Small deviations in $p$-variation for multidimensional Lévy processes

Abstract

Let $Z$ be an $\mathbb{R}^{d}$-valued Lévy process with strong finite $p$-variation for some $p < 2$. We prove that the “decompensated” process $\Tilde{Z}$ obtained from $Z$ by annihilating its generalized drift has a small deviations property in $p$-variation. This property means that the null function belongs to the support of the law of $\Tilde{Z}$ with respect to the $p$-variation distance. Thanks to the continuity results of T. J. Lyons/D. R. E. Williams [19], [35], this allows us to prove a support theorem with respect to the $p$-Skorohod distance for canonical SDE’s driven by $Z$ without any assumption on $Z$, improving the results of H. Kunita [15]. We also give a criterion ensuring the small deviation property for $Z$ itself, noticing that the characterization under the uniform distance, which we had obtained in [26], no more holds under the $p$-variation distance.