Normalized SSEP macroscopic fluctuation action

Fix an integer $d\ge 1$ and let ${\mathbb{T}}^d=(\mathbb{R}\mathrm{/}\mathbb{Z}{)}^d$ carry normalized product Haar measure $dx$, with $dx({\mathbb{T}}^d)=1$. Coordinates and time are dimensionless in the diffusive normalization $D=1$. For $r\in (0,1)$ set the mobility function $\chi (r):=r(1-r)$. For any finite interval $[a,b]$ with $a<b$, let $\rho :[a,b]\times {\mathbb{T}}^d\to \mathbb{R}$ be a scalar density and let $j:[a,b]\times {\mathbb{T}}^d\to {\mathbb{R}}^d$ be a vector current. Require both fields to be smooth up to the time endpoints and periodic in space, and require that for some $\epsilon \in (0,1\mathrm{/}2)$, $\epsilon \le \rho (t,x)\le 1-\epsilon$ for every $(t,x)$. Call $(\rho ,j)$ admissible when it also satisfies the continuity equation ${\mathrm{\partial }}_t\rho +\mathrm{\nabla }\cdot j=0$. Define its dimensionless action by

${\mathcal{I}}_{[a,b]}(\rho ,j):=\frac{1}{4}{\int }_a^b{\int }_{{\mathbb{T}}^d}\frac{\mathrm{\mid }j(t,x)+\mathrm{\nabla }\rho (t,x){\mathrm{\mid }}^2}{\rho (t,x)(1-\rho (t,x))}dxdt\mathrm{.}$

This is the macroscopic fluctuation functional in the stated normalized SSEP coefficient convention. It is a definition of a functional and does not assert a microscopic path-space large-deviation principle or a probability estimate.