Ergodic Theory of Complex Random Dynamics

这个项目的目标是无穷维随机动力系统的遍历理论。主要目标是分析与随机介质相关的随机过程中的稳定统计模式，如随机拉格朗日系统，定向聚合物，随机增长模型和随机流体动力学模型。遍历性是随机稳定性的一种形式，意味着系统的长期统计特性对初始状态不敏感。特别是，它使与复杂系统相关的测量变得有意义。确定性和随机动力系统中的长期统计模式可以通过平稳分布来描述，而该项目的目标正是描述这些平稳状态。它建立在最近由PI在Burgers方程和相关系统的遍历理论中取得的进展所创造的势头之上。伯格斯方程的核心作用是因为它是模拟从交通到宇宙大尺度结构形成的各种现象的基础。它同时是一个流体动力学模型，一个增长模型，一个守恒定律，并密切相关的随机控制，定向聚合物，和一个重要的KPZ普遍性类模型的统计力学。该计划将扩大对Burgers方程和相关的复杂随机动力系统在非紧环境中的统计模式的理解，研究随机偏微分方程或类似模型在无界区域中的遍历性质要求发展新的数学技术。该项目将使用和加强数学统计力学，概率论和动力学的现代方法，为第一次通过渗透和最后一次通过渗透模型开发的技术，晶格动物，浓度不等式，随机控制，吉布斯系综的热力学极限和相互作用粒子系统的流体动力学极限，沿着变分方法和分析技术，如不等式和嵌入，典型的偏微分方程。静态分布的描述和分析将通过样本测量，一个力一个解决方案的原则，随机单边拉格朗日极小和他们的聚合物对应。

This project is targeted at ergodic theory of infinite-dimensional random dynamical systems. The main goal is the analysis of steady statistical patterns in stochastic processes associated with random media such as random Lagrangian systems, directed polymers, random growth models, and models of stochastic hydrodynamics. Ergodicity is a form of stochastic stability that means that the long-term statistical properties of the system are not sensitive to the initial state. In particular, it makes measurements related to complex systems meaningful. Long-term statistical patterns in deterministic and random dynamical systems can be described via stationary distributions, and it is the description of these stationary regimes that the project aims at. It builds around the momentum that has been created by recent progress by the PI in ergodic theory of the Burgers equation and related systems. The central role of the Burgers equation is due to the fact that it is fundamental in modeling a variety of phenomena from traffic to the formation of large scale structure of the Universe. It is simultaneously a fluid dynamics model, a growth model, a conservation law, and is tightly related to stochastic control, directed polymers, and an important KPZ universality class of models of statistical mechanics. The proposed program will extend the understanding of statistical patterns for the Burgers equation and related complex random dynamical systems in noncompact settings.Studying ergodic properties of stochastic partial differential equations or similar models in unbounded domains calls for development of new mathematical techniques. The project will use and enhance modern methods of mathematical statistical mechanics, probability theory, and dynamics, techniques developed for first-passage percolation and last passage percolation models, lattice animals, concentration inequalities, stochastic control, thermodynamic limits for Gibbs ensembles and hydrodynamic limits for interacting particle systems, along with variational approach and analysis techniques such as inequalities and embeddings, typical for partial differential equations. The description and analysis of stationary distributions will be obtained via sample measures, one force one solution principles, random one-sided Lagrangian minimizers and their polymer counterparts.