Mathematical Analysis of Infinite Dimensional Stochastic Models

在项目计划的基础上进行研究，取得了以下研究成果。Fleming-Viot过程在种群遗传学中起着重要的作用，我们得到了两个重要的结果。首先，我们考虑了具有突变和无界选择的模型作为遗传因素。在这种情况下，甚至没有证明我们与平稳分布的唯一性问题一起解决的扩散过程的适定性。这项工作是与S.N.Ethier(美国)共同完成的。解决了离散时间马尔可夫链模型的扩散逼近问题。其次，解决了具有突变和选择的Fleming-Viot过程的可逆性问题，即刻画了过程具有可逆分布的突变算子。这项工作是与李振辉(中国)和L.Yau(美国)共同完成的。(Shiga)2.我们考虑了随机游走者在临时和空间变化的随机环境…中的生存概率问题进一步，得到了小参数代数的生存概率的精确渐近性。为了证明这一点，我们对随机游动模型的对偶对象--线性随机偏微分方程组进行了详细的分析。这一结果似乎是与T.Furuya的共同工作。定向聚合物模型在数学上与这个问题密切相关，我们得到了一些关于随机配分函数在低维情况下的渐近行为的重要结果，这比高维情况更难。(志贺)3.对于一个机械的多粒子系统，内山建立了流体动力学极限，并确定了它的流体动力学方程，即在这种情况下的扩散方程。对于可视为马尔可夫系统的余有限富氏群中的动力系统，Morita发展了转移算子的微扰分析，解决了一些与数论有关的遍历问题。在数学金融学的启发下，高冈得到了连续局部鞅一致可积的充要条件。较少

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