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A class of stochastic evolutions that scale to the porous medium equation

Abstract

A class of reversible Markov jump processes on a periodic lattice is described and a result about their scaling behavior stated: Under diffusion scaling, the empirical measure converges to a solution of the porous medium equation on thed-dimensional torus. The process can be viewed as a randomly interacting configuration of sticks that evolves through exchanges of stick pieces between nearest neighbors through a zero-range pressure mechanism, with conservation of total stick length.

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Feng, S., Iscoe, I. & Seppäläinen, T. A class of stochastic evolutions that scale to the porous medium equation. J Stat Phys 85, 513–517 (1996). https://doi.org/10.1007/BF02174218

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Key Words

  • Porous medium equation
  • hydrodynamic scaling limit