arXiv subjects

Lorenzo Cristofaro

Publications and source records attributed to Lorenzo Cristofaro.

9 recordsLinked to original sources

arXiv2026

Shot-noise processes with logarithmic response function and their scaling limits

We consider shot-noise processes with an impulse response written in terms of the logarithm of the ratio between current and event time (instead of the usual absolute time difference). We study its finite-time properties as well as its weak convergence, under appropriate scaling and with general assumptions on the dependence of noises on event times. The limiting process coincides with the so-called Hadamard fractional Brownian motion (introduced in Beghin, Cristofaro, Polito (2026)), which represents a middle ground between standard Brownian motion and fractional Brownian motion. It shares with the former the one-dimensional distribution (i.e. Gaussian with the same first two moments), while possessing the long-memory property (within a certain parameter range) of the latter, though with smaller intensity. Therefore, we identify a natural probabilistic scheme based on shot-noise processes whose scaling limit is the Hadamard fractional Brownian motion, thereby providing a concrete stochastic finite-time counterpart of this process.

math.PR↗

arXiv2026

Generalized random processes related to Hadamard operators and Le Roy measures

The definition of generalized random processes in Gel'fand sense allows to extend well-known stochastic models, such as the fractional Brownian motion, and study the related fractional pde's, as well as stochastic differential equations in distributional sense. By analogy with the construction (in the infinite-dimensional white-noise space) of the latter, we introduce two processes defined by means of Hadamard-type fractional operators. When used to replace the time derivative in the governing p.d.e.'s, the Hadamard-type derivatives are usually associated with ultra-slow diffusions. On the other hand, in our construction, they directly determine the memory properties of the so-called Hadamard fractional Brownian motion (H-fBm) and its long-time behaviour. Still, for any finite time horizon, the H-fBm displays a standard diffusing feature. We then extend the definition of the H-fBm from the white noise space to an infinite dimensional grey-noise space built on the Le Roy measure, so that our model represents an alternative to the generalized grey Brownian motion. In this case, we prove that the one-dimensional distribution of the process satisfies a heat equation with non-constant coefficients and fractional Hadamard time-derivative. Finally, once proved the existence of the distributional derivative of the above defined processes and derived an integral formula for it, we construct an Ornstein-Uhlenbeck type process and evaluate its distribution.

math.PR↗

arXiv2025

Generalized Wright Analysis in Infinite Dimensions

This paper investigates a broad class of non-Gaussian measures, $ μ_Ψ$, associated with a family of generalized Wright functions, $_mΨ_q$. First, we study these measures in Euclidean spaces $\mathbb{R}^d$, then define them in an abstract nuclear triple $\mathcal{N}\subset\mathcal{H}\subset\mathcal{N}'$. We study analyticity, invariance properties, and ergodicity under a particular group of automorphisms. Then we show the existence of an Appell system which allows the extension of the non-Gaussian Hilbert space $L^2(μ_Ψ)$ to the nuclear triple consisting of test functions' and distributions' spaces, $(\mathcal{N})^{1}\subset L^2(μ_Ψ)\subset(\mathcal{N})_{μ_Ψ}^{-1}$. Furthermore, thanks to the definition of two transformations, $S_{μ_Ψ}$ and $T_{μ_Ψ}$, we study Donsker's delta as an element within $(\mathcal{N})_{μ_Ψ}^{-1}$ applying the integral equations fulfilled by $_mΨ_q$.

math.PR↗

arXiv2025

The Poisson Multiplication Formula

We establish necessary and sufficient conditions implying that the product of $m\geq 2$ Poisson functionals, living in a finite sum of Wiener chaoses, is square-integrable. Our conditions are expressed in terms of iterated add-one cost operators, and are obtained through the use of a novel family of Poincaré inequalities for almost surely finite random variables, generalizing the recent findings by Trauthwein (2024). When specialized to the case of multiple Wiener-Itô integrals, our results yield general multiplication formulae on the Poisson space under minimal conditions, naturally expressed in terms of partitions and diagrams. Our work addresses several questions left open in a seminal work by Surgailis (1984), and completes a line of research initiated in Döbler and Peccati (2018).

math.PR↗

arXiv2023

A class of infinite-dimensional Gaussian processes defined through generalized fractional operators

The generalization of fractional Brownian motion in infinite-dimensional white and grey noise spaces has been recently carried over, following the Mandelbrot-Van Ness representation, through Riemann-Liouville type fractional operators. Our aim is to extend this construction by means of general fractional derivatives and integrals, which we define through Bernstein functions. According to the conditions satisfied by the latter, some properties of these processes (such as continuity, local times, variance asymptotics and persistence) are derived. On the other hand, they are proved to display short- or long-range dependence, if obtained by means of a derivative or an integral, respectively, regardless of the Bernstein function chosen. Moreover, this kind of construction allows us to define the corresponding noise and to derive an Ornstein-Uhlenbeck type process, as solution of an integral equation.

math.PR↗

arXiv2023

Renewal processes linked to fractional relaxation equations with variable order

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 $t\geq0$. In particular, we use the operator introduced by Scarpi in the Seventies and later reformulated in the regularized Caputo sense in Garrappa et al. (2021), inside the framework of the so-called general fractional calculus. The model obtained extends the well-known time-fractional Poisson process of fixed order $α\in (0,1)$ and tries to overcome its limitation consisting in the constancy of the derivative order (and therefore of the memory degree of the interarrival 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 analysed and its limiting behavior established.

math.PR↗

arXiv2023

Characterization and Analysis of Generalized Grey Incomplete Gamma Noise

The grey incomplete gamma distributions was established by one of the authors in a previous publication. In this article we use the Kondratiev characterization theorem to identify those via a suitable Laplace transform with holomorphic functions with suitable properties. We establish theorems for the integration and convergence of sequences of these distributions. As direct applications of these analytic tools we give the examples of Donsker's delta function, the local time, identify the time-derivative of the process as a suitable distribution and define the Gamma grey Ornstein-Uhlenbeck process.

math.PR↗

arXiv2022

A fractional approach to study the pure-temporal Epidemic Type Aftershock Sequence (ETAS) process for earthquakes modeling

In statistical seismology, the Epidemic Type Aftershocks Sequence (ETAS) model is a branching process used world-wide to forecast earthquake intensity rates and reproduce many statistical features observed in seismicity catalogs. In this paper, we describe a fractional differential equation that governs the earthquake intensity rate of the pure temporal ETAS model by using the Caputo fractional derivative and we solve it analytically. We highlight that the tools and special functions of fractional calculus simplify the classical methods employed to obtain the intensity rate and let us describe the change of solution decay for large times. We also apply and discuss the theoretical results to the Japanese catalog in the period 1965-2003.

physics.geo-ph↗

arXiv2022

Non-Gaussian Measures in Infinite Dimensional Spaces: the Gamma-Grey Noise

In the context of non-Gaussian analysis, Schneider [27] introduced grey noise measures, built upon Mittag-Leffler functions; analogously, grey Brownian motion and its generalizations were constructed (see, for example, [25], [6], [7], [8]). 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.

math.PR↗