Mirko D’Ovidio

Time-Changed Processes Governed by Space-Time Fractional Telegraph Equations

In this work, we construct compositions of vector processes of the form , t > 0, , β ∈ (0, 1], , whose distribution is related to space-time fractional n-dimensional telegraph equations. We present within a unifying framework the pde connections of n-dimensional isotropic stable processes S2βn whose random time is represented by the inverse , t > 0, of the superposition of independent positively skewed stable processes, , t > 0, (H2ν1, Hν2, independent stable subordinators). As special cases for n = 1, and β = 1, we examine the telegraph process T at Brownian time B ([14]) and establish the equality in distribution , t > 0. Furthermore the iterated Brownian motion ([2]) and the two-dimensional motion at finite velocity with a random time are investigated. For all these processes, we present their counterparts as Brownian motion at delayed stable-distributed time.

On the fractional counterpart of the higher-order equations

In this work, we study the solutions of some fractional higher-order equations. Special cases in which time-fractional derivatives take integer values are also examined and the explicit solutions are presented. Such solutions can be expressed by means of the transition laws of stable subordinators and their inverse processes. In particular, we establish connections between fractional and higher-order equations.

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 equation of vibrations of plates is considered and the case of circular vibrating disks CR is investigated by applying the methods of planar orthogonally reflecting Brownian motion within CR. The analysis of the fractional version (of order ν) of the Fresnel equation is also performed and, in detail, some specific cases, like ν=1/2, 1/3, 2/3, are analyzed. By means of the fundamental solution of the Fresnel equation, a pseudo-process F(t), t>0 with real sign-varying density is constructed and some of its properties examined. The composition of F with reflecting Brownian motion B yields the law of biquadratic heat equation while the composition of F with the first passage time Tt of B produces a genuine probability law strictly connected with the Cauchy 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 Bhp(t),t>0 are examined and their probability laws derived. The higher-order partial differential equations governing the distributions of and are obtained and discussed. Processes of the form R[gamma](Tt),t>0,Bhp(Tt), t>0 where 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 (where 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 , (of respective dimensions 2,3) we study Bhp(Tt),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).

Solutions of fractional logistic equations by Euler’s numbers

Abstract In this paper, we solve in the convergence set, the fractional logistic equation making use of Euler’s numbers. To our knowledge, the answer is still an open question. The key point is that the coefficients can be connected with Euler’s numbers, and then they can be explicitly given. The constrained of our approach is that the formula is not valid outside the convergence set. The idea of the proof consists to explore some analogies with logistic function and Euler’s numbers, and then to generalize them in the fractional case.

Spectral densities related to some fractional stochastic differential equations

In this paper we consider fractional higher-order stochastic differential equations of the form \begin{align*} \left( \mu + c_\alpha \frac{d^\alpha}{d(-t)^\alpha} \right)^\beta X(t) = \mathcal{E}(t) , \quad t\geq 0,\; \mu>0,\; \beta>0,\; \alpha \in (0,1) \cup \mathbb{N} \end{align*} where

From Sturm-Liouville problems to fractional and anomalous diffusions

\mathcal{E}(t)

Composition of Processes and Related Partial Differential Equations

is a Gaussian white noise. We derive stochastic processes satisfying the above equations of which we obtain explicitly the covariance functions and the spectral functions.