On pathwise uniqueness for multidimensional McKean--Vlasov equations

TL;DR

This paper proves pathwise uniqueness for multi-dimensional McKean--Vlasov equations under moderate regularity conditions, requiring Dini-continuous drift and Lipschitz continuous, nondegenerate diffusion.

Contribution

It establishes pathwise uniqueness for multidimensional McKean--Vlasov equations with less restrictive regularity assumptions on the coefficients.

Findings

Pathwise uniqueness holds under Dini-continuous drift.

Diffusion coefficients must be Lipschitz and uniformly nondegenerate.

Results extend classical uniqueness results to broader coefficient regularities.

Abstract

Pathwise uniqueness for multi-dimensional stochastic McKean--Vlasov equation is established under moderate regularity conditions on the drift and diffusion coefficients. Both drift and diffusion depend on the marginal measure of the solution. For pathwise uniqueness, the drift is assumed to be Dini-continuous in the state variable, while the diffusion must be Lipschitz, continuous in time and uniformly nondegenerate. The setting is classical McKean--Vlasov, that is, coefficients of the equation are represented as integrals over the marginal distributions of the process.