FRG: Collaborative Research: Stochastics and Dynamics: Asymptotic problems

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。同时考虑确定性和随机因素的数学模型在科学技术中变得越来越重要。一般来说，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状态是很自然的。我们将研究具有守恒律的系统的确定性和随机扰动，特别是具有多井哈密顿量的哈密顿系统的扰动。将考虑受噪声扰动的系统的亚稳态和随机共振。本文将研究椭圆型偏微分方程奇异摄动的新问题以及拟线性抛物型方程，这些问题将导致动力系统的一类新的随机摄动。将考虑聚合物的数学模型和导致异常粒子输运的模型。我们计划研究一些有限维系统的渐近问题，特别是随机偏微分方程。我们将发展随机过程、动力系统和偏微分方程的渐近分析新方法。将描述与亚稳态、PDE的奇异扰动、随机介质中的粒子运动和波动有关的新效应。该研究涉及数学的许多分支，并在物理学、生物学和工程学中有各种应用。渐近方法能够而且应该在教育新一代研究人员方面发挥重要作用。我们计划与研究生和博士后合作，就这些主题举办研讨会和组织会议。

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.