Asymptotic Problems in Random Dynamics

为了对物理、生命科学、排队系统、金融等领域中研究的各种自然现象的行为进行充分的建模，必须考虑噪声的影响。这个项目的中心是随机动力系统的长期行为的分析，可以通过回答以下问题来描述：对大量测量序列进行平均的基本程序是否有意义？结果取决于什么，又是如何实现的？回答这些问题的基础是研究系统演化的统计平稳机制。这个项目的目标是研究在从交通到神经元系统再到宇宙大尺度结构的各种应用中出现的几类随机动力系统的统计平稳区域的存在/不存在、它们的描述和行为：(I)随机强迫的Hamilton-Jacobi方程；(Ii)具有多个不稳定的动力系统的小随机扰动；(Iii)具有随机开关的系统。对于一般的随机强迫的Hamilton-Jacobi方程，PI建议用单侧作用量极小值及其正的温度对应项，定向聚合物的热力学单边极限来描述全局行为。为此，PI建议将定向聚合物的概念从二次哈密顿量扩展到一般哈密顿量。PI将解决的其他问题还有：局域化/离域化、特征指数、与由整体解定义的单调变换流相关的重整化群、该重整化群和简化离散模型的不动点、激波强度和年龄的统计。小噪声扰动将主要研究具有异宿网络的动力系统，该网络由多个由异宿轨道相互连接的不稳定平衡点组成，时间尺度比文献中已有的研究要长得多。这将允许得出关于不变分布的行为的结论，并探讨齐化问题。为此，将详细研究网络中不太可能的转变的机制和相关的时间尺度。对所涉及的随机变量的小规模分析将涉及Malliavin微积分。对于具有随机切换的系统，前人的工作证明了在广义Hörmander型亚椭圆度条件下，这些系统存在唯一的不变测度，并且这个测度是绝对连续的。得到不变密度的进一步正则性，研究不变密度的光滑性和奇异性是一个困难的问题。PI和合著者的最新进展提供了用于接近一般亚椭圆交换系统的工具。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The influence of noise must be taken into account for adequate modeling of the behavior of various natural phenomena studied in physics, life sciences, queuing systems, finance, etc. This project is centered around the analysis of long-term behavior of random dynamical systems that can be described by answering the following questions: does the fundamental procedure of averaging large sequences of measurements make sense? what do the results depend on and how? Answering these questions is based on studying statistically stationary regimes of the system's evolution. This project is targeted at existence/nonexistence of such statistically stationary regimes, their description and behavior for several types of random dynamical systems arising in various applications from traffic to neuronal systems to the large-scale structure of the Universe: (i) randomly forced Hamilton-Jacobi equations; (ii) small random perturbations of dynamical systems with multiple instabilities; (iii) systems with random switching.For general randomly forced Hamilton-Jacobi equations, the PI proposes to obtain a description of the global behavior in terms of one-sided action minimizers and their positive temperature counterparts, thermodynamic one-sided limits for directed polymers. For this, the PI proposes an extension of the notion of the directed polymer from quadratic to general Hamiltonians. Further questions that the PI will address are: localization/delocalization, characteristic exponents, the renormalization group associated with the flow of monotone transformations defined by global solutions, the fixed points of this renormalization group and simplified discrete models, statistics of shock magnitudes and ages. Small noisy perturbations will mainly be studied for dynamical systems with heteroclinic networks, consisting of multiple unstable equilibria connected to each other by heteroclinic orbits, on much longer time scales than those already studied in the literature. This will allow to make conclusions about the behavior of invariant distributions and approach homogenization questions. For this, the mechanism of unlikely transitions in the network and the associated time scales will be studied in detail. The small scale analysis of the involved random variables will involve Malliavin calculus. For systems with random switchings, the previous work showed that under broad Hörmander-type hypoellipticity conditions, these systems have a unique invariant measure and this measure is absolutely continuous. Obtaining further regularity of the invariant density, studying its smoothness and the character of singularities turned out to be a hard problem. The recent progress by the PI and coauthors provides tools that will be used to approach general hypoelliptic switching systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.