Invariant Gibbs measures for $(1+1)$-dimensional wave maps into Lie groups

TL;DR

This paper proves the almost sure global well-posedness and invariance of the Gibbs measure for (1+1)-dimensional wave maps into Lie groups, using a novel approximation and geometric analysis.

Contribution

It introduces a new finite-dimensional approximation with Killing renormalization and establishes invariance of the Gibbs measure for geometric wave equations.

Findings

First invariance result for geometric wave equations

Global convergence of the approximation method

Analysis of interactions using geometric and probabilistic tools

Abstract

We discuss the $(1+1)$-dimensional wave maps equation with values in a compact Lie group. The corresponding Gibbs measure is given by a Brownian motion on the Lie group, which plays a central role in stochastic geometry. Our main theorem is the almost sure global well-posedness and invariance of the Gibbs measure for the wave maps equation. It is the first result of this kind for any geometric wave equation. Our argument relies on a novel finite-dimensional approximation of the wave maps equation which involves the so-called Killing renormalization. The main part of this article then addresses the global convergence of our approximation and the almost invariance of the Gibbs measure under the corresponding flow. The proof of global convergence requires a carefully crafted Ansatz which includes modulated linear waves, modulated bilinear waves, and mixed modulated objects. The interactions between the different objects in our Ansatz are analyzed using an intricate combination of analytic, geometric, and probabilistic ingredients. In particular, geometric aspects of the wave maps equation are utilized via orthogonality, which has previously been used in the deterministic theory of wave maps at critical regularity. The proof of almost invariance of the Gibbs measure under our approximation relies on conservative structures, which are a new framework for the approximation of Hamiltonian equations, and delicate estimates of the energy increment.