Asymptotic Problems in Random Dynamics

为了对物理学、生命科学、排队系统、金融等领域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。这个项目是围绕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：平均大测量序列的基本程序有意义吗？结果取决于什么，如何依赖？回答这些问题的基础是研究系统演化的统计平稳状态。这个项目的目标是存在/不存在这样的统计平稳制度，他们的描述和行为的几种类型的随机动力系统产生的各种应用从交通到神经元系统到宇宙的大尺度结构：(i)随机强迫汉密尔顿-雅可比方程；（ii）具有多重不稳定性的动力系统的小随机扰动；（iii）随机切换系统。对于一般的随机强迫Hamilton-Jacobi方程，PI建议根据单侧作用极小值及其正温度对应物，即定向聚合物的热力学单侧极限来获得全局行为的描述。为此，PI提出将定向聚合物的概念从二次元扩展到一般哈密顿量。PI将解决的进一步问题是：局部化/非局部化，特征指数，与由全局解定义的单调变换流相关的重整化群，该重整化群的不动点和简化的离散模型，冲击震级和年龄的统计。小噪声扰动将主要研究具有异斜网络的动力系统，这些网络由多个不稳定平衡通过异斜轨道相互连接组成，其时间尺度比文献中已经研究的要长得多。这将使我们能够得出关于不变分布的行为的结论，并解决同质化问题。为此，将详细研究网络中不可能转变的机制和相关的时间尺度。所涉及的随机变量的小规模分析将涉及到Malliavin演算。对于具有随机开关的系统，前人的研究表明，在广义Hörmander-type次椭圆条件下，这些系统具有唯一不变测度，且该测度是绝对连续的。得到不变量密度的进一步正则性，研究其平滑性和奇异性是一个难题。PI和合作者最近的进展提供了工具，将用于接近一般的半椭圆开关系统。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.