Asymptotic behavior of the measure-valued branching process with immigration (Q687898)

The authors study a class of measure-valued branching processes with immigration defined as \(Y_ t=X_ t+I_ t\), \(t\geq 0\), where \(X_ t\) satisfies the usual branching property and \(I_ t\) with \(I_ 0=0\) is independent of \(X_ t\). A law of large numbers for \(t\to\infty\) is proved in the case of super-Brownian motion on \(\mathbb{R}^ d\), \(d\geq 3\). Equilibrium distributions and spatial transformations are also discussed.