Absolute continuity of interacting measure-valued branching processes and its occupation-time processes (Q5955852)

Let \(X_t\) be the interaction measured-valued branching \(a\)-symmetric stable process over \(\mathbb{R}^d\) \((1< a< 2)\) contructed by \textit{S. Méléard} and \textit{S. Roelly} [Stochastics Stochastics Rep. 44, No. 1/2, 103-121 (1993; Zbl 0786.60065)]. The authors present: Theorem 1. \(X_t\), the interacting measure-valued branching \(a\)-symmetric stable process on \(\mathbb{R}\), contains three properties (see this note). Theorem 2. If \(d\leq 3\), then \(Y_t\), the occupation-time process of the interacting measure-valued branching Brownian motion on \(\mathbb{R}^t\), is almost surely absolutely continuous with respect to \(dx\) for \(t\geq 0\). For other details see the authors references.