Authors: Gérard Le Caër
Publish Date: 2010/07/11
Volume: 140, Issue: 4, Pages: 728-751
Abstract
A constrained diffusive random walk of n steps in ℝ d and a random flight in ℝ d which are equivalent were investigated independently in recent papers J Stat Phys 127813 2007 J Theor Probab 20769 2007 and J Stat Phys 1311039 2008 The n steps of the walk are independent and identically distributed random vectors of exponential length and uniform orientation Conditioned on the sum of their lengths being equal to a given value l closedform expressions for the distribution of the endpoint of the walk were obtained altogether for any n for d=124 Uniform distributions of the endpoint inside a ball of radius l were evidenced for a walk of three steps in 2D and of two steps in 4DThe previous walk is generalized by considering step lengths which have independent and identical gamma distributions with a shape parameter q0 Given the total walk length being equal to 1 the step lengths have a Dirichlet distribution whose parameters are all equal to q The walk and the flight above correspond to q=1 Simple analytical expressions are obtained for any d≥2 and n≥2 for the endpoint distributions of two families of walks whose q are integers or halfintegers which depend solely on d These endpoint distributions have a simple geometrical interpretation Expressed for a twostep planar walk whose q=1 it means that the distribution of the endpoint on a disc of radius 1 is identical to the distribution of the projection on the disc of a point M uniformly distributed over the surface of the 3D unit sphere Five additional walks with a uniform distribution of the endpoint in the inside of a ball are found from known finite integrals of products of powers and Bessel functions of the first kind They include four different walks in ℝ3 two of two steps and two of three steps and one walk of two steps in ℝ4 Pearson–Liouville random walks obtained by distributing the total lengths of the previous Pearson–Dirichlet walks according to some specified probability law are finally discussed Examples of unconstrained random walks whose step lengths are gamma distributed are more particularly considered
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