walk model. This result appeared as ajoint work with T.Furuoya.Directed polymer model is a closely related with this problem in mathematical context, and we get some significant results on asymptotical behaviorof the random partition function in low dimensional case, which is harder than higher dimensional case. (Shiga)3. For a mechanical many particle system Uchiyama established the hydrodynamic limit and identified its hydrodynamic equation, that is a diffusion equation in this situation.4. For a dynamical system in cofinite Fuchsian groups which can be regarded as a Markov system, Morita developed a perterbational analysis of the transfer operator and solved some ergodic problem that is related to number theory.5. Motivated by mathematical finance Takaoka obtained a neccesary and suffucient condition for a continuous local martingale to be uniformly integrable. Less