Equivalence of operator-splitting schemes for the integration of the Langevin equation

TL;DR

This paper demonstrates the mathematical equivalence of different operator-splitting schemes used for integrating the Langevin equation, focusing on the directed percolation process, and extends this equivalence to a broader class of problems.

Contribution

It provides a formal proof of the equivalence between two common splitting schemes for Langevin equation integration, generalizing the result to an entire class of operator-splitting methods.

Findings

The two splitting schemes are mathematically equivalent via an automorphism.

An extended class of equivalent operator-splitting integrations is defined.

The results apply to the directed percolation process and similar problems.

Abstract

We investigate the equivalence of different operator-splitting schemes for the integration of the Langevin equation. We consider a specific problem, so called the directed percolation process, which can be extended to a wider class of problems. We first give a compact mathematical description of the operator-splitting method and introduce two typical splitting schemes that will be useful in numerical studies. We show that the two schemes are essentially equivalent through the map that turns out to be an automorphism. An associated equivalent class of operator-splitting integrations is also defined by generalizing the specified equivalence.