Markov Process

New York University, New York NY

Investigators

Abstract

0104343 Varadhan The focus of this project is interacting particle systems and their scaling limits. When we have a large system of particles interacting through local interactions, the evolution of such a system when it starts far away from equilibrium poses several questions. Density being often the only conserved quantity, there is usually some sort of a transport equation appearing as the scaling limit. Superposed on it is the motion of individual particles, which is usually an inhomogeneous Markov process after the interactions are somehow averaged in the scaling limit. Laws of large numbers as well as fluctuations and large deviations from them are some of the interesting problems in this context. In addition the coefficients arising in the limiting descriptions are functions of density and or other parameters and the regularity of their dependence on the parameters is also of importance. There are a large class of physical processes where the rules of behavior are prescribed at the level of individual units. These rules concern the nature of the interaction between individual units that could involve some randomness as well. But one needs to make predictions of the collective behavior of the units at a much larger scale. This project deals with the derivation of the rules of collective behavior from models of interactions at the level of individual units. In the physical sciences examples of such problems that have been successfully studied include the rules of fluid flow that are derived from the laws of interaction between molecules that make up the fluid. Similar problems in the social sciences, for instance one of predicting macroeconomic behavior based on models of economic exchange between individuals, have not been adequately addressed.