# The chemical birth-death process with additive noise

Source: [The chemical birth-death process with additive noise](<https://www.sota2.com/research/paper/68fa3ad31e209131ce8cdb41>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T00:16:40.418Z

Authors: John J. Vastola

Published: 2019-10-21T02:25:12.000Z

## Abstract

The chemical birth-death process, whose chemical master equation (CME) is exactly solvable, is a paradigmatic toy problem often used to get intuition for how stochasticity affects chemical kinetics. In a certain limit, it can be approximated by an Ornstein-Uhlenbeck-like process which is also exactly solvable. In this paper, we use this system to showcase eight qualitatively different ways to exactly solve continuous stochastic systems: (i) integrating the stochastic differential equation; (ii) computing the characteristic function; (iii) eigenfunction expansion; (iv) using ladder operators; (v) the Martin-Siggia-Rose-Janssen-De Dominicis path integral; (vi) the Onsager-Machlup path integral; (vii) semiclassically approximating the Onsager-Machlup path integral; and (viii) approximating the solution to the corresponding CME.

## Recorded benchmark results

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1910.09117>)

[PDF](<https://arxiv.org/pdf/1910.09117.pdf>)