On the Abstract Subordinated Exit Equation - Hassen Mejri; Ezzedine Mliki (Abstract and Applied Analysis)
Let ℙ=(𝑃𝑡)𝑡>0 be a 𝐶0-contraction semigroup on a real Banach space ℬ. A ℙ-exit law is a ℬ-valued function 𝑡∈]0,∞[→𝜑𝑡∈ℬ satisfying the functional equation: 𝑃𝑡𝜑𝑠=𝜑𝑡+𝑠, 𝑠,𝑡>0. Let 𝛽 be a Bochner subordinator and let ℙ𝛽 be the subordinated semigroup of ℙ (in the Bochner sense) by means of 𝛽. Under some regularity assumption, it is proved in this paper that each ℙ𝛽-exit law is subordinated to a unique ℙ-exit law.