Comment on "On Uniqueness of SDE Decomposition in A-type Stochastic Integration" [arXiv:1603.07927v1]

TL;DR

This paper critiques recent claims about the uniqueness of stochastic differential equation (SDE) decomposition, arguing that the boundary conditions necessary for uniqueness are not properly defined and that some counterexamples are not valid within the original framework.

Contribution

It clarifies the conditions under which SDE decomposition is unique and refutes claims that certain examples disprove this, emphasizing the importance of proper boundary conditions.

Findings

Boundary conditions are not sufficiently specified for uniqueness.

Counterexamples rely on implicit assumptions outside the original framework.

Gradient expansion method fails in certain cases.

Abstract

The uniqueness issue of SDE decomposition theory proposed by Ao and his co-workers has recently been discussed. A comprehensive study to investigate connections among different landscape theories [J. Chem. Phys. 144, 094109 (2016)] has pointed out that the decomposition is generally not unique, while Ao et al. (arXiv:1603.07927v1) argues that such conclusions are "incorrect" because of the missing boundary conditions. In this comment, we will combine literatures research and concrete examples to show that the concrete and effective boundary conditions have not been proposed to guarantee the uniqueness, hence the arguments in [arXiv:1603.07927v1] are not sufficient. Moreover, we show that the "uniqueness" of the O-U process decomposition referred by YTA paper is unable to serve as a counterexample to ZL's result since additional assumptions have been made implicitly beyond the original SDE decomposition framework, which cannot be applied to general nonlinear cases. Some other issues such as the failure of gradient expansion method will also be discussed. Our demonstration contributes to better understanding of the relevant papers as well as the SDE decomposition theory.