Small ball estimates in p-variation for stable processes

Let {Zt, t ≥ 0} be a strictly stable process on R with index α ∈ (0, 2]. We prove that for every p > α, there exists γ = γ(α, p) and K = K(α, p) ∈ (0,+∞) such that lim ε↓0 ε logP [||Z||p ≤ ε] = −K, where ||Z||p stands for the strong p-variation of Z on [0, 1]. The critical exponent γ(α, p) takes a different shape according as |Z| is a subordinator and p > 1, or not. The small ball constant K(α, p) is explicitly computed when p ≤ 1, and a lower bound on K(α, 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(α, p) in terms of the Brownian small ball constant under the (1/p)-Hölder semi-norm. Along the way, we remark that the positive random variable ||Z||pp is not necessarily stable when p > 1, which gives a negative answer to an old question of P. E. Greenwood [9].