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

  • Stefan AdamsEmail author
  • Nicolas Dirr
  • Mark A. Peletier
  • Johannes Zimmer


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.


Diffusion Equation Particle System Brownian Particle Empirical Measure Large Deviation Principle 
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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Stefan Adams
    • 1
    Email author
  • Nicolas Dirr
    • 2
  • Mark A. Peletier
    • 3
  • Johannes Zimmer
    • 4
  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK
  2. 2.School of MathematicsCardiff UniversityCardiff, WalesUK
  3. 3.Department of Mathematics and Computer Science and Institute of Complex Molecular SystemsTechnische Universiteit EindhovenEindhovenThe Netherlands
  4. 4.Department of Mathematics SciencesUniversity of BathBathUK

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