Analysis of Infinite-dimensional Markovian Models

我们进行了这个项目，并取得了以下成果。1. Fleming-Viot过程是一类重要的测度值马尔可夫过程，是群体遗传学的基本模型。在此基础上，我们解决了一个公开问题，即确定Fleming-Viot过程具有时间可逆平稳分布的情形。2.在选择强度是无界的情况下，我们证明了过程的适定性和平稳分布的唯一性。这个问题比有界选择的情况要困难得多。3.相互作用扩散系统是相互作用马氏过程理论的一个重要分支，在噪声空间独立的情况下得到了很好的发展。我们考虑了空间相关噪声，得到了一些遍历性结果。4.具有高斯白色噪声势的热方程定义了一个函数值扩散，从随机聚合物模型的观点来看，这是重要的。对于这个方程，我们开发了渐近分析的时刻的解决方案。

We performed this project and obtained the following results. 1. Fleming-Viot proccesess form an important class of measure-valued Markov processes and well-applied as basic models to population genetics. Concerning these we resolved an open problem to determine the situation that the Fleming-Viot processes have time reversible stationary distribution. 2. In the case that selective intensity is unbounded we proved well-posedness of the processes and uniqueness of stationary distribution. This problem is much harder than the bounded selection case. 3. Interacting diffusion systems form an important class of the theory of interacting Markov processes, which have been well developed when the noises are spatially independent. We considered spatially correlated noises, and obtained some ergodic results. 4. Heat equation with Gausian white noise potential defines a function-valued diffusion, which is important from a view point of random polymer model. For this equation we developed asymptotic analysis of moments of the solution.