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2014 A compact containment result for nonlinear historical superprocess approximations for population models with trait-dependence
Sandra Kliem
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Electron. J. Probab. 19: 1-13 (2014). DOI: 10.1214/EJP.v19-3506

Abstract

We consider an approximating sequence of interacting population models with branching, mutation and competition. Each individual is characterized by its trait and the traits of its ancestors. Birth- and death-events happen at exponential times. Traits are hereditarily transmitted unless mutation occurs. The present model is an extension of the model used in [Méléard and Tran, EJP, 2012], where for large populations with small individual biomasses and under additional assumptions, the diffusive limit is shown to converge to a nonlinear historical superprocess. The main goal of the present article is to verify a compact containment condition in the more general setup of Polish trait-spaces and general mutation kernels that allow for a dependence on the parent's trait. As a by-product, a result on the paths of individuals is obtained. An application to evolving genealogies on marked metric measure spaces is mentioned where genealogical distance, counted in terms of the number of births without mutation, can be regarded as a trait. Because of the use of exponential times in the modeling of birth- and death-events the analysis of the modulus of continuity of the trait-history of a particle plays a major role in obtaining appropriate bounds.

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Sandra Kliem. "A compact containment result for nonlinear historical superprocess approximations for population models with trait-dependence." Electron. J. Probab. 19 1 - 13, 2014. https://doi.org/10.1214/EJP.v19-3506

Information

Accepted: 18 October 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1329.60305
MathSciNet: MR3272330
Digital Object Identifier: 10.1214/EJP.v19-3506

Subjects:
Primary: 60J80
Secondary: 60G57 , 60J68 , 60K35

Keywords: compact containment , evolution model , exponential rates , interacting particle systems , measure-valued processes on càdlàg functions , Nonlinear historical superprocess , Polish space , tightness

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Vol.19 • 2014
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