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性因素的数学模型在科学和技术中变得越来越重要。这些模型通常相当复杂。通常，它们包括许多表征系统的参数（扩散系数，化学反应速率，时间尺度等）。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，特别是具有多阱哈密顿量的哈密顿系统的扰动。将考虑受噪声扰动系统的亚稳性和随机共振。本文将研究椭圆型偏微分方程的奇异摄动问题以及拟线性抛物型方程的奇异摄动问题，从而得到一类新的动力系统的随机摄动。聚合物的数学模型和模型导致异常粒子传输将被考虑。我们计划研究无穷维系统的一些渐近问题，特别是随机偏微分方程的渐近问题。我们将发展随机过程、动力系统和偏微分方程的渐近分析的新方法。新的影响有关的亚稳性，奇异摄动的偏微分方程的，粒子和波动的运动在随机介质中将被描述。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。渐近方法可以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，举办研讨会并组织有关这些主题的会议。

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.