Asymptotic Problems for Stochastic Processes & PDE's

随机过程的渐近问题

• 批准号: 9504177
• 负责人: Mark Freidlin
• 金额: $ 20.21万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-06-01 至 2000-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9504177&HistoricalAwards=false
• 关键词: Asymptotic; Problems; Stochastic; Processes; &amp

9504177 Freidlin Abstract We study random perturbations of dynamical systems with conservation laws. The evolution of the first integral leads to a diffusion process on a graph corresponding to the first integral. This allows progress in a number of PDE problems with a small parameter. The two-dimensional Navier-Stocks equation with a large Reynolds number, for exampke, belongs to this class of problems. We study also large deviations for the diffusion-transmutation processes. These results help to describe the asymptotic behavior of wave fronts and other patterns in reaction-diffusion equations. Mathematical models of real processes sometimes,turn out to be too complicated for analysis. But very often some of the parameters included in the model are small (or large) in comparison with the others. This allows us to simplify the model and often to obtain the solution in a form convenient for applications. We study small random perturbations of dynamical systems, large scale approximation in reaction- diffusion models and other problems related to an interplay between random and deterministic factors.

摘要研究具有守恒律的动力系统的随机摄动。第一个积分的演化导致了与第一个积分对应的图上的扩散过程。这允许在许多具有小参数的PDE问题中取得进展。例如，具有大雷诺数的二维Navier-Stocks方程就属于这类问题。我们还研究了扩散-嬗变过程的大偏差。这些结果有助于描述反应扩散方程中波前和其他模式的渐近行为。实际过程的数学模型有时会变得过于复杂，难以分析。但是，与其他参数相比，模型中包含的一些参数通常很小（或很大）。这使我们能够简化模型，并经常以方便应用程序的形式获得解决方案。我们研究动力系统的小随机扰动，反应扩散模型的大尺度近似以及其他与随机因素和确定性因素之间相互作用有关的问题。