Asymptotic Problems for Stochastic Processes and PDE's

随机过程和偏微分方程的渐近问题

• 批准号: 0103589
• 负责人: Mark Freidlin
• 金额: $ 9.76万
• 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: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0103589&HistoricalAwards=false
• 关键词: Asymptotic; Problems; Stochastic; Processes; PDE

Several classes of asymptotic problems are considered. The averaging principle for deterministic and stochastic perturbations, asymptotic problems for reaction-diffusion equations, and problems related to stochastic resonance are among them. The long-time evolution of perturbed systems with conservation laws, even in the case of purely deterministic perturbations, leads, in general, to stochastic process on complexes defined by the conservation laws. So the classical averaging principle (say, for deterministic perturbations of integrable Hamiltonian systems when the Hamiltonian has many critical points) should be treated in the stochastic framework. In this research small diffusion asymptotics for reaction-diffusion in an incompressible 2D-fluid, which is closely related to the averaging for Hamiltonian systems, is studied. Another class of problems concerns the large deviation theory and stochastic resonance. A number of new effects such as large amplitude oscillations and stabilization induced by the small noise in autonomous systems are considered.The asymptotic approach is one of the most powerful tools of applied mathematics. In particular, the averaging principle plays the leading role when systems combining multi-scale processes are considered. Such problems arise in mechanics, in material sciences, in biophysics, and in other areas. This research does not just consider problems concerning the mathematical justification of the averaging principle, but also describes new applications and new effects. In recent years stochastic-resonance-type effects, which first appeared in the theory of long-time evolution of the climate, have attracted the attention of specialists in many areas of physics, engineering, and biology. The mathematical theory of these effects is

考虑了几类渐近问题。其中包括确定性和随机扰动的平均原理，反应扩散方程的渐近问题，以及与随机共振有关的问题。具有守恒定律的扰动系统的长期演化，即使在纯确定性扰动的情况下，一般也会导致由守恒定律定义的复体上的随机过程。因此，经典的平均原理（例如，当可积哈密顿系统有许多临界点时的确定性扰动）应该在随机框架中处理。本文研究了不可压缩二维流体中反应扩散的小扩散渐近性，它与哈密顿系统的平均密切相关。另一类问题涉及大偏差理论和随机共振。考虑了自治系统中由小噪声引起的大振幅振荡和稳定等新效应。渐近方法是应用数学中最强大的工具之一。当考虑多尺度过程组合的系统时，平均原理起主导作用。这样的问题出现在力学、材料科学、生物物理学和其他领域。本研究不仅考虑了平均原理的数学证明问题，而且描述了新的应用和新的效果。近年来，随机共振型效应首次出现在气候长期演化理论中，引起了物理学、工程学和生物学等诸多领域专家的关注。这些效应的数学理论是