THE NONISOTROPIC 2-DIMENSIONAL RANDOM-WALK

A formula is obtained for the joint probability density function of the angle and length of the resultant of an N-step non-isotropic random walk in two dimensions for arbitrary step angle and radius probability density and for any fixed number of steps. The problem is attacked by applying the theory of generalized functions concentrated on smooth manifolds. The analysis is presented initially for the case where only the angles are random. The characteristic function is defined for the walk in terms of angular and radial frequencies and the inversion is obtained in terms of a sum of Hankel transforms. The Hankel transform sum is transformed by showing that it can be interpreted in terms of the motions of the two-dimensional Euclidean plane corresponding to the rotations and translations resulting from a sequence of fixed steps. This transformation results in an expression involving integrations over two manifolds defined by delta functions. The properties of the manifolds defined by the delta functions am then considered and this results in some simplification of the formulae. The analysis is then generalized to the case where both the phase and length of each step in the walk are random. Finally, seven examples am presented including the general two-step walk and three walks which lead to generalized K density functions for the resultant.