arXiv subjects

Enzo Orsingher

Publications and source records attributed to Enzo Orsingher.

At least 19 recordsLinked to original sources

arXiv2026

Integrals of some compound processes

By means of the theory of compound (non-homogenoeus) Poisson processes we define a general framework which includes most of the generalizations of the Poisson processes, both in the Skellam sense and in the space-fractional sense. We prove that Bernstein subordination is the only time-changing leading to a compound Poisson process. We also consider the case of subordination with inverse Bernstein subordinators. Then, we focus on the integrals of compound processes. We give an explicit representation of the fractional integral of compound Poisson process, showing that for fixed t it is distributed as a compound Poisson random variable. We then extend this result to more general integral forms. Furthermore, we obtain some limit results, explicit forms of the iterated integrals and their governing equation. Finally, we study the integral of compound renewal processes in the Fourier-Laplace domain.

math.PR↗

arXiv2026

Point processes of the Poisson-Skellam family

We study a general non-homogeneous Skellam-type process with jumps of arbitrary fixed sizes. We express this process in terms of a linear combination of Poisson processes and study several properties, including the summation of independent processes of the same family, some possible decompositions (which present particularly interesting characteristics) and the limit behaviors. A compound Poisson representation and a discrete approximation are also presented. Then, we study the fractional integral of the process as well as the iterated integral of the running average. Finally, we consider some time-changed versions related to Lévy subordinators, connected to the Bernstein functions, and to the inverses of stable subordinators.

math.PR↗

arXiv2026

Dirichlet problems and exit distributions for the telegraph process and its planar extensions

In this paper, we study boundary-value problems describing the exit distribution of finite-velocity random motions from prescribed domains. For the standard telegraph process, with and without drift, we derive the Dirichlet problems governing the exit point and mean exit time from a closed interval. We then extend the analysis to a planar finite-velocity model with orthogonal directions, for which we obtain the associated Laplace and Poisson-type equations for the exit distribution and mean exit time. In the special case of an infinite strip, explicit solutions are obtained. In all cases, we show that our equations and results converge, in the hydrodynamic limit, to the corresponding ones for Brownian motion.

math.PR↗

arXiv2026

One-dimensional and planar random motions with variable propagation speeds

In this paper, we study univariate and planar random motions with variable propagation speeds. We first consider motions with space-varying velocity, which can be reduced to constant-velocity motions by means of suitable nonlinear transformations. We examine a special case of a motion which is confined within the unit interval. To provide a general expression of the moments of this process, we introduce a new family of polynomials which generalize the classical Euler polynomials. We then examine a planar extension of this process which moves along orthogonal directions. A process with velocity depending on the direction is also examined, and its mean conditional on the initial direction and the number of direction changes is given in terms of confluent hypergeometric functions. We conjecture that, in the hydrodynamic limit, this process is absorbed at a point in an arbitrarily small time. We finally study a motion with time-dependent velocity. We prove that this process can be represented as an integral with respect to a standard telegraph process, and we obtain its covariance function explicitly. Moreover, we show that this process behaves as a Itô integral with respect to Brownian motion in the hydrodynamic limit.

math.PR↗

arXiv2025

On the distribution of the telegraph meander and its properties

In this paper we present the distribution of the telegraph meander, a random function obtained by conditioning the telegraph process to stay above the zero level. The reflection principle for finite-velocity random motions allows the law of the telegraph meander to be expressed in terms of the spatial derivative of the law of the telegraph process with initial negative velocity. As a result, we are able to obtain the characteristic function, the moments and the hyperbolic equation that governs the law of the telegraph meander. Furthermore, we prove that Brownian meander is the weak limit of the telegraph meander.

math.PR↗

arXiv2025

Interacting point processes

We study two different types of vector point processes with interacting components, introducing a migration-type effect. The first case concerns two groups which modify their states with rate functions depending on time only. This yields a representation of the vector process in terms of independent non-homogeneous Skellam processes. In the general case, the decomposition involves independent Poisson processes. The second model is a birth-death-migration vector process. In the case of the linear death-migration we show that, for a fixed time instant, the vector is equal in distribution to the sum of two independent Multinomial random variables. As a byproduct we derive the distribution of a pure migration process. Finally, we study the described vector processes time-changed with the inverse of Bernstein subordinator, establishing a general result concerning the relatioship between fractional difference-differential equations and the probability mass function of a wider class of point processes.

math.PR↗

arXiv2024

Higher-order fractional equations and related time-changed pseudo-processes

We study Cauchy problems of fractional differential equations in both space and time variables by expressing the solution in terms of ``stochastic composition" of the solutions to two simpler problems. These Cauchy sub-problems respectively concern the space and the time differential operator involved in the main equation. We provide some probabilistic and pseudo-probabilistic applications, where the solution can be interpreted as the pseudo-transition density of a time-changed pseudo-process. To extend our results to higher order time-fractional problems, we introduce pseudo-subordinators as well as its pseudo-inverse. Finally, we present our results in the case of more general differential operators and we interpret the results by means of a linear combination of pseudo-subordinators and its inverse.

math.PR↗

arXiv2024

General Airy-type equations, heat-type equations and pseudo-processes

We present a systematic study of higher-order Airy-type differential equations providing the explicit form of the solutions, deriving their power series expansions and a probabilistic interpretation. Under suitable convergence hypotheses, we compute their integral on the real line and, by means of complex integration, we provide alternative explicit forms. We then focus on the differential equations governing their derivatives, their products, their convolutions and higher-order Scorer type equations. Then, we study higher-order heat-type fractional Cauchy, showing that their fundamental solutions can be expressed in terms of Airy-type functions and their convolutions, recovering as special cases several results of appeared in previous papers. Furthermore, pseudo-processes theory permits us to give nice interpretations of the results, extending them in the case of equations involving different fractional operators and study the moments of the solutions.

math.PR↗

arXiv2024

On a planar random motion with asymptotically correlated components

We study a planar random motion $\big(X(t),\,Y(t)\big)$ with orthogonal directions, where the direction switches are governed by a homogeneous Poisson process. At each Poisson event, the moving particle turns clockwise or counterclockwise according to a rule which depends on the current direction. We prove that the components of the vector $\big(X(t),\,Y(t)\big)$ can be represented as linear combinations of two independent telegraph processes with different intensities. The exact distribution of $\big(X(t),\,Y(t)\big)$ is then obtained both in the interior of the support and on its boundary, where a singular component is present. We show that, in the hydrodynamic limit, the process behaves as a planar Brownian motion with correlated components. The distribution of the time spent by the process moving vertically is then studied. We obtain its exact distribution and discuss its hydrodynamic limit. In particular, in the limiting case, the process $\big(X(t),\,Y(t)\big)$ spends half of the time moving vertically.

math.PR↗

arXiv2024

Planar random motions in a vortex

We study a planar random motion $\big(X(t),Y(t)\big)$ with orthogonal directions which can turn clockwise, counterclockwise and reverse its direction each with a different probability. The support of the process is given by a time-varying square and the singular distributions on the boundary and the diagonals of the square are obtained. In the interior of the support, we study the hydrodynamic limit of the distribution. We then investigate the time $T(t)$ spent by the process moving vertically and the joint distribution of $\big(T(t),Y(t)\big)$. We prove that, in the hydrodynamic limit, the process $\big(X(t),Y(t)\big)$ spends half the time moving vertically.

math.PR↗

arXiv2023

Analysis of fractional Cauchy problems with some probabilistic applications

In this paper we give an explicit solution of Dzherbashyan-Caputo-fractional Cauchy problems related to equations with derivatives of order $νk$, for $k$ non-negative integer and $ν>0$. The solution is obtained by connecting the differential equation with the roots of the characteristic polynomial and it is expressed in terms of Mittag-Leffler-type functions. Under the some stricter hypothesis the solution can be expressed as a linear combination of Mittag-Leffler functions with common fractional order $ν$. We establish a probabilistic relationship between the solutions of differential problems with order $ν/m$ and $ν$, for natural $m$. Finally, we use the described method to solve fractional differential equations arising in the fractionalization of partial differential equations related to the probability law of planar random motions with finite velocities.

math.PR↗

arXiv2022

Pseudoprocesses related to higher-order equations of vibrations of rods

In this paper we study Fresnel pseudoprocesses whose signed measure density is a solution to a higher-order extension of the equation of vibrations of rods. We also investigate space-fractional extensions of the pseudoprocesses related to the Riesz operator. The measure density is represented in terms of generalized Airy functions which include the classical Airy function as a particular case. We prove that the Fresnel pseudoprocess time-changed with an independent stable subordinator produces genuine stochastic processes. In particular, if the exponent of the subordinator is chosen in a suitable way, the time-changed pseudoprocess is identical in distribution to a mixture of stable processes. The case of a mixture of Cauchy distributions is discussed and we show that the symmetric mixture can be either unimodal or bimodal, while the probability density function of an asymmetric mixture can possibly have an inflection point.

math.PR↗

arXiv2022

Stable distributions and pseudo-processes related to fractional Airy functions

In this paper we study pseudo-processes related to odd-order heat-type equations composed with Lévy stable subordinators. The aim of the article is twofold. We first show that the pseudo-density of the subordinated pseudo-process can be represented as an expectation of damped oscillations with generalized gamma distributed parameters. This stochastic representation also arises as the solution to a fractional diffusion equation, involving a higher-order Riesz-Feller operator, which generalizes the odd-order heat-type equation. We then prove that, if the stable subordinator has a suitable exponent, the time-changed pseudo-process becomes a genuine Lévy stable process. This result permits us to obtain a power series representation for the probability density function of an arbitrary asymmetric stable process of exponent $ν>1$ and skewness parameter $β$, with $0<\lvertβ\lvert<1$. The methods we use in order to carry out our analysis are based on the study of a fractional Airy function which emerges in the investigation of the higher-order Riesz-Feller operator.

math.PR↗

arXiv2022

Random motions in $\mathbb{R}^3$ with orthogonal directions

This paper is devoted to the detailed analysis of three-dimensional motions in $\mathbb{R}^3$ with orthogonal directions switching at Poisson times and moving with constant speed $c>0$. The study of the random position at an arbitrary time $t>0$ on the surface of the support, forming an octahedron $S_{ct}$, is completely carried out on the edges $E_{ct}$ and faces $F_{ct}$. In particular, the motion on the faces $F_{ct}$ is analysed by means of a transformation which reduces it to a three-directions planar random motion. This permits us to obtain an integral representation on $F_{ct}$ in terms of integral of products of first order Bessel functions. The investigation of the distribution of the position $p=p(t,x,y,z)$ inside $S_{ct}$ implied the derivation of a sixth-order partial differential equation governing $p$ (expressed in terms of the products of three D'Alembert operators). A number of results, also in explicit form, concern the time spent on each direction and the position reached by each coordinates as the motion devolpes. The analysis is carried out when the incoming direction is orthogonal to the ongoing one and also when all directions can be uniformely choosen at each Poisson event. If the switches are governed by homogeneus Poisson process many explicit results are obtained.

math.PR↗

arXiv2022

Hitting distribution of a correlated planar Brownian motion in a disk

In this paper we study the hitting probability of a circumference $C_R$ for a correlated Brownian motion $\underline{B}(t)=\left(B_1(t), B_2(t)\right)$, $ρ$ being the correlation coefficient. The analysis starts by first mapping the circle $C_R$ into an ellipse $E$ with semiaxes depending on $ρ$ and transforming the differential operator governing the hitting distribution into the classical Laplace operator. By means of two different approaches (one obtained by applying elliptic coordinates) we obtain the desired distribution as a series of Poisson kernels.

math.PR↗

arXiv2021

Stochastic dynamics of generalized planar random motions with orthogonal directions

We study planar random motions with finite velocities, of norm $c>0$, along orthogonal directions and changing at the instants of occurrence of a non-homogeneous Poisson process with rate function $λ(t),\ t\ge0$. We focus on the distribution of the current position $\bigl(X(t), Y(t)\bigr),\ t\ge0$, in the case where the motion has orthogonal deviations and where also reflection is admitted. In all the cases the process is located within the closed square $S_{ct}=\{(x,y)\in \mathbb{R}^2\,:\,|x|+|y|\le ct\}$ and we obtain the probability law inside $S_{ct}$, on the edge $\partial S_{ct}$ and on the other possible singularities, by studying the partial differential equations governing all the distributions examined. A fundamental result is that the vector process $\bigl(X(t), Y(t)\bigr)$ is probabilistically equivalent to a linear transformation of two (independent or dependent) one-dimensional symmetric telegraph processes with rate function proportional to $λ(t)$ and velocity $c/2$. Finally, we extend the results to a wider class of orthogonal-type evolutions.

math.PR↗

arXiv2021

Models of space-time random fields on the sphere

We study general models of random fields associated with non-local equations in time and space. We discuss the properties of the corresponding angular power spectrum and find asymptotic results in terms of random time changes.

math.PR↗

arXiv2020

On the Exact Distributions of the Maximum of the Asymmetric Telegraph Process

In this paper we present the distribution of the maximum of the asymmetric telegraph process in an arbitrary time interval $[0,t]$ under the conditions that the initial velocity $V(0)$ is either $c_1$ or $-c_2$ and the number of changes of direction is odd or even. For the case $V(0) = -c_2$ the singular component of the distribution of the maximum displays an unexpected cyclic behavior and depends only on $c_1$ and $c_2$, but not on the current time $t$. We obtain also the unconditional distribution of the maximum for either $V(0) = c_1$ or $V(0) = -c_2$ and its expression has the form of series of Bessel functions. We also show that all the conditional distributions emerging in this analysis are governed by generalized Euler-Poisson-Darboux equations. We recover all the distributions of the maximum of the symmetric telegraph process as particular cases of the present paper. We underline that it rarely happens to obtain explicitly the distribution of the maximum of a process. For this reason the results on the range of oscillations of a natural process like the telegraph model make it useful for many applications.

math.PR↗