Theorems Jordan-Kinderlehrer-Otto Convergence Theorem for the Fokker-Planck Equation Attributions

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Statement

Let $n\in\mathbb N$. Let $V\in C^2(\mathbb R^n;\mathbb R)$, let $\lambda\in\mathbb R$, and assume that, for every $x\in\mathbb R^n$ and every $\xi\in\mathbb R^n$,

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\begin{align*} \xi^\top J(\nabla V)_x\xi\ge \lambda |\xi|^2. \end{align*}

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Assume also that there are constants $a>0$ and $C>0$ such that, for every $x\in\mathbb R^n$,

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\begin{align*} a|x|^2-C\le V(x)\le C(1+|x|^2) \end{align*}

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and

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\begin{align*} |\nabla V(x)|\le C(1+|x|). \end{align*}

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Define the free energy functional $\mathcal F:\mathcal P_2(\mathbb R^n)\to(-\infty,+\infty]$ as follows. If $\rho=u\mathcal L^n$ for a Borel density $u:\mathbb R^n\to[0,\infty)$ satisfying $u\log u\in L^1(\mathbb R^n)$, then

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

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For all other $\rho\in\mathcal P_2(\mathbb R^n)$, set $\mathcal F[\rho]=+\infty$.

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Let $\rho_0\in\mathcal P_2(\mathbb R^n)$ satisfy $\mathcal F[\rho_0]<\infty$. For each $\tau>0$, define a sequence $(\rho_k^\tau)_{k\ge0}$ in $\mathcal P_2(\mathbb R^n)$ by $\rho_0^\tau=\rho_0$ and, for each $k\ge0$, by choosing a minimizer

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\begin{align*} \rho_{k+1}^\tau\in\operatorname*{argmin}_{\rho\in\mathcal P_2(\mathbb R^n)}\left\{\mathcal F[\rho]+\frac{1}{2\tau}W_2^2(\rho,\rho_k^\tau)\right\}. \end{align*}

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Define the piecewise-constant interpolation $\bar\rho_\tau:[0,\infty)\to\mathcal P_2(\mathbb R^n)$ by $\bar\rho_\tau(0)=\rho_0$ and

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\begin{align*} \bar\rho_\tau(t)=\rho_k^\tau \end{align*}

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for $k\in\mathbb N$ and $t\in((k-1)\tau,k\tau]$.

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Assume the standard JKO compactness theorem applies to $\mathcal F$ under the preceding growth hypotheses: bounded energy sublevels have uniformly integrable second moments, and the moment-plus-increment bounds for the JKO interpolations imply subsequential pointwise $W_2$ convergence to a locally absolutely continuous curve. Then for every sequence $(\tau_j)_{j\ge1}$ in $(0,\infty)$ with $\tau_j\downarrow0$, there are a subsequence, still denoted $(\tau_j)_{j\ge1}$, and a locally absolutely continuous curve

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\begin{align*} \rho:[0,\infty)\to(\mathcal P_2(\mathbb R^n),W_2) \end{align*}

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such that

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\begin{align*} W_2(\bar\rho_{\tau_j}(t),\rho(t))\to0 \end{align*}

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for every $t\ge0$.

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For $\mathcal L^1$-a.e. $t\ge0$ with $\mathcal F[\rho(t)]<\infty$, write $\rho(t)=u(t,\cdot)\mathcal L^n$ for its Borel density. The curve $\rho$ is a distributional solution of the Fokker-Planck equation

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

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with initial datum $\rho_0$, in the sense that for every $\varphi\in C_c^\infty([0,\infty)\times\mathbb R^n)$,

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\begin{align*} \int_0^\infty\int_{\mathbb R^n}\left(\partial_t\varphi(t,x)+\Delta\varphi(t,x)-\nabla V(x)\cdot\nabla\varphi(t,x)\right)\,d\rho(t)(x)\,d\mathcal L^1(t)+\int_{\mathbb R^n}\varphi(0,x)\,d\rho_0(x)=0. \end{align*}

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Define the relative Fisher information $\mathcal I_V:\mathcal P_2(\mathbb R^n)\to[0,+\infty]$ as follows. If $\rho=u\mathcal L^n$, if the distributional vector field $\nabla u+u\nabla V$ is represented by a locally integrable Borel vector field on $\mathbb R^n$, and if

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\begin{align*} \int_{\mathbb R^n}\frac{|\nabla u(x)+u(x)\nabla V(x)|^2}{u(x)}\,d\mathcal L^n(x)<\infty, \end{align*}

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with the integrand interpreted as $0$ on the set where $u=0$ and $\nabla u+u\nabla V=0$, then

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\begin{align*} \mathcal I_V[\rho]=\int_{\mathbb R^n}\frac{|\nabla u(x)+u(x)\nabla V(x)|^2}{u(x)}\,d\mathcal L^n(x). \end{align*}

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For all other $\rho\in\mathcal P_2(\mathbb R^n)$, set $\mathcal I_V[\rho]=+\infty$.

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Assume, as supplied by the JKO slope-identification input, that $\mathcal F$ is proper, lower semicontinuous, $\lambda$-displacement convex, and has strong upper gradient satisfying $|\partial\mathcal F|^2=\mathcal I_V$ along every limiting curve obtained above. Then for all $0\le s\le t<\infty$,

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\begin{align*} \mathcal F[\rho(t)]+\frac{1}{2}\int_s^t |\rho'|_{W_2}^2(r)\,d\mathcal L^1(r)+\frac{1}{2}\int_s^t\mathcal I_V[\rho(r)]\,d\mathcal L^1(r)\le \mathcal F[\rho(s)], \end{align*}

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where $|\rho'|_{W_2}$ denotes the metric derivative of $\rho$ as a curve in $(\mathcal P_2(\mathbb R^n),W_2)$.

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