Mathematical Analysis of Infinite Dimensional Stochastic Models

无限维随机模型的数学分析

• 批准号: 09640246
• 负责人: SHIGA Tokuzo
• 金额: $ 1.92万
• 依托单位: TOKYO INSTITUTE OF TECHNOLOGY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640246/
• 关键词: infinite dimensional diffusion; Flemig-Viot process; genetical model; unbounded selection; time-reversibility; stochastic partial differential equation; random walk in random environment; directed polymer model; 可逆定常分布; 分数冪モーメント; ランダムポリマーモデル; ラ

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

根据项目计划进行研究，我们得到了以下研究结果：弗莱明-维奥过程在群体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，我们考虑了突变和无界选择作为遗传因素的模型。在这种情况下，它甚至没有证明扩散过程的适定性，我们将其与平稳分布的唯一性问题一起解决。这项工作是与s.n.e ethier（美国）联合进行的。进一步解决了离散时间马尔可夫链模型的扩散逼近问题。其次，我们解决了具有突变和选择的Fleming-Viot过程的可逆性问题，即表征了该过程具有可逆分布的突变算子。这项工作是与李志辉（中国）和邱丽丽（美国）共同完成的。(志贺)2。研究了随机行走者在临时和空间变化的随机环境下的生存概率问题，得到了小参数区域的生存概率的精确渐近解。为了证明这一点，我们对随机漫步模型的对偶对象线性随机偏微分方程进行了详细的分析。定向聚合物模型在数学上与该问题密切相关，我们得到了低维情况下随机配分函数的渐近行为的一些有意义的结果，这比高维情况更难。(志贺)3。对于一个力学多粒子系统，Uchiyama建立了水动力极限，并确定了它的水动力方程，即在这种情况下的扩散方程。对于可视为马尔可夫系统的有限Fuchsian群中的动力系统，Morita发展了传递算子的越位分析，并解决了一些与数论有关的遍历问题。在数学金融学的激励下，高冈得到了连续局部鞅一致可积的一个充要条件。少

项目成果

M.Taniguchi: "Multiple existence and linear stability of equilibrium balls in a nonlinear free boundary problem" Quart.Appl.Math.掲載予定.

M.Taniguchi：“非线性自由边界问题中平衡球的多重存在性和线性稳定性”Quart.Appl.Math.待出版。

Yuji Nomura: "Absence of diffusion near the bottom of the spectrum for a random Schroedinger operator on L^2(R^3)" J.Math.Kyoto Univ.Vol.37. 639-687 (1997)

Yuji Nomura：“L^2(R^3) 上的随机薛定谔算子在光谱底部附近不存在扩散”J.Math.Kyoto Univ.Vol.37。

T.Furuoya and T.Shiga: "Sample Lyapunov exponent for a class of linear Markovian systems over Z^d." Osaka J.Math.35. 35-72 (1998)

T.Furuoya 和 T.Shiga：“Z^d 上一类线性马尔可夫系统的样本 Lyapunov 指数。”

K.Uchiyama: "Wiener's test for the random walks with mean zero and finite variance." Ann.Probab.26. 368-376 (1998)

K.Uchiyama：“均值为零且方差有限的随机游走维纳检验。”

K.Uchiyama: "Green's functions for random walks on Z^N" Proc.London Math.Soc.,. 77. 215-240 (1998)

K.Uchiyama：“Z^N 上随机游走的格林函数”Proc.London Math.Soc.，。