Measure-valued branching diffusions: Immigrations, excursions and limit theorems
Let \(M_\rho (S)\) be a suitable set of \(\sigma\)-finite measures on a locally compact metric space \(S\) and let \(\{\mathbb{P}_\mu : \mu \in M_\rho (S)\}\) be a realization of the measure-valued branching diffusion (MBD) on the path space \(C([0, \infty)\), \(M_\rho (S))\). An immigration diffusion \(\{\mathbb{Q}_\mu : \mu \in M_\rho (S)\}\) is defined by \(\mathbb{Q}_\mu = \mathbb{P}_\mu * \mathbb{Q}_0\) for a probability \(\mathbb{Q}_0\) on \(C([0, \infty)\), \(M_\rho (S))\), where \(*\) denotes the convolution operation. The first result is a characterization of the range of the probability \(\mathbb{Q}_0\), which is given by a 1-1 correspondence between the \(\mathbb{Q}_0\) and the entrance law for the underlying semigroup of the MBD. In particular, when the underlying process is an absorbing Brownian motion (ABM) in a smooth domain, the immigration consists of two parts, one is a measure supported by the interior domain and the other is a measure supported by the boundary. The latter one involves excursions of the ABM from the boundary. Secondly the immigration diffusion process of the super ABM over \((0, \infty)\) is discussed, for which a stochastic partial differential equation is derived. When the immigration measure has compact support, so does the immigration process. A limit theorem for the range of the immigration process is proved. Central limit theorems of immigration processes are discussed in the third part. Assuming that the underlying process is a Lévy process in \(\mathbb{R}^d\), one observes a ``clustering-diffusive dichotomy'' in the central limit theorems. More precisely, if the symmetrization of the underlying process is recurrent, then the limiting Gaussian field is spatially uniform, while if the symmetrization is transient, the limiting Gaussian field is spatially fluctuating.
scientific article; zbMATH DE number 836413