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

Abstract

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