Mathematical Sciences: Markov Processes

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ABSTRACT We study complex systems modeled by interacting particles, or, more generally, by Markov Processes with a high dimensional state space. In the presence of conservation laws, the Markov Process will have several invariant measures. If the interaction is local then the evolution will take the system quickly to a local equilibrium parametrized by the local averages of the conserved quantities. These parameters that are functions of space and time will evolve according to some evolution law after suitable rescaling. There are several examples of this phenomenon, known as hydrodynamic scaling: the transition from Hamiltonian Equations for particles to Euler Equations for a fluid, Nonlinear diffusion equations for the bulk diffusion of various interacting particle systems etc. The project investigates rigorously this transition for several classes of models. One of the central problems in the use of mathematical models in the physical as well as social sciences, is the problem of scaling. One has a complex system consisting of a large number of interacting components which is evolving in time. Its properties are specified at the level of individual components. However one is interested in how certain small number of measurements made on the large or global scale evolve in time. The aim of this research is to make a mathematically valid link between the two in certain specific physical models.