Generalized Wiener Process and Kolmogorov's Equation for Diffusion induced by Non-Gaussian Noise Source

TL;DR

This paper develops a generalized framework for modeling non-Gaussian noise in diffusion processes, deriving corresponding Kolmogorov and Fokker-Planck equations, and analyzing stationary distributions for anomalous diffusion.

Contribution

It introduces a generalized Wiener process for non-Gaussian noise and derives new Kolmogorov equations directly from the Langevin equation, expanding the theoretical understanding of non-Gaussian stochastic processes.

Findings

Derived Kolmogorov's equation for non-Gaussian processes

Obtained Fokker-Planck and fractional equations for various diffusion types

Calculated stationary distributions for specific anomalous diffusion cases

Abstract

We show that the increments of generalized Wiener process, useful to describe non-Gaussian white noise sources, have the properties of infinitely divisible random processes. Using functional approach and the new correlation formula for non-Gaussian white noise we derive directly from Langevin equation, with such a random source, the Kolmogorov's equation for Markovian non-Gaussian process. From this equation we obtain the Fokker-Planck equation for nonlinear system driven by white Gaussian noise, the Kolmogorov-Feller equation for discontinuous Markovian processes, and the fractional Fokker-Planck equation for anomalous diffusion. The stationary probability distributions for some simple cases of anomalous diffusion are derived.