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1 results

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

  • Part of
  • Wenming Hong, Shengli Liang
  • Journal:
    Advances in Applied Probability , First View
    Published online by Cambridge University Press:
    02 December 2024, pp. 1-35
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  • 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 a time-inhomogeneous random environment (RWRE), which is related to 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 in itself.