Contributions to the Stieltjes moment problem and to the intertwining of Markov semigroups

Abstract

Each chapter of this thesis is represented as a metaphorical meeting between mathematicians and their ideas: Stieltjes meet Gauss; Berg meet Urbanik; Dynkin meet Jacobi; Dynkin meet Villani. In the "Stieltjes meet Gauss'' part we provide some new criteria for the determinacy problem of the Stieltjes moment problem, starting with a Tauberian type criterion for indeterminacy that is expressed purely in terms of the asymptotic behavior of the moment sequence (and its extension to imaginary lines). Under an additional assumption this provides a converse to the classical Carleman's criterion, thus yielding an equivalent condition for determinacy. We also provide a criterion for determinacy that only involves the large asymptotic behavior of the distribution (or of the density if it exists), which can be thought of as an Abelian counterpart to the previous Tauberian type result. This latter criterion generalizes Hardy's condition for determinacy, and under some further assumptions yields a converse to Pedersen's refinement of Krein's celebrated theorem. The proofs utilize non-classical Tauberian results for moment sequences that are analogues of the ones developed by Feigin and Yaschin and Balkema et al. for the bi-lateral Laplace transform in the context of asymptotically parabolic functions, which generalize the classical Gaussian setting. We illustrate these results by studying the time-dependent moment problem for the law of a process whose logarithm is a Lévy process, which is a generalization of the log-normal distribution. Along the way, we derive the large asymptotic behavior of the density of spectrally-negative Lévy processes having a Gaussian component, which may be of independent interest. We continue the study of this time-dependent moment problem in the "Berg meet Urbanik'' part where we focus on Berg-Urbanik semigroups, a class of multiplicative convolution semigroups on $\R_+$ that is in bijection with the set of Bernstein functions. Berg and Durán proved that the law of such semigroups is determinate (at least) up to time $t=2$, and for the Bernstein function $\phi(u)=u$ Berg made the striking observation that for time $t>2$ the law of this semigroup is indeterminate. We extend these works by estimating the threshold time $\scr{T}_\phi \in [2,\infty]$ that it takes for the law of such Berg-Urbanik semigroups to transition from determinacy to indeterminacy in terms of simple properties of the underlying Bernstein function $\phi$, such as its Blumenthal-Getoor index. In particular, we show that $\scr{T}_\phi = 2$ for any Bernstein function $\phi$ with a drift component, thereby generalizing Berg's result to this entire class. One of the several strategies we implement to deal with the different cases relies on the non-classical Abelian type criterion mentioned above. To implement this approach we provide detailed information regarding distributional properties of the semigroup such as existence and smoothness of a density, and, the large asymptotic behavior for all $t > 0$ of this density along with its successive derivatives, which are original results in the Lévy process literature. In the "Dynkin meet Jacobi'' part we introduce and study non-local Jacobi operators, which generalize the classical (local) Jacobi operator. We show that these operators extend to the generator of an ergodic Markov semigroup with an invariant probability measure and study its spectral and convergence properties. In particular, we give a series expansion of the semigroup in terms of explicitly defined polynomials, which are counterparts of the classical Jacobi orthogonal polynomials. In addition, we give a complete characterization of the spectrum of the non-self-adjoint generator and semigroup. We show that the variance decay of the semigroup is hypocoercive in the sense of Villani, with explicit constants, which provides a natural generalization of the spectral gap estimate. After a random warm-up time the semigroup also decays exponentially in entropy, and is both hypercontractive and ultracontractive. Our proofs hinge on the development of intertwining relations---a notion for Markov semigroups introduced by Dynkin---between local and non-local Jacobi operators/semigroups, with the local Jacobi operator/semigroup serving as a reference object for transferring properties to the non-local ones. Finally, in the "Dynkin meet Villani'' part, we offer an original and comprehensive spectral theoretical approach to the study of convergence to equilibrium, and in particular of the hypocoercivity phenomenon, for contraction semigroups in Hilbert spaces. Here we utilize intertwining to transfer spectral information from a known, reference semigroup $\tilde{P} = (e^{-t\tilde{\A}})_{t \geq 0}$ to a target semigroup $P$ that is the object of study. This allows us to obtain conditions under which $P$ satisfies a hypocoercive estimate with exponential decay rate given by the spectral gap of $\tilde{\A}$. Along the way we also develop a functional calculus involving the non-self-adjoint resolution of identity induced by the intertwining relations. We apply these results to degenerate, hypoelliptic Ornstein-Uhlenbeck semigroups on $\R^d$, and non-local Jacobi semigroups on $[0,1]^d$; in both cases we obtain hypocoercive estimates and are able to explicitly identify the hypocoercive constants. All work in this thesis was done in collaboration with P. Patie, and the work in Chapter 4 was done with the additional collaboration of P. Cheridito and A. Srapionyan. The contents of Chapters 2--5 have been submitted to peer-reviewed journals as follows: - Non-classical Tauberian and Abelian type criteria for the moment problem, arXiv:1804.10721 [math.PR], 19pp., 2018; - The log-Lévy moment problem via Berg-Urbanik semigroups, Studia Math., accepted, 41pp., 2019; - On non-local ergodic Jacobi semigroups: spectral theory, convergence-to-equilibrium and contractivity, arXiv:1905.07832 [math.PR], 40pp., 2019; - A spectral theoretical approach for hypocoercivity applied to some degenerate hypoelliptic, and non-local operators, arXiv:1905.07042 [math.PR], 22pp., 2019.