Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes

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

We study the quasi-stationary behavior of multidimensional processes absorbed when one of the coordinates vanishes. Our results cover competitive or weakly cooperative Lotka-Volterra birth and death processes and Feller diffusions with competitive Lotka-Volterra interaction. To this aim, we develop original non-linear Lyapunov criteria involving two Lyapunov functions, which apply to general Markov processes.

Mots clés: stochastic Lotka-Volterra systems; multitype population dynamics; multidimensional birth and death process; multidimensional Feller diffusions; process absorbed on the boundary; quasi-stationary distribution; uniform exponential mixing property; Lyapunov function

Domaines: Probabilités [math.PR]

Nicolas Champagnat, Denis Villemonais. Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes. 2017. ⟨hal-01503697v1⟩