  In this note we derive large-scale regularity properties of solutions to
second-order linear elliptic equations with random coefficients on the half-
space with homogeneous Neumann boundary data; it is a companion to
arXiv:1604.02717 in which the situation for homogeneous Dirichlet boundary data
was addressed. Similarly to arXiv:1604.02717, the results in this contribution
are expressed in terms of a first-order Liouville principle. It follows from an
excess-decay that is shown through means of a stochastic
homogenization-inspired Campanato iteration. The core of this contribution is
the construction of a sublinear half-space- adapted corrector/vector potential
pair that, in contrast to arXiv:1604.02717, is adapted to the Neumann boundary
data.
