Opuscula Math. 31, no. 2 (), 195-207
http://dx.doi.org/10.7494/OpMath.2011.31.2.195
Opuscula Mathematica

# 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 $$\mathbb{P}^{\beta}$$.
Keywords: sub-Markovian semigroup, exit law, subordinator, Bernstein function, Bochner subordination.
Mathematics Subject Classification: 47A50, 47D03, 39B42.