Conditioned superprocesses and their weighted occupation times
Consider a superprocess \(X\) (an \(M_ F(E)\)-valued Markov process with Laplace functional, where \(M_ F(E)\) denotes the space of finite measures on a locally compact separable metric space \(E\) with its Borel \(\sigma\)-field). Because of the fact that the initial measure \(\mu\) of \(X_ t\) is finite, the process \(X_ t\) will die in finite time, \({\mathbf P}^ \mu\)-a.s., in other words, \(X_ t = 0\) for sufficiently large \(t\). So-called conditioned superprocesses, obtained by starting \(X\) at an initial measure \(\mu \in M_ F(E) \setminus \{0\}\) and conditioning on the event that the process stays alive forever, are considered. Then the asymptotic behavior of the weighted occupation time (whether the total amount of mass which ever visits a fixed set is finite or infinite) is studied. Necessary and sufficient conditions, based on the asymptotics of the underlying motion, for the total occupation time to be infinite are given. Some special cases are investigated. It is also proved that, when properly scaled, the occupation times for conditioned super-Brownian motion on \({\mathbf R}^ d\) converge to a function of a local time when \(d \leq 3\).
scientific article; zbMATH DE number 724095
scientific article