Let $ \widetilde{X} $ be a locally finite Gromov hyperbolic graph whose Gromov boundary consists of infinitely many points and with a cocompact isometric action of a discrete group $ \Gamma $. We show the uniform Ancona inequality for the Brownian motion which implies that the $ \lambda $-Martin boundary coincides with the Gromov boundary for any $ \lambda \in [0, \lambda_0], $ in particular at the bottom of the spectrum $ \lambda_0 $.

中文翻译：

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Gromov双曲度量图的布朗运动的马丁边界

令$ \ widetilde {X} $为局部有限的Gromov双曲图，其Gromov边界由无限多个点组成，并且具有离散组$ \ Gamma $的共同压缩等距作用。我们显示了布朗运动的一致Ancona不等式，这意味着$ \ lambda $ -Martin边界与[$，\ lambda_0]中的任何$ \ lambda \，尤其是频谱底部$的Gromov边界重合。 \ lambda_0 $。