Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of L\'evy processes

We provide the increasing eigenfunctions associated to spectrally negative self-similar Feller semigroups, which have been introduced by Lamperti. These eigenfunctions are expressed in terms of a new family of power series which includes, for instance, the modified Bessel functions of the first kind and some generalizations of the Mittag-Leffler function. Then, we show that some specific combinations of these functions are Laplace transforms of self-decomposable or infinitely divisible distributions concentrated on the positive line with respect to the main argument, and, more surprisingly, with respect to a parameter of the process. In particular, this generalizes a result of Hartman (1976) obtained for the increasing solution of the Bessel differential equation. Finally, we compute, for some cases, the associated decreasing eigenfunctions and derive the Laplace transform of the exponential functionals of some spectrally negative L\'evy processes with a negative first moment

Spectral expansions of non-self-adjoint generalized Laguerre semigroups

We provide the spectral expansion in a weighted Hilbert space of a substantial class of invariant non-self-adjoint and non-local Markov operators which appear in limit theorems for positive-valued Markov processes. We show that this class is in bijection with a subset of negative definite functions and we name it the class of generalized Laguerre semigroups. Our approach, which goes beyond the framework of perturbation theory, is based on an in-depth and original analysis of an intertwining relation that we establish between this class and a self-adjoint Markov semigroup, whose spectral expansion is expressed in terms of the classical Laguerre polynomials. As a by-product, we derive smoothness properties for the solution to the associated Cauchy problem as well as for the heat kernel. Our methodology also reveals a variety of possible decays, including the hypocoercivity type phenomena, for the speed of convergence to equilibrium for this class and enables us to provide an interpretation of these in terms of the rate of growth of the weighted Hilbert space norms of the spectral projections. Depending on the analytic properties of the aforementioned negative definite functions, we are led to implement several strategies, which require new developments in a variety of contexts, to derive precise upper bounds for these norms.Comment: 162page

Bernstein-gamma functions and exponential functionals of Levy Processes

We study the equation $M_\Psi(z+1)=\frac{-z}{\Psi(-z)}M_\Psi(z), M_\Psi(1)=1$ defined on a subset of the imaginary line and where $\Psi$ is a negative definite functions. Using the Wiener-Hopf method we solve this equation in a two terms product which consists of functions that extend the classical gamma function. These functions are in a bijection with Bernstein functions and for this reason we call them Bernstein-gamma functions. Via a couple of computable parameters we characterize of these functions as meromorphic functions on a complex strip. We also establish explicit and universal Stirling type asymptotic in terms of the constituting Bernstein function. The decay of $|M_{\Psi}(z)|$ along imaginary lines is computed. Important quantities for theoretical and applied studies are rendered accessible. As an application we investigate the exponential functionals of Levy Processes whose Mellin transform satisfies the recurrent equation above. Although these variables have been intensively studied, our new perspective, based on a combination of probabilistic and complex analytical techniques, enables us to derive comprehensive and substantial properties and strengthen several results on the law of these random variables. These include smoothness, regularity and analytical properties, large and small asymptotic behaviour, including asymptotic expansions, bounds, and Mellin-Barnes representations for the density and its successive derivatives. We also study the weak convergence of exponential functionals on a finite time horizon when the latter expands to infinity. As a result of new factorizations of the law of the exponential functional we deliver important intertwining relation between members of the class of positive self-similar semigroups. The derivation of our results relies on a mixture of complex-analytical and probabilistic techniques

The log-L\'evy moment problem via Berg-Urbanik semigroups

We consider the Stieltjes moment problem for the Berg-Urbanik semigroups which form a class of multiplicative convolution semigroups on $\mathbb{R}_+$ that is in bijection with the set of Bernstein functions. Berg and Dur\'an proved that the law of such semigroups is moment 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 moment 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 moment determinacy to moment indeterminacy in terms of simple properties of the underlying Bernstein function $\phi$, such as its Blumenthal-Getoor index. One of the several strategies we implement to deal with the different cases relies on a non-classical Abelian type criterion for the moment problem, recently proved by the authors. 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. In particular, these results, which are original in the L\'evy processes literature, may be of independent interests.Comment: Studia Mathematic

Spectral decomposition of fractional operators and a reflected stable semigroup

In this paper, we provide the spectral decomposition in Hilbert space of the $\mathcal{C}_0$-semigroup $P$ and its adjoint \hatP having as generator, respectively, the Caputo and the right-sided Riemann-Liouville fractional derivatives of index $1<\alpha<2$. These linear operators, which are non-local and non-self-adjoint, appear in many recent studies in applied mathematics and also arise as the infinitesimal generators of some substantial processes such as the reflected spectrally negative $\alpha$-stable process. Our approach relies on intertwining relations that we establish between these semigroups and the semigroup of a Bessel type process whose generator is a self-adjoint second order differential operator. In particular, from this commutation relation, we characterize the positive real axis as the continuous point spectrum of $P$ and provide a power series representation of the corresponding eigenfunctions. We also identify the positive real axis as the residual spectrum of the adjoint operator \hatP and elucidates its role in the spectral decomposition of these operators. By resorting to the concept of continuous frames, we proceed by investigating the domain of the spectral operators and derive two representations for the heat kernels of these semigroups. As a by-product, we also obtain regularity properties for these latter and also for the solution of the associated Cauchy problem.Comment: in pres

Cauchy Problem of the non-self-adjoint Gauss-Laguerre semigroups and uniform bounds of generalized Laguerre polynomials

We propose a new approach to construct the eigenvalue expansion in a weighted Hilbert space of the solution to the Cauchy problem associated to Gauss-Laguerre invariant Markov semigroups that we introduce. Their generators turn out to be natural non-self-adjoint and non-local generalizations of the Laguerre differential operator. Our methods rely on intertwining relations that we establish between these semigroups and the classical Laguerre semigroup and combine with techniques based on non-harmonic analysis. As a by-product we also provide regularity properties for the semigroups as well as for their heat kernels. The biorthogonal sequences that appear in their eigenvalue expansion can be expressed in terms of sequences of polynomials, and they generalize the Laguerre polynomials. By means of a delicate saddle point method, we derive uniform asymptotic bounds that allow us to get an upper bound for their norms in weighted Hilbert spaces. We believe that this work opens a way to construct spectral expansions for more general non-self-adjoint Markov semigroups.Comment: 33 page

Intertwining, Excursion Theory and Krein Theory of Strings for Non-self-adjoint Markov Semigroups

In this paper, we start by showing that the intertwining relationship between two minimal Markov semigroups acting on Hilbert spaces implies that any recurrent extensions, in the sense of It\^o, of these semigroups satisfy the same intertwining identity. Under mild additional assumptions on the intertwining operator, we prove that the converse also holds. This connection, which relies on the representation of excursion quantities as developed by Fitzsimmons and Getoor, enables us to give an interesting probabilistic interpretation of intertwining relationships between Markov semigroups via excursion theory: two such recurrent extensions that intertwine share, under an appropriate normalization, the same local time at the boundary point. Moreover, in the case when one of the (non-self-adjoint) semigroup intertwines with the one of a quasi-diffusion, we obtain an extension of Krein's theory of strings byshowing that its densely defined spectral measure is absolutely continuous with respect to the measure appearing in the Stieltjes representation of the Laplace exponent of the inverse local time. Finally, we illustrate our results with the class of positive self-similar Markov semigroups and also the reflected generalized Laguerre semigroups. For the latter, we obtain their spectral decomposition and provide, under some conditions, a perturbed spectral gap estimate for its convergence to equilibrium

A transformation for L\'evy processes with one-sided jumps and applications

The aim of this work is to extend and study a family of transformations between Laplace exponents of L\'evy processes which have been introduced recently in a variety of different contexts by Patie, Kyprianou and Patie, and, Gnedin, as well as in older work of Urbanik . We show how some specific instances of this mapping prove to be useful for a variety of applications