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Essential m-dissipativity for generators of infinite-dimensional non-linear degenerate diffusion processes
Benedikt Eisenhuth, Martin Grothaus · 2021-04-10 · via math.PR updates on arXiv.org

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 $αf-Nf=g$, $α\in (0,\infty)$, generalizing some results of Da Prato and Lunardi. Second we prove essential m-dissipativity for generators $(L_Φ,\mathcal{F}C_b^{\infty})$ of infinite-dimensional non-linear degenerate diffusion processes. We emphasize that the essential m-dissipativity of $(L_Φ,\mathcal{F}C_b^{\infty})$ 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 diffusion equations. Furthermore, the essential m-dissipativity of $(L_Φ,\mathcal{F}C_b^{\infty})$ 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.

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