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Construction of a Gibbs measure for the zonal Dirac equation

Anne-Sophie de Suzzoni, Cyril Malézé

TL;DR

The paper develops a framework to construct Gibbs measures for a zonal Dirac equation on the sphere with a Hartree-type nonlinearity. By exploiting the warped-product structure of $\mathbb S^d$ and a zonal spectral basis, it defines a Wick-renormalized, truncation-dependent dynamics and proves the associated energy and nonlinearity converge in an appropriate probabilistic sense. It then uses tightness and Skorokhod representations to obtain a random weak solution whose law is a Gibbs measure $\rho_\infty$, shown as the limit of invariant truncated measures, though invariance of the limit is not established due to nonuniqueness. This work extends Gibbs-measure techniques to Dirac-type PDEs on compact manifolds under zonal symmetry and provides a controlled renormalization framework for such nonlinear dispersive systems.

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.

Construction of a Gibbs measure for the zonal Dirac equation

TL;DR

The paper develops a framework to construct Gibbs measures for a zonal Dirac equation on the sphere with a Hartree-type nonlinearity. By exploiting the warped-product structure of and a zonal spectral basis, it defines a Wick-renormalized, truncation-dependent dynamics and proves the associated energy and nonlinearity converge in an appropriate probabilistic sense. It then uses tightness and Skorokhod representations to obtain a random weak solution whose law is a Gibbs measure , shown as the limit of invariant truncated measures, though invariance of the limit is not established due to nonuniqueness. This work extends Gibbs-measure techniques to Dirac-type PDEs on compact manifolds under zonal symmetry and provides a controlled renormalization framework for such nonlinear dispersive systems.

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.
Paper Structure (8 sections, 18 theorems, 166 equations)

This paper contains 8 sections, 18 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. The zonal Dirac equation and Gibbs measures
  3. Main result
  4. Definition of the zonal Dirac operator
  5. Wick renormalization and Gibbs measure
  6. Renormalisation of the energy
  7. Renormalized nonlinearity
  8. Existence of a solution for the renormalized Dirac equation

Key Result

Theorem 1.1

Let $s<-\frac{1}{2}$ and $q>12d$. We set $p\in [2,\infty)$ such that $\frac{1}{p} + \frac{1}{q} = \frac{1}{2}$. Let $W$ be a non negative integral operator whose kernel is non negative, which is continuous from $L^{p/2}(\varmathbb S^d,\varmathbb C)$ into $L^q(\varmathbb S^d,\varmathbb C)$ and which

Theorems & Definitions (42)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1: Convention for $\Gamma$ matrices
  • Proposition 2.1
  • proof
  • Proposition 2.2: Dirac operator on $\varmathbb S^d$
  • Remark 2.1
  • Definition 2.2: Zonal functions
  • ...and 32 more