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

We prove the existence of an invariant measure μμ for the transition semigroup PtPt associated with the fast diffusion porous media equation in a bounded domain O⊂RdO⊂Rd, perturbed by a Gaussian noise. The Kolmogorov infinitesimal generator NN of PtPt in L2(H−1(O),μ)L2(H−1(O),μ) is characterized as the closure of a second-order elliptic operator in H−1(O)H−1(O). Moreover, we construct the Sobolev space W1,2(H−1(O),μ)W1,2(H−1(O),μ) and prove that D(N)⊂W1,2(H−1(O),μ)D(N)⊂W1,2(H−1(O),μ).