Natural cutoff in Levy flights caused by dissipative nonlinearity

Levy flight models are often used to describe stochastic processes in complex systems. However, due to the occurrence of diverging position and/or velocity fluctuations Levy flights are physically problematic if describing the dynamics of a particle of finite mass. Here we show that the velocity distribution of a random walker subject to Levy noise can be regularized by nonlinear friction, leading to a natural cutoff in the velocity distribution and finite velocity variance.