Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes

Pierre, Patie

Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 3, pp. 667-684.

Nous commençons par caractériser les fonctions propres croissantes, au sens strict, de la famille d'opérateurs intégro-différentiels (0.1), pour tout α>0, γ≥0, f une function définie sur ${\Re }^{+}$ et suffissament régulière, et où les coefficients $b\in \Re$, σ≥0 et la mesure ν, qui satisfait la condition d'intégrabilité ∫₀^∞(1∧r²)ν(dr)<+∞, sont données, de manière unique, par la distribution d'une variable aléatoire infiniment divisible et spectralement négative dont on écrit ψ son exposant caractéristique. L^(γ) est le générateur infinitésimal d'un processus positif Fellerien α-auto-similaire, introduit par Lamperti [Z. Wahrsch. Verw. Gebiete 22 (1972) 205-225]. Les fonctions propres sont définies en terme d'une nouvelle famille de séries entières qui contient, par exemple, les fonctions de Bessel modifiées du premier ordre et des généralisations des fonctions de Mittag-Leffler. Nous continuons par montrer que des combinaisons particulières de ces séries entières correspondent à des transformées de Laplace de variables aléatoires positives auto-décomposables ou infiniment divisibles, par rapport à la valeur propre associée mais aussi par rapport au paramètre ψ(γ), ce qui est plus surprenant. En particulier, ceci généralise un résultat de Hartman [Ann. Sc. Norm. Super. Pisa Cl. Sci. IV-III (1976) 267-287] sur les fonctions de Bessel modifiées. Finalement, nous calculons, dans certains cas, les fonctions propres décroissantes, ce qui nous permet de caractériser la loi, par le biais de sa transformée de Laplace, de la fonctionnelle exponentielle de certains processus de Lévy spectralement négatifs ayant un premier moment négatif.

We first characterize the increasing eigenfunctions associated to the following family of integro-differential operators, for any α, x>0, γ≥0 and f a smooth function on ${\Re }^{+}$,

$$\begin{array}{c}\hfill {𝐋}^{\left(\gamma \right)}f\left(x\right)=x^{-\alpha }(\frac{\sigma }{2}{x}^2{f}^{\text{'}\text{'}}\left(x\right)+(\sigma \gamma +b)x{f}^{\text{'}}\left(x\right)\\ \hfill +{\int }_0^{\infty }\left(f\left({\mathrm{e}}^{-r}x\right)-f\left(x\right)\right){\mathrm{e}}^{-r\gamma }+x{f}^{\text{'}}\left(x\right)r{𝕀}_{\{r\le 1\}}\nu \left(\mathrm{d}r\right)),(0.1)\end{array}$$

where the coefficients $b\in \Re$, σ≥0 and the measure ν, which satisfies the integrability condition ∫₀^∞(1∧r²)ν(dr)<+∞, are uniquely determined by the distribution of a spectrally negative, infinitely divisible random variable, with characteristic exponent ψ. L^(γ) is known to be the infinitesimal generator of a positive α-self-similar Feller process, which has been introduced by Lamperti [Z. Wahrsch. Verw. Gebiete 22 (1972) 205-225]. The 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 the parameter ψ(γ). In particular, this generalizes a result of Hartman [Ann. Sc. Norm. Super. Pisa Cl. Sci. IV-III (1976) 267-287] 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 Lévy processes with a negative first moment.

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