On the Bochner subordination of exit laws

Abstract

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 $\mathbb{P}^{\beta}$.