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 $$\mathcal {F}C_b^{\infty }$$ F C 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 $$\alpha f-Nf=g$$ α f - N f = g , $$\alpha \in (0,\infty )$$ α ∈ ( 0 , ∞ ) , generalizing some results of Da Prato and Lunardi. Second, we prove essential m-dissipativity for generators $$(L_{\Phi },\mathcal {F}C_b^{\infty })$$ ( L Φ , F C b ∞ ) of infinite-dimensional degenerate diffusion processes. We emphasize that the essential m-dissipativity of $$(L_{\Phi },\mathcal {F}C_b^{\infty })$$ ( L Φ , F C 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 $$(L_{\Phi },\mathcal {F}C_b^{\infty })$$ ( L Φ , F C b ∞ ) and $$(N,\mathcal {F}C_b^{\infty })$$ ( N , F C 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.