$\gamma>0$ then $P^{n}(\phi^n) \to P(\phi)$.

In view of the variational definition , this lemma readily implies that the Dirichlet form ${{\mathcal E}}$ is lower semicontinuous on sublevel sets of the entropy. Analogously, recalling , the kinematic term ${{\mathcal R}}$ is lower semicontinuous on the sets $\big\{
(P,\Theta) \in {{\mathcal S}}_\mathrm{be} \colon \sup_{ t\in [0,T]} {{\mathcal H}}(P_t)
\le \ell \big\}$, $\ell \in {{\mathbb R}}_+$.

We start by proving uniqueness, and in particular by showing that if $(P^1, \Theta^1)$ and $(P^2, \Theta^2)$ are solutions then $P^1=P^2$. Assume by contradiction that there exists $t\in (0,T]$ such that $P^1_t\neq P^2_t$ and let $(\bar P, \bar \Theta)=\frac 1 2 \big (P^1, \Theta^1 \big)+\frac 1 2 (P^2, \Theta^2)$. In view of Remark \[t:rem\], by the convexity of $\mathcal E$, $\mathcal R$ and the strict convexity of ${{\mathcal H}}$ $${{\mathcal H}}(\bar P_t)+\int_0^t \!{\mathop{}\!\mathrm{d}}s\, {{\mathcal E}} (\bar P_s) + {{\mathcal R}} ^{0,t}(\bar P, \bar \Theta)< {{\mathcal H}} (Q),$$ which by provides the desired contradiction. Uniqueness is now concluded by observing that, for a given $P\in C_{\mathrm{be}}\big([0,T]; {{\mathcal P}}({{\mathbb T}}^d\times{{\mathcal V}})\big)$, the map $\Theta\mapsto {{\mathcal R}}(P, \Theta)$ is strictly convex, so that we can repeat the argument above with $P^1=P^2=P$ and deduce $\Theta^1=\Theta^2$.

Postponing the proof of the existence, 