Contacts
mura@bo.infn.it
anto.mura@gmail.com
Antonio Mura phd thesis @ fracalmo.org
Contacts
mura@bo.infn.it
anto.mura@gmail.com
Presentata da:
Relatore:
Antonio
Mura
Prof. Francesco
Mainardi
Coordinatorne:
Prof. Fabio Ortolani
This work provides a forward step in the study and
comprehension of the relationships between stochastic processes and a certain
class of integral-partial differential equation, which can be used in order to
model anomalous diffusion and transport in statistical physics. In the first
part, we brought the reader through the fundamental notions of
probability and stochastic processes, stochastic
integration and stochastic differential equations as well. In
particular, within the study of H-sssi processes, we focused on
fractional Brownian motion (fBm) and
its discrete-time increment process, the fractional Gaussian noise
(fGn), which provide examples of
non-Markovian Gaussian processes.
The fGn, together with
stationary FARIMA processes, is widely used in the modeling and estimation of
long-memory, or long-range dependence
(LRD). Time series manifesting long-range
dependence, are often observed in nature especially in physics, meteorology,
climatology, but also in hydrology, geophysics, economy and many others. We
deepely studied LRD, giving many real data examples, providing statistical
analysis and introducing parametric methods of estimation. Then, we introduced
the theory of fractional integrals and derivatives, which indeed turns
out to be very appropriate for studying and modeling systems with long-memory
properties. After having introduced the basics concepts, we provided many
examples and applications. For instance, we investigated the relaxation equation
with distributed order time-fractional derivatives, which describes
models characterized by a strong memory component and can be used to model
relaxation in complex systems, which deviates from the classical exponential
Debye pattern. Then, we focused in the study of generalizations of the standard
diffusion equation, by passing through the preliminary study of the fractional
forward drift equation. Such generalizations have been obtained by using
fractional integrals and derivatives of distributed orders. In order to find a
connection between the anomalous diffusion described by these equations and the
long-range dependence, we introduced and studied the generalized grey
Brownian motion (ggBm), which is actually
a parametric class of H-sssi processes, which have indeed
marginal probability density function evolving in time according to a partial
integro-differential equation of fractional type. The ggBm is of course
Non-Markovian. All around the work, we have remarked many times that, starting
from a master equation of a probability density function
f(x,t), it is always possible to
define an equivalence class of stochastic processes with the same marginal
density function f(x,t). All these
processes provide suitable stochastic models for the starting equation.
Studying the ggBm, we just focused on a subclass made up of processes with
stationary increments. The ggBm has been defined canonically in
the so called grey noise space. However, we have been able to provide a
characterization notwithstanding the underlying probability space. We also
pointed out that that the generalized grey Brownian motion is a direct
generalization of a Gaussian process and in particular it generalizes Brownian
motion and fractional Brownian motion as well. Finally, we introduced and
analyzed a more general class of diffusion type equations related to certain
non-Markovian stochastic processes. We started from the forward drift equation,
which have been made non-local in time by the introduction of a suitable chosen
memory kernel
K(t). The
resulting non-Markovian equation has been interpreted in a natural way as the
evolution equation of the marginal density function of a random time process
l(t). We
then consider the subordinated process
Y(t)=X(l(t))
where X(t)
is a Markovian diffusion. The corresponding time-evolution of the marginal
density function of
Y(t) is
governed by a non-Markovian Fokker-Planck equation which involves the same
memory kernel
K(t).
We developed several applications and derived the exact solutions. Moreover, we
considered different stochastic models for the given equations, providing path
simulations.