PC Seminar: Polynomial Characterization and Existence of QSDs for Markov Processes

Carsten Wiuf (Copenhagen) gives this seminar. Welcome to join!

Abstract: This talk is concerned with quasi-stationary distributions (QSDs) on $\N_0$ for a specific class of continuous time Markov chains (CTMCs), characterised by a finite, but otherwise arbitrary, set of jump vectors. A QSD describes the long time behaviour of the CTMC before absorption.

Examples arise in ecology, epidemiology, cellular biology and chemistry. Birth-death processes are special examples with jumps of size $1$ and $-1$.

Let $(X_t)_{t\ge 0}$ be a CTMC on $\N_0$ that eventually is absorbed into a trapping set $A\subseteq\N_0$. A probability distribution $\nu$ with support on $A^c=\N_0\!\setminus\! A$ is a QSD for $(X_t)_{t\ge 0}$, if

$$\PP_\nu(X_t\in B\, |\, t<\tau_A)=\nu(B),\quad B\subseteq A^c,\quad\text{for all}\quad t\ge 0,$$ where $ \tau_A=\inf\{t\ge0 \colon X_t\in A\}$ is the time until absorption, and $\PP_{\nu}$ is the distribution of the process before absorption with initial distribution $\nu$.

In this context, we characterise recursively all left invariant probability measures $\nu$ of the $q$-matrix of the process in terms of a finite number of generating terms $\nu(J)=(\nu(i_1),\ldots,\nu(i_d))$, $J=\{i_1,\ldots,i_d\}$, where $J$ is independent of the corresponding eigenvalue $-\theta\in\R$. This is to say, $\nu(n)=\sum_{j\in J}R_j(n,\theta)\nu(j)$, where

the coefficients $R_j(n,\theta)$ are given recursively in $n$. Based on this and relying on Perron-Frobenius theory for infinite matrices, we prove existence and uniqueness of an extremal QSD for Kingman's parameter $\theta_K>0$ (also called the decay parameter), provided the CTMC is ultimately absorbed. Furthermore, we show the existence of a series of polynomials of increasing degree, $ \rho_n(\theta)$, such that $\theta(n)\downarrow\theta_K$, where $\theta(n)$ is the smallest real root of $ \rho_n(\theta)$. These results mimic results for birth-death processes, where the polynomials are further known to be orthogonal (seminal work by Karlin and McGregor). In special cases of our setting, the polynomials $R_j(n,\theta)$, $n\in\N_0$, form sequences of $d$-orthogonal polynomials, where $d$ is the number of elements of $J$, extending the birth-death case where $d=1$.

We further discuss numerial means to find the generator for an invariant measure and to determine Kingman's parameter. The results are illustrated with numerical examples inspired by stochastic reaction network theory.

This is joint work with Chuang Xu (University of Hawaii) and Mads Chr Hansen (formerly at University of Copenhagen).

This is a seminar in our seminar series on Probability and Combinatorics (PC).