Speaker: Tresnia Berah
Title: Selection principles for quasi-stationary distributions and reinforcement processes
Abstract: (https://arxiv.org/abs/2606.25857) Let P be a sub-Markov matrix on a finite set S. We study a reinforced process that evolves according to P until absorption and is then resurrected at a point sampled from its weighted past occupation measure ??, with positive weights w? and cumulative weights W? = w? + ? + w?. Whereas the irreducible case leads to almost sure convergence towards the unique quasi-stationary distribution (QSD), we consider reducible kernels, for which several QSDs may coexist. Under generic assumptions on P and suitable assumptions on the weights and initial condition, there are finitely many QSDs and the asymptotic selection is governed by the summability of 1/W?. If the series ? 1/W? diverges, then ?? converges almost surely towards the QSD associated with the largest Perron value. If the series converges, then ?? converges almost surely towards a QSD, and each QSD is selected with positive probability. For polynomial weights w? = 1 and w? = k^q for k ? 1, the first regime corresponds to ?1 ? q ? 0 and the second to q > 0.