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LU-type factorizations for birth--death processes and their Darboux transformations

José Arcia-Manoleskos, Manuel Domínguez de la Iglesia

TL;DR

This work develops a comprehensive LU--UL factorization framework for the infinitesimal generator of birth--death processes on $\mathbb{N}_0$, and analyzes when the corresponding Darboux transform yields another birth--death generator. It provides explicit coefficient recursions, probabilistic interpretations in terms of hitting and occupation times, and spectral data via orthogonal polynomials and their Christoffel/Darboux transforms. The theory is illustrated on classical models such as the M/M/1 queue, the M/M/\infty queue, and linear birth--death processes across three regimes, highlighting how absorption, recurrence, and invariant measures behave under the Darboux transformation. The results offer a structured method to generate and analyze families of related birth--death processes with controlled spectral properties and stochastic dynamics.

Abstract

We study LU-type factorizations of the infinitesimal generator of a birth--death process on $\mathbb{N}_0$. Our goal is to characterize those factorizations whose Darboux transformations (that is, inverting the order of the factors) yield new infinitesimal generators of birth--death processes. Two types are considered: lower--upper (LU), which is unique and upper--lower (UL), which involves a free parameter. For both cases, we determine the conditions under which such factorizations can occur, derive explicit formulas for their coefficients, and provide a probabilistic interpretation of the factors. The spectral properties and associated orthogonal polynomials of the Darboux transformations are also analyzed. Finally, the general results are applied to classical examples such as the $M/M/1$ and $M/M/\infty$ queues and to different cases of linear birth--death processes.

LU-type factorizations for birth--death processes and their Darboux transformations

TL;DR

This work develops a comprehensive LU--UL factorization framework for the infinitesimal generator of birth--death processes on , and analyzes when the corresponding Darboux transform yields another birth--death generator. It provides explicit coefficient recursions, probabilistic interpretations in terms of hitting and occupation times, and spectral data via orthogonal polynomials and their Christoffel/Darboux transforms. The theory is illustrated on classical models such as the M/M/1 queue, the M/M/\infty queue, and linear birth--death processes across three regimes, highlighting how absorption, recurrence, and invariant measures behave under the Darboux transformation. The results offer a structured method to generate and analyze families of related birth--death processes with controlled spectral properties and stochastic dynamics.

Abstract

We study LU-type factorizations of the infinitesimal generator of a birth--death process on . Our goal is to characterize those factorizations whose Darboux transformations (that is, inverting the order of the factors) yield new infinitesimal generators of birth--death processes. Two types are considered: lower--upper (LU), which is unique and upper--lower (UL), which involves a free parameter. For both cases, we determine the conditions under which such factorizations can occur, derive explicit formulas for their coefficients, and provide a probabilistic interpretation of the factors. The spectral properties and associated orthogonal polynomials of the Darboux transformations are also analyzed. Finally, the general results are applied to classical examples such as the and queues and to different cases of linear birth--death processes.
Paper Structure (13 sections, 9 theorems, 193 equations)

This paper contains 13 sections, 9 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. LU and UL factorizations and the Darboux transformation
  4. LU factorization
  5. Probabilistic interpretation of the coefficients of the LU factorization
  6. UL factorization
  7. Probabilistic interpretation of the coefficients of the UL factorization
  8. The $M/M/1$ queue
  9. The $M/M/\infty$ queue
  10. Linear birth--death processes
  11. Case $\lambda<\mu$
  12. Case $\lambda>\mu$
  13. Case $\lambda=\mu$

Key Result

Lemma 2.1

Let $\{X_t, t\geq0\}$ be the birth--death process with infinitesimal generator $\mathcal{A}$ given by QQmm with $\mu_0>0$. For some $n\in\mathbb{N}_0$, let $p(i)=\mathbb{P}_i(\tau_{n}<\tau_{-1}), i=-1,0,\ldots n$. Then the function $p$ is harmonic with respect to $\mathcal{A}$ and can be written as where $Q_n(0)$ is the sequence of polynomials generated by the three-term recurrence relation QQs e

Theorems & Definitions (24)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • ...and 14 more