On Subordination of Convolution Semigroups

Abstract

Let μ := (μt)t > 0 be a convolution semigroup on ℝd endowed with its Lebesgue measure λ. Let ℳ(μ) be the set of all positive Borel measures ω on ℝd for which μt * ω ≪ λ, for each t > 0. Let EX(μ) be the set of all exit laws of μ, i.e.the set of Borel functions φ :]0, ∞[× ℝd ↦ [0, ∞] which verifies the functional equation (by putting φt := φ(t, .)) [Formula: see text] Under some λ-regularity assumptions, it is proved in this paper that EX(μ) can be identified with ℳ(μ). Moreover, let μβ be the convolution semigroup obtained by subordination of μ by a subordinator β (i.e. [Formula: see text] where β is a convolution semigroup on [0,∞[). It is also proved in this paper that, an exit law ψ of μβ is subordinated to some exit law of μ, if and only if ω belongs to ℳ(μ) where ω ∈ ℳ(μβ) is the associated measure to ψ by the preceding identification.