In this article we extend the modern, powerful and simple abstract Hilbert space strategy for proving hypocoercivity that has been developed originally by Dolbeault, Mouhot and Schmeiser in [DMS10]. As well-known, hypocoercivity methods imply an exponential decay to equilibrium with explicit computable rate of convergence. Our extension is now made for studying the long-time behavior of some strongly continuous semigroup generated by a (degenerate) Kolmogorov backward operator L. Additionally, we introduce several domain issues into the framework. Necessary conditions for proving hypocoercivity need then only to be verified on some fixed operator core of L. Furthermore, the setting is also suitable for covering existence and construction problems as required in many applications. The methods are applicable to various, different, Kolmogorov backward evolution problems. As a main part, we apply the extended framework to the (degenerate) spherical velocity Langevin equation. The latter can be seen as some kind of an analogue to the classical Langevin equation in case spherical velocities are required. This model is of important industrial relevance and describes the fiber lay-down in the production process of nonwovens. For the construction of the strongly continuous contraction semigroup we make use of modern hypoellipticity tools and pertubation theory.
We construct N -particle Langevin dynamics in R d or in a cuboid region with periodic boundary for a wide class of N -particle potentials and initial distributions which are absolutely continuous w.r.t. Lebesgue measure. The potentials are in particular allowed to have singularities and discontinuous gradients (forces). An important point is to prove an L p -uniqueness of the associated non-symmetric, non-sectorial degenerate elliptic generator. Analyzing the associated functional analytic objects, we also give results on the long-time behaviour of the dynamics, when the invariant measure is finite: Firstly, we prove the weak mixing property whenever it makes sense (i.e. whenever { < ∞} is connected). Secondly, for a still quite large class of potentials we also give a rate of convergence of time averages to equilibrium when starting in the equilibrium distribution. In particular, all results apply to N -particle systems with pair interactions of Lennard-Jones type.
For a contraction C 0 -semigroup on a separable Hilbert space, the decay rate is estimated by using the weak Poincaré inequalities for the symmetric and anti-symmetric part of the generator. As applications, non-exponential convergence rate is characterized for a class of degenerate diffusion processes, so that the study of hypocoercivity is extended. Concrete examples are presented.
We construct an infinite dimensional analysis with respect to nonGaussian measures of Mittag-Leffler type which we call Mittag-Leffler measures. It turns out that the well-known Wick ordered polynomials in Gaussian analysis cannot be generalized to this non-Gaussian case. Instead of using Wick ordered polynomials we prove that a system of biorthogonal polynomials, called Appell system, is applicable to the Mittag-Leffler measures. Therefore we are able to introduce a test function and a distribution space. As an application we construct Donsker's delta in a non-Gaussian setting as a weak integral in the distribution space.
Mittag-Leffler analysis is an infinite dimensional analysis with respect to non-Gaussian measures of Mittag-Leffler type which generalizes the powerful theory of Gaussian analysis and in particular white noise analysis. In this paper we further develop the Mittag-Leffler analysis by characterizing the convergent sequences in the distribution space. Moreover we provide an approximation of Donsker's delta by square integrable functions. Then we apply the structures and techniques from Mittag-Leffler analysis in order to show that a Green's function to the time-fractional heat equation can be constructed using generalized grey Brownian motion (ggBm) by extending the fractional Feynman-Kac formula [Sch92]. Moreover we analyse ggBm, show its differentiability in a distributional sense and the existence of corresponding local times. (2010): Primary: 46F25, 60G22. Secondary: 26A33, 33E12.
Mathematics Subject ClassificationRemark 2.1. N is a perfect space, i.e. every bounded and closed set in N is compact. As a consequence strong and weak convergence coincide in both N and N ′ , see page 73 in [GV64] and Section I.6.3 and I.6.4 in [GS68].Example 2.2. Consider the white noise setting, where N = S(R) is the space of Schwartz test functions, H = L 2 (R, dx) and N ′ = S ′ (R) are the tempered distributions. S(R) is dense in L 2 (R, dx) and can be represented as the projective limit of certain Hilbert spaces H p , p > 0, with norms denoted by |·| p , see e.g. [Kuo96]. Thus the white noise setting is an example for the nuclear triple described above.
The Mittag-Leffler measureAs Mittag-Leffler measures µ β , 0 < β < 1, we denote the probability measures on N ′ whose characteristic functions are given via Mittag-Leffler functions. The Mittag-Leffler function was introduced by Gösta Mittag-Leffler in [ML05] and we also consider a generalization first appeared in [Wim05]. Definition 2.3. For 0 < β < ∞ the Mittag-Leffler function is an entire function defined by its power series E β (z) := ∞ n=0 z n Γ(βn + 1), z ∈ C.
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