Harnack inequality and applications for stochastic generalized porous media equations

TL;DR

This paper establishes a dimension-free Harnack inequality and strong Feller property for stochastic generalized porous media equations, leading to explicit bounds and properties of the associated semigroup.

Contribution

It introduces novel coupling and Girsanov transformation techniques to derive key inequalities and properties for these stochastic equations.

Findings

Proved dimension-free Harnack inequality for the semigroup.

Derived explicit upper bounds for the $L^p$-norm of the density.

Established hypercontractivity, ultracontractivity, and compactness of the semigroup.

Abstract

By using coupling and Girsanov transformations, the dimension-free Harnack inequality and the strong Feller property are proved for transition semigroups of solutions to a class of stochastic generalized porous media equations. As applications, explicit upper bounds of the $L^p$-norm of the density as well as hypercontractivity, ultracontractivity and compactness of the corresponding semigroup are derived.