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<p>We consider branching processes for structured populations: each individual is characterised by a type or trait which belongs to a general measurable state space. We focus on the supercritical recurrent case, where the population may survive and grow and the trait distribution converges to a probability measure. The branching process is then expected to be driven by the positive triplet of first eigenvalue problem of the first moment semigroup. Under the assumption of convergence of the renormalized semigroup in weighted total variation norm, we prove strong convergence of the renormalized empirical measure and non-degeneracy of the limit of the martingale. Convergence is obtained under an <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(L\log L \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo>log</mo> <mi>L</mi> </mrow> </math></EquationSource> </InlineEquation> condition which provides a Kesten–Stigum result in infinite dimension and relaxes the uniform convergence of the first moment semigroup in the work of Asmussen and Hering in 1976. The techniques of proof combine families of martingales and contraction properties and the truncation procedure of Asmussen and Hering. These results unify part of the literature and capture new situations, as illustrated by absorbed branching diffusion, the house of cards model and some growth-fragmentation processes.</p>

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Strong law of large numbers and \(L\log L\) condition for supercritical branching processes

  • 期刊论文

  • 发表日期：

    2026年8月1日

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  • Vincent Bansaye,
  • Tresnia Berah,
  • Bertrand Cloez

摘要

We consider branching processes for structured populations: each individual is characterised by a type or trait which belongs to a general measurable state space. We focus on the supercritical recurrent case, where the population may survive and grow and the trait distribution converges to a probability measure. The branching process is then expected to be driven by the positive triplet of first eigenvalue problem of the first moment semigroup. Under the assumption of convergence of the renormalized semigroup in weighted total variation norm, we prove strong convergence of the renormalized empirical measure and non-degeneracy of the limit of the martingale. Convergence is obtained under an \(L\log L \) L log L condition which provides a Kesten–Stigum result in infinite dimension and relaxes the uniform convergence of the first moment semigroup in the work of Asmussen and Hering in 1976. The techniques of proof combine families of martingales and contraction properties and the truncation procedure of Asmussen and Hering. These results unify part of the literature and capture new situations, as illustrated by absorbed branching diffusion, the house of cards model and some growth-fragmentation processes.