Stochastic completeness and $L^1$-Liouville property for second-order elliptic operators

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.