arXiv subjects

Enzo Orsingher

Publications and source records attributed to Enzo Orsingher.

66 records · Page 4Linked to original sources

arXiv2011

The Space-Fractional Poisson Process

In this paper we introduce the space-fractional Poisson process whose state probabilities $p_k^α(t)$, $t>0$, $α\in (0,1]$, are governed by the equations $(\mathrm d/\mathrm dt)p_k(t) = -λ^α(1-B)p_k^α(t)$, where $(1-B)^α$ is the fractional difference operator found in the study of time series analysis. We explicitly obtain the distributions $p_k^α(t)$, the probability generating functions $G_α(u,t)$, which are also expressed as distributions of the minimum of i.i.d.\ uniform random variables. The comparison with the time-fractional Poisson process is investigated and finally, we arrive at the more general space-time fractional Poisson process of which we give the explicit distribution.

math.PR↗

arXiv2011

Randomly Stopped Nonlinear Fractional Birth Processes

We present and analyse the nonlinear classical pure birth process $\mathpzc{N} (t)$, $t>0$, and the fractional pure birth process $\mathpzc{N}^ν(t)$, $t>0$, subordinated to various random times, namely the first-passage time $T_t$ of the standard Brownian motion $B(t)$, $t>0$, the $α$-stable subordinator $\mathpzc{S}^α(t)$, $α\in (0,1)$, and others. For all of them we derive the state probability distribution $\hat{p}_k (t)$, $k \geq 1$ and, in some cases, we also present the corresponding governing differential equation. We also highlight interesting interpretations for both the subordinated classical birth process $\hat{\mathpzc{N}} (t)$, $t>0$, and its fractional counterpart $\hat{\mathpzc{N}}^ν(t)$, $t>0$ in terms of classical birth processes with random rates evaluated on a stretched or squashed time scale. Various types of compositions of the fractional pure birth process $\mathpzc{N}^ν(t)$ have been examined in the last part of the paper. In particular, the processes $\mathpzc{N}^ν(T_t)$, $\mathpzc{N}^ν(\mathpzc{S}^α(t))$, $\mathpzc{N}^ν(T_{2ν}(t))$, have been analysed, where $T_{2ν}(t)$, $t>0$, is a process related to fractional diffusion equations. Also the related process $\mathpzc{N}(\mathpzc{S}^α({T_{2ν}(t)}))$ is investigated and compared with $\mathpzc{N}(T_{2ν}(\mathpzc{S}^α(t))) = \mathpzc{N}^ν(\mathpzc{S}^α(t))$. As a byproduct of our analysis, some formulae relating Mittag--Leffler functions are obtained.

math.PR↗

arXiv2011

Vibrations and fractional vibrations of rods, plates and Fresnel pseudo-processes

Different initial and boundary value problems for the equation of vibrations of rods (also called Fresnel equation) are solved by exploiting the connection with Brownian motion and the heat equation. The analysis of the fractional version (of order $ν$) of the Fresnel equation is also performed and, in detail, some specific cases, like $ν=1/2$, 1/3, 2/3, are analyzed. By means of the fundamental solution of the Fresnel equation, a pseudo-process $F(t)$, $t>0$ with real sign-varying density is constructed and some of its properties examined. The equation of vibrations of plates is considered and the case of circular vibrating disks $C_R$ is investigated by applying the methods of planar orthogonally reflecting Brownian motion within $C_R$. The composition of F with reflecting Brownian motion $B$ yields the law of biquadratic heat equation while the composition of $F$ with the first passage time $T_t$ of $B$ produces a genuine probability law strictly connected with the Cauchy process.

math.PR↗

arXiv2011

Hitting spheres on hyperbolic spaces

For a hyperbolic Brownian motion on the Poincaré half-plane $\mathbb{H}^2$, starting from a point of hyperbolic coordinates $z=(η, α)$ inside a hyperbolic disc $U$ of radius $\barη$, we obtain the probability of hitting the boundary $\partial U$ at the point $(\bar η,\bar α)$. For $\barη \to \infty$ we derive the asymptotic Cauchy hitting distribution on $\partial \mathbb{H}^2$ and for small values of $η$ and $\bar η$ we obtain the classical Euclidean Poisson kernel. The exit probabilities $\mathbb{P}_z\{T_{η_1}<T_{η_2}\}$ from a hyperbolic annulus in $\mathbb{H}^2$ of radii $η_1$ and $η_2$ are derived and the transient behaviour of hyperbolic Brownian motion is considered. Similar probabilities are calculated also for a Brownian motion on the surface of the three dimensional sphere. For the hyperbolic half-space $\mathbb{H}^n$ we obtain the Poisson kernel of a ball in terms of a series involving Gegenbauer polynomials and hypergeometric functions. For small domains in $\mathbb{H}^n$ we obtain the $n$-dimensional Euclidean Poisson kernel. The exit probabilities from an annulus are derived also in the $n$-dimensional case.

math.PR↗

arXiv2011

Fractional diffusion equations and processes with randomly varying time

In this paper the solutions $u_ν=u_ν(x,t)$ to fractional diffusion equations of order $0<ν\leq 2$ are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order $ν=\frac{1}{2^n}$, $n\geq 1,$ we show that the solutions $u_{1/2^n}$ correspond to the distribution of the $n$-times iterated Brownian motion. For these processes the distributions of the maximum and of the sojourn time are explicitly given. The case of fractional equations of order $ν=\frac{2}{3^n}$, $n\geq 1,$ is also investigated and related to Brownian motion and processes with densities expressed in terms of Airy functions. In the general case we show that $u_ν$ coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions $u_ν$ and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.

math.PR↗

arXiv2011

On a fractional linear birth--death process

In this paper, we introduce and examine a fractional linear birth--death process $N_ν(t)$, $t>0$, whose fractionality is obtained by replacing the time derivative with a fractional derivative in the system of difference-differential equations governing the state probabilities $p_k^ν(t)$, $t>0$, $k\geq0$. We present a subordination relationship connecting $N_ν(t)$, $t>0$, with the classical birth--death process $N(t)$, $t>0$, by means of the time process $T_{2ν}(t)$, $t>0$, whose distribution is related to a time-fractional diffusion equation. We obtain explicit formulas for the extinction probability $p_0^ν(t)$ and the state probabilities $p_k^ν(t)$, $t>0$, $k\geq1$, in the three relevant cases $λ>μ$, $λ<μ$, $λ=μ$ (where $λ$ and $μ$ are, respectively, the birth and death rates) and discuss their behaviour in specific situations. We highlight the connection of the fractional linear birth--death process with the fractional pure birth process. Finally, the mean values $\mathbb{E}N_ν(t)$ and $\operatorname {\mathbb{V}ar}N_ν(t)$ are derived and analyzed.

math.PR↗

arXiv2011

Fractional pure birth processes

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 $\mathcal{N}_ν(t)$ 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 $\mathcal{N}_ν(t)=\mathcal{N}(T_{2ν}(t))$, 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.

math.PR↗

arXiv2010

Equations of Mathematical Physics and Compositions of Brownian and Cauchy processes

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.

math.PR↗

arXiv2010

Composition of processes and related partial differential equations

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.

math.PR↗

arXiv2010

Bessel processes and hyperbolic Brownian motions stopped at different random times

Iterated Bessel processes R^γ(t), t>0, γ>0 and their counterparts on hyperbolic spaces, i.e. hyperbolic Brownian motions B^{hp}(t), t>0 are examined and their probability laws derived. The higher-order partial differential equations governing the distributions of I_R(t)=_1R^γ(_2R^γ(t)), t>0 and J_R(t) =_1R^γ(|_2R^γ(t)|^2), t>0 are obtained and discussed. Processes of the form R^γ(T_t), t>0, B^{hp}(T_t), t>0 where T_t=\inf{s: B(s)=t} are examined and numerous probability laws derived, including the Student law, the arcsin laws (also their asymmetric versions), the Lamperti distribution of the ratio of independent positively skewed stable random variables and others. For the process R^γ(T^μ_t), t>0 (where T^μ_t = \inf{s: B^μ(s)=t} and B^μis a Brownian motion with drift μ) the explicit probability law and the governing equation are obtained. For the hyperbolic Brownian motions on the Poincaré half-spaces H^+_2, H^+_3 we study B^{hp}(T_t), t>0 and the corresponding governing equation. Iterated processes are useful in modelling motions of particles on fractures idealized as Bessel processes (in Euclidean spaces) or as hyperbolic Brownian motions (in non-Euclidean spaces).

math.PR↗

arXiv2010

Some Darling-Siegert relationships connected with random flights

We derive in detail four important results on integrals of Bessel functions from which three combinatorial identities are extracted. We present the probabilistic interpretation of these identities in terms of different types of random walks, including asymmetric ones. This work extends the results of a previous paper concerning the Darling-Siegert interpretation of similar formulas emerging in the analysis of random flights.

math.PR↗

arXiv2009

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

We consider some fractional extensions of the recursive differential equation governing the Poisson process, by introducing combinations of different fractional time-derivatives. We show that the so-called "Generalized Mittag-Leffler functions" (introduced by Prabhakar [20]) 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 tending to infinite. On the other hand, near the origin the behavior of the law of the interarrival times drastically changes for the parameter fractional parameter varying in the interval (0,1). For integer values of the parameter, these models can be viewed as a higher-order Poisson processes, connected with the standard case by simple and explict relationships.

math.PR↗