Probability Theory and Related Fields

, Volume 151, Issue 1–2, pp 1–43 | Cite as

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

Article

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.

Keywords

Singular degenerate porous media type equation Probabilistic representation 

Mathematics Subject Classification (2000)

60H30 60H10 60G46 35C99 58J65 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Viorel Barbu
    • 1
  • Michael Röckner
    • 2
    • 3
  • Francesco Russo
    • 4
    • 5
  1. 1.University A1.I. CuzaIasiRomania
  2. 2.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  3. 3.Department of Mathematics and StatisticsPurdue UniversityW. LafayetteUSA
  4. 4.INRIA Rocquencourt, Equipe MathFi and Cermics Ecole des Ponts, Domaine de VoluceauLe Chesnay CedexFrance
  5. 5.Ecole Nationale Supérieure des Techniques Avancées (ENSTA, ParisTech)ParisFrance

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