Small ball estimates in p-variation for stable processes

Let {Z(t), t greater than or equal to 0} be a strictly stable process on R with index alpha is an element of (0, 2]. We prove that for every p > alpha, there exists gamma = gamma(alpha,p) and K = K-alpha,K-p is an element of (0, +infinity) such that lim(epsilondown arrow0) epsilon(gamma) log P[parallel toZparallel to(p) less than or equal to epsilon] = -K where parallel toZparallel to(p) stands for the strong p-variation of Z on [0,1]. The critical exponent gamma(alpha,p), takes a different shape according as \Z\ is a subordinator and p > 1, or not. The small ball constant K-alpha,K-p is explicitly computed when p less than or equal to 1, and a lower bound on K-alpha,K-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,K-p in terms of the Brownian small ball constant under the (1/p)-Hoder semi-norm. Along the way, we remark that the positive random variable parallel toZparallel to(p)(p) is not necessarily stable when p > 1, which gives a negative answer to an old question of P. E. Greenwood.