On the Bochner subordination of exit laws

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