From a large-deviations principle to the Wasserstein gradient flow : a new micro-macro passage

We study the connection between a system of many independent Brownian particles on one hand and the deterministic diffusion equation on the other. For a fixed time step h > 0, a large-deviations rate functional Jh characterizes the behaviour of the particle system at t = h in terms of the initial distribution at t = 0. For the diffusion equation, a single step in the time-discretized entropy-Wasserstein gradient fl ow is characterized by the minimization of a functional Kh. We establish a new connection between these systems by proving that Jh and Kh are equal up to second order in h as h --> 0. This result gives a microscopic explanation of the origin of the entropy-Wasserstein gradient fl ow formulation of the diffusion equation. Simultaneously, the limit passage presented here gives a physically natural description of the underlying particle system by describing it as an entropic gradient flow.