Asymptotic Problems for Stochastic Processes and Differential Equations

随机过程和微分方程的渐近问题

• 批准号: 0803287
• 负责人: Mark Freidlin
• 金额: $ 18万
• 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: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0803287&HistoricalAwards=false
• 关键词: Asymptotic; Problems; Stochastic; Processes; Differential

Several classes of asymptotic problems will be considered. The first class concerns metastability and stochastic resonance for multiscale systems. The general approach to metastability and stochastic resonance in the framework of large deviations was developed in our previous papers. In the case of multiscale systems these notions should be modified. In particular, new effects appear in systems which are close to Hamiltonian and in systems with a small delay. Another class consists of new problems for PDE's with a small parameter and related stochastic processes. To describe the limiting behavior in these classical problems one should consider equations on graphs or open book spaces related to the original problam. Reaction-Diffusion-Advection equations in incompressible fluids, in strongly unisotropic media, and in narrow tubes will be considered. The last class of problems concerns the averaging principle for systems with many degrees of freedom and multiwell Hamiltonians. It turns out that even in pure deteministic systems the slow motion can be, in a sense, stochastic.Small perturbations of a system which often have negligible effects, when the system considered on a short time interval, can become crucial on larger timescale. The study of such effects is one of our main goals. We consider long time influence of small stochastic and deterministic perturbations and, under certain conditions, describe the long-time evolution of the system which essentially depends on the perturbations. Such kind of problems arise in numerous applications from physics and biology to economics and sociology. We are developing new mathematical approches to these problems.

本文将讨论几类渐近问题。第一类涉及多尺度系统的亚稳态和随机共振。我们在以前的论文中发展了大偏差框架下亚稳态和随机共振的一般方法。在多尺度系统的情况下，这些概念应该加以修改。特别是在接近哈密顿量的系统和具有小延迟的系统中，会出现新的效应。另一类是小参数偏微分方程的新问题和相关的随机过程。为了描述这些经典问题的极限行为，我们应该考虑与原始问题相关的图上方程或开书空间。将考虑不可压缩流体、强各向同性介质和窄管中的反应-扩散-平流方程。最后一类问题涉及具有多自由度和多井哈密顿量的系统的平均原理。事实证明，即使在纯决定论系统中，从某种意义上说，慢动作也可能是随机的。一个系统的小扰动通常在短时间内的影响可以忽略不计，但在更大的时间尺度上可能变得至关重要。研究这些影响是我们的主要目标之一。我们考虑了小的随机和确定性扰动的长时间影响，并在一定条件下描述了系统的长时间演化，而系统的演化本质上取决于这些扰动。从物理学、生物学到经济学、社会学的许多应用中都出现了这类问题。我们正在开发新的数学方法来解决这些问题。