Small deviations in p-variation for stable processes

Let $\{Z_t,t\ge 0\}$ be a strictly stable process on $\mathbb{R}$ with index $\alpha \in (0,2]$. We prove that for every $p>\alpha$, there exists $\gamma = \gamma (\alpha, p)$and$\k = \k (\alpha, p)\in (0, +\infty)$suchthat$$\lim_{\ee\downarrow 0}\ee^{\gamma}\log\pb\lcr ||Z||{p}\leq \ee \rcr = - \k,$$where$||Z||{p}$standsforthestrong$p$-variationof$Z$on$[0,1]$. The critical exponent $\gamma (\alpha ,p)$ takes a different shape according as $\mathrm{\mid }Z\mathrm{\mid }$ is a subordinator and $p>1$, or not. The small ball constant $\k (\alpha, p)$isexplicitlycomputedwhen$p \leq 1$,andalowerboundon$\k (\alpha, p)$ is easily obtained in the general case. In the symmetric case and when $p>2$, we can also give an upper bound on $\text{k}(\alpha ,p)$ in terms of the Brownian small ball constant under the $(1\mathrm{/}p)$-H"older semi-norm. Along the way, we remark that the positive random variable $\mathrm{\mid }\mathrm{\mid }Z\mathrm{\mid }{\mathrm{\mid }}_p^p$ is not necessarily stable when $p>1$, which gives a negative answer to an old question of P.~E.~Greenwood.