Langevin equation with super-heavy-tailed noise

TL;DR

This paper extends the Langevin equation framework to include super-heavy-tailed noise, deriving an exact generalized Fokker-Planck equation, analyzing probabilistic states, and validating results through numerical simulations.

Contribution

It introduces a novel Langevin approach with super-heavy-tailed noise, deriving an exact generalized Fokker-Planck equation and establishing the connection between absorption rates and noise distribution.

Findings

Two probabilistic states: survived and absorbed

Exact solution of the generalized Fokker-Planck equation

Numerical scheme confirms theoretical predictions

Abstract

We extend the Langevin approach to a class of driving noises whose generating processes have independent increments with super-heavy-tailed distributions. The time-dependent generalized Fokker-Planck equation that corresponds to the first-order Langevin equation driven by such a noise is derived and solved exactly. This noise generates two probabilistic states of the system, survived and absorbed, that are equivalent to those for a classical particle in an absorbing medium. The connection between the rate of absorption and the super-heavy-tailed distribution of the increments is established analytically. A numerical scheme for the simulation of the Langevin equation with super-heavy-tailed noise is developed and used to verify our theoretical results.