Propagation of chaos in Fisher information
Jules Grass, Christophe Poquet, Arnaud Guillin
TL;DR
This work proves sharp propagation of chaos in Fisher information for mean-field diffusions with smooth interactions by deriving a new Fisher-information lemma that controls the evolution of the information between two diffusion laws. Using the BBGKY hierarchy, the authors obtain coupled differential inequalities for the relative entropy and the Fisher information across k-particle marginals, with a diffusion–drift decomposition that yields a dissipative structure. The main result shows that, under suitable smoothness and Hessian-boundedness assumptions, H_t^k + I_t^k ≤ M_T k^2/N^2, and the Gaussian example confirms the optimality of the rate I_t^k = O(k^2/N^2). Regularity estimates ensure the necessary hypotheses persist for the particle system and its limit, enabling the method to handle both torus and R^d settings and suggesting potential extensions to singular interactions and uniform-in-time results.
Abstract
We present a new method for proving sharp local propagation of chaos in Fisher Information for particles with smooth interaction and drift. We rely on a new Lemma computing the Fisher Information of two diffusion processes with smooth drifts and fine estimates on the hessian of the law of the solution of the McKean-Vlasov equation. It allows us to obtain a new propagation of chaos in Fisher information, generalizing Lacker's seminal work by using the BBGKY hierarchy to obtain a system of differential inequalities satisfied by both the relative entropy and the Fisher Information of k particles. We also show with a simple Gaussian example that our decay rate is optimal.