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arXiv · 2306.15204

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

Wenming Hong·Shengli Liang
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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.

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Wenming Hong, Shengli Liang. 2023-06-27. Convergence of the derivative martingale for the branching random walk in time-inhomogeneous random environment. https://doi.org/10.1017/apr.2024.55

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