  We consider a non-linear, one-dimensional wave equation system with
finite-dimensional stochastic driving terms and with weak dissipation. A
stationary process that solves the system is used to model steady-state
non-equilibrium heat flow through a non-linear medium. We show existence and
uniqueness of invariant measures for the system modified with ultraviolet
cutoffs, and we obtain estimates for the field covariances with respect to
these measures, estimates that are uniform in the cutoffs. Finally, we discuss
the limit of these measures as the ultraviolet cutoffs are removed.
