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Probability Theory and Related Fields

, Volume 173, Issue 3–4, pp 999–1062 | Cite as

An invariance principle for branching diffusions in bounded domains

  • Ellen PowellEmail author
Article

Abstract

We study branching diffusions in a bounded domain D of \(\mathbb {R}^d\) in which particles are killed upon hitting the boundary \(\partial D\). It is known that any such process undergoes a phase transition when the branching rate \(\beta \) exceeds a critical value: a multiple of the first eigenvalue of the generator of the diffusion. We investigate the system at criticality and show that the associated genealogical tree, when the process is conditioned to survive for a long time, converges to Aldous’ Continuum Random Tree under appropriate rescaling. The result holds under only a mild assumption on the domain, and is valid for all branching mechanisms with finite variance, and a general class of diffusions.

Mathematics Subject Classification

60J60 60J80 

Notes

Acknowledgements

I would particularly like to thank Nathanaël Berestycki, for suggesting this problem and for many helpful discussions. I am also grateful to the anonymous referee for numerous useful comments and suggestions.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Eidgenossische Technische Hochschule ZurichZurichSwitzerland

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