Conservativeness of time changed processes and Liouville property for Schr\"odinger operators

TL;DR

This paper links the conservativeness of time-changed stochastic processes to the Liouville property of Schrödinger operators, providing criteria based on potential decay rates at infinity or boundary.

Contribution

It introduces a new criterion connecting process conservativeness with the Liouville property for Schrödinger operators, offering necessary and sufficient conditions.

Findings

Established a criterion for the Liouville property via process conservativeness.

Derived conditions based on potential decay rates at infinity or boundary.

Connected stochastic process properties with spectral characteristics of Schrödinger operators.

Abstract

We establish a criterion for the Liouville property for Schr\"odinger operators via the conservativeness of time changed processes. Using this criterion, we obtain necessary and sufficient conditions for the Liouville property for some Schr\"odinger operators in terms of the decay rates of the potentials at infinity/boundary.