Recurrence and transience of branching diffusion processes on Riemannian manifolds

• Alexander Grigor'yan ^[2] ; Mark Kelbert ^[1]
  1. [1] University of Wales

     University of Wales

     Castle, Reino Unido

     
  2. [2] Imperial College
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 31, Nº. 1, 2003, págs. 244-284
• Idioma: inglés
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• Resumen

  • We relate the recurrence and transience of a branching diffusion process on a Riemannian manifold M to some properties of a linear elliptic operator onM (including spectral properties). There is a trade-off between the tendency of the transient Brownian motion to escape and the birth process of the new particles. If the latter has a high enough intensity then it may override the transience of the Brownian motion, leading to the recurrence of the branching process, and vice versa. In the case of a spherically symmetric manifold, the critical intensity of the population growth can be found explicitly.