gradient flow of the entropy is recovered from by projecting on the density. Indeed, by the Benamou-Brenier lemma [@BeBr], we deduce that if $(\mu,J)$ is a solution to the heat equation according to Definition \[he\], then $\mu=(\mu_t)_{t\in[0,T]}$ satisfies $$\label{hemu}
  H(\mu_T)+\int_0^T \!{\mathop{}\!\mathrm{d}}t\, \Big\{ E(\mu_t)+ 
  \frac 12 \big| \dot \mu_t  \big|^2  \Big\} \leq H(\nu)$$ where $ \big| \dot \mu_t  \big|$ is the metric derivative of $t\mapsto \mu_t$ with respect to the Wasserstein-$2$ distance, namely $\big| \dot \mu_t  \big| =\lim_{h\to 0}
\mathrm{d}_{W_2}\big(\mu_{t+h},\mu_t\big)/h$, where $\mathrm{d}_{W_2}$ denotes the Wasserstein-$2$ distance on ${{\mathcal P}}({{\mathbb T}}^d)$.

Conversely, let $\mu$ be a solution to satisfying $\mu_0=\nu$. Introduce the functional $J^\mu$ on $C^1\big([0,T]\times
{{\mathbb T}}^d;{{\mathbb R}}^d\big)$ defined by $J^\mu(w) = \int_0^T\!{\mathop{}\!\mathrm{d}}t\,\mu_t \big(\nabla\cdot D w_t\big)$. Since $\int_0^T\!{\mathop{}\!\mathrm{d}}t \, E(\mu_t) \le H(\nu)$, the functional $J^\mu$ extends to an element of ${{\mathcal M}}\big( [0,T]\times {{\mathbb T}}^d;{{\mathbb R}}^d\big)$, still denoted by $J^\mu$. Using again the Benamou-Brenier lemma it is then straightforward to check that the pair $(\mu,J^\mu)$ is a solution to the heat equation in the sense of Definition \[he\].

The previous remarks, together with the existence and uniqueness result for the formulation in [@Gi], imply the following statement.

\[t:uniq\] For each $\nu\in{{\mathcal P}}({{\mathbb T}}^d)$, with $H(\nu)< \infty$, there exists a unique solution 