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

In this paper we construct models obtained by suitably combining Brownian motions and telegraphs in such a way that their transition functions satisfy higher-order parabolic or hyperbolic equations of different types. Equations with time-varying coefficients are also derived by considering processes endowed either with drift or with suitable modifications of their structure. Finally the distribution of the maximum of the iterated Brownian motion (along with some other properties) is presented.

Iterated processes and their applications to higher order differential equations

Random motions governed by third-order equations

Let v(x, t) denote the displacement of an infinitely long, idealized string performing damped vibrations caused by white noise. Upper and lower bounds for the distribution of max s v(x, s) and max x v(x, t) are presented. The results are obtained by adapting Levy-type inequalities and exploiting a connection of v(x, t) with the Ornstein-Uhlenbeck process through Slepian's theorem. The case of forced-damped vibrations is also analysed. Finally, a section is devoted to the case of a semi-infinite string performing damped vibrations.

Damped vibrations excited by white noise

This paper is devoted to the analysis of random motions on the line and in the space R^d (d > 1) performed at finite velocity and governed by a non-homogeneous Poisson process with rate \lambda(t). The explicit distributions p(x,t) of the position of the randomly moving particles are obtained solving initial-value problems for the Euler- Poisson-Darboux equation when \lambda(t) = \alpha/t, t > 0. We consider also the case where \lambda(t) = \lambda coth \lambda t and \lambda(t) = \lambda tanh \lambda t, where some Riccati differential equations emerge and the explicit distributions are obtained for d = 1. We also examine planar random motions with random velocities by projecting random flights in R^d onto the plane. Finally the case of planar motions with four orthogonal directions is considered and the corresponding higher-order equations with time-varying coefficients obtained.

Random flights related to the Euler-Poisson-Darboux equation

We obtain the explicit distribution of the position of a particle performing a cyclic, minimal, random motion with constant velocity c in Rn. The n+1 possible directions of motion as well as the support of the distribution form a regular hyperpolyhedron (the first one having constant sides and the other expanding with time t), the geometrical features of which are here investigated.

The distribution is obtained by using order statistics and is expressed in terms of hyper-Bessel functions of order n+1. These distributions are proved to be connected with (n+1)th order p.d.e. which can be reduced to Bessel equations of higher order.

Some properties of the distributions obtained are examined. This research has been inspired by a conjecture formulated in Orsingher and Sommella [E. Orsingher, A.M. Sommella, A cyclic random motion in R3 with four directions and finite velocity, Stochastics Stochastics Rep. 76 (2) (2004) 113–133] which is here proved to be false.

/pdf/minimal-cyclic-random-motion-in-rn-and-hyper-bessel-3lvmc24lht.pdf

Enzo Orsingher

Papers

Iterated processes and their applications to higher order differential equations

Random motions governed by third-order equations

Damped vibrations excited by white noise

Random flights related to the Euler-Poisson-Darboux equation

Minimal cyclic random motion in Rn and hyper-bessel functions