Entropy-Cost Inequalities for McKean-Vlasov SDEs with Singular Interactions

TL;DR

This paper establishes entropy-cost inequalities for McKean-Vlasov SDEs with singular interactions, including Coulomb and Riesz kernels, providing well-posedness and regularity results through new probability distances and hyperbound estimates.

Contribution

It introduces a novel probability distance to measure singular interactions and characterizes the path space of solutions using local hyperbound estimates.

Findings

Proves well-posedness of McKean-Vlasov SDEs with singular kernels.

Derives regularity estimates for solutions.

Establishes entropy-cost inequalities for these equations.

Abstract

For a class of McKean-Vlasov stochastic differential equations with singular interactions, which include the Coulomb/Riesz/Biot-Savart kernels as typical examples (Examples 2.1 and 2.2), we derive the well-posedness and regularity estimates by establishing the entropy-cost inequality. To measure the singularity of interactions, we introduce a new probability distance induced by local integrable functions, and estimate this distance for the time-marginal laws of solutions by using the Wasserstein distance of initial distributions. A key point of the study is to characterize the path space of time-marginal distributions for the solutions, by using local hyperbound estimates on diffusion semigroups.