A Liouville Theorem for Domains in Graphs

We show a Liouville theorem for domains in graphs with Dirichlet boundary conditions.

More specifically, we characterize the non-existence of non-zero bounded harmonic functions.

Since Dirichlet boundary conditions give rise to Laplacians with a positive killing term, we can first characterize the validity of the Liouville theorem by the fact that the Green operator applied to the killing terms is equal to 1, or in other words, that the constant function $1$ is a potential.

Secondly, we derive a characterization in terms of stochastic completeness at infinity and and total loss of heat.

Thirdly, we give a characterization in terms of a Green formula for superharmonic potentials.

Finally, we investigate the Liouville property in terms of recurrence and transience of the graph without killing term.

As an application, we consider subsets of the Euclidean space such as cones and percolation clusters, and weakly spherically symmetric graphs.