arXiv subjects

Enzo Orsingher

Publications and source records attributed to Enzo Orsingher.

At least 37 records · Page 2Linked to original sources

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↗

arXiv2020

Estimates for distribution of suprema of solutions to higher-order partial differential equations with random initial conditions

In the paper we consider higher-order partial differential equations from the class of linear dispersive equations. We investigate solutions to these equations subject to random initial conditions given by harmonizable $φ$-sub-Gaussian processes. The main results are the bounds for the distributions of the suprema for solutions. We present the examples of processes for which the assumptions of the general result are verified and bounds are written in the explicit form. The main result is also specified for the case of Gaussian initial condition.

math.PR↗

arXiv2020

On the distribution of the maximum of the telegraph process

In this paper we present the distribution of the maximum of the telegraph process in the cases where the initial velocity is positive or negative with an even and an odd number of velocity reversals. For the telegraph process with positive initial velocity a reflection principle is proved to be valid while in the case of an initial leftward displacement the conditional distributions are perturbed by a positive probability of never visiting the half positive axis. Various relationships are established among the mentioned four classes of conditional distributions of the maximum. The unconditional distributions of the maximum of the telegraph process are obtained for positive and negative initial steps as well as their limiting behaviour. Furthermore the cumulative distributions and the general moments of the conditional maximum are presented.

math.PR↗

arXiv2020

Random flights connecting Porous Medium and Euler-Poisson-Darboux equations

In this paper we consider the Porous Medium Equation and establish a relationship between its Kompanets-Zel'dovich-Barenblatt solution $u(\xd,t), \xd\in \mathbb R^d,t>0$ and random flights. The time-rescaled version of $u(\xd,t)$ is the fundamental solution of the Euler-Poisson-Darboux equation which governs the distribution of random flights performed by a particle whose displacements have a Dirichlet probability distribution and choosing directions uniformly on a $d$-dimensional sphere (see, e.g., \cite{dgo}). We consider the space-fractional version of the Euler-Poisson-Darboux equation and present the solution of the related Cauchy problem in terms of the probability distributions of random flights governed by the classical Euler-Poisson-Darboux equation. Furthermore, this research is also aimed at studying the relationship between the solutions of a fractional Porous Medium Equation and the fractional Euler-Poisson-Darboux equation. A considerable part of the paper is devoted to the analysis of the probabilistic tools of the solutions of the fractional equations. Also the extension to higher-order Euler-Poisson-Darboux equation is considered and the solutions interpreted as compositions of laws of pseudoprocesses.

math.PR↗

arXiv2019

On the sojourn time of a Generalized Brownian meander

In this paper we study the sojourn time on the positive half-line up to time $ t $ of a drifted Brownian motion with starting point $ u $ and subject to the condition that $ \min_{ 0\leq z \leq l} B(z)> v $, with $ u > v $. This process is a drifted Brownian meander up to time $ l $ and then evolves as a free Brownian motion. We also consider the sojourn time of a bridge-type process, where we add the additional condition to return to the initial level at the end of the time interval. We analyze the weak limit of the occupation functional as $ u \downarrow v $. We obtain explicit distributional results when the barrier is placed at the zero level, and also in the special case when the drift is null.

math.PR↗

arXiv2019

The last zero crossing of an iterated Brownian motion with drift

In this paper we consider the iterated Brownian motion $ ^{μ_1}_{μ_2}\!I(t) = B_1^{μ_1} ( | B_{2}^{μ_2} (t)|) $ where $B_j^{μ_j} , j=1,2$ are two independent Brownian motions with drift $μ_j$. Here we study the last zero crossing of $ ^{μ_1}_{μ_2}\!I(t) $ and for this purpose we derive the last zero-crossing distribution of the drifted Brownian motion. We derive also the joint distribution of the last zero crossing before $ t $ and of the first passage time through the zero level of a Brownian motion with drift $ μ$ after $ t $. All these results permit us to derive explicit formulas for ${^I_μT_0} = \sup \{ s < \max_{0\leq z\leq t} |B_2(z)| : B_1^μ(s) = 0 \}$. Also the iterated zero-crossing $ {^{μ_1} T}_{0, {^{μ_2} T}_{0,t}} $ is analyzed and extended to the case where the level of nesting is arbitrary.

math.PR↗

arXiv2019

Some results on the Brownian meander with drift

In this paper we study the drifted Brownian meander, that is a Brownian motion starting from $ u $ and subject to the condition that $ \min_{ 0\leq z \leq t} B(z)> v $ with $ u > v $. The limiting process for $ u \downarrow v $ is analyzed and the sufficient conditions for its construction are given. We also study the distribution of the maximum of the meander with drift and the related first-passage times. The representation of the meander endowed with a drift is provided and extends the well-known result of the driftless case. The last part concerns the drifted excursion process the distribution of which coincides with the driftless case.

math.PR↗

arXiv2019

Drifted Brownian motions governed by fractional tempered derivatives

Fractional equations governing the distribution of reflecting drifted Brownian motions are presented. The equations are expressed in terms of tempered Riemann--Liouville type derivatives. For these operators a Marchaud-type form is obtained and a Riesz tempered fractional derivative is examined, together with its Fourier transform.

math.PR↗

arXiv2017

A note on Hadamard fractional differential equations with varying coefficients and their applications in probability

In this paper we show several connections between special functions arising from generalized COM-Poisson-type statistical distributions and integro-differential equations with varying coefficients involving Hadamard-type operators. New analytical results are obtained, showing the particular role of Hadamard-type derivatives in connection with a recently introduced generalization of the Le Roy function. We are also able to prove a general connection between fractional hyper-Bessel-type equations involving Hadamard operators and Le Roy functions.

math.PR↗

arXiv2017

On semi-Markov processes and their Kolmogorov's integro-differential equations

Semi-Markov processes are a generalization of Markov processes since the exponential distribution of time intervals is replaced with an arbitrary distribution. This paper provides an integro-differential form of the Kolmogorov's backward equations for a large class of homogeneous semi-Markov processes, having the form of an abstract Volterra integro-differential equation. An equivalent evolutionary (differential) form of the equations is also provided. Fractional equations in the time variable are a particular case of our analysis. Weak limits of semi-Markov processes are also considered and their corresponding integro-differential Kolmogorov's equations are identified.

math.PR↗

arXiv2017

Asymptotic behaviour of non-isotropic random walks with heavy tails

A random flight on a plane with non-isotropic displacements at the moments of direction changes is considered. In the case of exponentially distributed flight lengths a Gaussian limit theorem is proved for the position of a particle in the scheme of series when jump lengths and non-isotropic displacements tend to zero. If the flight lengths have a folded Cauchy distribution the limiting distribution of the particle position is a convolution of the circular bivariate Cauchy distribution with a Gaussian law.

math.PR↗

arXiv2016

Time-inhomogeneous jump processes and variable order operators

In this paper we introduce non-decreasing jump processes with independent and time non-homogeneous increments. Although they are not Lévy processes, they somehow generalize subordinators in the sense that their Laplace exponents are possibly different Bernštein 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.

math.PR↗

arXiv2015

Motion among random obstacles on a hyperbolic space

We consider the motion of a particle along the geodesic lines of the Poincarè 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.

math-ph↗

arXiv2015

Space-time fractional equations and the related stable processes at random time

In this paper we consider the general fractional equation \sum_{j=1}^m λ_j \frac{\partial^{ν_j}}{\partial t^{ν_j}} w(x_1,..., x_n ; t) = -c^2 (-Δ)^βw(x_1,..., x_n ; t), for ν_j \in (0,1], β\in (0,1] with initial condition w(x_1,..., x_n ; 0)= \prod_{j=1}^n δ(x_j). The solution of the Cauchy problem above coincides with the distribution of the n-dimensional process \bm{S}_n^{2β} \mathcal{L} c^2 {L}^{ν_1,..., ν_m} (t) \r, t>0, where \bm{S}_n^{2β} is an isotropic stable process independent from {L}^{ν_1,..., ν_m}(t) which is the inverse of {H}^{ν_1,..., ν_m} (t) = \sum_{j=1}^m λ_j^{1/ν_j} H^{ν_j} (t), t>0, with H^{ν_j}(t) independent, positively-skewed stable r.v.'s of order ν_j. The problem considered includes the fractional telegraph equation as a special case as well as the governing equation of stable processes. The composition \bm{S}_n^{2β} (c^2 {L}^{ν_1,..., ν_m} (t)), t>0, supplies a probabilistic representation for the solutions of the fractional equations above and coincides for β= 1 with the n-dimensional Brownian motion at the time {L}^{ν_1,..., ν_m} (t), t>0. The iterated process {L}^{ν_1,..., ν_m}_r (t), t>0, inverse to {H}^{ν_1,..., ν_m}_r (t) =\sum_{j=1}^m λ_j^{1/ν_j} _1H^{ν_j} (_{2}H^{ν_j} (_3H^{ν_j} (... _{r}H^{ν_j} (t)...))), t>0, permits us to construct the process \bm{S}_n^{2β} (c^2 {L}^{ν_1,..., ν_m}_r (t)), t>0, the distribution of which solves a space-fractional generalized telegraph equation. For r \to \infty and β= 1 we obtain a distribution which represents the n-dimensional generalisation of the Gauss-Laplace law and solves the equation \sum_{j=1}^m λ_j w(x_1,..., x_n) = c^2 \sum_{j=1}^n \frac{\partial^2}{\partial x_j^2} w(x_1,..., x_n).

math.PR↗

arXiv2015

Fractional diffusions with time-varying coefficients

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 $L^α$ into Erdélyi--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.

math.PR↗

arXiv2015

Spectral functions related to some fractional stochastic differential equations

In this paper we consider fractional higher-order stochastic differential equations of the form \begin{align*} \left( μ+ c_α\frac{d^α}{d(-t)^α} \right)^βX(t) = \mathcal{E}(t) , \quad t\geq 0,\; μ>0,\; β>0,\; α\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.

math.PR↗

arXiv2015

Population models at stochastic times

In this article, we consider time-changed models of population evolution $\mathcal{X}^f(t)=\mathcal{X}(H^f(t))$, where $\mathcal{X}$ is a counting process and $H^f$ is a subordinator with Laplace exponent $f$. In the case $\mathcal{X}$ 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 $\mathcal{X}$ 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. A special attention is devoted to the case where birth and death rates coincide; the sojourn times are also analysed.

math.PR↗

arXiv2014

Fractional spherical random fields

In this paper we study the solutions of different forms of fractional equations on the unit sphere $\mathbb{S}_{1}^{2}$ $\subset \mathbb{R}^{3}$ possessing the structure of time-dependent random fields. We study the correlation functions of the random fields emerging in the analysis of the solutions of the fractional equations and examine their long-range behaviour.

math.PR↗