Fixed-mean action quasi-potential

Fix $d\ge 1$, $\bar{\rho }\in (0,1)$, and normalized product Haar measure $dx$ on ${\mathbb{T}}^d$. Let $\rho :{\mathbb{T}}^d\to (0,1)$ be a smooth target profile whose values are bounded away from $0$ and $1$. A connection of duration $T>0$ from the homogeneous profile $\bar{\rho }$ to $\rho$ is a smooth pair $(u,j)$ on $[0,T]\times {\mathbb{T}}^d$, smooth up to the time endpoints and periodic in space, such that for some ${\epsilon }_T\in (0,1\mathrm{/}2)$ one has ${\epsilon }_T\le u(t,x)\le 1-{\epsilon }_T$, ${\int }_{{\mathbb{T}}^d}u(t,x)dx=\bar{\rho }$ for all $t$, ${\mathrm{\partial }}_{t}u+\mathrm{\nabla }\cdot j=0$, $u(0,x)=\bar{\rho }$, and $u(T,x)=\rho (x)$. Define

${\mathcal{V}}_{\bar{\rho }}(\rho ):=\inf \{{\mathcal{I}}_{[0,T]}(u,j):T>0,(u,j)\text{ is a connection of duration }T\text{ from }\bar{\rho }\text{ to }\rho \},$

with the infimum of the empty set defined to be $+\mathrm{\infty }$. The action is the functional defined in the preceding item.

Normalized SSEP macroscopic fluctuation action