Convolution semigroups associated with measure-valued branching processes
Suppose that \(E\) is a Lusin topological space and \(M (E)\) is the set of finite Borel measures on \(E\) endowed with the weak convergence topology. Let \(X\) be a regular measure-valued branching process (MB-process) with state space \(M(E)\) and transition semigroup \((Q_t)_{t \geq 0}\). Let \((N_t)_{t \geq 0}\) be a family of probability measures on \(M(E)\). We call \((N_t)_{t \geq 0}\) a (skew) convolution semigroup associated with \(X\) or \( (Q_t)_{t \geq 0}\) provided \(N_{r + 1} = (N_r Q_t)*N_t\) for all \(r,t \geq 0\), where ``\(*\)'' denotes the convolution operation. This relation holds if and only if \(Q^N_t (\mu, \cdot) : = Q_t (\mu, \cdot) * N_t\) defines a Markov semigroup \((Q^N_t)_{t \geq 0}\) on \(M(E)\). If \(Y\) is a Markov process in \(M(E)\) having transition semigroup \((Q^N_t)_{t \geq 0}\), we call it an immigration process associated with \(X\). It is proved that \((N_t)_{t \geq 0}\) is such a convolution semigroup if and only if there is an infinitely divisible probability entrance law \((K_t)_{t > 0}\) for \((Q_t)_{t \geq 0}\) such that \(\log L_{N_t} = \int^t_0 [\log L_{K_s}] ds\) for all \(t \geq 0\), where \(L_{N_t}\) and \(L_{K_s}\) denote the Laplace functionals of \(N_t\) and \(K_s\), respectively. It is pointed out that the immigration process associated with a Borel right MB-process does not always have a right continuous realization. See the following paper for some developments of the study.
infinitely divisible probability entrance law
scientific article; zbMATH DE number 883576