Small deviations for fractional stable processes

Let {R-t, 0 <= t <= 1} be a symmetric alpha-stable Riemann-Liouville process with Hurst parameter H > 0. Consider a translation invariant, beta-self-similar, and p-pseudo-additive functional semi-norm vertical bar vertical bar (.) vertical bar vertical bar. We show that if H > beta + 1/p and gamma = (H - beta - 1/p)(-1), then lim epsilon(gamma)(epsilon down arrow 0) logP[vertical bar vertical bar R vertical bar vertical bar <= epsilon] = -K epsilon (-infinity, 0), with K finite in the Gaussian case alpha = 2. If alpha < 2, we prove that K is finite when R is continuous and H > beta + 1/p + 1/alpha. We also show that under the above assumptions, lim epsilon(gamma)(epsilon down arrow 0) logP[vertical bar vertical bar X vertical bar vertical bar <= epsilon] = -K epsilon (-infinity, 0), where X is the linear a-stable fractional motion with Hurst parameter H E (0, 1) (if a = 2, then X is the classical fractional Brownian motion). These general results cover many cases previously studied in the literature, and also prove the existence of new small deviation constants, both in Gaussian and non-Gaussian frameworks. (c) 2004 Elsevier SAS. All rights reserved.