An occupation time approach for convergence of measure-valued processes, and the death process of a branching system
Let \({\mathcal M}_ p (R^ d)\) be a space of Radon measures \(\mu\) on \(R^ d\) such that \(\int (1 + | x|^ 2)^{-p} d\mu < \infty\) and let \(D(R_ +, {\mathcal M}_ p (R^ d))\) be a space of right-continuous with left limits functions from \(R_ +\) into \({\mathcal M}_ p (R^ d)\), endowed with the Skorokhod topology. The \((d,\alpha, \beta, b)\)- superprocess \(X = \{X(t), t \geq 0\}\) is a time-homogeneous Markov process with paths in \(D(R_ +, {\mathcal M}_ p (R^ d))\) whose transition Laplace functional is given by some nonlinear differential equation. It is constructed a family of time and \(n\) dependent Markov branching particle systems on \(R^ d\), and then particle processes \(N^ n = \{N^ n (t)\), \(t \geq 0\}\) of the particle systems which are the sum of the delta-functions of locations of the particles present at time \(t\) and death processes \(D^ n = \{D^ n (t)\), \(t \geq 0\}\) of the particle systems which are the sum of the delta-functions of locations of the particles that have died up to time \(t\). Then it is defined \(X^ n = n^{-1} N^ n\) and \(G^ n = m_ n ^{-1} D^ n\), where \(m_ n = cn^{1 + \beta}\) (it is given particle mass \(n^{-1}\) and \(m_ n^ {- 1}\)). The main result is: If \(X^ n(0) \to X(0)\), then for each \(t > 0\) \[ \left(X^ n (t), \int_ 0^ t X^ n(s) ds,\;G^ n(t)\right)\to\left(X(t), \int_ 0^ t X(s) ds, G(t)\right) \text{ as } n \to \infty, \] and it is given the Laplace functional of the limit as a solution of some nonlinear differential equation.
scientific article; zbMATH DE number 681919
scientific article