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 underline 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 Brownain motion and fractional Brownain 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.
| Identifer | oai:union.ndltd.org:unibo.it/oai:amsdottorato.cib.unibo.it:846 |
| Date | 12 June 2008 |
| Creators | Mura, Antonio <1978> |
| Contributors | Mainardi, Francesco |
| Publisher | Alma Mater Studiorum - Università di Bologna |
| Source Sets | Università di Bologna |
| Language | English |
| Detected Language | English |
| Type | Doctoral Thesis, PeerReviewed |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
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