## Bernoulli

• Bernoulli
• Volume 25, Number 4A (2019), 2649-2695.

### Uniform sampling in a structured branching population

Aline Marguet

#### Abstract

We are interested in the dynamic of a structured branching population where the trait of each individual moves according to a Markov process. The rate of division of each individual is a function of its trait and when a branching event occurs, the trait of the descendants at birth depends on the trait of the mother and on the number of descendants. In this article, we explicitly describe the penalized Markov process, named auxiliary process, corresponding to the dynamic of the trait of a “typical” individual by giving its associated infinitesimal generator. We prove a Many-to-One formula and a Many-to-One formula for forks. Furthermore, we prove that this auxiliary process characterizes exactly the process of the trait of a uniformly sampled individual in a large population approximation. We detail three examples of growth-fragmentation models: the linear growth model, the exponential growth model and the parasite infection model.

#### Article information

Source
Bernoulli, Volume 25, Number 4A (2019), 2649-2695.

Dates
Revised: July 2018
First available in Project Euclid: 13 September 2019

https://projecteuclid.org/euclid.bj/1568362039

Digital Object Identifier
doi:10.3150/18-BEJ1066

Mathematical Reviews number (MathSciNet)
MR4003561

Zentralblatt MATH identifier
07110108

#### Citation

Marguet, Aline. Uniform sampling in a structured branching population. Bernoulli 25 (2019), no. 4A, 2649--2695. doi:10.3150/18-BEJ1066. https://projecteuclid.org/euclid.bj/1568362039

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