In the Hammersley harness processes the real-valued height at each site i in
Z^d is updated at rate 1 to an average of the neighboring heights plus a
centered random variable (the noise). We construct the process "a la Harris"
simultaneously for all times and boxes contained in Z^d. With this
representation we compute covariances and show L^2 and almost sure time and
space convergence of the process. In particular, the process started from the
flat configuration and viewed from the height at the origin converges to an
invariant measure. In dimension three and higher, the process itself converges
to an invariant measure in L^2 at speed t^{1-d/2} (this extends the convergence
established by Hsiao). When the noise is Gaussian the limiting measures are
Gaussian fields (harmonic crystals) and are also reversible for the process.