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.
"Probabilistic representation for solutions of an irregular porous media type equation." Ann. Probab. 38 (5) 1870 - 1900, September 2010. https://doi.org/10.1214/10-AOP526