TITLE: Liouville Theorem
QUESTION [6 upvotes]: Liouville theorem for superharmonic functions states that

Any bounded function $f:\mathbb R^n\to\mathbb R$ admitting an inequality $\Delta f\leq 0$ on $\mathbb R^n$ is a constant function. 

Here $\Delta$ is a Laplacian. I wonder what are the extension of this theorem to other class of operators, i.e. what are necessary and what are sufficient conditions for this theorem to hold. 
I am especially interested if there is such a theorem for a discrete Laplacian.

REPLY [2 votes]: I thought that discrete Liouville theorem for the lattice $\mathbb{Z}^2$ was proved by Heilbronn (On discrete harmonic functions. - Proc. Camb. Philos. Soc. , 1949, 45, 194-206). But recently I knew from  Alexander Khrabrov that there is an older article of Capoulade with almost the same result (Sur quelques proprietes des fonctions harmoniques et des fonctions preharmoniques, - Mathematica (Cluj), 6 (1932), 146-151.) 
Unfortunately the last article is not available for me.