The aim of the article is to construct solutions to second order in time stochastic partial differential equations and to show hypocoercivity of their corresponding transition semigroups. More generally, we analyze infinite-dimensional nonlinear stochastic differential equations in terms of their infinitesimal generators. In the first part of this article we use resolvent methods developed by Beznea, Boboc and Röckner to construct µ Φ -standard right processes with infinite lifetime and weakly continuous paths providing weak solutions to infinite-dimensional Langevin dynamics with invariant measure µ Φ . The second part deals with the general abstract Hilbert space hypocoercivity method, first described by Dolbeaut, Mouhout and Schmeiser and made rigorous in the Kolmogorov backwards setting by Grothaus and Stilgenbauer. In order to apply the method to infinite-dimensional Langevin dynamics we use an essential m-dissipativity statement for infinite-dimensional Ornstein-Uhlenbeck operators, perturbed by the gradient of a potential, with possible unbounded diffusion operators as coefficients and corresponding regularity estimates. Furthermore, essential m-dissipativity of a non-sectorial Kolmogorov backward operator associated to the dynamic and Poincaré inequalities for measures with densities w.r.t. infinitedimensional non-degenerate Gaussian measures are substantial. Deriving a stochastic representation of the semigroup generated by the Kolmogorov backward operator as the transition semigroup of the µ Φ -standard right process enables us to show an L 2exponential ergodic result for the weak solution. In the end we apply our results to explicit infinite-dimensional degenerate diffusion equations.

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.

The aim of this article is to construct solutions to second order in time stochastic partial differential equations and to show hypocoercivity of the corresponding transition semigroups. More generally, we analyze non-linear infinite-dimensional degenerate stochastic differential equations in terms of their infinitesimal generators. In the first part of this article we use resolvent methods developed by Beznea, Boboc and Röckner to construct diffusion processes with infinite lifetime and explicit invariant measures. The processes provide weak solutions to infinite-dimensional Langevin dynamics. The second part deals with a general abstract Hilbert space hypocoercivity method, developed by Dolbeaut, Mouhot and Schmeiser. In order to treat stochastic (partial) differential equations, Grothaus and Stilgenbauer translated these concepts to the Kolmogorov backwards setting taking domain issues into account. To apply these concepts in the context of infinite-dimensional Langevin dynamics we use an essential m-dissipativity result for infinite-dimensional Ornstein–Uhlenbeck operators, perturbed by the gradient of a potential. We allow unbounded diffusion operators as coefficients and apply corresponding regularity estimates. Furthermore, essential m-dissipativity of a non-sectorial Kolmogorov backward operator associated to the dynamic is needed. Poincaré inequalities for measures with densities w.r.t. infinite-dimensional non-degenerate Gaussian measures are studied. Deriving a stochastic representation of the semigroup generated by the Kolmogorov backward operator as the transition semigroup of a diffusion process enables us to show an $$L^2$$ L 2 -exponential ergodicity result for the weak solution. Finally, we apply our results to explicit infinite-dimensional degenerate diffusion equations.