arXiv subjects

L. Beghin

Publications and source records attributed to L. Beghin.

5 recordsLinked to original sources

arXiv2026

Beta-weighted non-local differential operators and related stochastic processes

In this work we introduce a class of non-local differential operators defined through a beta-weighted averaging of the ordinary derivative. We investigate their analytical properties and establish connections with the Caputo and Erd\'elyi-Kober operators. Differential equations involving the beta-weighted derivative are studied by Mellin transform methods, leading to solutions represented in terms of Barnes G-functions and a new class of G-hypergeometric functions. We also analyze asymptotic properties, Laplace transforms, and the second-order equation involving the sequential beta-weighted derivative. Finally, we present stochastic applications of these results, showing that continuous-time random walks, with waiting times characterized by the beta-weighted derivative, converge to Brownian motions time-changed by a scaled inverse stable subordinator. We compare this anomalous-diffusion model with a time-changed Brownian motion whose one-dimensional distribution solve a heat-type equation with beta-weighted derivative.

math.PR↗

arXiv2023

Fox-H densities and completely monotone generalized Wright functions

Due to their flexibility, Fox-$H$ functions are widely studied and applied to many research topics, such as astrophysics, mechanical statistic, probability, etc. Well-known special cases of Fox-$H$ functions, such as Mittag-Leffler and Wright functions, find a wide application in the theory of stochastic processes, anomalous diffusions and non-Gaussian analysis. In this paper, we focus on certain explicit assumptions that allow us to use the Fox-$H$ functions as densities. We then provide a subfamily of the latter, called Fox-$H$ densities with all moments finite, and give their Laplace transforms as entire generalized Wright functions. The class of random variables with these densities is proven to possess a monoid structure. We present eight subclasses of special cases of such densities (together with their Laplace transforms) that are particularly relevant in applications, thanks to their probabilistic interpretation. To analyze the existence conditions of Fox-$H$ functions red as well as their sign, we derive asymptotic results and their analytic extension.

math.PR↗

arXiv2019

Integro-differential equations linked to compound birth processes with infinitely divisible addends

Stochastic modelling of fatigue (and other material's deterioration), as well as of cumulative damage in risk theory, are often based on compound sums of independent random variables, where the number of addends is represented by an independent counting process. We consider here a cumulative model where, instead of a renewal process (as in the Poisson case), a linear birth (or Yule) process is used. This corresponds to the assumption that the frequency of \textquotedblleft damage" increments accelerates according to the increasing number of \textquotedblleft damages". We start from the partial differential equation satisfied by its transition density, in the case of exponentially distributed addends, and then we generalize it by introducing a space-derivative of convolution type (i.e. defined in terms of the Laplace exponent of a subordinator). Then we are concerned with the solution of integro-differential equations, which, in particular cases, reduce to fractional ones. Correspondingly, we analyze the related cumulative jump processes under a general infinitely divisible distribution of the (positive) jumps. Some special cases (such as the stable, tempered stable, gamma and Poisson) are presented.

math.PR↗

arXiv2016

Space-fractional versions of the negative binomial and Polya-type processes

In this paper, we introduce a space fractional negative binomial (SFNB) process by subordinating the space fractional Poisson process to a gamma subordinator. Its one-dimensional distributions are derived in terms of generalized Wright functions and their governing equations are obtained. It is a Lévy process and the corresponding Lévy measure is given. Extensions to the case of distributed order SFNB process, where the fractional index follows a two-point distribution, is analyzed in detail. The connections of the SFNB process to a space fractional Polya-type process is also pointed out. Moreover, we define and study a multivariate version of the SFNB obtained by subordinating a $d$-dimensional space-fractional Poisson process by a common independent gamma subordinator. Some applications of the SFNB process to the studies of population's growth and epidemiology are pointed out. Finally, we discuss an algorithm for the simulation of the SFNB process.

math.PR↗

arXiv2015

Population processes sampled at random times

In this paper we study the iterated birth process of which we examine the first-passage time distributions and the hitting probabilities. Furthermore, linear birth processes, linear and sublinear death processes at Poisson times are investigated. In particular, we study the hitting times in all cases and examine their long-range behavior. The time-changed population models considered here display upward (Birth process) and downward jumps (death processes) of arbitrary size and, for this reason, can be adopted as adequate models in ecology, epidemics and finance situations, under stress conditions.

math.PR↗