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

TL;DR

This paper investigates the relationship between stochastic completeness and the $L^1$-Liouville property for second-order elliptic operators on noncompact manifolds, showing how these properties can be manipulated via a positive function and exploring their behavior under operator products.

Contribution

It demonstrates the existence of a positive function transforming an elliptic operator to make the manifold stochastically incomplete and establishes the equivalence of $L^1$-Liouville properties under this transformation, also analyzing their interaction in operator products.

Findings

Existence of a positive function $ ho$ making the manifold stochastically incomplete for $P_ ho$.

Equivalence of $L^1$-Liouville property for $P$ and $P_ ho$.

Analysis of stochastic completeness and $L^1$-Liouville property in skew product 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, \[ \int_{M} k_{P_{\rho}}^{M}(x, y, t) \ {\rm d}y < 1 \qquad \forall (x, t) \in M \times (0, \infty), \] 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.