Construction of a Gibbs measure for the zonal Dirac equation

TL;DR

This paper develops a framework for constructing Gibbs measures for the Dirac equation on the sphere with Hartree-type nonlinearity, demonstrating existence of solutions with prescribed statistical properties.

Contribution

It introduces a novel approach to define Gibbs measures for a zonal Dirac equation model on the sphere, extending probabilistic methods to this setting.

Findings

Established existence of Gibbs measures for the zonal Dirac equation

Proved existence of weak solutions with prescribed statistical law

Applied compactness arguments to demonstrate measure invariance

Abstract

We propose a framework to construct Gibbs measures for the Dirac equation. We consider the Dirac equation on the sphere with a "Hartree-type" nonlinearity. We consider a zonal model, that is the analog of a spherically symmetric model but on the sphere. We build a Gibbs measure for this model. With a compactness argument, we prove the existence of a random variable that is a weak solution to the Dirac equation and whose law is the Gibbs measure at all times.