Analytic solution to space-fractional Fokker-Planck equations for tempered-stable L\'{e}vy distributions with spatially linear, time-dependent drift

TL;DR

This paper derives explicit time-dependent solutions for space-fractional Fokker-Planck equations driven by tempered-stable Lévy noise with spatially linear and time-dependent drift, expanding understanding of non-steady stochastic systems.

Contribution

It provides the first analytic solutions for these complex equations with general linear, time-dependent drift and tempered-stable Lévy noise.

Findings

Explicit time-dependent solutions derived

Applicable to systems with non-steady drift

Enhances modeling of anomalous diffusion processes

Abstract

We derive analytic solutions for the full time dependence of space-fractional Fokker-Planck equations corresponding to stochastic Langevin equations with additive tempered-stable L\'{e}vy noise terms. The drift terms are generalised to be spatially linear, but may contain arbitrary time dependence such that no steady-state solution is available, even for the deterministic system.