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Ancestral lineages for a branching annihilating random walk

Pascal Oswald

Abstract

We study ancestral lineages of individuals of a stationary discrete-time branching annihilating random walk (BARW) on the $d$-dimensional lattice $\mathbb{Z}^d$. Each individual produces a Poissonian number of offspring with mean $μ$ which then jump independently to a uniformly chosen site with a fixed distance $R$ of their parent. By interpreting the ancestral lineage of such an individual as a random walk in a dynamical random environment, we obtain a law of large numbers and a functional central limit theorem for the ancestral lineage.

Ancestral lineages for a branching annihilating random walk

Abstract

We study ancestral lineages of individuals of a stationary discrete-time branching annihilating random walk (BARW) on the -dimensional lattice . Each individual produces a Poissonian number of offspring with mean which then jump independently to a uniformly chosen site with a fixed distance of their parent. By interpreting the ancestral lineage of such an individual as a random walk in a dynamical random environment, we obtain a law of large numbers and a functional central limit theorem for the ancestral lineage.
Paper Structure (11 sections, 10 theorems, 68 equations, 1 figure)

This paper contains 11 sections, 10 theorems, 68 equations, 1 figure.

Table of Contents

  1. Model and main result
  2. Abstract conditions and proof of Theorem \ref{['thm:main']}
  3. Conditions on the environment
  4. Conditions on the random walk
  5. Renormalisation construction
  6. Properties of pm
  7. Local densities and goodness
  8. Concentration and comparison of local densities with deterministic profiles
  9. Proof of Proposition \ref{['prop:main_proposition']}
  10. Proof of Proposition \ref{['prop:speed_bound']}
  11. Funding

Key Result

Proposition 1.2

Let $\mu\in (1,e^2)$.

Figures (1)

  • Figure 1: Sketch of the density profiles $\zeta_{k}^{R,\pm}$ (in orange) from BARW for dimension $d=1$. The bottom shows the profiles $\zeta_0^{R,\pm}$, which then expand to $\zeta_k^{R,\pm}$ in the top image. The profiles are chosen such that the distance of the profiles to $\theta_\mu$ is smaller than $\varepsilon_\mathrm{FP}$ in the central constant part of the profiles.

Theorems & Definitions (22)

  • Remark 1.1
  • Proposition 1.2: Survival and complete convergence, BARW
  • Theorem 1.3
  • Proposition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof : Proof of Theorem \ref{['thm:main']}
  • Lemma 3.1
  • proof
  • Remark 3.2
  • ...and 12 more