Small Ball Estimates in p-Variation for Stable Processes

Let Z t , t ≥ 0 be a strictly stable process on $${\mathbb{R}}$$ with index α ∈ (0, 2]. We prove that for every p > α, there exists γ = γα, p and $$K = K_{\alpha ,p} \in (0, + \infty)$$ such that $${\mathop {\lim }\limits_{\varepsilon \downarrow 0}} \varepsilon ^\gamma \log \;{\mathbb{P}}[||Z||_p \leqslant \varepsilon ] = - K,$$ where || Z|| p stands for the strong p-variation of Z on [0,1]. The critical exponent γα p , takes a different shape according as | Z| is a subordinator and p > 1, or not. The small ball constant $$K_{\alpha ,p}$$ is explicitly computed when p > 1, and a lower bound on $$K_{\alpha ,p}$$ is easily obtained in the general case. In the symmetric case and when p > 2, we can also give an upper bound on $$K_{\alpha ,p}$$ in terms of the Brownian small ball constant under the (1/p)-Höder semi-norm. Along the way, we remark that the positive random variable $$||Z||_p^p$$ is not necessarily stable when p > 1, which gives a negative answer to an old question of P. E. Greenwood.10