Liouville Theorem 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.
 A: If you are interested in the case for the discrete Laplacian, check out the paper of Rigoli, Salvatori, and Vignati titled "Liouville properties on graphs" (DOI: 10.1112/S0025579300012031). 
Among the results proven is the following:

Let $G$ be a graph and let $q$ be an arbitrary point in $G$. Let $u$ be a $p$-subharmonic function on $G$ for $p > 1$. Suppose that for all $R$ sufficiently large $$ \sup_{B_R(q)} u \lesssim \frac{(R\log R)^{(p-1)/p}}{|S_R(q)|^{1/p}} $$ and $$ |S_R(q)| \lesssim (R\log R)^{p-1} $$ where $S_R(q) = B_R(q) \setminus B_{R-1}(q) $ is the "sphere of the radius $R$", then $u$ is constant. 

The requirement on the volume growth rate of balls of radius $R$ is typical: this is not just the case for graphs. Liouville theorems for non-compact, complete Riemannian manifolds are usually proven under the assumption of a lower bound on the Ricci curvature, which can be used to prove volume growth bounds on the Riemannian manifold (the simplest example being the Bishop-Gromov theorem).
A: 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.
