Global well-posedness of the stochastic Abelian-Higgs equations in two dimensions

TL;DR

This paper establishes the global well-posedness of stochastic Abelian-Higgs equations in two dimensions using a novel covariant approach involving heat kernel estimates and a monotonicity formula.

Contribution

It introduces a new covariant framework and techniques to analyze stochastic gauge field equations, advancing mathematical understanding of their well-posedness.

Findings

Proves global existence and uniqueness of solutions.

Develops covariant stochastic objects and estimates.

Introduces a covariant monotonicity formula.

Abstract

We prove the global well-posedness of the stochastic Abelian-Higgs equations in two dimensions. The proof is based on a new covariant approach, which consists of two parts: First, we introduce covariant stochastic objects. The covariant stochastic objects and their multi-linear interactions are controlled using covariant heat kernel estimates. Second, we control nonlinear remainders using a covariant monotonicity formula, which is inspired by earlier work of Hamilton.