CAREER: Ergodicity and Random Media

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

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.