On the Abstract Subordinated Exit Equation

Abstract

Let $\mathbb{\mathbb{P}}=(P_t{)}_{t>0}$ be a $C_0$-contraction semigroup on a real Banach space $\mathcal{ℬ}$. A $\mathbb{\mathbb{P}}$-exit law is a $\mathcal{ℬ}$-valued function $t\in ]0,\infty [\to {\phi }_t\in \mathcal{ℬ}$ satisfying the functional equation: $P_t{\phi }_s={\phi }_{t+s}$, $s,t>0$. Let $\beta$ be a Bochner subordinator and let ${\mathbb{\mathbb{P}}}^{\beta }$ be the subordinated semigroup of $\mathbb{\mathbb{P}}$ (in the Bochner sense) by means of $\beta$. Under some regularity assumption, it is proved in this paper that each ${\mathbb{\mathbb{P}}}^{\beta }$-exit law is subordinated to a unique $\mathbb{\mathbb{P}}$-exit law.