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We concern the analysis of the long time behavior of interfaces in systemswith two components. Each component evolves according to 1-d Allen-Cahnequation with Neumann boundary conditions, perturbed by small space-time whitenoise and with symmetric double well potential in the interval$[-\epsilon^{-1},\epsilon^{-1}]$. The two components interact with each otherby an attractive linear force. Instantons are defined as the stationarysolution of the Allen-Cahn equation without noise which connects two purephases. We prove that for time $t=\epsilon^{-1}$, in the limit $\epsilon\rightarrow 0$, when initial states are close to an instanton, two componentsstay close to the same instanton, whose center moves as a Brownian motion.

Brownian fluctuations of the interface in a system with two linear attracted components and white noises