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Tightness of the maximum of branching random walk in random environment and zero-crossings of solutions to discrete parabolic differential equations

Jiří Černý, Flavio Dalessi

TL;DR

This work proves quenched tightness for the maximum of a branching random walk in a bounded i.i.d. random environment (BRWRE) by centering around quenched medians, without extra environmental assumptions beyond ellipticity. A central analytic tool is the monotonicity of zero-crossings for solutions to the discrete parabolic equation $\partial_t u(t,x)=\tfrac{1}{2}\Delta_d u(t,x)-\kappa(t,x)u(t,x)$, which is connected to the BRWRE through a randomized F-KPP representation. The authors develop discrete analogs of wave-propagation lemmas, construct tilted measures to control large-deviation behavior, and use annihilating-particle constructions to transfer PDE monotonicity to probabilistic tightness. Overall, the paper extends quenched tightness results to the discrete setting and clarifies the role of zero-crossings in controlling BRWRE fluctuations, with implications for BRWRE and BBMRE analyses in general environments.

Abstract

We study branching random walk on $\mathbb{Z}$ in a bounded i.i.d. random environment. For this process, we prove that, for almost every realization of the environment, the distributions of the maximally displaced particle (re-centered around their medians) are tight. This extends the result of arXiv:2408.01555 , where tightness was established in the annealed sense, and of arXiv:2212.12390 , where a similar quenched result was proved for branching Brownian motion in random environment. Our proof relies on studying certain discrete-space linear PDEs and establishing that the number of zero-crossings of their solutions is non-increasing in time. We observe that our technique requires no additional assumptions on the environment, in contrast to arXiv:2408.01555 , arXiv:2212.12390 .

Tightness of the maximum of branching random walk in random environment and zero-crossings of solutions to discrete parabolic differential equations

TL;DR

This work proves quenched tightness for the maximum of a branching random walk in a bounded i.i.d. random environment (BRWRE) by centering around quenched medians, without extra environmental assumptions beyond ellipticity. A central analytic tool is the monotonicity of zero-crossings for solutions to the discrete parabolic equation , which is connected to the BRWRE through a randomized F-KPP representation. The authors develop discrete analogs of wave-propagation lemmas, construct tilted measures to control large-deviation behavior, and use annihilating-particle constructions to transfer PDE monotonicity to probabilistic tightness. Overall, the paper extends quenched tightness results to the discrete setting and clarifies the role of zero-crossings in controlling BRWRE fluctuations, with implications for BRWRE and BBMRE analyses in general environments.

Abstract

We study branching random walk on in a bounded i.i.d. random environment. For this process, we prove that, for almost every realization of the environment, the distributions of the maximally displaced particle (re-centered around their medians) are tight. This extends the result of arXiv:2408.01555 , where tightness was established in the annealed sense, and of arXiv:2212.12390 , where a similar quenched result was proved for branching Brownian motion in random environment. Our proof relies on studying certain discrete-space linear PDEs and establishing that the number of zero-crossings of their solutions is non-increasing in time. We observe that our technique requires no additional assumptions on the environment, in contrast to arXiv:2408.01555 , arXiv:2212.12390 .
Paper Structure (11 sections, 13 theorems, 116 equations)

This paper contains 11 sections, 13 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Tightness of the maximum of branching random walk in random environment
  3. Monotonicity of the number of zero-crossings
  4. Branching random walk and the randomized F-KPP equation
  5. The randomized F-KPP equation
  6. Proof of Theorem \ref{['thm:tightness']} assuming Lemmas \ref{['lemma:wave_time']} and \ref{['lemma:wave']}
  7. Tilting and exponential change of measure
  8. Proofs of Lemmas \ref{['lemma:wave_time']} and \ref{['lemma:wave']}
  9. Proof of Theorem \ref{['Thm:zero-crossings']}
  10. Killed random walk and the Feynman--Kac formula
  11. Annihilating particles

Key Result

Theorem 2.1

For $\mathbb P$-almost every realization of the environment, the family $(M(t)- m(t) )_{t \geq 0}$ is tight under $\mathtt{P}_0^\xi$.

Theorems & Definitions (33)

  • Theorem 2.1
  • Theorem 3.1
  • Proposition 4.1: Proposition 7.1 in CD20
  • Remark 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Claim 4.5
  • proof
  • Claim 4.6
  • proof : Proof
  • ...and 23 more