The Annals of Probability

Probabilistic representation for solutions of an irregular porous media type equation

Philippe Blanchard, Michael Röckner, and Francesco Russo

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We consider a porous media type equation over all of ℝd, d=1, with monotone discontinuous coefficient with linear growth, and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. The interest in such singular porous media equations is due to the fact that they can model systems exhibiting the phenomenon of self-organized criticality. One of the main analytic ingredients of the proof is a new result on uniqueness of distributional solutions of a linear PDE on ℝ1 with not necessarily continuous coefficients.

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Ann. Probab., Volume 38, Number 5 (2010), 1870-1900.

First available in Project Euclid: 17 August 2010

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Primary: 60H30: Applications of stochastic analysis (to PDE, etc.) 60H10: Stochastic ordinary differential equations [See also 34F05] 60G46: Martingales and classical analysis 35C99: None of the above, but in this section 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60]

Singular porous media type equation probabilistic representation self-organized criticality (SOC)


Blanchard, Philippe; Röckner, Michael; Russo, Francesco. Probabilistic representation for solutions of an irregular porous media type equation. Ann. Probab. 38 (2010), no. 5, 1870--1900. doi:10.1214/10-AOP526.

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  • [1] Alt, H. W. (2002). Lineare Funktionalanalysis: Eine Anwendungsorientierte Einführung, 5th ed. Springer, Berlin.
  • [2] Bak, P. (1996). How Nature Works: The Science of Self-Organized Criticality. Copernicus, New York.
  • [3] Bantay, P. and Janosi, I. M. (1992). Avalanche dynamics from anomalous diffusion. Phys. Rev. Lett. 68 2058–2061.
  • [4] Barbu, V. (1976). Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Leiden.
  • [5] Barbu, V. (1993). Analysis and Control of Nonlinear Infinite-dimensional Systems. Mathematics in Science and Engineering 190. Academic Press, San Diego, CA.
  • [6] Barbu, V., Röckner, M. and Russo, F. (2010). Probabilistic representation for solutions of a porous media equation: The irregular degenerate case. Probab. Theory Related Fields. To appear.
  • [7] Barlow, M. T. and Yor, M. (1982). Semimartingale inequalities via the Garsia–Rodemich–Rumsey lemma, and applications to local times. J. Funct. Anal. 49 198–229.
  • [8] Benachour, S., Chassaing, P., Roynette, B. and Vallois, P. (1996). Processus associés à l’équation des milieux poreux. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 793–832.
  • [9] Benilan, P., Brezis, H. and Crandall, M. G. (1975). A semilinear equation in L1(RN). Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 523–555.
  • [10] Bénilan, P. and Crandall, M. G. (1981). The continuous dependence on φ of solutions of ut−Δφ(u)=0. Indiana Univ. Math. J. 30 161–177.
  • [11] Bogachev, V. I. (2007). Measure Theory, Vol. I, II. Springer, Berlin.
  • [12] Bogachev, V. I., Krylov, N. V. and Röckner, M. (2001). On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions. Comm. Partial Differential Equations 26 2037–2080.
  • [13] Brezis, H. (1977). Opérateurs Maximaux Monotones et Semigroupes de Contraction dans les Espaces de Hilbert. North-Holland, Amsterdam.
  • [14] Brézis, H. and Crandall, M. G. (1979). Uniqueness of solutions of the initial-value problem for ut−Δφ(u)=0. J. Math. Pures Appl. (9) 58 153–163.
  • [15] Crandall, M. G. and Evans, L. C. (1975). On the relation of the operator /∂s+/∂τ to evolution governed by accretive operators. Israel J. Math. 21 261–278.
  • [16] Evans, L. C. (1977). Nonlinear evolution equations in an arbitrary Banach space. Israel J. Math. 26 1–42.
  • [17] Figalli, A. (2008). Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. J. Funct. Anal. 254 109–153.
  • [18] Graham, C., Kurtz, T. G., Méléard, S., Protter, P. E., Pulvirenti, M. and Talay, D. (1996). Probabilistic Models for Nonlinear Partial Differential Equations. Lecture Notes in Math. 1627. Springer, Berlin.
  • [19] Jourdain, B. (2000). Probabilistic approximation for a porous medium equation. Stochastic Process. Appl. 89 81–99.
  • [20] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [21] Le Bris, C. and Lions, P. L. (2008). Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients. Comm. Partial Differential Equations 33 1272–1317.
  • [22] McKean, H. P. Jr. (1967). Propagation of chaos for a class of non-linear parabolic equations. In Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967) 41–57. Air Force Office Sci. Res., Arlington, VA.
  • [23] Rozkosz, A. and Slominski, L. (1993). On weak solutions of one-dimensional SDEs with time-dependent coefficients. Stochastics Stochastics Rep. 42 199–208.
  • [24] Senf, T. (1993). On one-dimensional stochastic differential equations without drift and with time-dependent diffusion coefficients. Stochastics Stochastics Rep. 43 199–220.
  • [25] Showalter, R. E. (1997). Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Mathematical Surveys and Monographs 49. Amer. Math. Soc., Providence, RI.
  • [26] Stein, E. M. (1970). Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series 30. Princeton Univ. Press, Princeton, NJ.
  • [27] Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 233. Springer, Berlin.
  • [28] Sznitman, A.-S. (1991). Topics in propagation of chaos. In École D’Été de Probabilités de Saint-Flour XIX—1989. Lecture Notes in Math. 1464 165–251. Springer, Berlin.