Small deviations for fractional stable processes

Abstract

Let { R t , 0 ⩽ t ⩽ 1 } be a symmetric α-stable Riemann–Liouville process with Hurst parameter H > 0 . Consider a translation invariant, β-self-similar, and p-pseudo-additive functional semi-norm ‖ ⋅ ‖ . We show that if H > β + 1 / p and γ = ( H − β − 1 / p ) −1 , then lim ɛ ↓ 0 ɛ γ log P [ ‖ R ‖ ⩽ ɛ ] = − K ∈ [ − ∞ , 0 ) , with K finite in the Gaussian case α = 2 . If α < 2 , we prove that K is finite when R is continuous and H > β + 1 / p + 1 / α . We also show that under the above assumptions, lim ɛ ↓ 0 ɛ γ log P [ ‖ X ‖ ⩽ ɛ ] = − K ∈ ( − ∞ , 0 ) , where X is the linear α-stable fractional motion with Hurst parameter H ∈ ( 0 , 1 ) (if α = 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.