Statistical Description of Stochastic Dynamical Systems

这个建议是一个由三个部分组成的研究计划，旨在更好地理解随机动力系统。在第一部分“无限维的随机方程”中，我建议研究一些与流体动力湍流和其他非平衡统计力学问题有关的原型随机偏微分方程。重点是超越这些方程更标准的存在/唯一性陈述，并考虑更难的问题，如：典型的解是什么样子？它的概率密度函数的性质是什么？在提案的第二部分“有效随机建模”中，我考虑了大型系统，其中变量可以分为两组，在不同的时间尺度上进化。利用马尔可夫过程的奇异摄动技术，通过消去快速变量，推导出单独慢变量的有效方程。最后，在提案的第三部分，题为“过渡路径-字符串方法”中，我考虑了这样的系统，由于系统中典型能量障碍与可用热能之间的差异，动力学在亚稳态周围经过长时间的等待，然后从一个状态突然跳到另一个状态。这些系统的有效动力学是由亚稳态之间的转变途径和这些转变发生的速率决定的。这里的目标是开发和实现有效的数值算法来计算这些路径和速率。尽管计算机性能得到了迅速的提高，但在可预见的将来，许多科学和工程领域的问题还不能直接进行数值模拟。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，涉及到如此多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，虽然不可能对这些例子中的动态进行完整的描述，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……）进行平均来适当地定义，这些集合以更有规则的方式演变，从而提供有关系统的最有用的信息。本提案的主要目标是改进这些大系统的统计分析技术，包括粗粒度变量及其满足的方程的识别。通常强调的技术是真正的计算工具——无论是分析的还是数值的——并且允许对解的性质进行具体和明确的研究。

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.