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Convergence of the derivative martingale for the branching random walk in time-inhomogeneous random environment

Wenming Hong, Shengli Liang

TL;DR

This work analyzes a discrete-time BRW on $\mathbb{R}$ in an i.i.d. time-inhomogeneous environment (BRWRE) and derives a sharp, necessary-and-sufficient criterion for the non-trivial limit of the derivative martingale $D_n:=\sum_{|u|=n} V(u)e^{-V(u)}$. The authors connect BRWRE to a time-inhomogeneous RWRE via a time-dependent many-to-one lemma, and they develop a quenched harmonic function $U(\xi,y)$ together with Tanaka-type decomposition for the RWRE conditioned to stay non-negative. A truncated derivative martingale $D^{(\beta)}_n$ and a spinal decomposition are then employed to establish $L^1$ convergence criteria and to characterize degeneracy vs. non-degeneracy of the limit in terms of the one-step functionals $Y:=\sum_{|u|=1} e^{-V(u)}$ and $Z:=\sum_{|u|=1} V(u) e^{-V(u)} 1_{\{V(u)\ge0\}}$, via the condition $\mathbb{E}[Y\log^2_+Y+Z\log_+Z]<\infty$. The results extend the classical BRW derivative-martingale theory to time-inhomogeneous environments and provide exact thresholds for the fixed-point structure of the associated smoothing transformation.

Abstract

Consider a branching random walk on the real line with a random environment in time (BRWRE). A necessary and sufficient condition for the non-triviality of the limit of the derivative martingale is formulated. To this end, we investigate the random walk in time-inhomogeneous random environment (RWRE), which related the BRWRE by the many-to-one formula. The key step is to figure out Tanaka's decomposition for the RWRE conditioned to stay non-negative (or above a line), which is interesting itself as well.

Convergence of the derivative martingale for the branching random walk in time-inhomogeneous random environment

TL;DR

This work analyzes a discrete-time BRW on in an i.i.d. time-inhomogeneous environment (BRWRE) and derives a sharp, necessary-and-sufficient criterion for the non-trivial limit of the derivative martingale . The authors connect BRWRE to a time-inhomogeneous RWRE via a time-dependent many-to-one lemma, and they develop a quenched harmonic function together with Tanaka-type decomposition for the RWRE conditioned to stay non-negative. A truncated derivative martingale and a spinal decomposition are then employed to establish convergence criteria and to characterize degeneracy vs. non-degeneracy of the limit in terms of the one-step functionals and , via the condition . The results extend the classical BRW derivative-martingale theory to time-inhomogeneous environments and provide exact thresholds for the fixed-point structure of the associated smoothing transformation.

Abstract

Consider a branching random walk on the real line with a random environment in time (BRWRE). A necessary and sufficient condition for the non-triviality of the limit of the derivative martingale is formulated. To this end, we investigate the random walk in time-inhomogeneous random environment (RWRE), which related the BRWRE by the many-to-one formula. The key step is to figure out Tanaka's decomposition for the RWRE conditioned to stay non-negative (or above a line), which is interesting itself as well.
Paper Structure (14 sections, 14 theorems, 160 equations)

This paper contains 14 sections, 14 theorems, 160 equations.

Table of Contents

  1. Introduction and main result
  2. Quenched harmonic function and conditioned random walk
  3. The many-to-one lemma
  4. Quenched harmonic function
  5. Random walk conditioned to stay above a line: quenched and annealed
  6. Quenched probability $\mathbb{P}^{+,(\beta)}_{\xi}\left(x;dy\right)$ and annealed renewal measure $\mathcal{R}(dy)$
  7. Quenched Tanaka's decomposition
  8. Application of Tanaka's decomposition
  9. Change of measure and spinal decomposition
  10. Truncated derivative martingale and change of probabilities
  11. Spinal decomposition of the time-inhomogeneous branching random walk
  12. Proof of Theorem \ref{['main result']}
  13. The sufficient condition
  14. The necessary condition

Key Result

Theorem 1.1

Under the assumptions $\left(ass1\right)$, $\left(boundary condition\right)$ and $\left(2 more moment\right)$, we have $\left(1\right)$ The derivative martingale $\left(D_{n},n\geq0\right)$ converges almost surely to a non-negative finite limit which we denote by $D_{\infty}$, i.e., $\left(2\right)$ For almost all $\xi$, $D_{\infty}$ is non-triviality if and only if more precisely, where $\log_

Theorems & Definitions (25)

  • Theorem 1.1
  • Lemma 2.1: Many-to-one
  • Proposition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5: Quenched Tanaka's decomposition
  • proof
  • Proposition 2.6: Annealed excursion distribution
  • proof
  • ...and 15 more