Hypocoercivity for non-linear infinite-dimensional degenerate stochastic differential equations

Abstract

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.

Infinite Dimensional Piecewise Deterministic Markov Processes

“…(iv) There are no refreshments. (v) This follows from (19) and Assumption 4.1 (3). (vi) There are no refreshments.…”

“…We shall prove Theorem 6.2 by applying the Abstract Hypocoercivity Theorem introduced in [16]. We shall use the formulation from [19] which has been developed for infinite dimensional stochastic differential equations. Assume that the operators (BS, C ) and (BA(1−Π), C ) are bounded and there exist constants…”

“…Note that [19] requires that the function Φ is convex however if Φ = Φ 1 + Φ 2 with Φ 1 convex and Φ 2 bounded then the Poincaré inequality still holds, this can be seen by first applying [19,Proposition 4] for Φ 1 and then by the same argument as in [4, Proposition 4.2.7] the Poincaré inequality holds for Φ. Therefore taking g = Πf for f ∈ L 2 µ we have that Assumption 6.5 (H3) holds.…”