Limit theorems for Markov processes indexed by continuous time Galton–Watson trees

Bansaye, Vincent; Delmas, Jean‐François; Marsalle, Laurence; Tran, Viet Chi

doi:10.1214/10-aap757

Cited by 62 publications

(123 citation statements)

References 57 publications

(121 reference statements)

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“…A similar connection lies at the basis of an algorithm to measure large deviation functions using a population dynamics [15,16]. In the mathematical literature on branching processes, relations of this kind are known as Many-to-One formulas [17]; they explain the existence of a statistical biais, when choosing uniformly one individual in a population as opposed to following a lineage.…”

Section: Introductionmentioning

confidence: 99%

Linking lineage and population observables in biological branching processes

2019

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Using a population dynamics inspired by an ensemble of growing cells, a set of fluctuation theorems linking observables measured at the lineage and population levels are derived. One of these relations implies inequalities comparing the population doubling time with the mean generation time at the lineage or population levels. We argue that testing these inequalities provides useful insights into the underlying mechanism controlling the division rate in such branching processes.

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Section: Introductionmentioning

confidence: 99%

Linking lineage and population observables in biological branching processes

2019

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“…We also refer to the work of Bansaye and al. [5] for law of large numbers theorems using Many-to-One formulas. On the right-hand side of (1.1) appears a Markov process with penalized (or rewarded) trajectories which describes the dynamic of the trait of a typical individual.…”

Section: Introductionmentioning

confidence: 99%

Uniform sampling in a structured branching population

Marguet¹

2019

Bernoulli

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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.

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“…for which ergodic behaviour can be obtained through coupling arguments. This auxiliary semigroup describes the trajectory of a typical particle and has been used recently for the study of branching Markov processes in discrete and continuous time [4,5,2,34] and processes killed at a boundary [11,18,35]. We come back in Appendix A on the link between these topics in probability and ergodic estimates for semigroups.…”

Section: Introductionmentioning

confidence: 99%

Ergodic Behavior of Non-conservative Semigroups via Generalized Doeblin’s Conditions

2019

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We provide quantitative estimates in total variation distance for positive semigroups, which can be non-conservative and non-homogeneous. The techniques relies on a family of conservative semigroups that describes a typical particle and Doeblin's type conditions inherited from [11] for coupling the associated process. Our aim is to provide quantitative estimates for linear partial differential equations and we develop several applications for population dynamics in varying environment. We start with the asymptotic profile for a growth diffusion model with time and space non-homogeneity. Moreover we provide general estimates for semigroups which become asymptotically homogeneous, which are applied to an age-structured population model. Finally, we obtain a speed of convergence for periodic semigroups and new bounds in the homogeneous setting. They are illustrated on the renewal equation. Contents 39References 39 2010 Mathematics Subject Classification. Primary 35B40; Secondary 47A35, 47D06, 60J80, 92D25.as t → ∞ and the ergodic behavior of the auxiliary semigroup. The proof of this Lemma is essentially an adaptation of the method in [11,12] that we extend to general semigroups in non-homogeneous environment, while they restrict their study to absorbed Markov processes. This more general semigroup setting allows us to capture a wider range of applications, like the renewal equation we consider in Section 3. Moreover, we go beyond the contraction of the auxiliary semigroup P (t) and characterize the asymptotic behavior of (M 0,t ) t≥0 , which is a novelty compared to the previous results. More precisely, for any initial time s ≥ 0, we propose conditions involving a coupling probability measure ν which guarantee the existence of a positive bounded function h s and a family of probabilities (γ t ) t≥0 such that when t → ∞ sup µ TV ≤1 µM s,t − µ(h s )ν(m s,t )γ t TV = o ν(m s,t ) .

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Limit theorems for Markov processes indexed by continuous time Galton–Watson trees

Cited by 62 publications

References 57 publications

Linking lineage and population observables in biological branching processes

Linking lineage and population observables in biological branching processes

Uniform sampling in a structured branching population

Ergodic Behavior of Non-conservative Semigroups via Generalized Doeblin’s Conditions

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