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On the algebra of equal-input matrices in time-inhomogeneous Markov flows

Michael Baake, Jeremy Sumner

TL;DR

This work addresses the problem of describing time-inhomogeneous Markov flows generated by equal-input type matrices, allowing noncommuting generators. It leverages the Peano–Baker series and Magnus expansion to derive explicit solutions for solvable non-commuting families, and reduces the dynamics to a low-dimensional vector ODE for the equal-input parameter when $Q(t)=Q_{\boldsymbol{q}(t)}$. The authors construct exact forward flows with a clean additive structure $M(t)=\mathbb{I}+A_0(t)+A_{\triangle}(t)$ and provide closed-form expressions for the associated real matrix logarithms $R(t)$, including a BCH-like formula within the equal-input class. They further extend the setup to two generalisations that preserve solvability: a weighted sum $Q(t)=\mu(t) Q_0 + C_{\boldsymbol{q}(t)}$ with a traceless perturbation, and a commuting subfamily $Q_0(t)$, yielding explicit expressions for $R(t)$ and conditions under which the forward flow remains within Markov/embeddable regimes. The results enhance understanding of non-stationary Markov processes with non-commuting generators and provide tractable models with explicit analytical control.

Abstract

Markov matrices of equal-input type constitute a widely used model class. The corresponding equal-input generators span an interesting subalgebra of the real matrices with zero row sums. Here, we summarise some of their amazing properties and discuss the corresponding Markov embedding problem, both homogeneous and inhomogeneous in time. In particular, we derive exact and explicit solutions for time-inhomogeneous Markov flows with non-commuting generator families of equal-input type and beyond.

On the algebra of equal-input matrices in time-inhomogeneous Markov flows

TL;DR

This work addresses the problem of describing time-inhomogeneous Markov flows generated by equal-input type matrices, allowing noncommuting generators. It leverages the Peano–Baker series and Magnus expansion to derive explicit solutions for solvable non-commuting families, and reduces the dynamics to a low-dimensional vector ODE for the equal-input parameter when . The authors construct exact forward flows with a clean additive structure and provide closed-form expressions for the associated real matrix logarithms , including a BCH-like formula within the equal-input class. They further extend the setup to two generalisations that preserve solvability: a weighted sum with a traceless perturbation, and a commuting subfamily , yielding explicit expressions for and conditions under which the forward flow remains within Markov/embeddable regimes. The results enhance understanding of non-stationary Markov processes with non-commuting generators and provide tractable models with explicit analytical control.

Abstract

Markov matrices of equal-input type constitute a widely used model class. The corresponding equal-input generators span an interesting subalgebra of the real matrices with zero row sums. Here, we summarise some of their amazing properties and discuss the corresponding Markov embedding problem, both homogeneous and inhomogeneous in time. In particular, we derive exact and explicit solutions for time-inhomogeneous Markov flows with non-commuting generator families of equal-input type and beyond.
Paper Structure (5 sections, 8 theorems, 77 equations)

This paper contains 5 sections, 8 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Inhomogeneous flow
  4. Two generalisations and their exact solutions
  5. Peano--Baker series and Magnus expansion

Key Result

Lemma 3.1

Assume that $Q(t) \in \mathcal{E}^{}_{0}$ for all $t\geqslant 0$, and that $Q(t)$ is piecewise continuous. Then, the Cauchy problem has a unique solution, where each $M(t)$ with $t\geqslant 0$ is a real matrix with row sum $1$ that has equal-input form. Further, if all $Q(t)$ are rate matrices, $\{ M(t) : t\geqslant 0 \}$ is a flow of equal-input Markov matrices.

Theorems & Definitions (15)

  • proof
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Proposition 4.2
  • proof
  • ...and 5 more