Density solutions to a class of integro-differential equations

Abstract

Abstract. We consider the integro-differential equation I α 0+ f = x m f on the half-line. We show that there exists a density solution, which is then unique and can be expressed in terms of the Beta distribution, if and only if m > α. These density solutions extend the class of generalized one-sided stable distributions introduced in [29] and more recently investigated in [27]. We study various analytical aspects of these densities, and we solve the open problems about infinite divisibility formulated in [27].

Moment problems related to Bernstein functions

“…-An easy consequence of (11) and (13) This result was obtained for m integer in Theorem 8.2 of [22], whose proof of the only if part is a consequence of Krein's condition and of the subexponential tail behaviour at infinity of the density of Y a,m , which is obtained therein by means of a certain class of special functions. It is shown in the Proposition of [12] that the latter subexponentiality property holds true for every m > 3a non necessarily an integer, so that one can conclude as in [22]. Overall, this proof is however more involved than the above HCM argument.…”

“…Moreover, we know by Corollary (b) in[12] that Y a,m has a HCM density for m 2a. A combination of Carleman's criterion and Theorem 7 in[15] shows that Y a,m is M-det if and only if m − a 2a 1 ⇐⇒ m 3a.…”

“…We conclude this paper with another example of strict dichotomy between moment-determinacy and moment-indeterminacy, in the spirit of Section 1. The framework is that of the r-gstable(a, m) laws recently studied in [12,22] (see also the references therein). These laws are well-defined if and only if 0 < a < m and they form a generalization of the inverse positive stable laws which correspond to the case m = 1.…”

“…These laws are well-defined if and only if 0 < a < m and they form a generalization of the inverse positive stable laws which correspond to the case m = 1. We refer to [12,22] where G is the double Gamma function (see (11) in [12]). It follows from the main Theorem in [12] that log Y a,m is infinitely divisible and the moment sequence {µ n } is hence always ID.…”

“…We refer to [12,22] where G is the double Gamma function (see (11) in [12]). It follows from the main Theorem in [12] that log Y a,m is infinitely divisible and the moment sequence {µ n } is hence always ID. It is also easy to see from the concatenation formula G(z + 1, τ ) = Γ(zτ Proof.…”