From a large-deviations principle to the Wasserstein gradient flow

S Adams, Nicolas Dirr, M Peletier, Johannes Zimmer

OAI: oai:purehost.bath.ac.uk:publications/3df649b7-ea45-4b27-896a-d71c65da28b8 • DOI: https://doi.org/10.1007/s00220-011-1328-4

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 → 0. 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.