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

Julien Berestycki; Leif Doering; Leonid Mytnik; Lorenzo Zambotti

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arXiv · 1111.4388

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

Julien Berestycki·Leif Doering·Leonid Mytnik·Lorenzo Zambotti

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Abstract

We study a class of self-similar jump type SDEs driven by H\"older-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.

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Julien Berestycki, Leif Doering, Leonid Mytnik, Lorenzo Zambotti. 2011-11-18. Hitting properties and non-uniqueness for SDE driven by stable processes. https://arxiv.org/abs/1111.4388

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