Date of Award

August 2024

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Chao Zhu

Committee Members

Richard Stockbridge, Jeb Willenbring, Peter Hinow, Lijing Sun

Keywords

contraction, coupling, exponential ergodicity, McKean-Vlasov stochastic differential equations, propagation of chaos

Abstract

In response to the pressing need of modeling, analyzing and applying complex systems with inherent distribution- and memory-dependent dynamical behaviours, this dissertation investigates both distribution- and memory-dependent stochastic differential equations. Following the establishment of the well-posedness of these stochastic differential equations, this dissertation is focused on asymptotic properties of the underlying processes. Under suitable conditions on the coefficients of the stochastic differential equations, this dissertation derives explicit quantitative contraction rates for the convergence in Wasserstein distance for McKean-Vlasov stochastic differential equations (MVSDEs) and McKean-Vlasov functional stochastic differential equations (MVFSDEs). The obtained contraction results for MVSDEs are further utilized to demonstrate a propagation of chaos uniformly over time. This propagation of chaos not only provides quantitative bounds on the convergence rate of interacting particle systems, but it also establishes exponential ergodicty for MVSDEs.

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