Showing papers by "Enzo Orsingher published in 2011"

Fractional diffusion equations and processes with randomly varying time

Abstract: In this paper the solutions $u_{ u}=u_{ u}(x,t)$ to fractional diffusion equations of order $0< u \leq 2$ are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order $ u =\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 $ u =\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 $u_{ u}$ coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions $u_{ u}$ and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.

On a fractional linear birth-death process

Abstract: 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 $p_k^ν(t), t>0, k≥0$. 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 $p_k^ν(t), t>0, k≥1$, 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 $\mathbb{E}N_{ u}(t)$ and $\operatorname{\mathbb{V}ar}N_{ u}(t)$ are derived and analyzed.

On a fractional linear birth--death process

Abstract: In this paper, we introduce and examine a fractional linear birth--death process $N_{ u}(t)$, $t>0$, whose fractionality is obtained by replacing the time derivative with a fractional derivative in the system of difference-differential equations governing the state probabilities $p_k^{ u}(t)$, $t>0$, $k\geq0$. We present a subordination relationship connecting $N_{ u}(t)$, $t>0$, with the classical birth--death process $N(t)$, $t>0$, by means of the time process $T_{2 u}(t)$, $t>0$, whose distribution is related to a time-fractional diffusion equation. We obtain explicit formulas for the extinction probability $p_0^{ u}(t)$ and the state probabilities $p_k^{ u}(t)$, $t>0$, $k\geq1$, in the three relevant cases $\lambda>\mu$, $\lambda<\mu$, $\lambda=\mu$ (where $\lambda$ and $\mu$ 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 $\mathbb{E}N_{ u}(t)$ and $\operatorname {\mathbb{V}ar}N_{ u}(t)$ are derived and analyzed.

Vibrations and Fractional Vibrations of Rods, Plates and Fresnel Pseudo-Processes

Abstract: 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 equation of vibrations of plates is considered and the case of circular vibrating disks CR is investigated by applying the methods of planar orthogonally reflecting Brownian motion within CR. 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 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 Tt of B produces a genuine probability law strictly connected with the Cauchy process.

Composition of Processes and Related Partial Differential Equations

Abstract: In this paper various types of compositions involving independent fractional Brownian motions \(B^{j}_{H_{j}}(t)\), t>0, j=1,2, are examined.

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 ℝ d , like, for example, (B 1(|C(t)|),…, B d (|C(t)|)) and (C 1(|C(t)|),…, C d (|C(t)|)), 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.

Flying randomly in $\mathbb{R}^d$ with Dirichlet displacements

Abstract: Random flights in $\mathbb{R}^d,d\geq 2,$ with Dirichlet-distributed displacements and uniformly distributed orientation are analyzed. The explicit characteristic functions of the position $\underline{\bf X}_d(t),\,t>0,$ when the number of changes of direction is fixed are obtained. The probability distributions are derived by inverting the characteristic functions for all dimensions $d$ of $\mathbb{R}^d$ and many properties of the probabilistic structure of $\underline{\bf X}_d(t),t>0,$ are examined. If the number of changes of direction is randomized by means of a fractional Poisson process, we are able to obtain explicit distributions for $P\{\underline{\bf X}_d(t)\in d\underline{\bf x}_d\}$ for all $d\geq 2$. A Section is devoted to random flights in $\mathbb{R}^3$ where the general results are discussed. The existing literature is compared with the results of this paper where in our view the classical Pearson's problem of random flights is resolved by suitably randomizing the step lengths. The random flights where changes of direction are governed by a homogeneous Poisson process are analyzed and compared with the model of Dirichlet-distributed displacements of this work.

Travelling Randomly on the Poincar\'e Half-Plane with a Pythagorean Compass

Abstract: A random motion on the Poincar\'e half-plane is studied. A particle runs on the geodesic lines changing direction at Poisson-paced times. The hyperbolic distance is analyzed, also in the case where returns to the starting point are admitted. The main results concern the mean hyperbolic distance (and also the conditional mean distance) in all versions of the motion envisaged. Also an analogous motion on orthogonal circles of the sphere is examined and the evolution of the mean distance from the starting point is investigated.

Compositions, Random Sums and Continued Random Fractions of Poisson and Fractional Poisson Processes

Abstract: In this paper we consider the relation between random sums and compositions of different processes. In particular, for independent Poisson processes $N_\alpha(t)$, $N_\beta(t)$, $t>0$, we show that $N_\alpha(N_\beta(t)) \overset{\text{d}}{=} \sum_{j=1}^{N_\beta(t)} X_j$, where the $X_j$s are Poisson random variables. We present a series of similar cases, the most general of which is the one in which the outer process is Poisson and the inner one is a nonlinear fractional birth process. We highlight generalisations of these results where the external process is infinitely divisible. A section of the paper concerns compositions of the form $N_\alpha(\tau_k^ u)$, $ u \in (0,1]$, where $\tau_k^ u$ is the inverse of the fractional Poisson process, and we show how these compositions can be represented as random sums. Furthermore we study compositions of the form $\Theta(N(t))$, $t>0$, which can be represented as random products. The last section is devoted to studying continued fractions of Cauchy random variables with a Poisson number of levels. We evaluate the exact distribution and derive the scale parameter in terms of ratios of Fibonacci numbers.

Randomly Stopped Nonlinear Fractional Birth Processes

Abstract: We present and analyse the nonlinear classical pure birth process $\mathpzc{N} (t)$, $t>0$, and the fractional pure birth process $\mathpzc{N}^ u (t)$, $t>0$, subordinated to various random times, namely the first-passage time $T_t$ of the standard Brownian motion $B(t)$, $t>0$, the $\alpha$-stable subordinator $\mathpzc{S}^\alpha(t)$, $\alpha \in (0,1)$, 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 $\hat{\mathpzc{N}}^ u (t)$, $t>0$ 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 $\mathpzc{N}^ u(t)$ have been examined in the last part of the paper. In particular, the processes $\mathpzc{N}^ u(T_t)$, $\mathpzc{N}^ u(\mathpzc{S}^\alpha(t))$, $\mathpzc{N}^ u(T_{2 u}(t))$, have been analysed, where $T_{2 u}(t)$, $t>0$, is a process related to fractional diffusion equations. Also the related process $\mathpzc{N}(\mathpzc{S}^\alpha({T_{2 u}(t)}))$ is investigated and compared with $\mathpzc{N}(T_{2 u}(\mathpzc{S}^\alpha(t))) = \mathpzc{N}^ u (\mathpzc{S}^\alpha(t))$. As a byproduct of our analysis, some formulae relating Mittag--Leffler functions are obtained.

Angular Processes Related to Cauchy Random Walks

Abstract: We study the angular process related to random walks in the Euclidean and in the non-Euclidean space where steps are Cauchy distributed. This leads to different types of nonlinear transformations of Cauchy random variables which preserve the Cauchy density. We give the explicit form of these distributions for all combinations of the scale and the location parameters. Continued fractions involving Cauchy random variables are analyzed. It is shown that the n-stage random variables are still Cauchy distributed with parameters related to Fibonacci numbers. This permits us to show the convergence in distribution of the sequence to the golden ratio.

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

Abstract: 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 $ u$) of the Fresnel equation is also performed and, in detail, some specific cases, like $ u=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.

Hitting spheres on hyperbolic spaces

Abstract: For a hyperbolic Brownian motion on the Poincare half-plane $\mathbb{H}^2$, starting from a point of hyperbolic coordinates $z=(\eta, \alpha)$ inside a hyperbolic disc $U$ of radius $\bar{\eta}$, we obtain the probability of hitting the boundary $\partial U$ at the point $(\bar \eta,\bar \alpha)$. For $\bar{\eta} \to \infty$ we derive the asymptotic Cauchy hitting distribution on $\partial \mathbb{H}^2$ and for small values of $\eta$ and $\bar \eta$ we obtain the classical Euclidean Poisson kernel. The exit probabilities $\mathbb{P}_z\{T_{\eta_1}

Higher-order Laplace equations and hyper-Cauchy distributions

Abstract: In this paper we introduce new distributions which are solutions of higher-order Laplace equations. It is proved that their densities can be obtained by folding and symmetrizing Cauchy distributions. Another class of probability laws related to higher-order Laplace equations is obtained by composing pseudo-processes with positively-skewed Cauchy distributions which produce asymmetric Cauchy densities in the odd-order case. A special attention is devoted to the third-order Laplace equation where the connection between the Cauchy distribution and the Airy functions is obtained and analyzed.

Angular processes related to Cauchy random walks

Abstract: We study the angular process related to random walks in the Euclidean and in the non-Euclidean space where steps are Cauchy distributed. This leads to different types of non-linear transformations of Cauchy random variables which preserve the Cauchy density. We give the explicit form of these distributions for all combinations of the scale and the location parameters. Continued fractions involving Cauchy random variables are analyzed. It is shown that the $n$-stage random variables are still Cauchy distributed with parameters related to Fibonacci numbers. This permits us to show the convergence in distribution of the sequence to the golden ratio.