The Growth of Additive Processes

Let Xt be any additive process in R d. There are finite indices δi, βi, i = 1, 2 and a function u, all of which are defined in terms of the characteristics of Xt, such that lim inf t→0 u(t) −1/η X * t = 0, if η > δ1, ∞, if η < δ2, lim sup t→0 u(t) −1/η X * t = 0, if η > β2, ∞, if η < β1, a.s., where X * t = sup 0≤s≤t |Xs|. When Xt is a Lévy process with X0 = 0, δ1 = δ2, β1 = β2 and u(t) = t. This is a special case obtained by Pruitt. When Xt is not a Lévy process, its characteristics are complicated functions of t. However, there are interesting conditions under which u becomes sharp to achieve δ1 = δ2, β1 = β2. 1. Introduction. A process X t with independent increments, rcll (right-continous with left limits) paths and values in R d is called additive if X t is continuous in probability and X 0 = 0. Additive processes represent a large family of nonhomogeneous processes and intersect the entirety of Feller processes at the class of Lévy processes. Pruitt [6] defined an index δ for each Lévy process X t with X 0 = 0 and showed that X t satisfies the Hölder conditions: lim inf t→0 t −1/η X * t = 0 or ∞ a.s. according as η > δ or η < δ, where X * t = sup 0≤s≤t |X s |. Its lim sup analogue was obtained by Blumenthal and Getoor [1] with an index β. Both results have their additive process counterparts. We define in terms of the characteristics of an additive process X t a nondecreasing continuous function u with u(0) = 0 and four finite indices δ i , β i , i = 1, 2 such that