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

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M2">\begin{document}$ P $\end{document}</tex-math></inline-formula> be a linear, second-order, elliptic operator with real coefficients defined on noncompact Riemannian manifold id="M3">\begin{document}$ M and satisfies id="M4">\begin{document}$ P1 = 0 in id="M5">\begin{document}$ $\end{document}</tex-math></inline-formula>. Assume further that id="M6">\begin{document}$ admits minimal positive Green function id="M7">\begin{document}$ We prove there exists smooth id="M8">\begin{document}$ \rho id="M9">\begin{document}$ such id="M10">\begin{document}$ is stochastically incomplete respect to the id="M11">\begin{document}$ P_{\rho} : \, $\end{document}</tex-math></inline-formula>, is,</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \int_{M} k_{P_{\rho}}^{M}(x, y, t) \ \,\mathrm{d}y < 1 \qquad \forall (x,t) \in \times (0, \infty), $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where id="M12">\begin{document}$ k_{P_{\rho}}^{M} denotes heat kernel associated id="M13">\begin{document}$ Moreover, id="M14">\begin{document}$ id="M15">\begin{document}$ L^1 $\end{document}</tex-math></inline-formula>-Liouville id="M16">\begin{document}$ if only id="M17">\begin{document}$ id="M18">\begin{document}$ id="M19">\begin{document}$ In addition, we study interplay between stochastic completeness id="M20">\begin{document}$ property of skew product two second-order operators.</p>