Random Flights in Higher Spaces

E. Orsingher () and A. Gregorio
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E. Orsingher: University of Rome “La Sapienza”
A. Gregorio: University of Milan

Journal of Theoretical Probability, 2007, vol. 20, issue 4, 769-806

Abstract: Abstract We consider in this paper random flights in ℝ d performed by a particle changing direction of motion at Poisson times. Directions are uniformly distributed on hyperspheres S 1 d . 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.

Keywords: Bessel functions; Random flights; Uniform distribution on hyperspheres; Legendre functions; Hyperspherical coordinates; Stirling’s formula; Hypergeometric functions (search for similar items in EconPapers)
Date: 2007
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10959-007-0093-y

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