Abstract
Let P = (P-t)(t >= 0) be a sub-Markovian semigroup on L-2 (m), let beta = (beta(t))(t >= 0) be a Bochner subordinator and let P-beta = (P-beta(t))(t >= 0) be the subordinated semigroup of P by means of beta, i.e. P-s(beta) : = integral(infinity)(0) P-r beta(s) (dr). Let phi : = (phi(t))(t > 0) be a P-exit law, i.e.
p(t) phi(s) = phi(s+t), s,t > 0
and let phi(beta)(t) : = integral(infinity)(0) phi(s) beta(t) (ds). Then phi(beta) : = (phi(beta)(t))(t>0) is a 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 P-beta-exit law psi : = (psi(t))(t > 0) is subordinated to a (unique) P-exit law phi (i.e. psi = phi(beta)) if and only if (P(t)u)(t > 0) subset of D (A(beta)), where u = integral(infinity)(0) e(-s) psi(s) ds and A(beta) is the L-2(m)-generator of P-beta.