From a Large-Deviations Principle to the Wasserstein Gradient Flow: A New Micro-Macro Passage

Stefan Adams, Nicolas Dirr, Mark A. Peletier, Johannes Zimmer

Research output: Contribution to journalArticlepeer-review

72 Scopus citations

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 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 flow 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 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.

Original languageEnglish
Pages (from-to)791-815
Number of pages25
JournalCommunications in Mathematical Physics
Volume307
Issue number3
DOIs
StatePublished - Nov 2011
Externally publishedYes

Fingerprint

Dive into the research topics of 'From a Large-Deviations Principle to the Wasserstein Gradient Flow: A New Micro-Macro Passage'. Together they form a unique fingerprint.

Cite this