Quasi-stationary distributions for continuous-time $\lambda$-recurrent jump processes

TL;DR

This paper investigates the existence and explicit representation of quasi-stationary distributions for continuous-time $\lambda$-recurrent jump processes, including cases with finite and infinite exit states.

Contribution

It provides an explicit formula for quasi-stationary distributions using the $Q$-matrix and offers conditions for their existence with infinite exit states.

Findings

Explicit representation of quasi-stationary distribution via $Q$-matrix

Conditions established for existence with infinite exit states

Representation of outside components by those within exit states

Abstract

For the continuous-time $\lambda$-recurrent jump process, the $\lambda$-recurrence assures the existence of quasi-stationary distribution when it has finite exit states (the states that have positive killing rates). And we give an explicit representation for this quasi-stationary distribution through $Q$-matrix, where the components of the quasi-stationary distribution outside the set $H$ of exit states can be represented by those within $H$. Sufficient condition is also provided for quasi-stationary distribution when the exit states are infinite.