Exponential convergence to quasi-stationary distribution for absorbed one-dimensional diffusions with killing

Nicolas Champagnat (1, 2) , Denis Villemonais (3, 1, 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 absorbed one-dimensional diffusion processes with killing on [0, ∞). We obtain criteria for the exponential convergence to a unique quasi-stationary distribution in total variation, uniformly with respect to the initial distribution. Our approach is based on probabilistic and coupling methods, contrary to the classical approach based on spectral theory results. Our general criteria apply in the case where ∞ is entrance and 0 either regular or exit, and are proved to be satisfied under several explicit assumptions expressed only in terms of the speed and killing measures. We also obtain exponential ergodicity results on the Q-process. We provide several examples and extensions, including diffusions with singular speed and killing measures, general models of population dynamics , drifted Brownian motions and some one-dimensional processes with jumps.

Mots clés: diffusions; one-dimensional diffusions with killing; quasi-stationary distribution; absorbed pro-cess; uniform exponential mixing property; Q-process; one dimensional processes with jumps

Domaines: Probabilités [math.PR]

Nicolas Champagnat, Denis Villemonais. Exponential convergence to quasi-stationary distribution for absorbed one-dimensional diffusions with killing. ALEA : Latin American Journal of Probability and Mathematical Statistics, 2017, 14, pp.177-199. ⟨10.30757/ALEA.v14-11⟩. ⟨hal-01217843⟩