Asymptotic Problems in Random Dynamics

对于大多数噪声驱动的随机动力系统来说，精确的长期预测是不可能的。然而，它们的长期统计特性往往是可以理解的。提高这种认识是本项目的主要目标。对于这类系统中的一些系统，长期统计不依赖于初始条件，并且对于一些系统，关于初始条件的某些信息被长时间保留。该项目旨在描述用于模拟物理现象的复杂随机动力学系统的这些行为：随机增长模型，前沿传播，流体动力学模型，与无序环境相互作用的聚合物链。对于具有多个固定状态的噪声系统，如非线性反应网络，该项目旨在研究这些状态之间的随机切换模式。该项目的教育和传播部分包括监督研究生和本科生参与研究，组织研讨会和会议并在会上发言。本计画的第一部份是发展随机环境中具随机哈密顿量之Hamilton-Jacobi方程及相关系统之遍历程式。基本对象是拉格朗日作用量的极小化，即由传统Gibbs形式给出的有向聚合和随机势中的广义Hamilton-Jacobi聚合。这个问题的循环包括寻找随机动态吸引子的解决方案在各种设置，有或没有边界条件，研究相关的无限单边极小和聚合物措施。在第二部分中，小随机扰动的动力系统与多个不稳定性组织成异宿网络将进行研究，重点是新兴的亚稳态图片多项式之间的不同制度的异宿循环的过渡时间。该项目的第三部分将研究具有多个不变测度的系统的非遍历平均模式，这些测度支持在显示亚稳定性的各种维度的流形上，即动态不受一个不变分布的支配，而是在各种状态之间间歇地切换，该奖项反映了NSF的法定使命，并通过使用基金会的学术价值和更广泛的影响评审标准。

Precise long-term predictions are impossible for most noise-driven random dynamical systems. However, their long-term statistical properties often can be understood. Improving this understanding is the main goal of this project. For some systems in this class, the long-term statistics do not depend on the initial conditions, and for some systems, certain information about the initial conditions is retained over long times. The project aims at describing these behaviors for complex random dynamical systems used to model phenomena in physics: random growth models, front propagation, fluid dynamics models, polymer chains interacting with disordered environments. For noisy systems with multiple stationary regimes such as nonlinear reaction networks, the project aims at studying patterns of random switching between those regimes. The project’s educational and dissemination components include supervising graduate and undergraduate participation in the research and organizing and speaking at seminars and conferences. The first part of the project is to develop the ergodic program for Hamilton-Jacobi equations with random Hamiltonians and related systems in random environments. The basic objects will be minimizers of Lagrangian action, that is, directed polymers given by the traditional Gibbs formalism and generalized Hamilton-Jacobi polymers in a random potential. This circle of problems includes finding random dynamic attractors for solutions in various settings, with or without boundary conditions, studying the associated infinite one-sided minimizers and polymer measures. In the second part, small random perturbations of dynamical systems with multiple instabilities organized into heteroclinic networks will be studied, with the focus on the emerging metastability picture with polynomial transition times between different regimes of heteroclinic cycling. The third part of the project will study patterns of nonergodic averaging for systems with multiple invariant measures supported on manifolds of various dimensions that display metastability, that is, the dynamics is not dominated by one invariant distribution but intermittently switches between various regimes, spending long times in each of them.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.