You can see below two numerical experiments implemented with DUNE. You may want to compare this animations with those of the deterministic Allen- Cahn.
Initial data: indicator function of a connected set
Initial data: random (IID Uniform([-0.2,0.2]) at each grid point)
The points quickly move to the nearest phase $\pm 1$, forming many connected components in each phase. The connected components are initially small, but join together according to 'stochastic mean curvature flow'. This behaves similarly to the deterministic case, but the boundaries are much rougher and the components may grow in directions contrary to mean curvature flow.
Higher resolution simulations
We can increase the resolution of the grid and see how this affects convergence. The approximation to the noise converges simultaneously with the grid, and this can be observed below.
Annihilation of phases
If we start with the initial data as the indicator function of a connected set, for example a square, we can study the time taken for the $+1$ phase to disappear. We compare this with the deterministic equation, and observe that the stochastic process takes significantly longer: after a number of simulations the stochastic process took on average $70.4\%$ longer. The deterministic process and a sample path of a stochastic process are shown below.