Hausdorff dimension of the image of additive processes

Let X t be any additive process in R d . There are two lower indices β T ′ , β T ″ and an upper index β T for T ∈ ( 0 , ∞ ) such that for any Borel set E ⊂ [ 0 , T ] , dim H X ( E ) ≥ ( β T ″ dim H E ) ∧ d , dim H X ( E ) ≥ β T ′ dim H E if β T ′ ≤ d , and dim H X ( E ) ≤ β T dim H E , where X ( E ) = { X s : s ∈ E } for E ∈ B ( R + ) and dim H denotes the Hausdorff dimension. When X t is a Lévy process, β T = β , β T ′ = β ′ , and β T ″ = β ″ , where β , β ′ , β ″ are Blumenthal and Getoor’s indices.