  Let $P$ be a linear, second-order, elliptic operator with real coefficients
defined on a noncompact Riemannian manifold $M$ and satisfies $P1=0$ in $M$.
Assume further that $P$ admits a minimal positive Green function in $M$. We
prove that there exists a smooth positive function $\rho$ defined on $M$ such
that $M$ is stochastically incomplete with respect to the operator $ P_{\rho}
:= \rho \, P $, that is,
  \[
  \int_{M} k_{P_{\rho}}^{M}(x, y, t) \ {\rm d}y < 1 \qquad \forall (x, t) \in M
\times (0, \infty),
  \]
  where $k_{P_{\rho}}^{M}$ denotes the minimal positive heat kernel associated
with $P_{\rho}$.
  Moreover, $M$ is $L^1$-Liouville with respect to $P_{\rho}$ if and only if
$M$ is $L^1$-Liouville with respect to $P$. In addition, we study the interplay
between stochastic completeness and the $L^1$-Liouville property of the skew
product of two second-order elliptic operators.
