Relaxation Equations with Stretched Non-local Operators: Renewals and Time-Changed Processes Luisa Beghin, Nikolai Leonenko, Jayme Vaz Journal of Theoretical Probability, 2026 We introduce and study renewal processes defined by means of extensions of the standard relaxation equation through “stretched” non-local operators (of order $$\\alpha $$ α and with parameter $$\\gamma $$ γ ). In a first case, we obtain a generalization of the fractional Poisson process, which displays either infinite or finite expected waiting times between arrivals, depending on the parameter $$\\gamma $$ γ . Therefore, the introduction in the operator of the non-homogeneous term driven by $$\\gamma $$ γ allows us to regulate the transition between different regimes of our renewal process. We then consider a second-order relaxation-type equation involving the same operator, under different sets of conditions on the constants involved; for a particular choice of these constants, we prove that the corresponding renewal process is linked to the first one by convex combination of its distributions. We also discuss alternative models related to the same equations and their time-changed representation, in terms of the inverse of a non-decreasing process which generalizes the $$\\alpha $$ α -stable Lévy subordinator.
Integro-differential equations linked to compound birth processes with infinitely divisible addends Luisa Beghin, Janusz Gajda, Aditya Maheshwari Mathematical Methods in the Applied Sciences, 2024 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.
Non-Gaussian Measures in Infinite Dimensional Spaces: the Gamma-Grey Noise Luisa Beghin, Lorenzo Cristofaro, Janusz Gajda Potential Analysis, 2024 In the context of non-Gaussian analysis, Schneider [29] introduced grey noise measures, built upon Mittag-Leffler functions; analogously, grey Brownian motion and its generalizations were constructed (see, for example, [6, 7, 9, 27]). In this paper, we construct and study a new non-Gaussian measure, by means of the incomplete-gamma function (exploiting its complete monotonicity). We label this measure Gamma-grey noise and we prove, for it, the existence of Appell system. The related generalized processes, in the infinite dimensional setting, are also defined and, through the use of the Riemann-Liouville fractional operators, the (possibly tempered) Gamma-grey Brownian motion is consequently introduced. A number of different characterizations of these processes are also provided, together with the integro-differential equation satisfied by their transition densities. They allow to model anomalous diffusions, mimicking the procedures of classical stochastic calculus.
Renewal processes linked to fractional relaxation equations with variable order Luisa Beghin, Lorenzo Cristofaro, Roberto Garrappa Journal of Mathematical Analysis and Applications, 2024 We introduce and study here a renewal process defined by means of a time-fractional relaxation equation with derivative order α(t) varying with time t≥0. In particular, we use the operator introduced by Scarpi in the seventies [1] and later reformulated in the regularized Caputo sense in [2], inside the framework of the so-called general fractional calculus. The obtained model extends the well-known time-fractional Poisson process of fixed order α∈(0,1) and tries to overcome its limitation consisting in the constancy of the derivative order (and therefore of the memory degree of the inter-arrival times) with respect to time. The variable order renewal process is proved to fall outside the usual subordinated representation, since it can not be simply defined as a Poisson process with random time (as happens in the standard fractional case). Finally a related continuous-time random walk model is analyzed and its limiting behavior established.
Stochastic applications of Caputo-type convolution operators with nonsingular kernels Luisa Beghin, Michele Caputo Stochastic Analysis and Applications, 2023 We consider here convolution operators, in the Caputo sense, with non-singular kernels. We prove that the solutions to some integro-differential equations with such operators (acting on the space variable) coincide with the transition densities of a particular class of Lévy subordinators (i.e. compound Poisson processes with non-negative jumps). We then extend these results to the case where the kernels of the operators have random parameters, with given distribution. This assumption allows greater flexibility in the choice of the kernel’s parameters and, consequently, of the jumps’ density function.
