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A Spine Approach to Branching Diffusions with Applications to Lp-Convergence of Martingales

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Séminaire de Probabilités XLII

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 1979))

Abstract

We present a modified formalization of the ‘spine’ change of measure approach for branching diffusions in the spirit of those found in Kyprianou [40] and Lyons et al. [44, 43, 41]. We use our formulation to interpret certain ‘Gibbs-Boltzmann’ weightings of particles and use this to give an intuitive proof of a general ‘Many-to-One’ result which enables expectations of sums over particles in the branching diffusion to be calculated purely in terms of an expectation of one ‘spine’ particle. We also exemplify spine proofs of the Lp-convergence (p ≥ 1) of some key ‘additive’ martingales for three distinct models of branching diffusions, including new results for a multi-type branching Brownian motion and discussion of left-most particle speeds.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Bath, BA2 7AY, Claverton Down, Bath, UK

    Robert Hardy & Simon C. Harris

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  1. Robert Hardy
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  2. Simon C. Harris
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Correspondence to Simon C. Harris .

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Editors and Affiliations

  1. Laboratoire de Mathématiques, Université de Versailles-St-Quentin, avenue des États-Unis 45, Versailles, 78035, France

    Catherine Donati-Martin

  2. Inst. Recherche, Mathématiques Avancées, Université Strasbourg I, rue René Descartes 7, Strasbourg CX, 67084, France

    Michel Émery

  3. Laboratoire de Mathematiques, Université de Versailles, av. des Etats-Unis 45, Versailles CX, 78035, France

    Alain Rouault

  4. CNRS UMR 6623, Lab. de Mathématiques, Université Besancon, route de Gray 16, Besancon CX, 25030, France

    Christophe Stricker

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Hardy, R., Harris, S.C. (2009). A Spine Approach to Branching Diffusions with Applications to Lp-Convergence of Martingales. In: Donati-Martin, C., Émery, M., Rouault, A., Stricker, C. (eds) Séminaire de Probabilités XLII. Lecture Notes in Mathematics(), vol 1979. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01763-6_11

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