Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

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Table Of Integrals Series And Products

Fractional Differential Equations

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

We study the fundamental solutions to time-fractional telegraph equations of order 2α. We are able to obtain the Fourier transform of the solutions for any α and to give a representation of their inverse, in terms of stable densities. For the special case α=1/2, we can show that the fundamental solution is the distribution of a telegraph process with Brownian time. In a special case, this becomes the density of the iterated Brownian motion, which is therefore the fundamental solution to a fractional diffusion equation of order 1/2 with respect to time.

Time-fractional telegraph equations and telegraph processes with brownian time

We present three different fractional versions of the Poisson process and some related results concerning the distribution of order statistics and the compound Poisson process. The main version is constructed by considering the difference-differential equation governing the distribution of the standard Poisson process, $ N(t),t>0$, and by replacing the time-derivative with the fractional Dzerbayshan-Caputo derivative of order $\nu\in(0,1]$. For this process, denoted by $\mathcal{N}_\nu(t),t>0,$ we obtain an interesting probabilistic representation in terms of a composition of the standard Poisson process with a random time, of the form $\mathcal{N}_\nu(t)= N(\mathcal{T}_{2\nu}(t)),$ $t>0$. The time argument $\mathcal{T}_{2\nu }(t),t>0$, is itself a random process whose distribution is related to the fractional diffusion equation. We also construct a planar random motion described by a particle moving at finite velocity and changing direction at times spaced by the fractional Poisson process $\mathcal{N}_\nu.$ For this model we obtain the distributions of the random vector representing the position at time $t$, under the condition of a fixed number of events and in the unconditional case. For some specific values of $\nu\in(0,1]$ we show that the random position has a Brownian behavior (for $\nu =1/2$) or a cylindrical-wave structure (for $\nu =1$).

/pdf/fractional-poisson-processes-and-related-planar-random-55gvqqe7pi.pdf

Fractional Poisson processes and related planar random motions

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

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.

/pdf/fractional-diffusion-equations-and-processes-with-randomly-3jd2ddhcud.pdf

Fractional diffusion equations and processes with randomly varying time.

https://www.dss.uniroma1.it/it/system/files/pubblicazioni/16-rt-orsingher-13-2011.pdf

Enzo Orsingher

Papers

Time-fractional telegraph equations and telegraph processes with brownian time

Fractional Poisson processes and related planar random motions

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

Fractional diffusion equations and processes with randomly varying time.

The space-fractional Poisson process