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Skeletal stochastic differential equations for superprocesses

Part of: Markov processes Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  23 November 2020

Dorottya Fekete ,
Joaquin Fontbona  and
Andreas E. Kyprianou
Dorottya Fekete*
Affiliation:
University of Exeter
Joaquin Fontbona*
Affiliation:
Universidad de Chile
Andreas E. Kyprianou*
Affiliation:
University of Bath
*
*Postal address: College of Engineering, Mathematics and Physical Sciences, Harrison Building, University of Exeter, North Park Road, Exeter EX4 4QF, UK. Email address: d.fekete@exeter.ac.uk
**Postal address: Center for Mathematical Modeling, DIM CMM, UMI 2807 UChile-CNRS, Universidad de Chile, Beauchef 851, Edificio Norte – Piso 7, Santiago, Chile. Email address: fontbona@dim.uchile.cl
***Postal address: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK. Email address: a.kyprianou@bath.ac.uk
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Abstract

It is well understood that a supercritical superprocess is equal in law to a discrete Markov branching process whose genealogy is dressed in a Poissonian way with immigration which initiates subcritical superprocesses. The Markov branching process corresponds to the genealogical description of prolific individuals, that is, individuals who produce eternal genealogical lines of descent, and is often referred to as the skeleton or backbone of the original superprocess. The Poissonian dressing along the skeleton may be considered to be the remaining non-prolific genealogical mass in the superprocess. Such skeletal decompositions are equally well understood for continuous-state branching processes (CSBP).

In a previous article [16] we developed an SDE approach to study the skeletal representation of CSBPs, which provided a common framework for the skeletal decompositions of supercritical and (sub)critical CSBPs. It also helped us to understand how the skeleton thins down onto one infinite line of descent when conditioning on survival until larger and larger times, and eventually forever.

Here our main motivation is to show the robustness of the SDE approach by expanding it to the spatial setting of superprocesses. The current article only considers supercritical superprocesses, leaving the subcritical case open.

Keywords

SuperprocessesSDEsskeletal decomposition

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 60G99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 4 , December 2020 , pp. 1111 - 1134
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Copyright
© Applied Probability Trust 2020

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