The central goal of the project will be to develop the ergodic theory of random transfer operators generated by random media systems. The principal questions in ergodic theory are existence, uniqueness and attraction properties of invariant measures for stochastic or deterministic dynamical systems. In the quenched random media setting, invariance is naturally replaced by skew-invariance and attraction has two natural counterparts, forward and pullback attraction. The main question is that of the existence of a global attracting skew-invariant positive solution for a cocycle generated by products of random linear cone-preserving operators. Therefore, one of the key points of the project is to develop a general analogue of the Perron--Frobenius theory for positive linear cocycles in noncompact settings. This approach will allow the asymptotic analysis for a wide class of infinite-dimensional stochastic systems and include the classical ergodic theory of Markov processes as a specific case. It is most promising in the situations where the random environment possesses certain localization properties. It is also aimed at the infinite-dimensional situations where the classical minorization conditions fail. Naturally, under this project several problems tightly related to the main topic will be studied: the problems of invariant measures for PDEs with random forcing and boundary conditions; localization issues for random directed polymers and associated parabolic models; universality classes for action-minimizing paths in random potential; the questions of regularity of transfer operators for infinite-dimensional systems and related techniques of infinite-dimensional Malliavin calculus and non-adapted stochastic analysis.
Random media problems and ergodic theory have been intensively explored by mathematicians and physicists for many years. In this project, these two fields are brought together. Ergodic theory studies statistical patterns arising in complex systems where it is hard or impossible to make exact predictions, and only probabilistic predictions make sense in the long run. There are many interesting and important practical situations including those from physics, biology and economy where the dynamics is determined by the randomness present in the environment which adds another layer of complication to the analysis. In particular, the statistical patterns themselves become random. The main and unifying goal of this project is to describe the random statistical patterns in the long-term behavior of these stochastic systems and understand qualitatively and quantitatively the mechanisms of their formation. The proposed activities include attracting students to this area of research beginning at the undergraduate level.
The main goal of this project was to develop tools for identifying and studying long term statistical properties of complex random dynamical systems arising in modeling various physical processes happening in random environments. This includes hydrodynamical systems and interface growth models described in terms of stochastic partial differential equations. Meaningful measurements of such systems are possible only if one is able to identify the statistically stationary regime the system is in. One of the main outcomes of this project is a complete description of stationary regimes for the stochastic Burgers equation in noncompact setting. Burgers equation is a basic hydrodynamic model. It has connections to multiple applications ranging from traffic modeling to the analysis of the large scale structure of the Universe. Nonetheless, this has been a long standing problem, and even conjectures about the long-term behavior of this had been extremely vague until the definitive answer that was given in the course of this project. An important side of this line of research turned out to be strengthening connections with other probabilistic models of mathematical physics like first passage percolation and last passage percolation, and other models that are believed to belong to the so called KPZ universality class. Similar results were obtained for a directed polymer model, where the uniqueness of a stationary regime turned out to be connected with dynamic localization properties. There are indications that this connection is typical for a broad class of systems in random environments. A definitive result on regularity of stationary regimes has been obtained for systems with random switchings also known as piecewise deterministic Markov processes. It was established that classical Hormander hypo-ellipticity conditions guarantee uniqueness and absolute continuity of invariant distributions for these systems having multiple applications to engineering and biology. Several outcomes of this project are related to the study of small random perturbations of dynamical systems, especially those admitting special structures called heteroclinic networks that consist of multiple unstable equilibria connected to each other. The search for this type of structure and its exploration is intensive in many fields of science, especially in neuroscience. Sometimes the role of fixed points connected by heteroclinic orbits is played by spatio-temporal coherent structures. The main result is that as the noise level tends to zero, the diffusion along the network converges to a jump process on the saddle points that may lack the Markov property, i.e., it may retain memory about its distant past. In particular, this result may be used to explain that only a limited vocabulary of neuron excitation patterns is usually observed. The main ideas of this reasearch are based on studying diffusion exit problems, where we are interested in the distribution of random locations of exit of the system from various domains and random times it takes to exit. The study of exit times motivated a psychological study of statistics of reaction times. It was shown that for most people the distribution of of decision times for decision problems with no a priori bias follows a very specific universal shape that can be explained via small noise exit problems. Other problems that were solved in the course of this project include: finding limiting branching statistics for large random trees and applying this to the study of RNA foldings; studying the limitng shape of large random trees as drawn on the plane; constructing solutions of nonlinear equations of mathematical physics using probabilistic tools. During this project two graduate students defended their Ph.D. Theses at Georgia Tech under the supervision of the PI. The PI ran a probability working seminar at Georgia Tech. Multiple undergraduate students worked with PI on computaional aspects of the problems described above. Besides the regular courses at Georgia Tech, the PI has taught several courses at various Summer Schools for graduate students in probability. Some of the lecture notes and video recording of these lectures are available online.
