Selection principles for quasi-stationary distributions and reinforcement processes

Abstract

Let \(P\) be a sub-Markov matrix on a finite set \(S\), representing the transition probabilities of a Markov chain on \(S\) absorbed at a cemetery point \(\partial\notin S\). We consider a reinforced process \((X_n,μ_n)\) defined as follows: \((X_n)\) behaves like a chain with kernel \(P\) until it dies, and when it dies at time \(n\), it is instantaneously ``resurrected&#39;&#39; at a point sampled according to its weighted past occupation measure \[ μ_n = \frac1{W_n} \left( w_0μ_0+\sum_{k=1}^n w_kδ_{X_k} \right), \qquad W_n=\sum_{k=0}^n w_k, \] where the positive weights $w_k$ satisfy certain technical assumptions, a typical example being given by $w_k = k^q$, with $q\geq -1$. When \(P\) is irreducible, the behaviour of \((μ_n)\) is well understood \cite{AFP}, \cite{bansaye2022non}: it converges almost surely toward the unique quasi-stationary distribution (QSD) of \(P\). The purpose of this paper is to investigate the general situation where \(P\) is not irreducible. Under generic assumptions on \(P\), there are finitely many QSDs. We prove that the asymptotic selection depends on the summability of the inverse cumulative weights \(1/W_n\). If \[ \sum_{n\geq0}\frac1{W_n}=\infty, \] then \((μ_n)\) almost surely converges toward the QSD associated with the largest Perron value. If instead \[ \sum_{n\geq0}\frac1{W_n}<\infty, \] then each QSD is selected with positive probability. In particular, for polynomial weights \(w_0=1\) and \(w_k=k^q\), \(k\geq1\), this gives almost sure selection of the QSD with largest Perron value for \(-1\leq q\leq0\), whereas each quasi-stationary distribution is selected with positive probability for \(q>0\).