Explicit solutions to fractional diffusion equations via Generalized Gamma Convolution

In this paper we deal with Mellin convolution of generalized Gamma densities which leads to integrals of modified Bessel functions of the second kind. Such convolutions allow us to explicitly write the solutions of the time-fractional diffusion equations involving the adjoint operators of a square Bessel process and a Bessel process

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

Fractional Poisson process with random drift

We study the connection between PDEs and L\'{e}vy processes running with clocks given by time-changed Poisson processes with stochastic drifts. The random times we deal with are therefore given by time-changed Poissonian jumps related to some Frobenious-Perron operators $K$ associated to random translations. Moreover, we also consider their hitting times as a random clock. Thus, we study processes driven by equations involving time-fractional operators (modelling memory) and fractional powers of the difference operator $I-K$ (modelling jumps). For this large class of processes we also provide, in some cases, the explicit representation of the transition probability laws. To this aim, we show that a special role is played by the translation operator associated to the representation of the Poisson semigroup

Delayed and rushed motions through time change

We introduce a definition of delayed and rushed processes in terms of lifetimes of base processes and time-changed base processes. Then, we consider time changes given by subordinators and their inverse processes. Our analysis shows that, quite surprisingly, time-changing with inverse subordinators does not necessarily imply delay of the base process. Moreover, time-changing with subordinators does not necessarily imply rushed base process.Comment: to appear on ALEA - Latin American Journal of Probability and Mathematical Statistic

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