On the abstract subordinated exit equation (Q1957575)

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