Abstract
We present some models of random motions with finite velocity in Euclidean spaces.
In R1 we discuss the telegraph process and give the main distributional results. We consider also the case of the asymmetric telegraph process and use relativistic transformations for its probabilistic analysis.
Random motions in the plane with infinite and with a finite number of possible directions of motion are discussed.
We then consider random flights in Rn where the particle performing the random motion changes direction at Poisson paced times and takes spherically uniform orientation at each change of direction.
Finally some fractional generalizations of these random motions are examined by applying the Mc. Bride theory of fractional powers of D’Alembert operators.
References
[1] A. De Gregorio, E. Orsingher, L. Sakhno. Motions with finite velocity analyzed with order statistics and differential equations, Theory of Probability and Mathematical Statistics, 71:63 – 79, 2005.
[2] A. Di Crescenzo. Exact transient analysis of a planar motion with three directions, Stoch. Stoch. Reports, 72: 175 – 189, 2002.
[3] R. Garra and E. Orsingher. Random flights governd by Klein-Gordon-type partial differential equations, Stochastic Processes and their Applications, 124: 2171 – 2187, 2014.
[4] R. Garra, E. Orsingher and F. Polito. Fractional Klein-Gordon equations and related stochastic processes, Published online in Journal of Statistical Physics, March 2014.
[5] A.D. Kolesnik, E. Orsingher. A planar random motion with an infinite number of directions controlled by the damped wave equation, Journal of Applied Probability, 42(4):1168–1182, 2005.
[6] A.C. McBride. Fractional Powers of a Class of Ordinary Differential Operators. Proceedings of the London Mathematical Society, 3(45):519–546, 1982.
[7] A.C. McBride. Fractional calculus and integral transforms of generalised functions. , Pitman, London, 1979.
[8] E. Orsingher, L. Beghin. Time-fractional telegraph equations and telegraph processes with Brownian time. Probability Theory and Related Fields, 128(1):141–160, 2004.
[9] E. Orsingher, De Gregorio. Random flights in higher spaces. Journal of Theoretical Probability, 20(4):769–806, 2007.
[10] W. Stadje. The exact probability distribution of a two-dimensional random walk. Journal of Statistical Physics, 46(1-2):207–216, 1987.
Brief Biography
I took my degree in 1970 in statistical and actuarial sciences. I became assistant in stochastic processes in February 1975 and associated professor in 1983 in the University of Rome La Sapienza in analysis.
I became full professor of probability theory in 1986 at the University of Salerno and starting from 1989 again at the University of Rome.
I have been visiting professor in many universities above all in Russia, China, Ukraine. The list of my talks can be found in my homepage as well as the journals for which I refereed and of which I am currently editor.
My main fields of scientific interest have been random motions at finite velocity, some types of random fields, pseudoprocesses, motions on hyperbolic spaces and fractional calculus and its applications to stochastic processes.
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