Unimodal and bimodal random motions of independent exponential steps

We consider random walks that arise from the repetition of independent, statistically identical steps, whose nature may be arbitrary. Such unimodal motions appear in a variety of contexts, including particle propagation, cell motility, swimming of micro-organisms, animal motion and foraging strategies. Building on general frameworks, we focus on the case where step duration is exponentially distributed. We explore systematically unimodal processes whose steps are ballistic, diffusive, cyclic or governed by rotational diffusion, and give the exact propagator in Fourier-Laplace domain, from which the moments and the diffusion coefficient are obtained. We also address bimodal processes, where two kinds of step are taken in turn, and show that the mean square displacement, the quantity of prime importance in experiments, is simply related to those of unimodal motions.