Chemical Kinetics, Markov Chains, and the Imaginary It\^{o} Interpretation

TL;DR

This paper explores the possibility of describing a chemical reaction process using a complex stochastic differential equation involving an imaginary term, addressing controversy in the physical literature.

Contribution

It provides partial evidence supporting the feasibility of representing the reaction with an SDE containing an imaginary component, bridging chemical kinetics and stochastic analysis.

Findings

Partial support for the imaginary Itô interpretation.

Discussion of the reaction as a Markov chain in continuous time.

Engagement with controversy in the literature.

Abstract

The abstract chemical reaction $$ A+A \to \emptyset, $$ understood as a Markov chain in continuous time, has been studied in the physical literature for several years. It has been claimed that this reaction can be described by means of the stochastic differential equation $$ d \phi = - \phi^2 dt + i \, \phi \, dW_t, $$ where $i$ is the imaginary unit. This affirmation is, at least, intriguing, and has led to controversy and criticisms in the literature. The goal of this work is to give partial evidence that such a description may be possible.