Showing papers by "Enzo Orsingher published in 2010"

Poisson-type processes governed by fractional and higher-order recursive differential equations

Abstract: We consider some fractional extensions of the recursive differential equation governing the Poisson process, i.e. $\partial_tp_k(t)=-\lambda(p_k(t)-p_{k-1}(t))$, $k\geq0$, $t>0$ by introducing fractional time-derivatives of order $ u,2 u,\ldots,n u$. We show that the so-called "Generalized Mittag-Leffler functions" $E_{\alpha,\beta^k}(x)$, $x\in\mathbb{R}$ (introduced by Prabhakar [24] )arise as solutions of these equations. The corresponding processes are proved to be renewal, with density of the intearrival times (represented by Mittag-Leffler functions) possessing power, instead of exponential, decay, for $t\to\infty$. On the other hand, near the origin the behavior of the law of the interarrival times drastically changes for the parameter $ u$ varying in $(0,1]$. For integer values of $ u$, these models can be viewed as a higher-order Poisson processes, connected with the standard case by simple and explict relationships.

Fractional pure birth processes

Abstract: We consider a fractional version of the classical nonlinear birth process of which the Yule–Furry model is a particular case. Fractionality is obtained by replacing the first order time derivative in the difference-differential equations which govern the probability law of the process with the Dzherbashyan–Caputo fractional derivative. We derive the probability distribution of the number $\mathcal{N}_{ u}(t)$ of individuals at an arbitrary time $t$. We also present an interesting representation for the number of individuals at time $t$, in the form of the subordination relation $\mathcal{N}_{ u}(t)=\mathcal{N}(T_{2 u}(t))$, where $\mathcal{N}(t)$ is the classical generalized birth process and $T_{2ν}(t)$ is a random time whose distribution is related to the fractional diffusion equation. The fractional linear birth process is examined in detail in Section 3 and various forms of its distribution are given and discussed.

Fractional pure birth processes

Abstract: We consider a fractional version of the classical nonlinear birth process of which the Yule--Furry model is a particular case. Fractionality is obtained by replacing the first order time derivative in the difference-differential equations which govern the probability law of the process with the Dzherbashyan--Caputo fractional derivative. We derive the probability distribution of the number $\mathcal{N}_{ u}(t)$ of individuals at an arbitrary time $t$. We also present an interesting representation for the number of individuals at time $t$, in the form of the subordination relation $\mathcal{N}_{ u}(t)=\mathcal{N}(T_{2 u}(t))$, where $\mathcal{N}(t)$ is the classical generalized birth process and $T_{2 u}(t)$ is a random time whose distribution is related to the fractional diffusion equation. The fractional linear birth process is examined in detail in Section 3 and various forms of its distribution are given and discussed.

Fractional Non-Linear, Linear and Sublinear Death Processes

Abstract: This paper is devoted to the study of a fractional version of non-linear , t>0, linear M ν(t), t>0 and sublinear $\mathfrak{M}^{ u}(t)$ , t>0 death processes. Fractionality is introduced by replacing the usual integer-order derivative in the difference-differential equations governing the state probabilities, with the fractional derivative understood in the sense of Dzhrbashyan–Caputo. We derive explicitly the state probabilities of the three death processes and examine the related probability generating functions and mean values. A useful subordination relation is also proved, allowing us to express the death processes as compositions of their classical counterparts with the random time process T 2ν(t), t>0. This random time has one-dimensional distribution which is the folded solution to a Cauchy problem of the fractional diffusion equation.

Moving randomly amid scattered obstacles

Abstract: We study a planar random motion at finite velocity performed by a particle which, at even-valued Poisson events, changes direction (each time chosen with uniform law in [0, 2π]). In other words this model assumes that the time between successive deviations is a Gamma random variable. It can also be interpreted as the motion of particles that can hazardously collide with obstacles of different size, some of which are capable of deviating the motion. We obtain the explicit densities of the random position under the condition that the number of deviations N(t) is known. We express as suitable combinations of distributions of the motion described by a particle changing direction at all Poisson events. The conditional densities of and are connected by means of a new discrete-valued random variable, whose distribution is expressed in terms of Beta integrals. The technique used in the analysis is based on rather involved properties of Bessel functions, which are derived and explored in detail in order to make th...

Composition of processes and related partial differential equations

Abstract: In this paper different types of compositions involving independent fractional Brownian motions B^j_{H_j}(t), t>0, j=1,$ are examined. The partial differential equations governing the distributions of I_F(t)=B^1_{H_1}(|B^2_{H_2}(t)|), t>0 and J_F(t)=B^1_{H_1}(|B^2_{H_2}(t)|^{1/H_1}), t>0 are derived by different methods and compared with those existing in the literature and with those related to B^1(|B^2_{H_2}(t)|), t>0. The process of iterated Brownian motion I^n_F(t), t>0 is examined in detail and its moments are calculated. Furthermore for J^{n-1}_F(t)=B^1_{H}(|B^2_H(...|B^n_H(t)|^{1/H}...)|^{1/H}), t>0 the following factorization is proved J^{n-1}_F(t)=\prod_{j=1}^{n} B^j_{\frac{H}{n}}(t), t>0. A series of compositions involving Cauchy processes and fractional Brownian motions are also studied and the corresponding non-homogeneous wave equations are derived.

A Brief Review on Some Fractional Point Processes

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Equations of Mathematical Physics and Compositions of Brownian and Cauchy processes

Abstract: We consider different types of processes obtained by composing Brownian motion $B(t)$, fractional Brownian motion $B_{H}(t)$ and Cauchy processes $% C(t)$ in different manners. We study also multidimensional iterated processes in $\mathbb{R}^{d},$ like, for example, $\left( B_{1}(|C(t)|),...,B_{d}(|C(t)|)\right) $ and $\left( C_{1}(|C(t)|),...,C_{d}(|C(t)|)\right) ,$ deriving the corresponding partial differential equations satisfied by their joint distribution. We show that many important partial differential equations, like wave equation, equation of vibration of rods, higher-order heat equation, are satisfied by the laws of the iterated processes considered in the work. Similarly we prove that some processes like $% C(|B_{1}(|B_{2}(...|B_{n+1}(t)|...)|)|)$ are governed by fractional diffusion equations.