A functional ergodic theorem for the occupation time process of a branching system
The author considers a system of particles in \(\mathbb{R}^d\) which starts from a Poisson random measure with the intensity being the Lebesgue measure. The particles move independently according to a spherically symmetric \(\alpha\)-stable Lévy motion. They undergo critical branching and the pgf of the branching mechanism is \(G(s)=s+(1+\beta)^{-1}(1-s)^{1+\beta}\) for some \(\beta\in (0,1]\). It is proved that in the critical dimension \(d=\alpha/\beta\) the rescaled occupation time process of the just described \((d,\alpha,\beta)\)-branching particle system converges in law on \(C([0,1], S^\prime(\mathbb{R}^d))\) to a non-trivial non-negative infinitely divisible process with explicitly given Laplace-Stieltjes transform.
\((d,\alpha, \beta)\)-branching particle system
scientific article; zbMATH DE number 5285352
scientific article