Hitting properties and non-uniqueness for SDE driven by stable processes

Julien Berestycki (1) , Leif Doering (1) , Leonid Mytnik , Lorenzo Zambotti (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We study a class of self-similar jump type SDEs driven by Hölder-continuous drift and noise coefficients. Using the Lamperti transformation for positive self-similar Markov processes we obtain a necessary and sufficient condition for almost sure extinction in finite time. We then show that for certain parameters pathwise uniqueness holds in a restricted sense, namely among solutions spending a Lebesgue-negligible amount of time at 0. A direct power transformation plays a key role.

Domaines: Probabilités [math.PR]

Julien Berestycki, Leif Doering, Leonid Mytnik, Lorenzo Zambotti. Hitting properties and non-uniqueness for SDE driven by stable processes. 2011. ⟨hal-00706796⟩