On the mean square displacement in Levy walks

Many physical and biological processes are modeled by "particles" undergoing L\'evy random walks. A feature of significant interest in these systems is the mean square displacement (MSD) of the particles. Long-time asymptotic approximations of the MSD have been established, via the Tauberian Theorem, for systems in which the distribution of the step durations is asymptotically a power law of infinite variance. We extend these results, using elementary analysis, and obtain closed-form expressions as well as power law bounds for the MSD in equilibrium, and representations of the MSD as sums of super-linear, linear, and sub-linear terms. We show that the super-linear components are determined by the mean and asymptotics of the step durations, but that the linear and sub-linear components (whose size has implications for the accuracy of the asymptotic approximation) depend on the entire distribution function.