  We study the kink motion for the one-dimensional stochastic Allen-Cahn
equation and its mass conserving counterpart. Using a deterministic slow
manifold, in the sharp interface limit for sufficiently small noise strength we
derive an explicit stochastic differential equation for the motion of the
interfaces, which is valid as long as the solution stays close to the manifold.
  On a relevant time-scale, where interfaces move at most by the minimal
allowed distance between interfaces, we show that the kinks behave
approximately like the driving Wiener-process projected onto the slow manifold,
while in the mass-conserving case they are additionally coupled via the mass
constraint.
