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Random Flights in Higher Spaces

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Abstract

We consider in this paper random flights in ℝd performed by a particle changing direction of motion at Poisson times. Directions are uniformly distributed on hyperspheres S d1 . We obtain the conditional characteristic function of the position of the particle after n changes of direction. From this characteristic function we extract the conditional distributions in terms of (n+1)−fold integrals of products of Bessel functions. These integrals can be worked out in simple terms for spaces of dimension d=2 and d=4. In these two cases also the unconditional distribution is determined in explicit form. Some distributions connected with random flights in ℝ3 are discussed and in some special cases are analyzed in full detail. We point out that a strict connection between these types of motions with infinite directions and the equation of damped waves holds only for d=2.

Related motions with random velocity in spaces of lower dimension are analyzed and their distributions derived.

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Authors and Affiliations

  1. Dipartimento di Statistica, Probabilità e Statistiche Applicate, University of Rome “La Sapienza”, Piazzale A. Moro 5, 00185, Rome, Italy

    E. Orsingher

  2. Dipartimento di Scienze Economiche, Aziendali e Statistiche, University of Milan, via Conservatorio 7, 20122, Milan, Italy

    A. De Gregorio

Authors
  1. E. Orsingher
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  2. A. De Gregorio
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Correspondence to E. Orsingher.

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Orsingher, E., De Gregorio, A. Random Flights in Higher Spaces. J Theor Probab 20, 769–806 (2007). https://doi.org/10.1007/s10959-007-0093-y

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  • DOI: https://doi.org/10.1007/s10959-007-0093-y

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