Statistical Description of Stochastic Dynamical Systems

这个建议是一个三部分的研究计划，旨在更好地了解随机动力系统。 在第一部分中，题为随机方程在无限维，我建议研究一些原型随机偏微分方程中出现的流体动力学湍流和非平衡统计力学的其他问题。 重点是超越这些方程的更标准的存在性/唯一性声明，并考虑更难的问题，如：一个典型的解决方案是什么样子的？ 它的概率密度函数的性质是什么？ 在第二部分的建议，题为有效的随机建模，我认为大型系统的变量可以分为两组在不同的时间尺度上发展。 慢变量单独的有效方程推导出消除的快速变量借用奇异摄动技术的马尔可夫过程。 最后，在第三部分的建议，题为过渡途径-字符串方法，我认为系统，由于在系统中的典型的能量障碍和可用的热能之间的差距，动力学进行长时间的等待期周围的亚稳态，然后突然跳跃从一个状态到另一个。 这些系统中的有效动力学由亚稳态之间的过渡路径和这些过渡发生的速率决定。 这里的目标是开发和实施有效的数值算法，计算这些路径和rates.尽管计算机性能的快速提高，许多问题的科学和工程利益将不适合直接数值模拟在可预见的未来。 例如，在流体动力学湍流、动力学临界现象、气候模拟、分子动力学、空间扩展系统中的相变中出现的典型问题涉及如此大量的变量在如此多的不同时空尺度上相互作用，以至于它们极大地压倒了直接的数值计算。 另一方面，虽然不可能完整描述这些示例中的动态，但也不一定有用。 实际上，人们通常只对一些粗粒度的变量感兴趣，这些变量通过对适当的集合（时间、空间等）进行平均来适当地定义，其以更有规律的方式演变，从而提供关于系统的最有用的信息。 本提案的主要目标是改进对这些大型系统进行统计分析的技术，包括确定粗粒度变量及其满足的方程。 共同的重点是技术，这是真正的计算工具-无论是分析或数值-并允许具体和明确的调查性质的解决方案。

This proposal is a three-part research program aimed at a better understanding of stochastic dynamical systems. In the first part, entitled Stochastic Equations in Infinite Dimensions, I propose to study some prototype stochastic partial differential equations which arise in connection with hydrodynamic turbulence and other problems in nonequilibrium statistical mechanics. The emphasis is on going beyond the more standard existence/uniqueness statements for these equations and considering harder questions like: What does a typical solution look like? What are the properties of its probability density function? In the second part of the proposal, entitled Effective Stochastic Modeling, I consider large systems where the variables can be separated into two groups evolving on different time scales. Effective equations for the slow variables alone are derived by elimination of the fast variables using techniques which borrow from singular perturbations techniques for Markov processes. Finally, in the third part of the proposal, entitled Transition Pathways -- String Method, I consider systems where, due to the disparity between typical energy barriers in the system and the thermal energy available, the dynamics proceed by long waiting periods around the metastable states followed by sudden jumps from one state to the other. The effective dynamics in these systems is determined by the transition pathways between the metastable states and the rates at which these transitions occur. The objective here is to develop and implement efficient numerical algorithms which compute these paths and rates.Despite the rapid improvement of computer performance, many problems of scientific and engineering interest will not be amenable to direct numerical simulations in the foreseeable future. Typical problems arising, for example, in hydrodynamic turbulence, dynamical critical phenomena, climate modeling, molecular dynamics, phase transition in spatially extended systems, involve such a large number of variables interacting on so many different space-time scales that they vastly overwhelm direct numerical computations. On the other hand, while a complete description of the dynamics in these examples is impossible, it is also not necessarily useful. Indeed one is typically interested only in some coarse-grained variables, suitably defined by means of averaging over appropriate ensembles (time, space, ...), which evolve in a more regular fashion and thereby provide the most useful information about the system. The main objective of the present proposal is to improve the techniques for such a statistical analysis of these large systems -- including the identification of the coarse-grained variables and the equations they satisfy. The common emphasis is on techniques which are truly computational tools -- either analytical or numerical -- and allow for concrete and explicit investigation of the properties of the solutions.