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

TL;DR

This paper establishes a necessary and sufficient condition for the non-trivial limit of the derivative martingale in a branching random walk with a time-inhomogeneous random environment, using advanced techniques in RWRE analysis.

Contribution

It introduces a new criterion for the derivative martingale's non-triviality in BRWRE and develops Tanaka's decomposition for RWRE conditioned to stay non-negative.

Findings

Derived a necessary and sufficient condition for martingale non-triviality.

Developed Tanaka's decomposition for RWRE conditioned to stay positive.

Linked BRWRE behavior to RWRE properties via many-to-one formula.

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.