Invariant measures and the Kolmogorov equation for the stochastic fast diffusion equation

Abstract

We prove the existence of an invariant measure μ for the transition semigroup P t associated with the fast diffusion porous media equation in a bounded domain O ⊂ R d , perturbed by a Gaussian noise. The Kolmogorov infinitesimal generator N of P t in L 2 ( H − 1 ( O ) , μ ) is characterized as the closure of a second-order elliptic operator in H − 1 ( O ) . Moreover, we construct the Sobolev space W 1 , 2 ( H − 1 ( O ) , μ ) and prove that D ( N ) ⊂ W 1 , 2 ( H − 1 ( O ) , μ ) .