The multifractal structure of stable occupation measure

Let X be a stable subordinator of index α and μ be the occupation measure of X. Denote d(μ,x) and d̄(μ,x) as the lower and upper local dimensions of μ. We obtain that the Hausdorff dimension of the set of the points where d̄(μ,x) = β is (2α2β) − α a.s. and the lower bound of packing dimension is 2α − β a.s. if α⩽β⩽2α. When β > 2α, the corresponding set is empty a.s.. And for a.s. Ω, the set of the points where d̄(μ,x) = α is the closure of X[0,1].