Uniform convergence of conditional distributions for absorbed one-dimensional diffusions

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

1 TOSCA - TO Simulate and CAlibrate stochastic models
2 IECL - Institut Élie Cartan de Lorraine

This article studies the quasi-stationary behaviour of absorbed one-dimensional diffusions. We obtain a necessary and sufficient condition 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. We provide several conditions ensuring this criterion, which apply to most practical cases. As a by-product, we prove that most strict local martingale diffusions are strict in a stronger sense: their expectation at any given positive time is actually uniformly bounded with respect to the initial position. We provide several examples and extensions, including the sticky Brownian motion and some one-dimensional processes with jumps. We also give exponential ergodicity results on the Q-process.

Mots clés: one dimensional processes with jumps; Q-process; absorbed process; one-dimensional diffusions; quasi-stationary distribution; strict local martingales; uniform exponential mixing property

Domaines: Probabilités [math.PR]

Nicolas Champagnat, Denis Villemonais. Uniform convergence of conditional distributions for absorbed one-dimensional diffusions. Advances in Applied Probability, 2018, 50 (1), pp.178-203. ⟨10.1017/apr.2018.9⟩. ⟨hal-01166960⟩