$${{\mathfrak D}}_n  (f)={{\mathfrak D}}_n^{\textrm{bulk}} (f)+{{\mathfrak D}}_n^{{\textrm{bnd}}}  (f),\label{eq:direxpl}$$ where $$\begin{aligned}
{{\mathfrak D}}_n^{\textrm{bulk}} (f)&=\sum_{x=1}^{n-2}\int \Big(1+\frac{E}{n^\gamma} \Big){\big(\eta(x)-\eta(x+1)\big)^2}\big(\nabla_{x,x+1} f(\eta)\big)^2\nu^n_\rho(d\eta)\\
{{\mathfrak D}}_n^{\textrm{bnd}} (f)&=
\int r_{0,1}(\eta)\big(f(\sigma^{1}\eta)
-f(\eta)\big)^2\nu^n_\rho(d\eta) \\ & \quad +
\int r_{n-1,n}(\eta)\big(f(\sigma^{n-1}\eta)
-f(\eta)\big)^2\nu^n_\rho(d\eta).\end{aligned}$$

The spaces of test functions {#ssec:test_functions}
----------------------------

Let ${{\mathcal C}}^\infty ([0, 1])$ be the space of real valued functions $\varphi : [0, 1] \to {{\mathbb R}}$ such that $\varphi$ is continuous in $[0,1]$ as well as all its derivatives. We denote by ${\rm d}^{k}\varphi$ the $k$–th derivative of $\varphi$, and for $k=1$ (resp. $k=2$) we simply denote it by $\nabla\varphi$ (resp. $\Delta\varphi$).

Before defining the *fluctuation fields* associated to our process, we first need to introduce the suitable space for test functions for each one of the fields that we will analyze. First of all, let ${{\mathcal S}}_{{\ensuremath{\operatorname{Dir}}}}$ and ${{\mathcal S}}_{{\ensuremath{\operatorname{Neu}}}}$ be the following spaces of functions: $$\begin{aligned}
{{\mathcal S}}_{{\ensuremath{\operatorname{Dir}}}} =&\Big\{ \varphi \in {{\mathcal C}}^\infty ([0, 1]) \; ; \;  {\rm d}^{2k}\varphi (0) ={\rm d}^{2k} \varphi
   (1) = 0, \text{ for any }k\in\mathbb{N}_0\Big\} \label{eq:space_test_funcDir}
\\
   {{\mathcal S}}_{\rm Neu} =& \Big\{\varphi \in {{\mathcal C}}^\infty([0,1]) \; ; \; {\rm d}^{2k+1} \varphi(0) ={\rm d}^{2k+1}  \varphi(1) = 0, \text{ for any } k\in\mathbb{N}_0 \Big\},\label{eq:space_test_funcNeu}
   \end{aligned}$$ both equipped with the family of seminorms $\big\{\sup_{u \in [0,1]} |{\rm 