Small deviations in p-variation for multidimensional Lévy processes

Let Z be an R-valued Lévy process with strong finite p-variation for some p < 2. We prove that the ”decompensated” process Z̃ 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 Z̃ with respect to the pvariation 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.