Remarks on ergodicity and invariant occupation measure in branching diffusions with immigra\-tion (Q2573628)

Let \(\{\eta_s\}\) be a spatially subcritical branching diffusion with immigration on \(\mathbb{R}^d\), constructed from the solution of a SDE, a space-dependent branching occurrence density, a space-dependent, local offspring law, a constant immigration occurrence density, and a constant immigration distribution. Denote by \(S\) the configuration space, by \(\Delta\) the void configuration, by \(x(A)\) the number of components of configuration \(x\) in \(A\), and by \(R\) the first jump time leading into \(\Delta\). Define the measures \(m(F)= {\mathbf E}_\Delta\int^R_0 1_F(\eta_s)\,ds\) on \(S\) and \(\overline m(A)= \int_S x(A)m(dx)\) on \(\mathbb{R}^d\). The authors give conditions in terms of model parameters, under which \(\eta_s\) does not explode in finite time, is Harris recurrent, admitting \(\Delta\) as a recurrent state, and such that both \(m\) and \(\overline m\) are finite. The invariant occupation measure \(\overline m\) is studied in detail, giving, in particular, conditions for the existence and smoothness of a Lebesgue density.