Let $n\in\mathbb N$, let $\mathbb N_0:=\{0\}\cup\mathbb N$, and let $\mathcal P_2(\mathbb R^n)$ be the set of Borel probability measures on $\mathbb R^n$ with finite second moment, equipped with $W_2$. Let $V\in C^2(\mathbb R^n)$ be bounded from below, and assume that there exist constants $a>0$, $b\in\mathbb R$, and $C>0$ such that, for every $x\in\mathbb R^n$, $V(x)\ge a|x|^2+b$ and $|\nabla V(x)|\le C(1+|x|)$. Define $\mathcal F:\mathcal P_2(\mathbb R^n)\to(-\infty,+\infty]$ by

\begin{align*} \mathcal F(\rho)=\int_{\mathbb R^n} r(x)\log r(x)\,d\mathcal L^n(x)+\int_{\mathbb R^n}V(x)\,d\rho(x) \end{align*}

when $\rho=r\,\mathcal L^n$, $r\log r\in L^1(\mathbb R^n)$, and $0\log0$ is interpreted as $0$, and by $\mathcal F(\rho)=+\infty$ otherwise. Let $\rho_0\in\mathcal P_2(\mathbb R^n)$ satisfy $\mathcal F(\rho_0)<+\infty$. Then, for every $\tau>0$, there exists a sequence $(\rho_k^\tau)_{k\in\mathbb N_0}\subset\mathcal P_2(\mathbb R^n)$ with $\rho_0^\tau=\rho_0$ such that, for every $k\in\mathbb N_0$, $\rho_{k+1}^\tau$ minimizes

\begin{align*} \rho\mapsto \mathcal F(\rho)+\frac{1}{2\tau}W_2^2(\rho,\rho_k^\tau) \end{align*}

over $\mathcal P_2(\mathbb R^n)$. For any such choice of minimizers, define $\bar\rho_\tau:[0,\infty)\to\mathcal P_2(\mathbb R^n)$ by $\bar\rho_\tau(0)=\rho_0$ and $\bar\rho_\tau(t)=\rho_k^\tau$ for $(k-1)\tau<t\le k\tau$. Then there exist a sequence $\tau_j\downarrow0$ and a narrowly continuous curve $\rho:[0,\infty)\to\mathcal P_2(\mathbb R^n)$ such that, for every $T>0$ and every $\varphi\in C_b(\mathbb R^n)$,

\begin{align*} \lim_{j\to\infty}\sup_{t\in[0,T]}\left|\int_{\mathbb R^n}\varphi(x)\,d\bar\rho_{\tau_j}(t)(x)-\int_{\mathbb R^n}\varphi(x)\,d\rho_t(x)\right|=0. \end{align*}

Moreover $\rho_t=r_t\,\mathcal L^n$ for $\mathcal L^1$-a.e. $t>0$, where $r:(0,\infty)\times\mathbb R^n\to[0,+\infty]$ is Borel measurable, and $r$ is a distributional solution of

\begin{align*} \partial_t r_t=\Delta r_t+\nabla\cdot(r_t\nabla V) \end{align*}

on $(0,\infty)\times\mathbb R^n$, meaning that for every $\zeta\in C_c^\infty((0,\infty)\times\mathbb R^n)$,

\begin{align*} \int_0^\infty\int_{\mathbb R^n} r_t(x)\left(\partial_t\zeta(t,x)+\Delta\zeta(t,x)-\nabla V(x)\cdot\nabla\zeta(t,x)\right)\,d\mathcal L^n(x)\,d\mathcal L^1(t)=0. \end{align*}