Asymptotic Problems for Stochastic Processes and PDE's

本项目将研究哈密顿系统的Freidlin摄动。这种扰动系统的长期行为，即使扰动是纯确定性的，也应该用一个与汉密尔顿函数相关的图上的随机过程来描述。一类偏微分方程的渐近问题，如平面上Navier-Stokes方程的小粘度渐近性问题或不可压缩流体中反应-扩散的小扩散渐近性问题，与哈密顿系统的随机扰动密切相关。概率方法在分析这些问题时很有用。本文研究了受随机噪声扰动的动力系统的长时间行为。这类问题出现在许多应用中，例如，宇宙物体的长时间行为或具有小粘度的流体的运动。这些扰动，即使很小，对于系统的长期行为也是必不可少的。研究人员在项目中开发的渐近方法是最有希望分析这类问题的方法。

9803522 Freidlin Perturbations of the Hamiltonian systems will be studied in this project. The long- time behavior of such perturbed systems, even if the perturbations are purely deterministic, should be described by a stochastic process on a graph related to the Hamilton function. A class of asymptotic problems for PDEs, such as the small viscosity asymptotics for the Navier-Stokes equations in the plane or small diffusion asymptotics for reaction - diffusion in an incompressible fluid, are closely related to random perturbations of the Hamiltonian systems. The probabilistic approach is useful in analyzing these problems. Long time behavior of dynamical systems perturbed by a stochastic noise is studied in this project. This type of problem arises in many applications, for example, the long time behavior of cosmic objects or the motion of a fluid with a small viscosity. The perturbations, even if they are small, become essential for the long-time behavior of the system. The asymptotic approach, which the investigator is developing in the project, is the most promising for analyzing this type of problem.