CAREER: Research and training in stochastic dynamics

该项目涉及两类随机动力学的分析。首先，PI将研究随机微分方程的解，这些方程在状态空间的两个或多个区域之间经历罕见的随机转换。这项工作的目标是了解这些转变的统计数据，特别是在小噪声状态下：转变发生的典型途径是什么？转换通常需要多长时间？这些问题与化学和分子动力学以及许多其他表现出亚稳态行为的物理系统的问题非常相关。PI将要研究的第二类随机动力学与随机相互作用粒子系统有关。要研究的具体系统包括随机运动、生长和选择，如进化模型、群体遗传学、适应动力学。PI将研究这种系统的连续极限和大时间极限。特别是，这将阐明相互作用粒子系统与连续体自由边界问题之间的新关系。当在一定尺度上观察时，许多物理和生物系统的行为似乎是随机的，或者受到小的随机波动的影响。这类系统的数学模型涉及到概率论。然而，利用这些数学模型来有效地预测长时间或大空间区域内的系统行为是非常困难的。因此，一个基本的科学和数学问题是理解一个空间或时间尺度上的随机动力学或相互作用如何影响另一个空间或时间尺度上的系统。该研究项目开发数学工具，用于预测和描述在微观水平上随机行为的某些系统的宏观行为。要研究的特定系统是由化学问题和生物进化模型驱动的。这些系统的一个共同特征是出现随机的，也许是罕见的转变：发生化学反应或细胞产生突变。该项目的教育部分包括对研究生和本科生进行概率、分析和应用交叉的培训，为他们在stem相关学科的职业生涯做好准备。

The project involves analysis of two types of stochastic dynamics. First, the PI will study solutions to stochastic differential equations which undergo rare, random transitions between two or more regions of the state space. The goal of this work is to understand the statistics of these transitions, especially in the small-noise regime: what are the typical pathways by which the transitions occur? what is the typical time required for a transition? These issues are very relevant to problems in chemistry and molecular dynamics, as well as many other physical systems exhibiting metastable behavior. The second type of stochastic dynamics which the PI will study has to do with stochastic interacting particle systems. Specific systems to be studied involve random motion, growth, and selection, as in models of evolution, population genetics, adaptive dynamics. The PI will study continuum limits and large-time limits for such systems. In particular, this will illuminate new relations between interacting particle systems and continuum free boundary problems.When viewed at a certain scale, many physical and biological systems seem to behave randomly or are influenced by small random fluctuations. Mathematical models of such systems involve probability theory. Nevertheless, it is very difficult to use these mathematical models to efficiently predict the system behavior over a long period of time or over a large spatial region. Therefore, a fundamental scientific and mathematical problem is to understand how random dynamics or interactions at one spatial or temporal scale influence a system at another spatial or temporal scale. This research project develops mathematical tools for predicting and describing the macroscopic behavior of certain systems which behave randomly at a microscopic level. The specific systems to be studied are motivated by problems in chemistry and by models of biological evolution. One common feature in these systems is the appearance of random, perhaps rare, transitions: a chemical reaction occurs or a cell produces a mutation. The educational component of the project includes the training of graduate and undergraduate students at the intersection of probability, analysis, and applications, preparing them for careers in STEM-related disciplines.