  Consider a discrete one-dimensional random surface whose height at a point
grows as a function of the heights at neighboring points plus an independent
random noise. Assuming that this function is equivariant under constant shifts,
symmetric in its arguments, and at least six times continuously differentiable
in a neighborhood of the origin, we show that as the variance of the noise goes
to zero, any such process converges to the Cole-Hopf solution of the 1D KPZ
equation under a suitable scaling of space and time. This proves an invariance
principle for the 1D KPZ equation, in the spirit of Donsker's invariance
principle for Brownian motion.
