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The branching random field

Published online by Cambridge University Press:  01 July 2016

Gail Ivanoff
Gail Ivanoff*
Affiliation:
University of Ottawa
*
∗Postal address: Faculty of Science and Engineering, Department of Mathematics, University of Ottawa, Ottawa, Ontario K1N 9B4, Canada.
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Abstract

The branching random field is studied under general branching and diffusion laws. Under a renormalization transformation it is shown that at finite fixed time the branching random field converges in law to a generalized Gaussian random field with independent increments. Very mild moment conditions are imposed on the branching process. Under more restrictive conditions on the branching and diffusion processes, the existence of a steady state distribution is proven in the critical case. A central limit theorem is proven for the renormalized steady state, but the limiting Gaussian random field no longer has independent increments. The covariance kernel is now a multiple of the potential kernel of the diffusion process.

Keywords

RANDOM FIELDPOINT PROCESSRANDOM MEASUREPROBABILITY GENERATING FUNCTIONALCUMULANT AND MOMENT DENSITY FUNCTIONSSIMPLE BRANCHING DIFFUSIONBRANCHING RANDOM FIELDSTEADY-STATE RANDOM FIELDGENERALIZED GAUSSIAN RANDOM FIELD
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 4 , December 1980 , pp. 825 - 847
Copyright
Copyright © Applied Probability Trust 1980 

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