Probability distribution function for systems driven by superheavy-tailed noise

Abstract

We develop a general approach for studying the cumulative probability distribution function of localized objects (particles) whose dynamics is governed by the first-order Langevin equation driven by superheavy-tailed noise. Solving the corresponding Fokker-Planck equation, we show that due to this noise the distribution function can be divided into two different parts describing the surviving and absorbing states of particles. These states and the role of superheavy-tailed noise are studied in detail using the theory of slowly varying functions.