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.
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Barbu, V., Röckner, M. & Russo, F. Probabilistic representation for solutions of an irregular porous media type equation: the degenerate case. Probab. Theory Relat. Fields 151, 1–43 (2011). https://doi.org/10.1007/s00440-010-0291-x
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DOI: https://doi.org/10.1007/s00440-010-0291-x