  We consider the stochastic Allen-Cahn equation driven by mollified space-time
white noise. We show that, as the mollifier is removed, the solutions converge
weakly to 0, independently of the initial condition. If the intensity of the
noise simultaneously converges to 0 at a sufficiently fast rate, then the
solutions converge to those of the deterministic equation. At the critical
rate, the limiting solution is still deterministic, but it exhibits an
additional damping term.
