A refined factorization of the exponential law

Let $\xi$ be a (possibly killed) subordinator with Laplace exponent $\phi$ and denote by $I_{\phi}=\int_0^{\infty}\mathrm{e}^{-\xi_s}\,\mathrm{d}s$, the so-called exponential functional. Consider the positive random variable $I_{\psi_1}$ whose law, according to Bertoin and Yor [Electron. Comm. Probab. 6 (2001) 95--106], is determined by its negative entire moments as follows: $$\mathbb {E}[I_{\psi_1}^{-n}]=\prod_{k=1}^n\phi(k),\qquad n=1,2,...$$ In this note, we show that $I_{\psi_1}$ is a positive self-decomposable random variable whenever the L\'{e}vy measure of $\xi$ is absolutely continuous with a monotone decreasing density. In fact, $I_{\psi_1}$ is identified as the exponential functional of a spectrally negative (sn, for short) L\'{e}vy process. We deduce from Bertoin and Yor [Electron. Comm. Probab. 6 (2001) 95--106] the following factorization of the exponential law ${\mathbf {e}}$: $$I_{\phi}/I_{\psi_1}\stackrel{\mathrm {(d)}}{=}{\mathbf {e}},$$ where $I_{\psi_1}$ is taken to be independent of $I_{\phi}$. We proceed by showing an identity in distribution between the entrance law of an sn self-similar positive Feller process and the reciprocal of the exponential functional of sn L\'{e}vy processes. As a by-product, we obtain some new examples of the law of the exponential functionals, a new factorization of the exponential law and some interesting distributional properties of some random variables. For instance, we obtain that $S(\alpha)^{\alpha}$ is a self-decomposable random variable, where $S(\alpha)$ is a positive stable random variable of index $\alpha\in(0,1)$.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ292 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

q-Invariant Functions for Some Generalizations of the Ornstein-Uhlenbeck Semigroup

We show that the multiplication operator associated to a fractional power of a Gamma random variable, with parameter q>0, maps the convex cone of the 1-invariant functions for a self-similar semigroup into the convex cone of the q-invariant functions for the associated Ornstein-Uhlenbeck (for short OU) semigroup. We also describe the harmonic functions for some other generalizations of the OU semigroup. Among the various applications, we characterize, through their Laplace transforms, the laws of first passage times above and overshoot for certain two-sided stable OU processes and also for spectrally negative semi-stable OU processes. These Laplace transforms are expressed in terms of a new family of power series which includes the generalized Mittag-Leffler functions.Comment: To appear in ALE

Self-similar Cauchy problems and generalized Mittag-Leffler functions

By observing that the fractional Caputo derivative can be expressed in terms of a multiplicative convolution operator, we introduce and study a class of such operators which also have the same self-similarity property as the Caputo derivative. We proceed by identifying a subclass which is in bijection with the set of Bernstein functions and we provide several representations of their eigenfunctions, expressed in terms of the corresponding Bernstein function, that generalize the Mittag-Leffler function. Each eigenfunction turns out to be the Laplace transform of the right-inverse of a non-decreasing self-similar Markov process associated via the so-called Lamperti mapping to this Bernstein function. Resorting to spectral theoretical arguments, we investigate the generalized Cauchy problems, defined with these self-similar multiplicative convolution operators. In particular, we provide both a stochastic representation, expressed in terms of these inverse processes and an explicit representation, given in terms of the generalized Mittag-Leffler functions, of the solution of these self-similar Cauchy problems

A Ciesielski-Taylor type identity for positive self-similar Markov processes

The aim of this note is to give a straightforward proof of a general version of the Ciesielski-Taylor identity for positive self-similar Markov processes of the spectrally negative type which umbrellas all previously known Ciesielski-Taylor identities within the latter class. The approach makes use of three fundamental features. Firstly a new transformation which maps a subset of the family of Laplace exponents of spectrally negative L\'evy processes into itself. Secondly some classical features of fluctuation theory for spectrally negative L\'evy processes as well as more recent fluctuation identities for positive self-similar Markov processes

Law of the absorption time of some positive self-similar Markov processes

Let X be a spectrally negative self-similar Markov process with 0 as an absorbing state. In this paper, we show that the distribution of the absorption time is absolutely continuous with an infinitely continuously differentiable density. We provide a power series and a contour integral representation of this density. Then, by means of probabilistic arguments, we deduce some interesting analytical properties satisfied by these functions, which include, for instance, several types of hypergeometric functions. We also give several characterizations of the Kesten's constant appearing in the study of the asymptotic tail distribution of the absorbtion time. We end the paper by detailing some known and new examples. In particular, we offer an alternative proof of the recent result obtained by Bernyk, Dalang and Peskir [Ann. Probab. 36 (2008) 1777--1789] regarding the law of the maximum of spectrally positive L\'{e}vy stable processes.Comment: Published in at http://dx.doi.org/10.1214/10-AOP638 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Smoothness of continuous state branching with immigration semigroups

In this work we develop an original and thorough analysis of the (non)-smoothness properties of the semigroups, and their heat kernels, associated to a large class of continuous state branching processes with immigration. Our approach is based on an in-depth analysis of the regularity of the absolutely continuous part of the invariant measure combined with a substantial refinement of Ogura's spectral expansion of the transition kernels. In particular, we find new representations for the eigenfunctions and eigenmeasures that allow us to derive delicate uniform bounds that are useful for establishing the uniform convergence of the spectral representation of the semigroup acting on linear spaces that we identify. We detail several examples which illustrate the variety of smoothness properties that CBI transition kernels may enjoy and also reveal that our results are sharp. Finally, our technique enables us to provide the (eventually) strong Feller property as well as the rate of convergence to equilibrium in the total variation norm

FIRST PASSAGE TIMES OVER STOCHASTIC BOUNDARIES FOR SUBDIFFUSIVE PROCESSES

Let X = ( X t ) t ≥ 0 \mathbb {X}=(\mathbb {X}_t)_{t\geq 0} be the subdiffusive process defined, for any t ≥ 0 t\geq 0 , by X t = X ℓ t \mathbb {X}_t = X_{\ell _t} where X = ( X t ) t ≥ 0 X=(X_t)_{t\geq 0} is a Lévy process and ℓ t = inf { s &gt; 0 ; K s &gt; t } \ell _t=\inf \{s&gt;0; \mathcal {K}_s&gt;t \} with K = ( K t ) t ≥ 0 \mathcal {K}=(\mathcal {K}_t)_{t\geq 0} a subordinator independent of X X . We start by developing a composite Wiener-Hopf factorization to characterize the law of the pair ( T a ( b ) , ( X − b ) T a ( b ) ) (\mathbb {T}_a^{({b})}, (\mathbb {X} - {b})_{\mathbb {T}_a^{({b})}}) where T a ( b ) = inf { t &gt; 0 ; X t &gt; a + b t } \begin{equation*}\mathbb {T}_a^{({b})} = \inf \{t&gt;0; \mathbb {X}_t &gt; a+ {b}_t \} \end{equation*} with a ∈ R a \in \mathbb {R} and b = ( b t ) t ≥ 0 {b}=({b}_t)_{t\geq 0} a (possibly degenerate) subordinator independent of X X and K \mathcal {K} . We proceed by providing a detailed analysis of the cases where either X \mathbb {X} is a self-similar or is spectrally negative. For the later, we show the fact that the process ( T a ( b ) ) a ≥ 0 (\mathbb {T}_a^{({b})})_{a\geq 0} is a subordinator. Our proofs hinge on a variety of techniques including excursion theory, change of measure, asymptotic analysis and on establishing a link between subdiffusive processes and a subclass of semi-regenerative processes. In particular, we show that the variable T a ( b ) \mathbb {T}_a^{({b})} has the same law as the first passage time of a semi-regenerative process of Lévy type, a terminology that we introduce to mean that this process satisfies the Markov property of Lévy processes for stopping times whose graph is included in the associated regeneration set.</p