39 research outputs found
(Non)Differentiability and Asymptotics for Potential Densities of Subordinators
For subordinators with positive drift we extend recent results on the
structure of the potential measures and the renewal densities. Applying Fourier
analysis a new representation of the potential densities is derived from which
we deduce asymptotic results and show how the atoms of the Levy measure
translate into points of (non)smoothness.Comment: 27 pages, appeared in Electronic Journal of Probability 201
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
defined on a subset of the imaginary line and where 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
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
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