FRG: Collaborative Research: Stochastics and Dynamics: Asymptotic problems

FRG：协作研究：随机学和动力学：渐近问题

• 批准号: 0854879
• 负责人: Jonathan Mattingly
• 金额: $ 17.63万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0854879&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Stochastics; Dynamics

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).Mathematical models taking both deterministic and stochastic factors into account are becoming increasingly important in science and technology. These models, as a rule, are rather complicated. Oftentimes, they include many parameters characterizing the system (diffusion coefficients, rates of chemical reactions, time scales, etc). The parameters often have different scales, so it is natural to consider various asymptotic regimes in these models. We will study deterministic and stochastic perturbations of systems with conservationlaws, in particular, perturbations of Hamiltonian systems with multi-well Hamiltonians. Metastability and stochastic resonance for systems perturbed by noise will be considered. New problems on singular perturbations of elliptic PDE's will be studied as well as quasi-linear parabolic equations which lead to a new class of stochastic perturbations of dynamical systems. Mathematical models of polymers and models leading to anomalous particle transport will be considered. We plan to study a number of asymptotic problems for inifinite-dimensional systems, in particular, for stochastic PDE's.We will develop new methods of asymptotic analysis for stochas-tic processes, dynamical systems and PDE's. New effects related to metastability, singular perturbations of PDE's, particle and wave motion in random media will be described. The research is related to many branches of mathematics and has various applications in physics, biology and engineering. Asymptotic methods can and should play an important role in educating the new generation of researchers. We plan to work with graduate students and postdocs, run seminars and organize conferences on these topics.

该奖项是根据2009年的《美国回收与再投资法》（公法111-5）资助的。将确定性和随机性因素同时考虑的数学模型在科学和技术中变得越来越重要。通常，这些模型相当复杂。通常，它们包括表征系统的许多参数（扩散系数，化学反应速率，时间尺度等）。这些参数通常具有不同的尺度，因此在这些模型中考虑各种渐近方案是很自然的。我们将研究具有保护爪的系统的确定性和随机扰动，尤其是对汉密尔顿系统具有多孔汉密尔顿人的扰动。将考虑对噪声扰动的系统的亚竞争性和随机共振。将研究有关椭圆PDE的奇异扰动的新问题以及准线性抛物线方程，从而导致新的动态系统随机扰动。将考虑聚合物和导致异常颗粒转运的模型的数学模型。我们计划研究无限二维系统的许多渐近问题，特别是为随机PDE'S.S.我们开发针对Stochas-TIC过程，动力学系统和PDE的新方法。将描述与亚稳定性，PDE的奇异扰动，随机介质中的粒子和波动运动有关的新效果。该研究与数学的许多分支有关，并且在物理，生物学和工程学方面具有各种应用。渐近方法可以并且应该在教育新一代研究人员中发挥重要作用。我们计划与研究生和博士后合作，举办研讨会并组织有关这些主题的会议。