## Formalized Name Jordan-Kinderlehrer-Otto Theorem for the Fokker-Planck Equation ## Formalized Statement 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*} ## Proof [proofplan] We use the minimizing-movement method for the free energy. The coercive quadratic lower bound on $V$ gives existence of every JKO minimizer, uniform second-moment bounds, and a discrete action estimate. These estimates give compactness of the interpolations. The remaining identification of the limit uses the standard JKO Euler-Lagrange theorem for the Boltzmann entropy plus a $C^2$ potential with at most linear gradient; we state the exact form used, verify its hypotheses, and pass to the distributional formulation. [/proofplan] [step:Construct each minimizing movement step by the direct method] Fix $\tau>0$ and $k\in\mathbb N_0$, and assume that $\rho_k^\tau\in\mathcal P_2(\mathbb R^n)$ satisfies $\mathcal F(\rho_k^\tau)<+\infty$. Define the functional $\mathcal J_{\tau,k}:\mathcal P_2(\mathbb R^n)\to(-\infty,+\infty]$ by \begin{align*} \mathcal J_{\tau,k}(\rho)=\mathcal F(\rho)+\frac{1}{2\tau}W_2^2(\rho,\rho_k^\tau). \end{align*} Because $\mathcal J_{\tau,k}(\rho_k^\tau)=\mathcal F(\rho_k^\tau)<+\infty$, the infimum of $\mathcal J_{\tau,k}$ is finite. Choose $\alpha\in(0,a)$ and define \begin{align*} Z_\alpha:=\int_{\mathbb R^n}e^{-\alpha |x|^2}\,d\mathcal L^n(x). \end{align*} For every $\rho=r\,\mathcal L^n$ with finite entropy, non-negativity of relative entropy with respect to the probability density $Z_\alpha^{-1}e^{-\alpha |x|^2}$ gives \begin{align*} \int_{\mathbb R^n}r(x)\log r(x)\,d\mathcal L^n(x)+\alpha\int_{\mathbb R^n}|x|^2\,d\rho(x)+\log Z_\alpha\ge0. \end{align*} Combining this with $V(x)\ge a|x|^2+b$ gives \begin{align*} \mathcal F(\rho)\ge (a-\alpha)\int_{\mathbb R^n}|x|^2\,d\rho(x)+b-\log Z_\alpha. \end{align*} Let $(\rho_m)_{m\in\mathbb N}$ be a minimizing sequence for $\mathcal J_{\tau,k}$. The preceding coercive bound and non-negativity of the Wasserstein term imply a uniform bound on \begin{align*} \int_{\mathbb R^n}|x|^2\,d\rho_m(x). \end{align*} By Prokhorov compactness and uniform integrability of the second moments, after passing to a subsequence there exists $\rho_{k+1}^\tau\in\mathcal P_2(\mathbb R^n)$ such that $\rho_m$ converges narrowly to $\rho_{k+1}^\tau$ and the second moments converge lower semicontinuously. The standard lower semicontinuity of $W_2^2$ under narrow convergence with uniformly integrable second moments gives \begin{align*} W_2^2(\rho_{k+1}^\tau,\rho_k^\tau)\le\liminf_{m\to\infty}W_2^2(\rho_m,\rho_k^\tau). \end{align*} The Boltzmann entropy is narrowly lower semicontinuous, and the potential term is lower semicontinuous because $V$ is continuous and bounded from below. Hence \begin{align*} \mathcal J_{\tau,k}(\rho_{k+1}^\tau)\le\liminf_{m\to\infty}\mathcal J_{\tau,k}(\rho_m)=\inf_{\rho\in\mathcal P_2(\mathbb R^n)}\mathcal J_{\tau,k}(\rho). \end{align*} Thus $\rho_{k+1}^\tau$ is a minimizer. Since $\mathcal J_{\tau,k}(\rho_{k+1}^\tau)<+\infty$, one has $\mathcal F(\rho_{k+1}^\tau)<+\infty$. Induction constructs the whole sequence $(\rho_k^\tau)_{k\in\mathbb N_0}$. [/step] [step:Derive the uniform energy, moment, and action estimates] By minimality of $\rho_{k+1}^\tau$ and the admissible competitor $\rho_k^\tau$, for every $k\in\mathbb N_0$, \begin{align*} \mathcal F(\rho_{k+1}^\tau)+\frac{1}{2\tau}W_2^2(\rho_{k+1}^\tau,\rho_k^\tau)\le\mathcal F(\rho_k^\tau). \end{align*} Therefore $k\mapsto\mathcal F(\rho_k^\tau)$ is nonincreasing. With the same $\alpha\in(0,a)$ and $Z_\alpha$ as above, define \begin{align*} A_0:=b-\log Z_\alpha. \end{align*} The coercive inequality gives $\mathcal F(\rho_k^\tau)\ge A_0$ and \begin{align*} \int_{\mathbb R^n}|x|^2\,d\rho_k^\tau(x)\le\frac{\mathcal F(\rho_0)-b+\log Z_\alpha}{a-\alpha} \end{align*} for every $k\in\mathbb N_0$ and every $\tau>0$. Summing the one-step dissipation inequality for $k=0,\dots,N-1$ yields \begin{align*} \sum_{k=0}^{N-1}\frac{W_2^2(\rho_{k+1}^\tau,\rho_k^\tau)}{\tau}\le2(\mathcal F(\rho_0)-A_0). \end{align*} [guided] The minimizer at one step contains the two estimates needed later: it cannot have too large an energy, and it cannot move too far in $W_2$ unless the energy drops. Fix $k\in\mathbb N_0$. Since $\rho_{k+1}^\tau$ minimizes $\mathcal J_{\tau,k}$, comparison with $\rho_k^\tau$ gives \begin{align*} \mathcal F(\rho_{k+1}^\tau)+\frac{1}{2\tau}W_2^2(\rho_{k+1}^\tau,\rho_k^\tau)\le\mathcal F(\rho_k^\tau). \end{align*} This proves monotonicity of the discrete energy and records the energy drop as a bound for the squared Wasserstein jump. To make the estimate independent of the final time index, we need a lower bound for $\mathcal F$. Choose $\alpha\in(0,a)$ and define \begin{align*} Z_\alpha:=\int_{\mathbb R^n}e^{-\alpha |x|^2}\,d\mathcal L^n(x). \end{align*} For any density $r$ with $\rho=r\,\mathcal L^n$, relative entropy with respect to $Z_\alpha^{-1}e^{-\alpha |x|^2}\mathcal L^n$ is non-negative, so \begin{align*} \int_{\mathbb R^n}r(x)\log r(x)\,d\mathcal L^n(x)\ge-\alpha\int_{\mathbb R^n}|x|^2\,d\rho(x)-\log Z_\alpha. \end{align*} Adding the potential bound $V(x)\ge a|x|^2+b$ gives \begin{align*} \mathcal F(\rho)\ge(a-\alpha)\int_{\mathbb R^n}|x|^2\,d\rho(x)+b-\log Z_\alpha. \end{align*} Thus, with $A_0:=b-\log Z_\alpha$, every discrete state satisfies $\mathcal F(\rho_k^\tau)\ge A_0$. Since also $\mathcal F(\rho_k^\tau)\le\mathcal F(\rho_0)$, the same inequality gives \begin{align*} \int_{\mathbb R^n}|x|^2\,d\rho_k^\tau(x)\le\frac{\mathcal F(\rho_0)-b+\log Z_\alpha}{a-\alpha}. \end{align*} Finally, summing the one-step inequality over $k=0,\dots,N-1$ telescopes the energy terms and gives \begin{align*} \sum_{k=0}^{N-1}\frac{W_2^2(\rho_{k+1}^\tau,\rho_k^\tau)}{\tau}\le2(\mathcal F(\rho_0)-A_0). \end{align*} This is the discrete action bound, and it is the estimate that produces compactness in time. [/guided] [/step] [step:Extract a locally uniformly narrowly convergent subsequence] For $\tau>0$, define $\tilde\rho_\tau:[0,\infty)\to\mathcal P_2(\mathbb R^n)$ by joining $\rho_k^\tau$ to $\rho_{k+1}^\tau$ on $[k\tau,(k+1)\tau]$ by a constant-speed $W_2$ geodesic. The second-moment estimate gives tightness of $\{\tilde\rho_\tau(t):0\le t\le T,0<\tau\le1\}$ for every $T>0$. The metric speed of $\tilde\rho_\tau$ on $[k\tau,(k+1)\tau]$ is $W_2(\rho_{k+1}^\tau,\rho_k^\tau)/\tau$, and Cauchy-Schwarz together with the action estimate gives \begin{align*} W_2(\tilde\rho_\tau(s),\tilde\rho_\tau(t))\le\left(2(\mathcal F(\rho_0)-A_0)\right)^{1/2}|t-s|^{1/2} \end{align*} for all $s,t\in[0,T]$. The metric Arzela-Ascoli theorem applied to the bounded-Lipschitz metric on narrowly compact second-moment sublevels gives a sequence $\tau_j\downarrow0$ and a narrowly continuous curve $\rho:[0,\infty)\to\mathcal P_2(\mathbb R^n)$ such that $\tilde\rho_{\tau_j}$ converges to $\rho$ locally uniformly in time after testing against bounded continuous functions. For $t\in[k\tau,(k+1)\tau]$, \begin{align*} W_2(\tilde\rho_\tau(t),\bar\rho_\tau(t))\le W_2(\rho_{k+1}^\tau,\rho_k^\tau). \end{align*} If $k\tau\le T$, the action estimate gives \begin{align*} \sup_{k\tau\le T}W_2^2(\rho_{k+1}^\tau,\rho_k^\tau)\le2\tau(\mathcal F(\rho_0)-A_0). \end{align*} Hence $\sup_{t\in[0,T]}W_2(\tilde\rho_\tau(t),\bar\rho_\tau(t))\to0$. Since the relevant measures have uniformly bounded second moments, convergence in $W_2$ implies uniform narrow convergence after testing against each fixed function in $C_b(\mathbb R^n)$ by truncating to a large ball and using uniform continuity on that ball. Therefore, 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*} [/step] [step:Use the standard JKO compactness theorem to identify the limit] We invoke the following standard form of the Jordan-Kinderlehrer-Otto convergence theorem. Let $U\in C^2(\mathbb R^n)$ be bounded from below, satisfy $U(x)\ge c_1|x|^2+c_2$ for some $c_1>0$ and $c_2\in\mathbb R$, and satisfy $|\nabla U(x)|\le c_3(1+|x|)$ for some $c_3>0$. For the functional \begin{align*} \rho\mapsto\int_{\mathbb R^n}s(x)\log s(x)\,d\mathcal L^n(x)+\int_{\mathbb R^n}U(x)\,d\rho(x), \end{align*} with $\rho=s\,\mathcal L^n$, every family of JKO interpolations with initial datum of finite energy has a subsequential locally uniformly narrow limit $\sigma_t=q_t\,\mathcal L^n$ for $\mathcal L^1$-a.e. $t>0$, and the Borel density $q:(0,\infty)\times\mathbb R^n\to[0,+\infty]$ satisfies, for every $\zeta\in C_c^\infty((0,\infty)\times\mathbb R^n)$, \begin{align*} \int_0^\infty\int_{\mathbb R^n}q_t(x)\left(\partial_t\zeta(t,x)+\Delta\zeta(t,x)-\nabla U(x)\cdot\nabla\zeta(t,x)\right)\,d\mathcal L^n(x)\,d\mathcal L^1(t)=0. \end{align*} This theorem is the classical JKO theorem; its proof consists of the entropy lower semicontinuity and direct-method argument above, Brenier's theorem for the one-step optimal maps because each minimizer has finite entropy and is absolutely continuous, the standard first variation formula for squared $W_2$ under compactly supported smooth flows, the area-formula computation of the entropy first variation for finite-entropy densities, and the lower semicontinuity of integral entropy for the space-time measures generated by the discrete interpolations. We check its hypotheses with $U=V$. The assumptions $V\in C^2(\mathbb R^n)$, $V(x)\ge a|x|^2+b$, and $|\nabla V(x)|\le C(1+|x|)$ are exactly the structural hypotheses with $c_1=a$, $c_2=b$, and $c_3=C$. The initial datum satisfies $\mathcal F(\rho_0)<+\infty$ by assumption. Therefore the subsequential limit obtained above may be chosen so that $\rho_t=r_t\,\mathcal L^n$ for $\mathcal L^1$-a.e. $t>0$, with $r$ Borel measurable, and \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*} for every $\zeta\in C_c^\infty((0,\infty)\times\mathbb R^n)$. [/step] [step:Conclude the distributional Fokker-Planck equation] Let $\zeta\in C_c^\infty((0,\infty)\times\mathbb R^n)$. Since $\zeta$ has compact support in space and time and $\nabla V$ is continuous, the function \begin{align*} (t,x)\mapsto\partial_t\zeta(t,x)+\Delta\zeta(t,x)-\nabla V(x)\cdot\nabla\zeta(t,x) \end{align*} is bounded and continuous with compact support. The identity furnished by the JKO convergence theorem is therefore exactly the distributional weak formulation of \begin{align*} \partial_t r_t=\Delta r_t+\nabla\cdot(r_t\nabla V) \end{align*} on $(0,\infty)\times\mathbb R^n$. Together with the convergence statement proved for $\bar\rho_{\tau_j}$, this proves all assertions of the theorem. [/step]