Recently I was reading the theory of elliptic operators and there was a statement like this: given an elliptic partial differential operator $L$ on $C^2(\mathbb{R}^n)$ the Liouville theorem accounts to the strong maximum principle (SMP). Unfortunately I cannot find now such place (and a paper in fact) - so I am curious what was the meaning of this statement.
That's why I wonder: is SMP sufficient for the Liouville theorem to hold for an elliptic operator $L$? From my side I tried to prove Liouville theorem using only SMP - but I haven't succeed.
I assume that SMP and Liouville theorem are known, but I can provide definitions if needed.