Beller Lectureship Talk: Levy Flights and Walks in Nature

Levy flights are Markovian random processes whose underlying jump length distribution exhibits the long-tailed form. Their probability density in a homogeneous environment is defined through the characteristic function, an immediate consequence being the divergence of the variance. As such, Levy flights are a natural generalization of Gaussian diffusion processes ensuing from the generalized central limit theorem. Despite this seemingly simple definition and their widespread field of applications, Levy processes are far from being completely understood. Here, we review recent work on Levy flights concerning the particular behavior of processes with diverging jump length distributions in regard to some of the fundamental properties of stochastic processes. In particular, we explore the behavior of Levy flights in external potentials, finding distinct multimodality of the probability density function and finite variance in steeper than harmonic potentials. We proceed to show that Levy flights display a universality in the first passage behavior, contradicting the naive result obtained from the method of images; moreover, for Levy flights, the first arrival turns out to differ from the problem of first passage. Next, we address the barrier crossing of Levy flights and show that the exponential survival behavior known from classical Kramers theory is preserved, while the activation behavior of the associated rate becomes non-Arrhenius. Finally, we explore the long-standing complication that Levy flights are `pathological' in the sense that their variance diverges, while the mass of the diffusing particle is non-zero and should therefore have a finite maximum velocity: We show that dissipative nonlinear friction in the dynamics causes a truncation of the Levy stable density of the velocity distribution. This leads to a new understanding of the physical nature of Levy flight processes as an approximation to a multitude of anomalous random processes.