Quasi-stationary distribution for multi-dimensional birth and death processes conditioned to survival of all coordinates

Nicolas Champagnat (1, 2) , Denis Villemonais (1, 3, 2)

1 TOSCA - TO Simulate and CAlibrate stochastic models
2 IECL - Institut Élie Cartan de Lorraine
3 ENSMN - Ecole Nationale Supérieure des Mines de Nancy

This article studies the quasi-stationary behaviour of multidimensional birth and death processes, modeling the interaction between several species, absorbed when one of the coordinates hits 0. We study models where the absorption rate is not uniformly bounded, contrary to most of the previous works. To handle this natural situation, we develop original Lyapunov function arguments that might apply in other situations with unbounded killing rates. We obtain the exponential convergence in total variation of the conditional distributions to a unique stationary distribution, uniformly with respect to the initial distribution. Our results cover general birth and death models with stronger intra-specific than inter-specific competition, and cases with neutral competition with explicit conditions on the dimension of the process.

Mots clés: strong intra-spe cific competition; neutral competition; Lyapunov function; Q -process; uniform exponen- tial mixing property; multidimensional birth and death process; process absorbe d on the boundary; quasi-stationary distribution

Domaines: Probabilités [math.PR]

Nicolas Champagnat, Denis Villemonais. Quasi-stationary distribution for multi-dimensional birth and death processes conditioned to survival of all coordinates. 2015. ⟨hal-01188172⟩