Enzo Orsingher

Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff's laws

In this paper we derive the explicit form of the probability law and of the associated flow function of a random motion governed by the telegraph equation. Connections of this law with the transition function of Brownian motion are explored. Lower bounds for the distribution of its maximum are obtained and some particular distributions of its maximum, conditioned by the number of velocity reversals, are presented. Finally some versions of motion admitting annihilation are proven to be connected with Kirchoffs laws of electrical circuits.

Fractional diffusion equations and processes with randomly varying time

In this paper the solutions u ν = u ν (x, t) to fractional diffusion equations of order 0 < v ≤ 2 are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order v = 1 2 n , n ≥ 1, we show that the solutions u 1/2 n correspond to the distribution of the n-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 v = 2 3 n , n ≥ 1, 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 u ν coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions u ν and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.

THE SPACE-FRACTIONAL TELEGRAPH EQUATION AND THE RELATED FRACTIONAL TELEGRAPH PROCESS

The space-fractional telegraph equation is analyzed and the Fourier transform of its fundamental solution is obtained and discussed. A symmetric process with discontinuous trajectories, whose transition function satisfies the space-fractional telegraph equation, is presented. Its limiting behaviour and the connection with symmetric stable processes is also examined.

Some universal limits for nonhomogeneous birth and death processes

In this paper we consider nonhomogeneous birth and death processes (BDP) with periodic rates. Two important parameters are studied, which are helpful to describe a nonhomogeneous BDP X = X(t), t≥ 0: the limiting mean value (namely, the mean length of the queue at a given time t) and the double mean (i.e. the mean length of the queue for the whole duration of the BDP). We find conditions of existence of the means and determine bounds for their values, involving also the truncated BDP XN. Finally we present some examples where these bounds are used in order to approximate the double mean.

Composition of Stochastic Processes Governed by Higher-Order Parabolic and Hyperbolic Equations

We consider compositions of stochastic processes that are governed by higherorder partial differential equations. The processes studied include compositions of Brownian motions, stable-like processes with Brownian time, Brownian motion whose time is an integrated telegraph process, and an iterated integrated telegraph process. The governing higher-order equations that are obtained are shown to be either of the usual parabolic type or, as in the last example, of hyperbolic type.

Probabilistic analysis of the telegrapher's process with drift by means of relativistic transformations

The telegraphers 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.

The space-fractional Poisson process

In this paper, we introduce the space-fractional Poisson process whose state probabilities pkα(t), t≥0, α∈(0,1], are governed by the equations (d/dt)pkα(t)=−λα(1−B)αpkα(t), where (1−B)α is the fractional difference operator found in the time series analysis. We explicitly obtain the distributions pkα(t), the probability generating functions Gα(u,t), which are also expressed as distributions of the minimum of i.i.d. uniform random variables. The comparison with the time-fractional Poisson process is investigated and finally, we arrive at the more general space–time-fractional Poisson process of which we give the explicit distribution.

Hyperbolic equations arising in random models

In this paper, models connected with hyperbolic partial differential equations are analysed. In particular a planar motion whose probability law is a solution of the equation of telegraphy is studied. Also the motion of a fluid-driven particle is considered and its probability distribution explicitly obtained. Linear transformations of relativistic nature are also analysed.

The arc-sine law and its analogs for processes governed by signed and complex measures

The question whether the classical arc-sine law of Paul Levy for the proportion of time spent by a Brownian particle on the positive half-line can be extended to generalized higher-order processes governed by signed and complex measures is studied. Both even and odd-order processes are considered, corresponding to heat-type equations with both real and imaginary coefficients. Finally, several mixtures of the earlier cases are analyzed as well.

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 N�(t) of individuals at an arbitrary time t. We also present an interesting representation for the number of individuals at time t, in the form of the subordination relation N�(t) = N(T2�(t)), where N(t) is the classical generalized birth process and T2�(t) 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.