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 $J_h$ 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 flow is characterized by the minimization of a functional $K_h$. We establish a new connection between these systems by proving that $J_h$ and $K_h$ are equal up to second order in $h$ as $h\to0$. This result gives a microscopic explanation of the origin of the entropy-Wasserstein gradient flow 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.