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Quenched CLT for ancestral lineages of logistic branching random walks

Matthias Birkner, Andrej Depperschmidt, Timo Schlüter

Abstract

We consider random walks in dynamic random environments which arise naturally as spatial embeddings of ancestral lineages in spatial locally regulated population models. In particular, as the main result, we prove the quenched central limit theorem for a random walk in dynamic random environment generated by time reversal of logistic branching random walks in a regime where the population density is sufficiently high. As an important tool we consider as auxiliary models random walks in dynamic random environments defined in terms of the time-reversal of oriented percolation. We show that the quenched central limit theorem holds if the influence of the random medium on the walks is suitably weak. The proofs of the quenched central limit theorems in these models rely on coarse-graining arguments and a construction of regeneration times for a pair of conditionally independent random walks in the same medium, combined with a coupling that relates them to a pair of independent random walks in two independent copies of the medium.

Quenched CLT for ancestral lineages of logistic branching random walks

Abstract

We consider random walks in dynamic random environments which arise naturally as spatial embeddings of ancestral lineages in spatial locally regulated population models. In particular, as the main result, we prove the quenched central limit theorem for a random walk in dynamic random environment generated by time reversal of logistic branching random walks in a regime where the population density is sufficiently high. As an important tool we consider as auxiliary models random walks in dynamic random environments defined in terms of the time-reversal of oriented percolation. We show that the quenched central limit theorem holds if the influence of the random medium on the walks is suitably weak. The proofs of the quenched central limit theorems in these models rely on coarse-graining arguments and a construction of regeneration times for a pair of conditionally independent random walks in the same medium, combined with a coupling that relates them to a pair of independent random walks in two independent copies of the medium.
Paper Structure (26 sections, 47 theorems, 425 equations, 3 figures)

This paper contains 26 sections, 47 theorems, 425 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Models and main results
  3. Logistic branching random walk
  4. Random walks in oriented percolation: auxiliary model
  5. General model
  6. Outline and proof strategies
  7. Random walks in oriented percolation: Proof of Theorem \ref{['thm:LLNuCLTmodel1']}
  8. Construction of simultaneous regeneration times for two walks
  9. Coupling pairs of walks with copies in independent environments
  10. Auxiliary results
  11. Quenched CLT along regeneration times
  12. Proof of Theorem \ref{['thm:LLNuCLTmodel1']} for $d \ge 2$
  13. Proof of Theorem \ref{['thm:LLNuCLTmodel1']} for $d=1$
  14. A more abstract setting
  15. Dimension $d=1$
  16. ...and 11 more sections

Key Result

Theorem 2.2

For $d\ge 1$, if the competition kernel $\lambda$ and the migration kernel $p$ satisfy Assumption ass:log_branch_walk, the ancestral lineage $X$ satisfies a quenched central limit theorem, i.e. for any continuous and bounded function $f$ on $\mathbb{R}^d$, we have where $\Phi(f)=\int_{\mathbb{R}^d}f(x)\Phi(dx)$ and $\Phi$ is a nontrivial centred $d$-dimensional normal distribution.

Figures (3)

  • Figure 1: Double cone with a double cone shell (grey), a time slice of the middle tube (blue), and a path of a random walk crossing the double cone shell from outside to inside (red).
  • Figure 2: A cross section of the cone shell including the middle tube $\mathcal{M}$ (blue) and a path $\gamma$ (red) crossing the cone shell from the outside to the inside of the double cone and hitting at least one point in $\mathcal{M}$ (blue dot).
  • Figure 3: Double cone and double cone shell in case $d=1$

Theorems & Definitions (103)

  • Theorem 2.2: Quenched CLT
  • Theorem 2.7: Quenched CLT for the auxiliary model
  • Remark 2.10
  • Lemma 2.11: Coupling with oriented percolation
  • Theorem 2.15: Quenched CLT for a more general class of environments
  • Remark 2.16: Theorem \ref{['thm:quenched_clt_log_branching']} as a corollary
  • Remark 2.17
  • Lemma 3.1: 2-cone analogue of Lemma 2.13 from BirknerCernyDepperschmidt2016
  • Remark 3.2
  • proof : Proof of Lemma \ref{['th:lemma2.13Analog']}
  • ...and 93 more