Weak Solutions to Vlasov-McKean Equations under Lyapunov-Type Conditions

TL;DR

This paper develops a Lyapunov-based framework to establish weak existence and uniqueness of law-dependent stochastic differential equations with irregular coefficients, extending previous results to more general settings.

Contribution

It introduces a novel approach using weighted total variation distances for irregular coefficients, broadening the scope of weak solution analysis for Vlasov-McKean equations.

Findings

Established weak uniqueness under minimal regularity assumptions.

Provided an abstract weak existence theorem for law-dependent SDEs.

Extended the Lyapunov approach to weighted total variation distances.

Abstract

We present a Lyapunov type approach to the problem of existence and uniqueness of general law-dependent stochastic differential equations. In the existing literature most results concerning existence and uniqueness are obtained under regularity assumptions of the coefficients w.r.t the Wasserstein distance. Some existence and uniqueness results for irregular coefficients have been obtained by considering the total variation distance. Here we extend this approach to the control of the solution in some weighted total variation distance, that allows us now to derive a rather general weak uniqueness result, merely assuming measurability and certain integrability on the drift coefficient and some non-degeneracy on the dispersion coefficient. We also present an abstract weak existence result for the solution of law-dependent stochastic differential equations with merely measurable coefficients, based on an approximation with law-dependent stochastic differential equations with regular coefficients under Lyapunov type assumptions.