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

Reference:

Adams, S., Dirr, N., Peletier, M. and Zimmer, J., 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.

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Official URL:

http://dx.doi.org/10.1007/s00220-011-1328-4

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 → 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.

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