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

We study wave maps with values in $S^d$, defined on the future light cone $\left\{\right|x|\le t\}\subset R^{1+1}$, with prescribed data at the boundary $\left\{\right|x|=t\}$. 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 a subsequence that converges in law in 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.