Comment on "Fokker-Planck equations for nonlinear dynamical systems driven by non-Gaussian L\'evy processes" [J. Math. Phys. 53, 072701 (2012)]

TL;DR

This paper critiques a previous derivation of Fokker-Planck equations for nonlinear systems driven by non-Gaussian Lévy processes, highlighting a false assumption but confirming the result under additional conditions.

Contribution

It identifies a critical flaw in the original derivation and clarifies the conditions under which the main result remains valid.

Findings

The original derivation used an incorrect assumption about Taylor series.

The main result is valid only with extra assumptions.

The critique clarifies the conditions for the original theorem's validity.

Abstract

In an article [J. Math. Phys. 53, 072701 (2012)] X. Sun and J. Duan presented Fokker-Planck equations for nonlinear stochastic differential equations with non-Gaussian L\'evy processes. In this comment we show a serious drawback in the derivation of their main result. In the proof of Theorem 1 in the aforementioned paper, a false assumption that each infinitely differentiable function with compact support is equal to its Taylor series, is used. We prove that although the derivation is incorrect, the result remains valid only if we add certain additional assumptions.