  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.
