Asymptotic Problems in Random Dynamics

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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.