New continuum approximations to random walks with fixed and shrinking step lengths

Abstract

We present new continuum approximations to random walks with fixed and shrinking step lengths. The main idea behind the new continuum approximation schemes is a recent suggestion by Keller that one must deal with partial differential equations involving second order time derivatives instead of first order time derivatives, if one wants to reduce the problem of infinite propagation speeds associated with the standard continuum approximation. We obtain the Fourier transforms of the probability distribution functions for random walks with fixed and geometrically shrinking steps.