  We present certain Liouville properties of eigenfunctions of second-order
elliptic operators with real coefficients, via an approach that is based on
stochastic representations of positive solutions, and criticality theory of
second-order elliptic operators. These extend results of Y. Pinchover to the
case of nonsymmetric operators of Schr\"odinger type. In particular, we provide
an answer to an open problem posed by Pinchover in [Comm. Math. Phys. 272
(2007), no. 1, 75-84, Problem 5]. In addition, we prove a lower bound on the
decay of positive supersolutions of general second-order elliptic operators in
any dimension, and discuss its implications to the Landis conjecture.
