Showing papers by "Enzo Orsingher published in 2015"
TL;DR: In this paper, the authors considered point processes Nf(t), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernstein functions f with Levy measure ν.
Abstract: In this paper we consider point processes Nf(t), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernstein functions f with Levy measure ν. We obtain the general expression of the probability generating functions Gf of Nf, the equations governing the state probabilities pkf of Nf, and their corresponding explicit forms. We also give the distribution of the first-passage times Tkf of Nf, and the related governing equation. We study in detail the cases of the fractional Poisson process, the relativistic Poisson process, and the gamma-Poisson process whose state probabilities have the form of a negative binomial. The distribution of the times τjlj of jumps with height lj (∑j=1rlj = k) under the condition N(t) = k for all these special processes is investigated in detail.
45 citations
TL;DR: In this paper, the authors analyzed the fractional Poisson process where the state probabilities p k ν k (t), t ≥ 0, are governed by time-fractional equations of order 0 < νk ≤ 1 depending on the number k of events that have occurred up to time t. They also introduced a different form of fractional state-dependent poisson process as a weighted sum of homogeneous Poisson processes.
Abstract: In this paper we analyse the fractional Poisson process where the state probabilities p k ν k (t), t ≥ 0, are governed by time-fractional equations of order 0 < ν k ≤ 1 depending on the number k of events that have occurred up to time t. We are able to obtain explicitly the Laplace transform of p k ν k (t) and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on ν k differs from that constructed from the fractional state equations (in the case of ν k = ν, for all k, they coincide with the time-fractional Poisson process). We also introduce a different form of fractional state-dependent Poisson process as a weighted sum of homogeneous Poisson processes. Finally, we consider the fractional birth process governed by equations with state-dependent fractionality.
29 citations
TL;DR: In this paper, the fractionalized diffusion equations governing the law of fractional Brownian motion BH(t) were analyzed and the solutions of these equations which are probability laws extending that of BH (t) are obtained.
Abstract: This paper is concerned with the fractionalized diffusion equations governing the law of the fractional Brownian motion BH(t). We obtain solutions of these equations which are probability laws extending that of BH(t). Our analysis is based on McBride fractional operators generalizing the hyper-Bessel operators L and converting their fractional power Lα into Erdelyi–Kober fractional integrals. We study also probabilistic properties of the random variables whose distributions satisfy space-time fractional equations involving Caputo and Riesz fractional derivatives. Some results emerging from the analysis of fractional equations with time-varying coefficients have the form of distributions of time-changed random variables.
27 citations
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TL;DR: In this article, the authors introduce non-decreasing jump processes with independent and time non-homogeneous increments, which generalize subordinators in the sense that their Laplace exponents are possibly different Bernstein functions for each time $t.
Abstract: In this paper we introduce non-decreasing jump processes with independent and time non-homogeneous increments. Although they are not Levy processes, they somehow generalize subordinators in the sense that their Laplace exponents are possibly different Bernstein functions for each time $t$. By means of these processes, a generalization of subordinate semigroups in the sense of Bochner is proposed. Because of time-inhomogeneity, two-parameter semigroups (propagators) arise and we provide a Phillips formula which leads to time dependent generators. The inverse processes are also investigated and the corresponding governing equations obtained in the form of generalized variable order fractional equations. An application to a generalized subordinate Brownian motion is also examined.
14 citations
TL;DR: In this article, the authors studied the fractionalized diffusion equations governing the law of fractional Brownian motion and obtained solutions of these equations which are probability laws extending that of $B_H(t).
Abstract: This paper is concerned with the fractionalized diffusion equations governing the law of the fractional Brownian motion $B_H(t)$. We obtain solutions of these equations which are probability laws extending that of $B_H(t)$. Our analysis is based on McBride fractional operators generalizing the hyper-Bessel operators $L$ and converting their fractional power $L^{\alpha}$ into Erd\'elyi--Kober fractional integrals. We study also probabilistic properties of the r.v.'s whose distributions satisfy space-time fractional equations involving Caputo and Riesz fractional derivatives. Some results emerging from the analysis of fractional equations with time-varying coefficients have the form of distributions of time-changed r.v.'s.
8 citations
TL;DR: In this article, the authors considered random flights in a sphere with center at the origin and radius R, where reflection is performed by means of circular inversion, and obtained the explicit probability distributions of the position of the moving particle when the number of changes of direction is fixed and equal to
Abstract: We consider random flights in $$\mathbb {R}^d$$
reflecting on the surface of a sphere $$\mathbb {S}^{d-1}_R,$$
with center at the origin and with radius R, where reflection is performed by means of circular inversion. Random flights studied in this paper are motions where the orientation of the deviations are uniformly distributed on the unit-radius sphere $$\mathbb {S}^{d-1}_1$$
. We obtain the explicit probability distributions of the position of the moving particle when the number of changes of direction is fixed and equal to $$n\ge 1$$
. We show that these distributions involve functions which are solutions of the Euler–Poisson–Darboux equation. The unconditional probability distributions of the reflecting random flights are obtained by suitably randomizing n by means of a fractional-type Poisson process. Random flights reflecting on hyperplanes according to the optical reflection form are considered and the related distributional properties derived.
5 citations
TL;DR: In this article, the authors introduced new distributions which are solutions of higher-order Laplace equations and proved that their densities can be obtained by folding and symmetrizing Cauchy distributions.
Abstract: In this paper we introduce new distributions which are solutions of higher-order Laplace equations. It is proved that their densities can be obtained by folding and symmetrizing Cauchy distributions. Another class of probability laws related to higher-order Laplace equations is obtained by composing pseudo-processes with positively skewed stable distributions which produce asymmetric Cauchy densities in the odd-order case. Special attention is devoted to the third-order Laplace equation where the connection between the Cauchy distribution and the Airy functions is obtained and analyzed.
4 citations
TL;DR: In this paper, the authors examined pseudoprocesses wrapped up on circles and derived their explicit signed density measures, which are obtained in the form of Poisson kernels, similar to the Von Mises (or Fisher) circular normal law and to the wrapped up law of Brownian motion.
Abstract: Pseudoprocesses, constructed by means of the solutions of higher-order heat-type equations, have been developed by several authors and many related functionals have been analysed by applying the Feynman–Kac functional or by means of the Spitzer identity. We here examine pseudoprocesses wrapped up on circles and derive their explicit signed density measures. By composing the circular pseudoprocesses with positively skewed stable processes, we arrive at genuine circular processes whose distribution is obtained in the form of Poisson kernels. The distribution of circular even-order pseudoprocesses is similar to the Von Mises (or Fisher) circular normal law and to the wrapped up law of Brownian motion. Time-fractional and space-fractional equations related to processes and pseudoprocesses on the unit radius circumference are introduced and analysed.
3 citations
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TL;DR: In this paper, the covariance functions and spectral functions of higher-order stochastic differential equations were derived explicitly for a Gaussian white noise model, and the spectral functions were obtained explicitly.
Abstract: In this paper we consider fractional higher-order stochastic differential equations of the form \begin{align*} \left( \mu + c_\alpha \frac{d^\alpha}{d(-t)^\alpha} \right)^\beta X(t) = \mathcal{E}(t) , \quad t\geq 0,\; \mu>0,\; \beta>0,\; \alpha \in (0,1) \cup \mathbb{N} \end{align*} where $\mathcal{E}(t)$ is a Gaussian white noise. We derive stochastic processes satisfying the above equations of which we obtain explicitly the covariance functions and the spectral functions.
1 citations
TL;DR: In this paper, the authors consider the case where a particle is specularly reflected when it hits randomly-distributed obstacles that are assumed to be motionless, and they show that their process converges to a Markovian process on the hyperbolic manifold.
Abstract: We consider the motion of a particle along the geodesic lines of the Poincare half-plane. The particle is specularly reflected when it hits randomly-distributed obstacles that are assumed to be motionless. This is the hyperbolic version of the well-known Lorentz Process studied by Gallavotti in the Euclidean context. We analyse the limit in which the density of the obstacles increases to infinity and the size of each obstacle vanishes: under a suitable scaling, we prove that our process converges to a Markovian process, namely a random flight on the hyperbolic manifold.
1 citations
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TL;DR: In this paper, the Fourier transforms of the solutions of some Cauchy problems for space-time fractional telegraph equations are provided, where the time derivatives are intended in the sense of Hilfer and Hadamard while the space fractional derivatives are meant in the meaning of Riesz-Feller.
Abstract: In this paper we consider space-time fractional telegraph equations, where the time derivatives are intended in the sense of Hilfer and Hadamard while the space fractional derivatives are meant in the sense of Riesz-Feller. We provide the Fourier transforms of the solutions of some Cauchy problems for these fractional equations. Probabilistic interpretations of some specific cases are also provided.
TL;DR: In this paper, the authors examined the first-passage time distributions and the hitting probabilities of the iterated birth process, linear birth processes, linear and sublinear death processes at Poisson times, and examined their long-range behavior.
Abstract: 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.