Superharmonic functions of Schrödinger operators and Hardy inequalities

Abstract

Given a Dirichlet form with generator ℒ and a measure 𝜇, we consider superharmonic functions of the Schrödinger operator ℒ + 𝜇. We probabilistically prove that the existence of superharmonic functions gives rise to the Hardy inequality. More precisely, the 𝐿²-Hardy inequality is derived from Itô's formula applied to the superharmonic function.