Random walks in dynamic random environments and ancestry under local population regulation

Birkner, Matthias; Černý, Jǐŕı; Depperschmidt, Andrej

doi:10.1214/16-ejp4666

Cited by 11 publications

(17 citation statements)

References 27 publications

(52 reference statements)

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“…This is an example of a random walk in a dynamic random environment (abbreviated by RWDRE), a class of models that have recently been the subject of intensive studies in the mathematical literature (see e.g. Andres et al, 2018, Avena et al, 2016, Birkner et al, 2016, Deuschel et al, 2017+, Hilário et al, 2015, Huveneers and Simenhaus, 2015, Redig and Völlering, 2013.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The contact process as seen from a random walk

Bethuelsen¹

2018

ALEA

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We consider a random walk on top of the supercritical contact process. In particular, we focus on the "contact process as seen from the random walk". Under the assumption that the infection rate of the contact process is large or the jump rate of the random walk is small, we show that this process has at most two extremal measures. Moreover, the convergence to these extremal measures is characterised by whether the contact process survives or dies out, similar to the complete convergence theorem known for the ordinary contact process. Using this, we prove a law of large numbers for the random walk. Our arguments apply to the processes considered on Z d , d ≥ 1, and under general assumptions on the jump probabilities of the random walk.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

The contact process as seen from a random walk

Bethuelsen¹

2018

ALEA

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“…Many challenging questions about the long-time behaviour of such continuous-mass interacting multi-type systems remain open. It is conceivable that the regeneration constructions for ancestral lineages we investigated in [6] and [7] might be adaptable to this context, and that this could enrich the pertinent 'tool box. '…”

Section: Conditioning On ηmentioning

confidence: 99%

Ancestral lineages in spatial population models with local regulation

Birkner,

Gantert

2019

Preprint

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We give a short overview on our work on ancestral lineages in spatial population models with local regulation. We explain how an ancestral lineage can be interpreted as a random walk in a dynamic random environment. Defining regeneration times allows to prove central limit theorems for such walks. We also consider several ancestral lineages in the same population and show for one prototypical example that in one dimension the corresponding system of coalescing walks converges to the Brownian web.

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“…We next state a lemma which is useful for most of our proofs. The proof and the statement is inspired by [5,Lemma 2.11]. For its statement, recall (1.5) and note that, as follows from the graphical representation,…”

Section: The Contact Processmentioning

confidence: 99%