  We consider the large-scale regularity of solutions to second-order linear
elliptic equations with random coefficient fields. In contrast to previous
works on regularity theory for random elliptic operators, our interest is in
the regularity at the boundary: We consider problems posed on the half-space
with homogeneous Dirichlet boundary conditions and derive an associated
$C^{1,\alpha}$-type large-scale regularity theory in the form of a
corresponding decay estimate for the homogenization-adapted tilt-excess. This
regularity theory entails an associated Liouville-type theorem. The results are
based on the existence of homogenization correctors adapted to the half-space
setting, which we construct - by an entirely deterministic argument - as a
modification of the homogenization corrector on the whole space. This adaption
procedure is carried out inductively on larger scales, crucially relying on the
regularity theory already established on smaller scales.
