On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions

Eva Loecherbach (1) , Dasha Loukianova (2)

1 AGM - UMR 8088 - Analyse, Géométrie et Modélisation
2 LAP - Laboratoire Analyse et Probabilités

We introduce a sequence of stopping times that allow to study an analogue of a life-cycle decomposition for a continuous time Markov process, which is an extension of the well-known splitting technique of Nummelin to the time-continuous case. As a consequence, we are able to give deterministic equivalents of additive functionals of the process and to state a generalisation of Chen's inequality. We apply our results to the problem of non-parametric kernel estimation of the drift of multi-dimensional recurrent, but not necessarily ergodic, diffusion processes.

Mots clés: Harris recurrence; Nummelin splitting; Continuous time Markov processes; Resolvents; Special functions; Additive functionals; Chacon-Ornstein theorem; Diffusion process; Nadaraya-Watson estimator

Domaines: Probabilités [math.PR]

Eva Loecherbach, Dasha Loukianova. On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions. Stochastic Processes and their Applications, 2008, 118 (8), pp.1301-1321. ⟨10.1016/j.spa.2007.09.003⟩. ⟨hal-00798486⟩