Asymptotic Problems in Random Dynamics

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.