Benedikt Eisenhuth, Martin Grothaus

• First essential m-dissipativity of an infinite-dimensional Ornstein-Uhlenbeck operator N, perturbed by the gradient of a potential, on a domain FC ∞ b of finitely based, smooth and bounded functions, is shown. Our considerations allow unbounded diffusion operators as coefficients. We derive corresponding second order regularity estimates for solutions f of the Kolmogorov equation ◂−▸αf−Nf=g, ◂+▸α∈(0,∞), generalizing some results of Da Prato and Lunardi. Second, we prove essential m-dissipativity for generators (◂,▸LΦ,FC ∞ b ) of infinite-dimensional degenerate diffusion processes. We emphasize that the essential m-dissipativity of (◂,▸LΦ,FC ∞ b ) is useful to apply general resolvent methods developed by Beznea, Boboc and Röckner, in order to construct martingale/weak solutions to infinite-dimensional non-linear degenerate stochastic differential equations. Furthermore, the essential m-dissipativity of (◂,▸LΦ,FC ∞ b ) and (◂,▸N,FC ∞ b ), as well as the regularity estimates are essential to apply the general abstract Hilbert space hypocoercivity method from Dolbeault, Mouhot, Schmeiser and Grothaus, Stilgenbauer, respectively, to the corresponding diffusions.