Superharmonic functions of Schrödinger operators and Hardy inequalities

Abstract

Given a Dirichlet form with generator $\mathcal{L}$ and a measure $\mu$, we consider superharmonic functions of the Schrödinger operator $\mathcal{L} + \mu$. We probabilistically prove that the existence of superharmonic functions gives rise to the Hardy inequality. More precisely, the $L^2$-Hardy inequality is derived from Itô's formula applied to the superharmonic function.