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Probabilistic representation for solutions of an irregular porous media type equation: the degenerate case
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  • Published: 26 May 2010

Probabilistic representation for solutions of an irregular porous media type equation: the degenerate case

  • Viorel Barbu1,
  • Michael Röckner2,3 &
  • Francesco Russo4,5 

Probability Theory and Related Fields volume 151, pages 1–43 (2011)Cite this article

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Abstract

We consider a possibly degenerate porous media type equation over all of \({\mathbb R^d}\) with d = 1, with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. This equation is motivated by some singular behaviour arising in complex self-organized critical systems. The main idea consists in approximating the equation by equations with monotone non-degenerate coefficients and deriving some new analytical properties of the solution.

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Author information

Authors and Affiliations

  1. University A1.I. Cuza, 6600, Iasi, Romania

    Viorel Barbu

  2. Fakultät für Mathematik, Universität Bielefeld, 33615, Bielefeld, Germany

    Michael Röckner

  3. Department of Mathematics and Statistics, Purdue University, W. Lafayette, IN, 47907, USA

    Michael Röckner

  4. INRIA Rocquencourt, Equipe MathFi and Cermics Ecole des Ponts, Domaine de Voluceau, Rocquencourt, B.P. 105, 78153, Le Chesnay Cedex, France

    Francesco Russo

  5. Ecole Nationale Supérieure des Techniques Avancées (ENSTA, ParisTech), Paris, France

    Francesco Russo

Authors
  1. Viorel Barbu
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  2. Michael Röckner
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  3. Francesco Russo
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Corresponding author

Correspondence to Francesco Russo.

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Barbu, V., Röckner, M. & Russo, F. Probabilistic representation for solutions of an irregular porous media type equation: the degenerate case. Probab. Theory Relat. Fields 151, 1–43 (2011). https://doi.org/10.1007/s00440-010-0291-x

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  • Received: 18 August 2009

  • Revised: 15 March 2010

  • Published: 26 May 2010

  • Issue Date: October 2011

  • DOI: https://doi.org/10.1007/s00440-010-0291-x

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Keywords

  • Singular degenerate porous media type equation
  • Probabilistic representation

Mathematics Subject Classification (2000)

  • 60H30
  • 60H10
  • 60G46
  • 35C99
  • 58J65
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