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

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