A trajectorial approach to relative entropy dissipation of McKean$\boldsymbol{-}$Vlasov diffusions: gradient flows and HWBI inequalities

TL;DR

This paper develops a trajectorial framework for analyzing the dissipation of relative entropy in McKean-Vlasov diffusions, connecting gradient flows and HWBI inequalities through stochastic analysis and Wasserstein calculus.

Contribution

It introduces a trajectorial approach to relative entropy dissipation for McKean-Vlasov diffusions, extending previous results to interacting systems and deriving new interpretations of gradient flows and HWBI inequalities.

Findings

Explicit computation of entropy dissipation rate along diffusion trajectories

New interpretation of gradient flow structure for granular media equation

Novel derivation of HWBI inequality using trajectorial methods

Abstract

We formulate a trajectorial version of the relative entropy dissipation identity for McKean$-$Vlasov diffusions, extending the results of the papers [FJ16,KST20a], which apply to non-interacting diffusions. Our stochastic analysis approach is based on time-reversal of diffusions and Lions' differential calculus over Wasserstein space. It allows us to compute explicitly the rate of relative entropy dissipation along every trajectory of the underlying diffusion via the semimartingale decomposition of the corresponding relative entropy process. As a first application, we obtain a new interpretation of the gradient flow structure for the granular media equation, generalizing the formulation in [KST20a] developed for the linear Fokker$-$Planck equation. Secondly, we show how the trajectorial approach leads to a new derivation of the HWBI inequality, which relates relative entropy (H), Wasserstein distance (W), barycenter (B) and Fisher information (I).