Probability distribution function for systems driven by superheavy-tailed noise

TL;DR

This paper presents a method to analyze the probability distribution of particles influenced by superheavy-tailed noise, revealing distinct surviving and absorbing states through solutions of the Fokker-Planck equation.

Contribution

It introduces a general approach to study systems driven by superheavy-tailed noise, highlighting the division of distribution functions into different particle states.

Findings

Distribution function splits into surviving and absorbing states

Superheavy-tailed noise causes particles to have distinct states

The approach uses the theory of slowly varying functions

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.