Stochastic incompleteness for graphs and weak Omori-Yau maximum principle
Consider on a graph \(V\) a Laplacian of type \[ \Delta u(x)={1\over\mu(x)}\sum_yb(x,y)(u(x)-u(y)). \] It is associated to a continuous-time Markov chain on \(V\), and one wonders whether \(V\) is stochastically complete (the Markov chain does not explode in finite time). A necessary and sufficient condition which is an analogue of the weak Omori-Yau condition is given, and some sufficient conditions for stochastic completeness are deduced. The case where \(\mu\equiv1\), and \(b(x,y)\) is 1 if \(x\) and \(y\) are neighbours and 0 otherwise is studied in more detail, with also criteria for stochastic incompleteness. This work unifies previously known criteria.
scientific article