A simple construction of the Anderson operator via its quadratic form in dimensions two and three

TL;DR

This paper introduces a straightforward method to construct the Anderson operator in two and three dimensions using quadratic forms and exponential transforms, bypassing complex regularity structures.

Contribution

It presents a novel, simplified construction of the Anderson operator via quadratic forms, avoiding advanced calculus techniques.

Findings

Constructs the Anderson operator using quadratic forms and exponential transforms.

Establishes positivity and irreducibility of the associated semigroup.

Proves the existence of a spectral gap for the operator.

Abstract

We provide a simple construction of the Anderson operator in dimensions two and three. This is done through its quadratic form. We rely on an exponential transform instead of the regularity structures or paracontrolled calculus which are usually used for the construction of the operator. The knowledge of the form is robust enough to deduce important properties such as positivity and irreducibility of the corresponding semigroup. The latter property gives existence of a spectral gap.