A stochastic Fokker-Planck equation and double probabilistic representation for the stochastic porous media type equation

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The purpose of the present paper consists in proposing and discussing a double probabilistic representation for a porous media equation in the whole space perturbed by a multiplicative colored noise. For almost all random realizations ω, one associates a stochastic differential equation in law with random coefficients, driven by an independent Brownian motion. The key ingredient is a uniqueness lemma for a linear SPDE of Fokker-Planck type with measurable bounded (possibly degenerated) random coefficients.