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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.

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    I think Byron's and George's answers [here](http://math.stackexchange.com/q/51926/) fit the bill of (part of) what you're looking for. See also (strong) [elliptic operators](http://en.wikipedia.org/wiki/Elliptic_operator), [maximum principle](http://en.wikipedia.org/wiki/Maximum_principle) and [Hopf maximum principle](http://en.wikipedia.org/wiki/Hopf_maximum_principle), so you have plenty of stuff to read while waiting for a serious answer.2011-08-26
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    [This paper](http://ddg.math.uni-goettingen.de/pub/laplacian.pdf) (found via google search) may be relevant.2011-08-26
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    @Theo: thank you, I've just taken a look, interesting. Matt E: thanks also,2011-08-26

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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).

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    cool, it seems that now I can patiently wait for a serious answer as Theo Buehler suggested.2011-08-26
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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.

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    Interesting, though I'm afraid I don't have an access to this paper either.2013-11-22
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    I've asked a question about "The origin of Discrete `Liouville's theorem'" at MathOverflow http://mathoverflow.net/questions/149621/the-origin-of-discrete-liouvilles-theorem2013-11-22