Essential m-dissipativity for Possibly Degenerate Generators of Infinite-dimensional Diffusion Processes

Abstract

First essential m-dissipativity of an infinite-dimensional Ornstein-Uhlenbeck operator N, perturbed by the gradient of a potential, on a domain mathcal {F}C_b^{infty } 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 alpha f-Nf=g, alpha in (0,infty ), generalizing some results of Da Prato and Lunardi. Second, we prove essential m-dissipativity for generators (L_{Phi },mathcal {F}C_b^{infty }) of infinite-dimensional degenerate diffusion processes. We emphasize that the essential m-dissipativity of (L_{Phi },mathcal {F}C_b^{infty }) 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 (L_{Phi },mathcal {F}C_b^{infty }) and (N,mathcal {F}C_b^{infty }), 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.