On the Bochner subordination of exit laws

Let $\PP=(P_t)_{t\ge 0}$ be a sub-Markovian semigroup on $L^2(m)$, let $\b=(\b_t)_{t\ge 0}$ be a Bochner subordinator and let $\PP^\beta=(P_t^\beta)_{t\ge 0}$ be the subordinated semigroup of $\PP$ by means of $\b$, i.e. $P^\b_s:=\int_0^\ii P_r\,\b_s(dr)$. Let $\varphi:=(\varphi_t)_{t>0}$ be a $\PP$-exit law, i.e. $$ P_t\varphi_s= \varphi_{s+t}, \qquad s,t>0$$ and let $\varphi^\b_t:=\int_0^\ii \varphi_s\,\b_t(ds)$. Then $\varphi^\b:=(\varphi_t^\b)_{t>0}$ is a $\PP^\b$-exit law whenever it lies in $L^2(m)$. This paper is devoted to the converse problem when $\b$ is without drift. We prove that a $\PP^\b$-exit law $\psi:=(\psi_t)_{t>0}$ is subordinated to a (unique) $\PP$-exit law $\varphi$ (i.e. $\psi=\varphi^\b$) if and only if $(P_tu)_{t>0}\ss D(A^\b)$, where $u=\int_0^\infty e^{-s} \psi_s ds$ and $A^\b$ is the $L^2(m)$-generator of $\PP^\b$.