The Annals of Probability

Recurrence and transience of branching diffusion processes on Riemannian manifolds

Alexander Grigor'yan and Mark Kelbert

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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.

Article information

Ann. Probab., Volume 31, Number 1 (2003), 244-284.

First available in Project Euclid: 26 February 2003

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Zentralblatt MATH identifier

Primary: 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Branching process Brownian motion Riemannian manifold recurrence transience maximum principle gauge


Grigor'yan, Alexander; Kelbert, Mark. Recurrence and transience of branching diffusion processes on Riemannian manifolds. Ann. Probab. 31 (2003), no. 1, 244--284. doi:10.1214/aop/1046294311.

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