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

Abstract

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,integral(M)kP(rho)(M)(x, y, t) dy < 1 for all (x, t) is an element of M x (0, infinity), where kP(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.