114 research outputs found
Random-time processes governed by differential equations of fractional distributed order
We analyze here different types of fractional differential equations, under
the assumption that their fractional order is random\ with
probability density We start by considering the fractional extension
of the recursive equation governing the homogeneous Poisson process
\ We prove that, for a particular (discrete) choice of , it
leads to a process with random time, defined as The distribution of the
random time argument can be
expressed, for any fixed , in terms of convolutions of stable-laws. The new
process is itself a renewal and
can be shown to be a Cox process. Moreover we prove that the survival
probability of , as well as its
probability generating function, are solution to the so-called fractional
relaxation equation of distributed order (see \cite{Vib}%).
In view of the previous results it is natural to consider diffusion-type
fractional equations of distributed order. We present here an approach to their
solutions in terms of composition of the Brownian motion with the
random time . We thus provide an
alternative to the constructions presented in Mainardi and Pagnini
\cite{mapagn} and in Chechkin et al. \cite{che1}, at least in the double-order
case.Comment: 26 page
Hitting spheres on hyperbolic spaces
For a hyperbolic Brownian motion on the Poincar\'e half-plane ,
starting from a point of hyperbolic coordinates inside a
hyperbolic disc of radius , we obtain the probability of
hitting the boundary at the point . For
we derive the asymptotic Cauchy hitting distribution on
and for small values of and we
obtain the classical Euclidean Poisson kernel. The exit probabilities
from a hyperbolic annulus in
of radii and 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 we obtain the Poisson kernel of
a ball in terms of a series involving Gegenbauer polynomials and hypergeometric
functions. For small domains in we obtain the -dimensional
Euclidean Poisson kernel. The exit probabilities from an annulus are derived
also in the -dimensional case
Probabilistic analysis of the telegrapher's process with drift by means of relativistic transformations
The telegrapher's process with drift is here examined and its distribution is
obtained by applying the Lorentz transformation. The related characteristic function as well as the distribution are also derived by solving an initial
value problem for the generalized telegraph equation
Ergodicity breaking in strong and network-forming glassy system
The temperature dependence of the non-ergodicity factor of vitreous GeO,
, as deduced from elastic and quasi-elastic neutron scattering
experiments, is analyzed. The data are collected in a wide range of
temperatures from the glassy phase, up to the glass transition temperature, and
well above into the undercooled liquid state. Notwithstanding the investigated
system is classified as prototype of strong glass, it is found that the
temperature- and the -behavior of follow some of the predictions
of Mode Coupling Theory. The experimental data support the hypothesis of the
existence of an ergodic to non-ergodic transition occurring also in network
forming glassy systems
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 ) of the Fresnel equation is also performed and, in
detail, some specific cases, like , 1/3, 2/3, are analyzed. By means
of the fundamental solution of the Fresnel equation, a pseudo-process ,
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 is investigated by applying the methods of
planar orthogonally reflecting Brownian motion within . The composition of
F with reflecting Brownian motion yields the law of biquadratic heat
equation while the composition of with the first passage time of
produces a genuine probability law strictly connected with the Cauchy process.Comment: 33 pages,8 figure
Randomly Stopped Nonlinear Fractional Birth Processes
We present and analyse the nonlinear classical pure birth process
\mathpzc{N} (t), , and the fractional pure birth process
\mathpzc{N}^\nu (t), , subordinated to various random times, namely the
first-passage time of the standard Brownian motion , , the
-stable subordinator \mathpzc{S}^\alpha(t), , and
others. For all of them we derive the state probability distribution , 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),
, and its fractional counterpart \hat{\mathpzc{N}}^\nu (t), 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}^\nu(t) have been examined in the last part of the
paper. In particular, the processes \mathpzc{N}^\nu(T_t),
\mathpzc{N}^\nu(\mathpzc{S}^\alpha(t)), \mathpzc{N}^\nu(T_{2\nu}(t)), have
been analysed, where , , is a process related to fractional
diffusion equations. Also the related process
\mathpzc{N}(\mathpzc{S}^\alpha({T_{2\nu}(t)})) is investigated and compared
with \mathpzc{N}(T_{2\nu}(\mathpzc{S}^\alpha(t))) = \mathpzc{N}^\nu
(\mathpzc{S}^\alpha(t)). As a byproduct of our analysis, some formulae
relating Mittag--Leffler functions are obtained
On random flights with non-uniformly distributed directions
This paper deals with a new class of random flights defined in the real space characterized
by non-uniform probability distributions on the multidimensional sphere. These
random motions differ from similar models appeared in literature which take
directions according to the uniform law. The family of angular probability
distributions introduced in this paper depends on a parameter which
gives the level of drift of the motion. Furthermore, we assume that the number
of changes of direction performed by the random flight is fixed. The time
lengths between two consecutive changes of orientation have joint probability
distribution given by a Dirichlet density function.
The analysis of is not an easy task, because it
involves the calculation of integrals which are not always solvable. Therefore,
we analyze the random flight obtained as
projection onto the lower spaces of the original random
motion in . Then we get the probability distribution of
Although, in its general framework, the analysis of is very complicated, for some values of , we can provide
some results on the process. Indeed, for , we obtain the characteristic
function of the random flight moving in . Furthermore, by
inverting the characteristic function, we are able to give the analytic form
(up to some constants) of the probability distribution of Comment: 28 pages, 3 figure
Customer experience challenges: bringing together digital, physical and social realms
This paper discusses important societal issues, such as individual and societal needs for privacy, security, and transparency. It sets out potential avenues for service innovation in these areas
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