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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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 17 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1282053774

Digital Object Identifier
doi:10.1214/10-AOP526

Mathematical Reviews number (MathSciNet)
MR2722788

Zentralblatt MATH identifier
1202.60111

Subjects
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]

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

Citation

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. https://projecteuclid.org/euclid.aop/1282053774


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