Redundant basis interpretation of Doi-Peliti method and an application

TL;DR

This paper introduces a redundant basis interpretation of the Doi-Peliti method, linking it to generating functions and enabling the derivation of various discrete processes, with practical application to stochastic differential equations like the noisy van der Pol system.

Contribution

It proposes a novel redundant basis interpretation for the Doi-Peliti method, extending its connection to generating functions and enhancing its applicability to stochastic differential equations.

Findings

Redundant basis yields accurate finite-state approximations.

Extended correspondence with generating function approach.

Numerical experiments validate the method's effectiveness.

Abstract

The Doi-Peliti method is effective for investigating classical stochastic processes, and it has wide applications, including field theoretic approaches. Furthermore, it is applicable not only to master equations but also to stochastic differential equations; one can derive a kind of discrete process from stochastic differential equations. A remarkable fact is that the Doi-Peliti method is related to a different analytical approach, i.e., generating function. The connection with the generating function approach helps to understand the derivation of discrete processes from stochastic differential equations. Here, a redundant basis interpretation for the Doi-Peliti method is proposed, which enables us to derive different types of discrete processes. The conventional correspondence with the generating function approach is also extended. The proposed extensions give us a new tool to study stochastic differential equations. As an application of the proposed interpretation, we perform numerical experiments for a finite-state approximation of the derived discrete process from the noisy van der Pol system; the redundant basis yields reasonable results compared with the conventional discrete process with the same number of states.