Time Series Path Integral Expansions for Stochastic Processes

TL;DR

This paper introduces a novel path integral expansion method for analyzing stochastic processes, enabling both analytical and numerical calculations of temporal effects across various models.

Contribution

It develops a unified framework using algebraic techniques and reproducing kernels to perform path integral expansions for birth-death and diffusion processes.

Findings

Provides a new series expansion differing from Dyson series.

Applies algebraic methods to birth-death processes with linear and quadratic rates.

Adapts techniques for diffusion process analysis.

Abstract

A form of time series path integral expansion is provided that enables both analytic and numerical temporal effect calculations for a range of stochastic processes. Birth-death processes with linear rates are analysed via coherent state Doi-Peliti techniques. The $\mathfrak{su}(1,1)$ Lie algebra is utilised to capture quadratic rate birth-death processes. The techniques are also adapted to diffusion processes. All methods rely on finding a suitable reproducing kernel associated with the underlying algebra to perform the expansion. The resulting series differ from those found in standard Dyson time series field theory techniques.