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.
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Vol. 15 (2001)
Published: 2001-09-28
10.1515/amsil
English