Approximate calculation of functional integrals arising from the operator approach

TL;DR

This paper introduces an approximate analytical method for calculating functional integrals in stochastic systems modeled by death-birth processes, utilizing operator formalism and eigenfunction expansion.

Contribution

It develops a novel approximation technique for functional integrals in stochastic processes using the operator approach and eigenfunction expansion.

Findings

Derived approximate probabilities for first and second states in pure birth process

Recast master equations into Schrödinger-like form using Doi-Peliti formalism

Proposed an eigenfunction expansion method for analytical treatment of functional integrals

Abstract

We apply the operator approach to a stochastic system belonging to a class of death-birth processes, which we introduce utilizing the master equation approach. By employing Doi- Peliti formalism we recast the master equation in the form of a Schr\"odinger-like equation. Therein appearing pseudo-Hamiltonian is conveniently expressed in a suitable Fock space, constructed using bosonic-like creation and annihilation operators. The kernel of the associated time evolution operator is rewritten using a functional integral, for which we propose an approximate method that allows its analytical treatment. The method is based on the expansion in eigenfunctions of the Hamiltonian generating given functional integral. In this manner, we obtain approximate values for the probabilities of the system being in the first and second states for the case of the pure birth process.