Essential m-dissipativity for generators of infinite-dimensional non-linear degenerate 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 non-linear 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 R\"ockner, in order to construct martingale/weak solutions to infinite-dimensional non-linear degenerate diffusion 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.