True speed on branching-process trees with random environments

Determine the almost-sure speed of a random walk in a random environment on a branching process in a varying environment with random edge weights under Condition B$'$, and establish whether it lies between the random-spherically-symmetric-tree speed \(v_{Tw}\) and the annealed one-step drift bound \(v_{BT_w}\), with equality to \(v_{BT_w}\) when offspring fluctuations vanish and strict inequality in general.

Background

Unlike the random spherically symmetric case, a branching process in a varying environment is not spherically symmetric, so the distance from the root is not a Markov chain and the birth–death reduction used to compute vTwv_{Tw} fails. The paper derives an annealed drift quantity vBTwv_{BT_w} and proves an inequality showing that it is at least as large as the corresponding spherically symmetric speed, but does not identify the actual speed. The authors explain that regeneration times and the stationary environment viewed from the particle may be needed to resolve the problem.

References

Accordingly, we expect $v_{BT_w}$ to be a strict heuristic upper bound rather than the actual speed, and we note that identifying the true speed remains an open problem. Under Condition B$'$, the RWRE on $BPVETw$ has an a.s. speed $v\in[v_{Tw},v_{BT_w}]$, with $v=v_{BT_w}$ in the degenerate case of vanishing offspring fluctuation, i.e., $\mathbb{V}{\mathbb{P}(d_n+)\to0$ as $n\to \infty$, and $v<v{BT_w}$ in general.

— Random walk in a non-homogeneous random environment on some random trees and the non-negative integers  (2609.30648 - Oraby et al., 25 Sep 2026) in Remark following Theorem 2 and Conjecture 2; reiterated in Section Conclusion