Doubly probabilistic representation for the stochastic porous media type equation

TL;DR

This paper introduces a novel doubly probabilistic framework for representing solutions to stochastic porous media equations with multiplicative colored noise, linking PDE solutions to stochastic differential equations with random coefficients.

Contribution

It proposes a new probabilistic representation for stochastic porous media equations involving random coefficients and colored noise, extending existing methods.

Findings

Establishes a connection between PDE solutions and stochastic differential equations with random coefficients.

Provides a framework for analyzing stochastic porous media equations in the whole space R^1.

Demonstrates the applicability of the representation to equations perturbed by multiplicative colored noise.

Abstract

The purpose of the present paper consists in proposing and discussing a doubly probabilistic representation for a stochastic porous media equation in the whole space R^1 perturbed by a multiplicative coloured noise. For almost all random realizations $\omega$, one associates a stochastic differential equation in law with random coefficients, driven by an independent Brownian motion.