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

AbstractWe prove the existence of an invariant measure μ for the transition semigroup Pt associated with the fast diffusion porous media equation in a bounded domain O⊂Rd, perturbed by a Gaussian noise. The Kolmogorov infinitesimal generator N of Pt in L2(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 W1,2(H−1(O),μ) and prove that D(N)⊂W1,2(H−1(O),μ)