×

Convergence of a ‘Gibbs-Boltzmann’ random measure for a typed branching diffusion. (English) Zbl 0985.60053

Azéma, J. (ed.) et al., Séminaire de Probabilités XXXIV. Berlin: Springer. Lect. Notes Math. 1729, 239-256 (2000).
Introduction: We consider certain ‘Gibbs-Boltzmann’ random measures which are derived from the positions of particles in the typed branching diffusion introduced by S. C. Harris and D. Williams [Astérisque 236, 133-154 (1996; Zbl 0857.60088)]. We prove that, as time progresses, these random measures almost surely converge to deterministic normal distributions (corresponding to the type distributions of the ‘dominant’ particles contributing to the measure at large times). The random measures considered are closely linked to some martingales of fundamental importance in the study of the long-term behaviour of the branching diffusion. The method of proof relies on a martingale expansion and the study of the behaviour of various families of martingales.
For the entire collection see [Zbl 0940.00007].

MSC:

60G57 Random measures
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K35 Interacting random processes; statistical mechanics type models; percolation theory

Citations:

Zbl 0857.60088
PDFBibTeX XMLCite
Full Text: Numdam EuDML