A solution to the reversible embedding problem for finite Markov chains

TL;DR

This paper introduces and fully solves the reversible embedding problem for finite Markov chains, providing conditions for existence, uniqueness, and a practical method for computation, with applications in physics and biochemistry.

Contribution

It defines the reversible embedding problem, proves its solution is unique when it exists, and offers a complete characterization and computational approach.

Findings

Reversible embedding, if it exists, is unique.

Provided necessary and sufficient conditions for existence.

Developed an effective method to compute the reversible embedding.

Abstract

The embedding problem for Markov chains is a famous problem in probability theory and only partial results are available up till now. In this paper, we propose a variant of the embedding problem called the reversible embedding problem which has a deep physical and biochemical background and provide a complete solution to this new problem. We prove that the reversible embedding of a stochastic matrix, if it exists, must be unique. Moreover, we obtain the sufficient and necessary conditions for the existence of the reversible embedding and provide an effective method to compute the reversible embedding. Some examples are also given to illustrate the main results of this paper.