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From a large-deviations principle to the Wasserstein Gradient Flow: a new micro-macro passage

Adams, Stefan, Dirr, Nicolas P., Peletier, Mark A. and Zimmer, Johannes 2011. From a large-deviations principle to the Wasserstein Gradient Flow: a new micro-macro passage. Communications in Mathematical Physics 307 (3) , pp. 791-815. 10.1007/s00220-011-1328-4

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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.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Publisher: Springer
ISSN: 0010-3616
Last Modified: 09 Mar 2020 16:00

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