On Uniqueness of "SDE Decomposition" in A-type Stochastic Integration

TL;DR

This paper proves the mathematical and physical uniqueness of the A-type stochastic integration framework for SDEs, enhancing understanding of its robustness and addressing recent questions about its validity.

Contribution

It provides rigorous demonstrations confirming the uniqueness of the A-type stochastic integration, clarifying its theoretical foundation and implications.

Findings

Mathematical proof of uniqueness of A-type stochastic integration.

Physical demonstration supporting the robustness of the framework.

Discussion on limitations of potential function derivation from steady states.

Abstract

An innovative theoretical framework for stochastic dynamics based on a decomposition of a stochastic differential equation (SDE) has been developed with an evident advantage in connecting deterministic and stochastic dynamics, as well as useful applications in physics, engineering, chemistry and biology. It introduces the A-type stochastic integration for SDE beyond traditional Ito's or Stratonovich's interpretation. Serious question on its uniqueness was recently raised. We provide here both mathematical and physical demonstrations that the uniqueness is guaranteed. Such discussion leads to a better understanding on the robustness of the novel framework. We also discuss the limitation of a related approach of obtaining potential function from steady state distribution.