scispace - formally typeset
Search or ask a question

Showing papers by "Enzo Orsingher published in 2016"


Journal ArticleDOI
TL;DR: In this article, 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.

27 citations


Journal ArticleDOI
TL;DR: In this paper, 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. Although they are not Levy processes, they somehow generalise subordinators, and by means of these processes, a generalization of subordinate semigroups, a two-parameter semigroup (propagators) arise and a Phillips formula which leads to time dependent generators.
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.

21 citations


Journal ArticleDOI
TL;DR: In this paper, the authors studied the correlation structures of the random fields emerging in the analysis of the solutions of two kinds of fractional equations displaying (Theorem 1) a long-range behaviour and(Theorem 2) a short-range behavior.

11 citations


Journal ArticleDOI
TL;DR: In this article, the authors examined the first-passage time distributions and the hitting probabilities of 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.

9 citations


Journal ArticleDOI
TL;DR: In this paper, the authors considered time-changed models of population evolution, where X f ( t ) = X ( H f( t )), where X is a counting process and H f is a subordinator with Laplace exponent f.
Abstract: In this paper we consider time-changed models of population evolution X f ( t ) = X ( H f ( t )), where X is a counting process and H f is a subordinator with Laplace exponent f . In the case where X is a pure birth process, we study the form of the distribution, the intertimes between successive jumps, and the condition of explosion (also in the case of killed subordinators). We also investigate the case where X represents a death process (linear or sublinear) and study the extinction probabilities as a function of the initial population size n 0 . Finally, the subordinated linear birth–death process is considered. Special attention is devoted to the case where birth and death rates coincide; the sojourn times are also analysed.

6 citations


Posted Content
TL;DR: In this article, the authors considered random motions on the line and on the plane with space-varying velocities and analyzed the explicit distribution of the position of the moving particle.
Abstract: Random motions on the line and on the plane with space-varying velocities are considered and analyzed in this paper. On the line we investigate symmetric and asymmetric telegraph processes with space-dependent velocities and we are able to present the explicit distribution of the position $\mathcal{T}(t)$, $t>0$, of the moving particle. Also the case of a non-homogeneous Poisson process (with rate $\lambda = \lambda(t)$) governing the changes of direction is analyzed in three specific cases. For the special case $\lambda(t)= \alpha/t$ we obtain a random motion related to the Euler-Poisson-Darboux (EPD) equation which generalizes the well-known case treated e.g. in Foong and Van Kolck (1992), Garra and Orsingher (2016) and Rosencrans (1973). A EPD--type fractional equation is also considered and a parabolic solution (which in dimension $d=1$ has the structure of a probability density) is obtained. Planar random motions with space--varying velocities and infinite directions are finally analyzed in Section 5. We are able to present their explicit distributions and for polynomial-type velocity structures we obtain the hyper and hypo-elliptic form of their support (of which we provide a picture).

3 citations


Journal ArticleDOI
TL;DR: In this article, the covariance functions and spectral densities of the stochastic processes satisfying the higher-order higher order differential equations were obtained explicitly for Gaussian white noise.
Abstract: In this paper we consider fractional higher-order stochastic differential equations of the form \[ \left ( \mu + c_\alpha \frac{d^\alpha } {dt^\alpha } \right )^\beta X(t) = \mathcal{E} (t) , \quad \mu >0,\; \beta >0,\; \alpha \in (0,1) \cup \mathbb{N} \] where $\mathcal{E} (t)$ is a Gaussian white noise. We obtain explicitly the covariance functions and the spectral densities of the stochastic processes satisfying the above equations.

2 citations


Posted Content
TL;DR: In this paper, the first hitting times of generalized Poisson processes were studied and the hitting probabilities of these processes were explicitly obtained and analyzed for the space-fractional Poisson process.
Abstract: This paper studies the first hitting times of generalized Poisson processes $N^f(t)$, related to Bernstein functions $f$. For the space-fractional Poisson processes, $N^\alpha(t)$, $t>0$ (corresponding to $f= x^\alpha$), the hitting probabilities $P\{T_k^\alpha<\infty\}$ are explicitly obtained and analyzed. The processes $N^f(t)$ are time-changed Poisson processes $N(H^f(t))$ with subordinators $H^f(t)$ and here we study $N\left(\sum_{j=1}^n H^{f_j}(t)\right)$ and obtain probabilistic features of these extended counting processes. A section of the paper is devoted to processes of the form $N(|\mathcal{G}_{H, u}(t)|)$ where $\mathcal{G}_{H, u}(t)$ are generalized grey Brownian motions. This involves the theory of time-dependent fractional operators of the McBride form. While the time-fractional Poisson process is a renewal process, we prove that the space-time Poisson process is no longer a renewal process.

2 citations


Journal ArticleDOI
TL;DR: In this article, 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, namely a random flight 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 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.

2 citations


Book ChapterDOI
29 May 2016
TL;DR: In this article, the authors considered random motions on the line and on the plane with space-varying velocities and analyzed the explicit distribution of the position of the moving particle.
Abstract: Random motions on the line and on the plane with space-varying velocities are considered and analyzed in this paper. On the line we investigate symmetric and asymmetric telegraph processes with space-dependent velocities and we are able to present the explicit distribution of the position \(\mathcal {T}(t)\), \(t>0\), of the moving particle. Also the case of a nonhomogeneous Poisson process (with rate \(\lambda = \lambda (t)\)) governing the changes of direction is analyzed in three specific cases. For the special case \(\lambda (t)= \alpha /t\), we obtain a random motion related to the Euler–Poisson–Darboux (EPD) equation which generalizes the well-known case treated, e.g., in (Foong, S.K., Van Kolck, U.: Poisson random walk for solving wave equations. Prog. Theor. Phys. 87(2), 285–292, 1992, [6], Garra, R., Orsingher, E.: Random flights related to the Euler-Poisson-Darboux equation. Markov Process. Relat. Fields 22, 87–110, 2016, [8], Rosencrans, S.I.: Diffusion transforms. J. Differ. Equ. 13, 457–467, 1973, [16]). A EPD-type fractional equation is also considered and a parabolic solution (which in dimension \(d=1\) has the structure of a probability density) is obtained. Planar random motions with space-varying velocities and infinite directions are finally analyzed in Sect. 5. We are able to present their explicit distributions, and for polynomial-type velocity structures we obtain the hyper- and hypoelliptic form of their support (of which we provide a picture).

1 citations