Contraction Rates For McKean-Vlassov Stochastic Differential Equations

Mr. Dan Noelck
University of Wisconsin-Milwaukee

This work focuses on the contraction rates for McKean-Vlasov stochastic differential equations (SDEs), McKean-Vlasov Stochastic differential delay equations (SDDEs), and path dependent McKean-Vlasov stochastic differential equations.
Under suitable conditions on the coefficients of the SDE, this dissertation derives explicit quantitative contraction rates for the convergence in Wasserstein distances of McKean-Vlasov SDEs using the coupling method. The contraction results are then used to prove a propagation of chaos uniformly in time, which
provides quantitative bounds on convergence rate of interacting particle systems, and establishes exponential ergodicity for McKean-Vlasov SDEs. The dissertation further develops suitable conditions on the coefficients of the McKean-Vlasov SDDE to obtain a contraction in Wasserstein distance using the coupling method again. These results are used to establish exponential ergodicity for McKean-Vlasov SDDEs. Last the dissertation obtains suitable conditions on the coefficients of the path dependent McKean-Vlasov SDE for a contraction in Wasserstein distance.

Advisor: Prof. Chao Zhu

Committee Members:
Profs. Lijing Sun, Jeb Willenbring, Richard Stockbridge, and Peter Hinow