  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.
