Abstract
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 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 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).
Funding Statement
B. Tschiderer acknowledges support by the Austrian Science Fund (FWF) under grant P28661, by the Vienna Science and Technology Fund (WWTF) through project MA16-021, and additionally appreciates travel support through the National Science Foundation (NSF) under grant NSF-DMS-14-05210. L.C. Yeung acknowledges support under grant NSF-DMS-20-04997.
Acknowledgments
We are grateful to Ioannis Karatzas and Walter Schachermayer for suggesting this problem and giving us generous advice. We thank Robert Fernholz, Miguel Garrido, Tomoyuki Ichiba, Donghan Kim, Kasper Larsen, and Mete Soner for helpful comments during the INTECH research meetings. Thanks also go to an associate editor and an anonymous referee for their valuable comments and suggestions.
Citation
Bertram Tschiderer. Lane Chun Yeung. "A trajectorial approach to relative entropy dissipation of McKean–Vlasov diffusions: Gradient flows and HWBI inequalities." Bernoulli 29 (1) 725 - 756, February 2023. https://doi.org/10.3150/22-BEJ1476
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