Ergodic Theory of Complex Random Dynamics

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This project is targeted at ergodic theory of infinite-dimensional random dynamical systems. The main goal is the analysis of steady statistical patterns in stochastic processes associated with random media such as random Lagrangian systems, directed polymers, random growth models, and models of stochastic hydrodynamics. Ergodicity is a form of stochastic stability that means that the long-term statistical properties of the system are not sensitive to the initial state. In particular, it makes measurements related to complex systems meaningful. Long-term statistical patterns in deterministic and random dynamical systems can be described via stationary distributions, and it is the description of these stationary regimes that the project aims at. It builds around the momentum that has been created by recent progress by the PI in ergodic theory of the Burgers equation and related systems. The central role of the Burgers equation is due to the fact that it is fundamental in modeling a variety of phenomena from traffic to the formation of large scale structure of the Universe. It is simultaneously a fluid dynamics model, a growth model, a conservation law, and is tightly related to stochastic control, directed polymers, and an important KPZ universality class of models of statistical mechanics. The proposed program will extend the understanding of statistical patterns for the Burgers equation and related complex random dynamical systems in noncompact settings.Studying ergodic properties of stochastic partial differential equations or similar models in unbounded domains calls for development of new mathematical techniques. The project will use and enhance modern methods of mathematical statistical mechanics, probability theory, and dynamics, techniques developed for first-passage percolation and last passage percolation models, lattice animals, concentration inequalities, stochastic control, thermodynamic limits for Gibbs ensembles and hydrodynamic limits for interacting particle systems, along with variational approach and analysis techniques such as inequalities and embeddings, typical for partial differential equations. The description and analysis of stationary distributions will be obtained via sample measures, one force one solution principles, random one-sided Lagrangian minimizers and their polymer counterparts.