Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Document Type

Dissertation

Abstract

Interacting particle systems (IPS) are used to rigorously derive partial differential equations modeling macroscopic behavior given by systems of stochastic differential equa- tions in physics, biology, and other sciences. These systems model the time evolution of n interacting particles with desired dynamics. In the n-system studied here, only two parti- cles may interact at a time (binary collisions). The particles are exchangeable; that is, the ordering of the particles does not play a role, and the interaction between the ith particle and the jth particle models an elastic collision. In this thesis, we use the method of inter- acting particle systems to find weak solutions to the weak formulation of the Boltzmann- Enskog equation perturbed by a Brownian motion. Conservation of energy is lost due to the addition of a Brownian motion. We solve the martingale problem corresponding to this n-system and prove a propagation of chaos result. The sequence of empirical measures of this n-system weakly converges to the law of a weak solution of a stochastic differential equation of the McKean-Vlasov type driven by a Poisson random measure.

Date

3-25-2025

Committee Chair

Sundar, Padmanabhan

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