Coupling by Change of Measure for Conditional McKean-Vlasov SDEs and Applications

TL;DR

This paper develops coupling methods via change of measure to establish inequalities and propagation of chaos results for conditional McKean-Vlasov SDEs, including systems with singular initial data and stochastic Hamiltonian systems.

Contribution

It introduces novel coupling techniques for conditional McKean-Vlasov SDEs, deriving log-Harnack inequalities and quantitative propagation of chaos under broader conditions.

Findings

Established log-Harnack inequality for non-degenerate conditional McKean-Vlasov SDEs.

Derived quantitative conditional propagation of chaos in relative entropy.

Extended results to conditional distribution dependent stochastic Hamiltonian systems.

Abstract

The couplings by change of measure are applied to establish log-Harnack inequality(equivalently the entropy-cost estimate) for conditional McKean-Vlasov SDEs and derive the quantitative conditional propagation of chaos in relative entropy for mean field interacting particle system with common noise. For the log-Harnack inequality, two different types of couplings will be constructed for non-degenerate conditional McKean-Vlasov SDEs with multiplicative noise. As to the quantitative conditional propagation of chaos in relative entropy, the initial distribution of interacting particle system is allowed to be singular with that of limit equation. The above results are also extended to conditional distribution dependent stochastic Hamiltonian system.