Showing papers by "Enzo Orsingher published in 2017"

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

Abstract: 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.

Some probabilistic properties of fractional point processes

Abstract: In this article, the first hitting times of generalized Poisson processes Nf(t), related to Bernstein functions f are studied. For the space-fractional Poisson processes, Nα(t), t > 0 (corresponding to f = xα), the hitting probabilities P{Tαk < ∞} are explicitly obtained and analyzed. The processes Nf(t) are time-changed Poisson processes N(Hf(t)) with subordinators Hf(t) and here we study and obtain probabilistic features of these extended counting processes. A section of the paper is devoted to processes of the form where 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.

Space–Time Fractional Equations and the Related Stable Processes at Random Time

Abstract: In this paper, we consider the general space–time fractional equation of the form $$\sum _{j=1}^m \lambda _j \frac{\partial ^{ u _j}}{\partial t^{ u _j}} w(x_1, \ldots , x_n ; t) = -c^2 \left( -\varDelta \right) ^\beta w(x_1, \ldots , x_n ; t)$$ , for $$ u _j \in \left( 0,1 \right] $$ and $$\beta \in \left( 0,1 \right] $$ with initial condition $$w(x_1, \ldots , x_n ; 0)= \prod _{j=1}^n \delta (x_j)$$ . We show that the solution of the Cauchy problem above coincides with the probability density of the n-dimensional vector process $$\varvec{S}_n^{2\beta } \left( c^2 \mathcal {L}^{ u _1, \ldots , u _m} (t) \right) $$ , $$t>0$$ , where $$\varvec{S}_n^{2\beta }$$ is an isotropic stable process independent from $$\mathcal {L}^{ u _1, \ldots , u _m}(t)$$ , which is the inverse of $$\mathcal {H}^{ u _1, \ldots , u _m} (t) = \sum _{j=1}^m \lambda _j^{1/ u _j} H^{ u _j} (t)$$ , $$t>0$$ , with $$H^{ u _j}(t)$$ independent, positively skewed stable random variables of order $$ u _j$$ . The problem considered includes the fractional telegraph equation as a special case as well as the governing equation of stable processes. The composition $$\varvec{S}_n^{2\beta } \left( c^2 \mathcal {L}^{ u _1, \ldots , u _m} (t) \right) $$ , $$t>0$$ , supplies a probabilistic representation for the solutions of the fractional equations above and coincides for $$\beta = 1$$ with the n-dimensional Brownian motion at the random time $$\mathcal {L}^{ u _1, \ldots , u _m} (t)$$ , $$t>0$$ . The iterated process $$\mathfrak {L}^{ u _1, \ldots , u _m}_r (t)$$ , $$t>0$$ , inverse to $$\mathfrak {H}^{ u _1, \ldots , u _m}_r (t) =\sum _{j=1}^m \lambda _j^{1/ u _j} \, _1H^{ u _j} \left( \, _2H^{ u _j} \left( \, _3H^{ u _j} \left( \ldots \, _{r}H^{ u _j} (t) \ldots \right) \right) \right) $$ , $$t>0$$ , permits us to construct the process $$\varvec{S}_n^{2\beta } \left( c^2 \mathfrak {L}^{ u _1, \ldots , u _m}_r (t) \right) $$ , $$t>0$$ , the density of which solves a space-fractional equation of the form of the generalized fractional telegraph equation. For $$r \rightarrow \infty $$ and $$\beta = 1$$ , we obtain a probability density, independent from t, which represents the multidimensional generalization of the Gauss–Laplace law and solves the equation $$\sum _{j=1}^m \lambda _j w(x_1, \ldots , x_n) = c^2 \sum _{j=1}^n \frac{\partial ^2}{\partial x_j^2} w(x_1, \ldots , x_n)$$ . Our analysis represents a general framework of the interplay between fractional differential equations and composition of processes of which the iterated Brownian motion is a very particular case.

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

Abstract: 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.

Planar piecewise linear random motions with jumps

Abstract: In this paper, we study persistent piecewise linear multidimensional random motions. Their velocities, switching at Poisson times, are uniformly distributed on a sphere. The changes of direction are accompanied with subsequent jumps of random length and of uniformly distributed orientation. In this paper, we obtain some useful properties and formulae of distributions of these processes. In particular, we get these distributions in the cases of jumps with Gaussian and exponential distributions of jump magnitudes.

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

Abstract: 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 a Volterra integro-differential equation. An equivalent evolutionary (differential) form of the equations is also provided. Weak limits of semi-Markov processes are also considered and their corresponding integro-differential Kolmogorov's equations are identified.

Some results on the generalized Bronwian meander

Abstract: 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.

Asymptotic behaviour of non-isotropic random walks with heavy tails

Abstract: 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.

Asymptotic behaviour of non-isotropic random walks with heavy tails

Abstract: 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.

Some results on the Brownian meander with drift

Abstract: 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.