Stochastic representation of fractional subdiffusion equation. The case of infinitely divisible waiting times, Levy noise and space-time-dependent coefficients

TL;DR

This paper provides a comprehensive stochastic representation of fractional subdiffusion equations with space-time-dependent coefficients, using subordinated Langevin equations driven by Brownian and Levy noises, addressing a longstanding problem in the field.

Contribution

It introduces a general stochastic process representation for fractional Fokker-Planck equations with variable coefficients, unifying subdiffusive dynamics under a broad ID framework.

Findings

Stochastic process representation for fractional subdiffusion with variable coefficients.

Solution to the stochastic representation problem in full generality.

Framework applicable to systems with Levy noise and space-time-dependent parameters.

Abstract

In this paper we analyze fractional Fokker-Planck equation describing subdiffusion in the general infinitely divisible (ID) setting. We show that in the case of space-time-dependent drift and diffusion and time-dependent jump coefficient, the corresponding stochastic process can be obtained by subordinating two-dimensional system of Langevin equations driven by appropriate Brownian and Levy noises. Our result solves the problem of stochastic representation of subdiffusive Fokker-Planck dynamics in full generality.