Stochastic incompleteness for graphs and weak Omori–Yau maximum principle

We prove an analogue of the weak Omori–Yau maximum principle and Khasʼminskiiʼs criterion for graphs in the general setting of Keller and Lenz. Our approach naturally gives the stability of stochastic incompleteness under certain surgeries of graphs. It allows to develop a unified approach to all known criteria of stochastic completeness/incompleteness, as well as to obtain new criteria.