Sur les lois de sortie des semi-groupes de convolution - Bachar, Imed (Annales Mathematicae Silesianae, 2001)
Let μ = (μt)t>o be a convolution semigroup on ℝd. An exit law for μ is a positive measurable function ϕ: ]0,∞[ × ℝd → [0,∞] which verifies the functional equation (by putting ϕt:=ϕ(t,·)) ∀s,t>0 : μs ⚹ ϕt = ϕs+t λ.a.e. where λ is the Lebesgue measure on ℝd. Following [1], we prove in this paper that the solutions of this equation are on the form ϕt = d(μ^t ⚹ β)/dλ λ.a.e. where μ^ := (μ^t)t>o is the reflected convolution semigroup of μ, β is a positive measure on ℝd such that μ^t ⚹ β ≪ λ, for evry t > 0. Moreover, we study the global solutions and their interpretations in terms of the negative definite function associated to μ.