topology. Let also $C_\mathrm{be}\big([0,T]; {{\mathcal P}} ({{\mathbb T}}^d\times {{\mathcal V}})\big)$ the set of paths $(P_t)_{t\in [0.T]}$ in $C\big([0,T]; {{\mathcal P}} ({{\mathbb T}}^d\times {{\mathcal V}})\big)$ such that $\sup_{t\in[0,T]}\mathcal H(P_t)<+\infty$ and let finally ${{\mathcal S}}_\mathrm{be} :=C_\mathrm{be} \big([0,T];
{{\mathcal P}} ({{\mathbb T}}^d \times {{\mathcal V}})\big) \times {{\mathcal M}}_\mathrm{a}
\big([0,T]\times {{\mathbb T}}^d \times {{\mathcal V}}\times {{\mathcal V}} \big)$

If $P\in {{\mathcal P}}({{\mathbb T}}^d\times {{\mathcal V}})$ has finite entropy, the Dirichlet form of the square root ${{\mathcal E}}$ can be defined by the variational formula $$\label{varD}
{{\mathcal E}}(P) := \sup_{\phi \in C_\mathrm{b}({{\mathbb T}}^d\times {{\mathcal V}})} 
\iint \! P({\mathop{}\!\mathrm{d}}x,{\mathop{}\!\mathrm{d}}v) \pi({\mathop{}\!\mathrm{d}}v') \, \sigma(v,v') 
\Big[ 1 - e^{\phi(x,v')-\phi(x,v)} \Big].$$ Note indeed the right hand side is well defined for any $\phi\in C_\mathrm{b}({{\mathbb T}}^d\times {{\mathcal V}})$ in view of Assumption \[t:asb\] and the basic entropy inequality $P(\psi ) \le {{\mathcal H}} (P) + \log \int\!{\mathop{}\!\mathrm{d}}x
\pi({\mathop{}\!\mathrm{d}}v)\, e^\psi$, $\psi\colon {{\mathbb T}}^d\times {{\mathcal V}}\to \mathbb R$. The representation corresponds to the Donsker-Varadhan large deviation for the empirical measure of the continuous time Markov chain on ${{\mathcal V}}$ with transition rates $\sigma(v',v)\pi({\mathop{}\!\mathrm{d}}v')$ [@DV]. Indeed, ${{\mathcal E}}(P)=\sup_{\phi}\{-P(e^{-\phi}{{\mathcal L}} e^\phi) \}$, where $$\label{def:L}
 {{\mathcal L}} g (v)=\int \pi({\mathop{}\!\mathrm{d}}v')\sigma(v',v)[g(v')-g(v)].$$

A variational representation for the kinematic term ${{\mathcal R}}$ is obtained by combining with the simple 