  We consider a porous media type equation over all of $\R^d$ 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. One of the main analytic
ingredients of the proof, is a new result on uniqueness of distributional
solutions of a linear PDE on $\R^1$ with non-continuous coefficients.
