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.Comment: 32 page

Higher-order Laplace equations and hyper-Cauchy distributions

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.Comment: 20 pages; 5 figures; Journal of Theoretical Probabilit

The distribution of the local time for "pseudo-processes" and its connections with fractional diffusion equations

We prove that the pseudoprocesses governed by heat-type equations of order $n\geq2$ have a local time in zero (denoted by $L_{0}^{n}(t)$) whose distribution coincides with the folded fundamental solution of a fractional diffusion equation of order $2(n-1)/n,n\geq2$: The distribution of $L_{0}^{n}(t)$ is also expressed in terms of stable laws of order $n/(n-1)$ and their form is analyzed. Furthermore, it is proved that the distribution of $L_{0}^{n}(t)$ is connected with a wave equation as $n\rightarrow\infty$. The distribution of the local time in zero for the pseudoprocess related to the Myiamoto’s equation is also derived and examined together with the corresponding telegraph-type fractional equation

Bessel processes and hyperbolic Brownian motions stopped at different random times

Iterated Bessel processes R(gamma) (t), t > 0, gamma > 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) = R(1)(gamma 1)(R(2)(gamma 2)(t)), t > 0 and J(R)(t) = R(1)(gamma 1) (R(2)(gamma 2) (t)(2)), t > 0 are obtained and discussed. Processes of the form R(gamma) (T(t)), t > 0, B(hp) (T(t)), t > 0 where T(t) = inf{s >= 0 : B(s) = t} are examined and numerous probability laws derived, including the Student law, the arcsine laws (also their asymmetric versions), the Lamperti distribution of the ratio of independent positively skewed stable random variables and others. For the random variable R(gamma)(T(t)(mu)), t > 0 (where T(t)(mu) = inf{s >= 0 : B(mu) (s) = t} and B(mu) is a Brownian motion with drift mu), the explicit probability law and the governing equation are obtained. For the hyperbolic Brownian motions on the Poincare half-spaces H(2)(+), H(3)(+) (of respective dimensions 2, 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). Crown Copyright (C) 2010 Published by Elsevier B.V. All rights reserved

On the Integral of Fractional Poisson Processes

In this paper we consider the Riemann--Liouville fractional integral $\mathcal{N}^{\alpha,\nu}(t)= \frac{1}{\Gamma(\alpha)} \int_0^t (t-s)^{\alpha-1}N^\nu(s) \, \mathrm ds$, where $N^\nu(t)$, $t \ge 0$, is a fractional Poisson process of order $\nu \in (0,1]$, and $\alpha > 0$. We give the explicit bivariate distribution $\Pr \{N^\nu(s)=k, N^\nu(t)=r \}$, for $t \ge s$, $r \ge k$, the mean $\mathbb{E}\, \mathcal{N}^{\alpha,\nu}(t)$ and the variance $\mathbb{V}\text{ar}\, \mathcal{N}^{\alpha,\nu}(t)$. We study the process $\mathcal{N}^{\alpha,1}(t)$ for which we are able to produce explicit results for the conditional and absolute variances and means. Much more involved results on $\mathcal{N}^{1,1}(t)$ are presented in the last section where also distributional properties of the integrated Poisson process (including the representation as random sums) is derived. The integral of powers of the Poisson process is examined and its connections with generalised harmonic numbers is discussed

Fractional diffusion equations and processes with randomly varying time

In this paper the solutions $u_{\nu}=u_{\nu}(x,t)$ to fractional diffusion equations of order $0<\nu \leq 2$ are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order $\nu =\frac{1}{2^n}$, $n\geq 1,$ 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 $\nu =\frac{2}{3^n}$, $n\geq 1,$ 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 $u_{\nu}$ coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions $u_{\nu}$ and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.Comment: Published in at http://dx.doi.org/10.1214/08-AOP401 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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

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 $\mathcal{N}_{\nu}(t)$ 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 $\mathcal{N}_{\nu}(t)=\mathcal{N}(T_{2\nu}(t))$, where $\mathcal{N}(t)$ is the classical generalized birth process and $T_{2\nu}(t)$ 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.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ235 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Pseudoprocesses related to space-fractional higher-order heat-type equations

In this paper we construct pseudo random walks (symmetric and asymmetric) which converge in law to compositions of pseudoprocesses stopped at stable subordinators. We find the higher-order space-fractional heat-type equations whose fundamental solutions coincide with the law of the limiting pseudoprocesses. The fractional equations involve either Riesz operators or their Feller asymmetric counterparts. The main result of this paper is the derivation of pseudoprocesses whose law is governed by heat-type equations of real-valued order $\gamma>2$. The classical pseudoprocesses are very special cases of those investigated here