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Rates of homogenization for fully-coupled McKean-Vlasov SDEs via the Cauchy-Problem on Wasserstein Space

Spring 2022
Zachary William Bezemek (BU)
Abstract: In many applications, an accurate model will have important features at multiple scales of time and/or space. There is a long history of analyzing such systems' effective behavior as the separation between these scales tends to infinity. The process of obtaining an effective equation in this limit is known as homogenization. It is well known that in the fully-coupled setting for stochastic differential equations (SDEs), one can obtain only convergence in distribution of the multiscale system to the homogenized system. There is great interest in obtaining rates of said convergence, and the prevailing method of doing so involves the study of certain Poisson equations and the Cauchy problem (Kolmogorov backward equation) associated to the homogenized system (R\"ockner and Xie 2021).
In this talk, we discuss extending this analysis to obtain rates of homogenization for fully-coupled multiscale McKean-Vlasov processes, i.e. stochastic differential equations whose coefficients depend on the distribution of the solution itself at each time. Equations of this form arise naturally from the many-particle limit of weakly-interacting diffusions, and there is immense interest in their properties in recent years due in part to the exploding popularity of Mean-field Games. Owing to the non-linearity of the Markov semigroup associated to McKean-Vlasov processes, the appropriate tool for studying rates of convergence is the Cauchy-Problem on the metric space of probability measures with finite second moment (Wasserstein Space), originally posed by Buckdahn, Li, Peng, and Rainer (2017). Hence, in the course of the proof we gain results on the regularity of Poisson-type of equations in their measure parameter and of the Cauchy-Problem on Wasserstein space that are of independent interest. We also gain as a corollary an extension of existing results on rates of homogenization for standard SDEs which allows for possibly non-linear test functionals of their Laws.