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

DOI10.1007/S00220-011-1328-4zbMATH Open1267.60106arXiv1004.4076OpenAlexW3098539775WikidataQ59873999 ScholiaQ59873999MaRDI QIDQ647376

Author name not available (Why is that?)

Publication date: 23 November 2011

Published in: (Search for Journal in Brave)

Abstract: 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 o 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.

Full work available at URL: https://arxiv.org/abs/1004.4076

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