In this note we prove an optimal volume growth condition for stochastic completeness of graphs under very mild assumptions. This is realized by proving a uniqueness class criterion for the heat equation which is an analogue to a corresponding result of Grigor'yan on manifolds. This uniqueness class criterion is shown to hold for graphs that we call globally local, i.e., graphs where we control the jump size far outside. The transfer from general graphs to globally local graphs is then carried out via so called refinements.

中文翻译：

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关于图的唯一性类、随机完备性和体积增长

在本笔记中，我们在非常温和的假设下证明了图的随机完整性的最佳体积增长条件。这是通过证明热方程的唯一性类标准来实现的，该标准类似于 Grigor'yan 在流形上的相应结果。这个唯一性等级标准被证明适用于我们称之为全局局部的图，即我们控制远处跳跃大小的图。然后通过所谓的细化执行从一般图到全局局部图的转换。