175 research outputs found
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 have a local time in zero (denoted by
) whose distribution coincides with the folded
fundamental solution of a fractional diffusion equation of order
: The distribution of is also
expressed in terms of stable laws of order and their
form is analyzed. Furthermore, it is proved that the distribution
of is connected with a wave equation as
. 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
, where , , is a
fractional Poisson process of order , and . We give
the explicit bivariate distribution , for , , the mean and the
variance . We study the
process for which we are able to produce explicit
results for the conditional and absolute variances and means. Much more
involved results on 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 to fractional diffusion
equations of order are analyzed and interpreted as densities of
the composition of various types of stochastic processes. For the fractional
equations of order , we show that the solutions
correspond to the distribution of the -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 , 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 coincides with the distribution of Brownian
motion with random time or of different processes with a Brownian time. The
interplay between the solutions 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
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 of individuals at an
arbitrary time . We also present an interesting representation for the
number of individuals at time , in the form of the subordination relation
, where is the
classical generalized birth process and 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 . The classical pseudoprocesses are very special
cases of those investigated here
On a fractional linear birth--death process
In this paper, we introduce and examine a fractional linear birth--death
process , , whose fractionality is obtained by replacing the
time derivative with a fractional derivative in the system of
difference-differential equations governing the state probabilities
, , . We present a subordination relationship
connecting , , with the classical birth--death process ,
, by means of the time process , , whose distribution is
related to a time-fractional diffusion equation. We obtain explicit formulas
for the extinction probability and the state probabilities
, , , in the three relevant cases ,
, (where and 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
and are derived and analyzed.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ263 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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