  We study the invariant distributions of Hammersley's serial harness process
in all dimensions and height fluctuations in one dimension. Subject to mild
moment assumptions there is essentially one unique invariant distribution, and
all other invariant distributions are obtained by adding harmonic functions of
the averaging kernel. We identify one Gaussian case where the invariant
distribution is i.i.d. Height fluctuations in one dimension obey the stochastic
heat equation with additive noise (Edwards-Wilkinson universality). We prove
this for correlated initial data subject to polynomial decay of strong mixing
coefficients, including process-level tightness in the Skorohod space of
space-time trajectories.

Probability