Kolmogorov equation associated to a stochastic Kuramoto–Sivashinsky equation

This paper investigates the relation between the Kolmogorov operator associated to a stochastic Kuramoto–Sivashinsky equation and the infinitesimal generator for the corresponding transition semigroup. We prove that the infinitesimal generator is the closure of Kolmogorov operator in the space of continuous functions with kth-polynomial growth with respect to π-convergence topology and the space L2(H,ν) respectively. The proof depends on various estimates on the solution, invariant measure and transition semigroup. As a product, we also obtains smoothing properties of the transition semigroup.