Exactly Solvable Birth and Death Processes

Abstract

Many examples of exactly solvable birth and death processes, a typical stationary Markov chain, are presented together with the explicit expressions of the transition probabilities. They are derived by similarity transforming exactly solvable `matrix' quantum mechanics, which is recently proposed by Odake and the author. The ($q$-)Askey-scheme of hypergeometric orthogonal polynomials of a discrete variable and their dual polynomials play a central role. The most generic solvable birth/death rates are rational functions of $q^x$ ($x$ being the population) corresponding to the $q$-Racah polynomial.

LaTeX, amsmath, amssymb, 24 pages, no figures

References in corpus (5)

• Orthogonal Polynomials from Hermitian Matrices
• Exactly solvable `discrete' quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation operators and coherent states
• Exact solution in the Heisenberg picture and annihilation-creation operators
• Deformed Fokker-Planck Equations
• Deformed multi-variable Fokker-Planck equations

Cited by in corpus (24)

• Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials
• Discrete Quantum Mechanics
• Multi-indexed (q-)Racah Polynomials
• Orthogonal Polynomials from Hermitian Matrices II
• Dual Christoffel transformations
• Multi-indexed Meixner and Little $q$-Jacobi (Laguerre) Polynomials
• First detection of threshold crossing events under intermittent sensing
• On a class of q-orthogonal polynomials and the q-Riemann Hilbert Problem
• Exactly solvable discrete time Birth and Death processes
• Bessel-like birth-death process
• Perturbations around the zeros of classical orthogonal polynomials
• Markov Chains Generated by Convolutions of Orthogonality Measures
• Exactly solvable inhomogeneous fermion systems
• Dual Polynomials of the Multi-Indexed ($q$-)Racah Orthogonal Polynomials
• On the Kemeny time for continuous-time reversible and irreversible Markov processes with applications to stochastic resetting and to conditioning towards forever-survival
• Asymptotics of Discrete $q$-Freud $\mathrm{II}$ orthogonal polynomials from the $q$-Riemann Hilbert Problem
• Casoratian Identities for the Discrete Orthogonal Polynomials in Discrete Quantum Mechanics with Real Shifts
• Quantum vs Classical Birth and Death Processes; Exactly Solvable Examples
• The $\mathfrak{su}(2)$ Krawtchouk oscillator model under the $\mathcal{C}\mathcal{P}$ deformed symmetry
• Analysis of non-reversible Markov chains via similarity orbit
• Asymptotic behaviours of q-orthogonal polynomials from a q-Riemann Hilbert Problem
• Time Series Path Integral Expansions for Stochastic Processes
• Multi-indexed Orthogonal Polynomials of a Discrete Variable and Exactly Solvable Birth and Death Processes
• Rahman polynomials