Selection principles for quasi-stationary distributions and reinforcement processes

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 $\mathrm{\partial }\mathrm{\notin }S$. We consider a reinforced process $(X_n,{\mu }_n)$ 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’’ at a point sampled according to its weighted past occupation measure $${\mu }_n=\frac{1}{W_n}(w_0{\mu }_0+\sum _{k=1}^n{w}_k{\delta }_{X_k}),W_n=\sum _{k=0}^n{w}_k,$$ where the positive weights $w_k$ satisfy certain technical assumptions, a typical example being given by $w_k=k^q$, with $q\ge -1$. When $P$ is irreducible, the behaviour of $({\mu }_n)$ 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 $1\mathrm{/}{W}_n$. If $$\sum _{n\ge 0}\frac{1}{W_n}=\mathrm{\infty },$$ then $({\mu }_n)$ almost surely converges toward the QSD associated with the largest Perron value. If instead $$\sum _{n\ge 0}\frac{1}{W_n}<\mathrm{\infty },$$ then each QSD is selected with positive probability. In particular, for polynomial weights $w_0=1$ and $w_k=k^q$, $k\ge 1$, this gives almost sure selection of the QSD with largest Perron value for $-1\le q\le 0$, whereas each quasi-stationary distribution is selected with positive probability for $q>0$.