A trajectorial approach to relative entropy dissipation of McKean-Vlasov diffusions: Gradient flows and HWBI inequalities

We formulate a trajectorial version of the relative entropy dissipation identity for McKean-Vlasov diffusions, extending recent results 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 explic-itly 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 a formulation developed recently 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).