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


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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  1. 1.

    M. Ekhaus and T. Seppäläinen, Stochastic dynamics macroscopically governed by the porous medium equation for isothermal flow,Ann. Acad. Sci. Fenn. Ser. A I Math., to appear.

  2. 2.

    S. Feng, I. Iscoe, and T. Seppäläinen, A microscopic mechanism for the porous medium equation, Preprint (1995).

  3. 3.

    M. Z. Guo, G. C. Papanicolaou, and S. R. S. Varadhan, Nonlinear diffusion limit for a system with nearest neighbor interactions,Common. Math. Phys. 118:31–59 (1988).

    Article  Google Scholar 

  4. 4.

    M. Muskat,The Flow of Homogeneous Fluids Through Porous Media (McGraw-Hill, New York, 1937).

    Google Scholar 

  5. 5.

    Y. Suzuki and K. Uchiyama, Hydrodynamic limit for a spin system on a multidimensional lattice,Prob. Theory Related. Fields 95:47–74 (1993).

    Article  Google Scholar 

  6. 6.

    J. L. Vazquez, An introduction to the mathematical theory of the porous medium equation, inShape Optimization and Free Boundaries, M. C. Delfour and G. Sabidussi, eds. (Kluwer, New York, 1992), pp. 261–286.

    Google Scholar 

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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).

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Key Words

  • Porous medium equation
  • hydrodynamic scaling limit