Stochastic systems with complex interactions and random environments

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This project studies several classes of stochastic processes that possess complicated interactions and in some cases also inhomogeneous or random environments: interacting particle systems, polymer models, and random motion in a random medium.The goal is to describe typical large scale behavior and to quantify deviations from the typical behavior.In polymer models the goal is to settle long-standing open problems on scaling exponents that describe the order of magnitude of the fluctuations of the molecule chain and the free energy.For a class of asymmetric zero range processes this project proposes to prove scaling properties of space-time correlations and current fluctuations that confirm KPZ-type behavior.For a single particle moving in a random environment this project studies large deviations, especially the question of when the rate functions for quenched and averaged large deviations coincide.For a large collection of particles moving in a dynamically evolving environment the goal is to prove distributional limit theorems for the current.This project investigates mathematical models that describe complex interactions and motion of particles in an irregularenvironment. These mathematical systems incorporate randomnessto model irregularity and unpredictability.The goal is to discover general mathematical principlesthat govern such systems. A key point is that thesesystems appear quite different at microscopic and macroscopic scales. So it is important to understand how different rules for small-scale interactions and motions lead to different large-scale systemwide behavior.Real-world phenomena that such mathematical studies can illuminate include the motion of vehicles on a freeway, packets making their way through a communication network, fluid particles in a tube, wetting transitions where fluid spreads in a porous medium, epidemics advancing among individuals in a population, or the fluctuations of a polymer chain in a fluid.Over the longer term understanding these complex interactions has profound implications for science and engineering and thereby for society. Models of the kind described in the proposal are intensely and concurrently studied by mathematicians, natural scientists, social scientists, and engineers.