On the optimal way to displace a stationary front

The stationary front in the title represents a planar interface which separates the two stable phases of a fluid. Due to stochastic forces, its position fluctuates and we would like to determine the most probable way in which a macroscopic displacement $R$ may occur in a given macroscopic time interval $T$.

Supposing planar symmetry, the problem becomes one dimensional and it is modelled by introducing a non local cost functional which must then be minimized over orbits which exhibit the desired displacement in the given time.

It is found that in a "sharp interface limit" the optimal behavior for $R$ small enough, is when the front moves with constant velocity $V=R/T$, the corresponding cost being $c V^2T$, $c$ a positive constant. However when $R$ increases past a critical value, the cost becomes smaller than $c V^2T$. The effect is caused by nucleations ahead of the moving front; there are critical values $R_n$, $n\ge 1$, so that, if $R\in (R_n,R_{n+1})$ then there are $n$ nucleations.