Solving McKean-Vlasov SDEs via relative entropy

TL;DR

This paper develops a robust solution theory for McKean-Vlasov SDEs and SPDEs using relative entropy, extending existing results to path-dependent interactions, singularities, fractional Brownian noise, and boundary-driven SPDEs.

Contribution

It introduces a new approach based on relative entropy for proving well-posedness of McKean-Vlasov equations, covering cases with singular interactions, fractional noise, and boundary SPDEs.

Findings

Proved weak existence and uniqueness for path-dependent interactions.

Extended results to singular interactions with Krylov-type conditions.

Constructed McKean-Vlasov SPDEs for heat, wave, and boundary noise cases.

Abstract

In this paper we explore the merit of relative entropy in proving weak well-posedness of McKean-Vlasov SDEs and SPDEs, extending the technique introduced in Lacker arxiv:2105.02983. In the SDE setting, we prove weak existence and uniqueness when the interaction is path dependent and only assumed to have linear growth. Meanwhile, we recover and extend the current results when the interaction has Krylov's $L_t^q-L_x^p$ type singularity for $\frac{d}{p}+\frac{2}{q}<1$, where $d$ is the dimension of space. We connect the aforementioned two cases which are traditionally disparate, and form a solution theory that is sufficiently robust to allow perturbations of sublinear growth at the presence of singularity, giving rise to the well-posedness of a new family of McKean-Vlasov SDEs. Our strategy naturally extends to the cases of a fractional Brownian driving noise $B^H$ for all $H\in\left(0,1\right)$, obtaining new results in each separate case $H\in\left(0,\frac{1}{2}\right)$ and $H\in\left(\frac{1}{2},1\right)$. In the SPDE setting, we construct McKean-Vlasov type SPDEs with bounded measurable coefficients from the prototype of stochastic heat equation in spatial dimension one, and we do the same construction for the stochastic wave equation and a SPDE with white noise acting only on the boundary. In addition, we generalize some quantitative propagation of chaos results for SDEs into the SPDE setting.