Hausdorff dimension of the image of additive processes

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