let $C\big([0,T]; {{\mathcal P}} ({{\mathbb T}}^d)\big)$ be the set of continuous paths on ${{\mathcal P}}({{\mathbb T}}^d)$ endowed with the topology of uniform convergence. Let also ${{\mathcal M}}\big([0,T]\times {{\mathbb T}}^d; {{\mathbb R}}^d \big)$ be the set of vector valued Radon measures on $[0,T]\times {{\mathbb T}}^d$ endowed with the weak\* topology. Set $S:=C\big([0,T]; {{\mathcal P}} ({{\mathbb T}}^d)\big)\times {{\mathcal M}}\big([0,T]\times
{{\mathbb T}}^d; {{\mathbb R}}^d \big)$ endowed with the product topology.

Given a positive $d\times d$ matrix $D$, the Fisher information $E\colon {{\mathcal P}}({{\mathbb T}}^d)\to[0,\infty]$ can be defined by the variational formula $$\label{varD-calore}
  E (\mu)=2\sup_{\phi\in C^2({{\mathbb T}}^d)}
  \Big\{ -\int \!{\mathop{}\!\mathrm{d}}\mu \, e^{-\phi}\nabla \cdot D \nabla e^\phi\Big\},$$ which implies its lower semicontinuity and convexity. The kinematic term $ R \colon S\to [0,\infty]$ admits the variational representation $$\label{varR-calore}
 R (\mu,J)=\sup_{w\in C([0,T]\times {{\mathbb T}}^d; {{\mathbb R}}^d)}
\Big\{J(w)-\frac 12 \int_0 ^T \!{\mathop{}\!\mathrm{d}}t\int \!{\mathop{}\!\mathrm{d}}\mu_t\,w\cdot Dw \Big\},$$ which implies its lower semicontinuity and convexity.

\[he\] Let $\nu\in{{\mathcal P}}({{\mathbb T}}^d)$ with $H(\nu)<+\infty$. A path $(\mu, J)\in
S$ is a solution of the heat equation with initial condition $\nu$ iff $\mu_0=\nu$ and $$\begin{aligned}
  \label{eqcont}
 &&\int_0^T \!{\mathop{}\!\mathrm{d}}t \,\mu_t(\partial_t\phi)   +J(\nabla \phi)=0,\qquad
 \phi\in C_c^1\big((0,T)\times {{\mathbb T}}^d  \big)\\
\label{ineq1-calore}
 && H(\mu_T)+\int_0^T \!{\mathop{}\!\mathrm{d}}t\, E(\mu_t)+ R (\mu, J) \leq H(\nu).\end{aligned}$$

The standard formulation of the heat equation as 