Exact probability distributions for noncorrelated random walk models

A stochastic model for the idealized locomotion of cells is studied. The cell is assumed to cover a polygonal line in ⁿ, the times between turns are exponentially distributed and independent of the directions, and the density of thenth directione does not depend on the (n–1)th directione. The resulting Markov process (X(t), D(t)) for position and direction of the motion at timet is studied by using the integrodifferential equation for the transition function. For example, the joint distribution of (X(t), D(t)) is derived in closed form ifn=2 orn=3 and all chosen directions (including the initial one) are uniformly distributed. For higher dimensions the combined Fourier-Laplace transform ofX(t) is given. The case of a fixed initial direction is also considered.