Mathematical Analysis of Infinite Dimensional Stochastic Models

根据项目计划进行研究，取得了以下研究成果. Fleming-Viot过程在群体遗传学中有着重要的作用，为此，我们得到了两个有意义的结果：第一，考虑了具有突变和无界选择的遗传因子模型;在这种情况下，它甚至没有证明适定性的扩散过程，我们解决了一起的唯一性问题的平稳分布。这项工作是与S.N.M.S.（美国）联合进行的。其次，我们解决了具有变异和选择的Fleming-Viot过程的可逆性问题，即刻画了使过程具有可逆分布的变异算子。这项工作是与Z.H.Li（中国）和L.Yau（美国）共同完成的。（滋贺）2.考虑了时空变化随机环境中随机步行者的生存概率问题 关于我们 并得到了小参数区域生存概率的精确渐近解。为了证明这一点，我们开发了一个详细的分析线性随机偏微分方程，这是随机游走模型的对偶对象。该结果是与T.Furuoya的联合工作。定向聚合物模型在数学背景下与该问题密切相关，并且我们在低维情况下随机配分函数的渐进行为方面得到了一些重要结果，这比高维情况更困难。（滋贺）3.对于一个力学的多质点系统，内山建立了其流体动力学极限，并确定了其流体动力学方程，即在这种情况下的扩散方程.对于上有限Fuchsian群中的动力系统，可以看作是一个马尔可夫系统，Morita发展了转移算子的扰动分析，并解决了一些与数论有关的遍历问题.受金融数学的启发，Takaoka得到了连续局部鞅一致可积的一个充分必要条件。少

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