Statistical mechanics of the wave maps equation in dimension 1+1

TL;DR

This paper investigates the statistical mechanics of wave maps in 1+1 dimensions, establishing well-posedness for boundary data, convergence of discretized solutions, and the probabilistic behavior with Brownian motion initial data.

Contribution

It introduces a discrete approximation framework for wave maps, proves convergence to the continuous solution, and analyzes the probabilistic structure with Brownian boundary data.

Findings

Discrete solutions converge to continuous wave maps as mesh size tends to zero.

Solutions with Brownian boundary data have accumulation points, enabling probabilistic interpretation.

Wave maps with Gibbs distribution initial data can be characterized through this framework.

Abstract

We study wave maps with values in S^d, defined on the future light cone {|x| <= t}, with prescribed data at the boundary {|x| = t}. Based on the work of Keel and Tao, we prove that the problem is well-posed for locally absolutely continuous boundary data. We design a discrete version of the problem and prove that for every absolutely continuous boundary data, the sequence of solutions of the discretised problem converges to the corresponding continuous wave map as the mesh size tends to 0. Next, we consider the boundary data given by the S^d-valued Brownian motion. We prove that the sequence of solutions of the discretised problems has an accumulation point for the topology of locally uniform convergence. We argue that the resulting random field can be interpreted as the wave-map evolution corresponding to the initial data given by the Gibbs distribution.