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Regularity results of the speed of biased random walks on Galton-Watson trees
Yuki Tokushige · 2018-11-13 · via math.PR updates on arXiv.org

We prove that the speed of $λ$-biased random walks on a supercritical Galton-Watson tree without leaves is differentiable when $λ\in(0,1)$, and give an expression of the derivative using a certain 2-dimensional Gaussian random variable. The proof heavily uses the renewal structure of Galton-Watson trees that was introduced by Lyons-Pemantle-Peres.

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