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It is well-known from the celebrated paper [JKO] that \eqref{FP} is the Wasserstein gradient flow of the relative entropy $$ \mathcal H_{\pi}(\rho)=\int_\Omega\frac{\rho(x)}{\pi(x)}\log\left(\frac{\rho(x)}{\pi(x)}\right) \pi(x) dx. $$ Here the Gibbs distribution $$ \pi(x)=\frac{1}{Z}e^{-V(x)} $$ is the unique stationary solution of \eqref{FP} ($Z>0$ is a normalizing factor so that $\int \pi=1$). This gradient flow structure can be formalized and made completely rigorous in the context of metric gradient flows and curves of maximal slope, see [AGS]. It is moreover known that, if $Hess V\geq \lambda$ for some $\lambda>0$, then the relative entropy $\rho\mapsto\mathcal H_\pi(\rho)$ is $\lambda$-geodesically convex (aka $\lambda$-displacement convex as introduced by R. McCann in [M]). This in turn can be exploited to prove long-time convergence $$ W_2(\rho_t,\pi) \leq C e^{-\lambda t} \qquad\mbox{and}\qquad \|\rho_t-\pi\|_{L^1}\leq C e^{-\lambda t}. $$ (here $W_2(\mu,\nu)$ is the quadratic Wasserstein distance between probability measures, and the various constants only depend on the intial entropy $\mathcal H_\pi(\rho_0)$ and $\lambda$) Roughly speaking, the proof is as follows: the entropy is dissipated along the evolution by the relative Fisher-information $$ \frac{d}{dt}\mathcal H_\pi(\rho_t)=-\mathcal I_\pi(\rho_t)\overset{def}{=}- \int_\Omega \left|\nabla\log\left(\frac{\rho_t(x)}{\pi(x)}\right)\right|^2 \rho_t(x) dx $$ The $\lambda$-convexity of $V$ guarantees next the Logarithmic-Sobolev Inequality \begin{equation} \label{LSI} \mathcal I_\pi(\rho)\geq 2\lambda\mathcal H_\pi(\rho), \tag{LSI} \end{equation} which by Grönwall's lemma immediately gives entropic decay $$ \mathcal H_\pi(\rho_t)\leq e^{-2\lambda t}\mathcal H_\pi(\rho_0). $$ On can then use a Talagrand inequality $W_2(\rho,\pi)\leq \sqrt{\frac{2\mathcal H_\pi(\rho)}{\lambda}}$ or a Csiszár-Kullback-Pinsker inequality $\|\rho-\pi\|_{L^1}\leq \sqrt{2\mathcal H_\pi(\rho)}$ to conclude. The key point is really here that the $\lambda$-convexity of the potential $V$ (or in other words the $\lambda$-displacement convexity of the relative entropy $\mathcal H_\pi(\cdot)$) is somehow equivalent to the Log-Sobolev inequality \eqref{LSI}. Of course this is very sketchy and there are various subtle points here, but let me remain formal for the sake of exposition.

Another and more standard way of deriving long-time convergence to the stationary measure is by purely linera spectral analysis. Indeed $\lambda_0=0$ is always eigenvalue of $-\Delta-\operatorname{div}( \cdot \nabla V)$ (being $\pi=\frac{1}{Z}e^{-V}$ in the kernel), but the next principal eigenvalue $\lambda_1>0$ should obviously quantify exponential convergence $\rho_t\to\pi$ in some Sobolev ($H^1$?) norm, as $t\to+\infty$.

I am asking this because for some projet of mine I am considering a degenerate version of the Fokker-Planck equation with diffusion $\Delta(\Theta(x)\rho)$, where $\Theta(x)\geq 0$ is a locally uniformly positive coefficient that vanishes on the boundary $\partial\Omega$. This makes the problem quite delicate (diffusion shuts down on the boundaries and no boundary conditions can be imposed), and not amenable to the above optimal transport machinery (at least not directly). After some careful manipulations (let me skip the details here) I managed to recast the problem in a "traditional form", but surprizingly enough I realized that at least in some completely explicit examples the optimal-transport rate $\lambda>0$ can be stricly smaller than the spectral gap $\lambda_1>0$, while I was expecting both to coincide. Actually I was expecting the opposite, since in the spirit of the above "trade-off from space to time"

It is well-known from the celebrated paper [JKO] that \eqref{FP} is the Wasserstein gradient flow of the relative entropy $$ \mathcal H_{\pi}(\rho)=\int_\Omega\frac{\rho(x)}{\pi(x)}\log\left(\frac{\rho(x)}{\pi(x)}\right) \pi(x) dx. $$ Here the Gibbs distribution $$ \pi(x)=\frac{1}{Z}e^{-V(x)} $$ is the unique stationary solution of \eqref{FP} ($Z>0$ is a normalizing factor so that $\int \pi=1$). This gradient flow structure can be formalized and made completely rigorous in the context of metric gradient flows and curves of maximal slope, see [AGS]. It is moreover known that, if $\text{Hess} V\geq \lambda$ for some $\lambda>0$, then the relative entropy $\rho\mapsto\mathcal H_\pi(\rho)$ is $\lambda$-geodesically convex (aka $\lambda$-displacement convex as introduced by R. McCann in [M]). This in turn can be exploited to prove long-time convergence $$ W_2(\rho_t,\pi) \leq C e^{-\lambda t} \qquad\mbox{and}\qquad \|\rho_t-\pi\|_{L^1}\leq C e^{-\lambda t}. $$ (here $W_2(\mu,\nu)$ is the quadratic Wasserstein distance between probability measures, and the various constants only depend on the initial entropy $\mathcal H_\pi(\rho_0)$ and $\lambda$) Roughly speaking, the proof is as follows: the entropy is dissipated along the evolution by the relative Fisher-information $$ \frac{d}{dt}\mathcal H_\pi(\rho_t)=-\mathcal I_\pi(\rho_t)\overset{def}{=}- \int_\Omega \left|\nabla\log\left(\frac{\rho_t(x)}{\pi(x)}\right)\right|^2 \rho_t(x) dx $$ The $\lambda$-convexity of $V$ guarantees next the Logarithmic-Sobolev Inequality \begin{equation} \label{LSI} \mathcal I_\pi(\rho)\geq 2\lambda\mathcal H_\pi(\rho), \tag{LSI} \end{equation} which by Grönwall's lemma immediately gives entropic decay $$ \mathcal H_\pi(\rho_t)\leq e^{-2\lambda t}\mathcal H_\pi(\rho_0). $$ On can then use a Talagrand inequality $W_2(\rho,\pi)\leq \sqrt{\frac{2\mathcal H_\pi(\rho)}{\lambda}}$ or a Csiszár-Kullback-Pinsker inequality $\|\rho-\pi\|_{L^1}\leq \sqrt{2\mathcal H_\pi(\rho)}$ to conclude. The key point is really here that the $\lambda$-convexity of the potential $V$ (or in other words the $\lambda$-displacement convexity of the relative entropy $\mathcal H_\pi(\cdot)$) is somehow equivalent to the Log-Sobolev inequality \eqref{LSI}. Of course this is very sketchy and there are various subtle points here, but let me remain formal for the sake of exposition.

Another and more standard way of deriving long-time convergence to the stationary measure is by purely linear spectral analysis. Indeed $\lambda_0=0$ is always eigenvalue of $-\Delta-\operatorname{div}( \cdot \nabla V)$ (being $\pi=\frac{1}{Z}e^{-V}$ in the kernel), but the next principal eigenvalue $\lambda_1>0$ should obviously quantify exponential convergence $\rho_t\to\pi$ in some Sobolev ($H^1$?) norm, as $t\to+\infty$.

I am asking this because for some project of mine I am considering a degenerate version of the Fokker-Planck equation with diffusion $\Delta(\Theta(x)\rho)$, where $\Theta(x)\geq 0$ is a locally uniformly positive coefficient that vanishes on the boundary $\partial\Omega$. This makes the problem quite delicate (diffusion shuts down on the boundaries and no boundary conditions can be imposed), and not amenable to the above optimal transport machinery (at least not directly). After some careful manipulations (let me skip the details here) I managed to recast the problem in a "traditional form", but surprisingly enough I realized that at least in some completely explicit examples the optimal-transport rate $\lambda>0$ can be strictly smaller than the spectral gap $\lambda_1>0$, while I was expecting both to coincide. Actually I was expecting the opposite, since in the spirit of the above "trade-off from space to time"

In a smooth, bounded and convex domain $\Omega\subset \mathbb R^d$ consider the usual linear Fokker-Planck equation with Neumann (some would say Robin) boundary conditions \begin{equation} \label{FP} \begin{cases} \partial_t \rho =\Delta\rho +\operatorname{div}(\rho\nabla V) & \mbox{for }t>0,x\in\Omega\\ (\nabla\rho +\rho\nabla V)\cdot\nu =0 & \mbox{on }\partial\Omega\\ \rho|_{t=0}=\rho_0 \end{cases} \tag{FP} \end{equation} where $V:\Omega\to\mathbb R$ is a given, smooth potential and $\nu$ is the outer unit normal on the boundary. The initial datum $\rho_0$ is a probabilidy density, $\rho_0\geq 0$ with $\int_\Omega\rho_0(x)dx=1$.

In a smooth, bounded and convex domain $\Omega\subset \mathbb R^d$ consider the usual linear Fokker-Planck equation with Neumann (some would say Robin) boundary conditions \begin{equation} \label{FP} \begin{cases} \partial_t \rho =\Delta\rho +\operatorname{div}(\rho\nabla V) & \mbox{for }t>0,x\in\Omega\\ (\nabla\rho +\rho\nabla V)\cdot\nu =0 & \mbox{on }\partial\Omega\\ \rho|_{t=0}=\rho_0 \end{cases} \tag{FP} \end{equation} where $V:\Omega\to\mathbb R$ is a given, smooth potential and $\nu$ is the outer unit normal on the boundary. The initial datum $\rho_0$ is a probability density, $\rho_0\geq 0$ with $\int_\Omega\rho_0(x)dx=1$.

It is well-known from the celebrated paper [JKO] that \eqref{FP} is the Wasserstein gradient flow of the relative entropy $$ \mathcal H_{\pi}(\rho)=\int_\Omega\frac{\rho(x)}{\pi(x)}\log\left(\frac{\rho(x)}{\pi(x)}\right) \pi(x) dx. $$ Here the Gibbs distribution $$ \pi(x)=\frac{1}{Z}e^{-V(x)} $$ is the unique stationary solution of \eqref{FP} ($Z>0$ is a normalizing factor so that $\int \pi=1$). This gradient flow structure can be formalized and made completely rigorous in the context of metric gradient flows and curves of maximal slope, see [AGS]. It is moreover known that, if $Hess V\geq \lambda$ for some $\lambda>0$, then the relative entropy $\rho\mapsto\mathcal H_\pi(\rho)$ is $\lambda$-geodesically convex (aka $\lambda$-displacement convex as introduced by R. McCann in [M]). This in turn can be exploited to prove long-time convergence $$ W_2(\rho_t,\pi) \leq C e^{-\lambda t} \qquad\mbox{and}\qquad \|\rho_t-\pi\|_{L^1}\leq C e^{-\lambda t}. $$ (here $W_2(\mu,\nu)$ is the quadratic Wasserstein distance between probability measures, and the various constants only depend on the intial entropy $\mathcal H_\pi(\rho_0)$ and $\lambda$) Roughly speaking, the proof is as follows: the entropy is dissipated along the evolution by the relative Fisher-information $$ \frac{d}{dt}\mathcal H_\pi(\rho_t)=-\mathcal I_\pi(\rho_t)\overset{def}{=} \int_\Omega \left|\nabla\log\left(\frac{\rho_t(x)}{\pi(x)}\right)\right|^2 \rho_t(x) dx $$ The $\lambda$-convexity of $V$ guarantees next the Logarithmic-Sobolev Inequality \begin{equation} \label{LSI} \mathcal I_\pi(\rho)\geq 2\lambda\mathcal H_\pi(\rho), \tag{LSI} \end{equation} which by Grönwall's lemma immediately gives entropic decay $$ \mathcal H_\pi(\rho_t)\leq e^{-2\lambda t}\mathcal H_\pi(\rho_0). $$ On can then use a Talagrand inequality $W_2(\rho,\pi)\leq \sqrt{\frac{2\mathcal H_\pi(\rho)}{\lambda}}$ or a Csiszár-Kullback-Pinsker inequality $\|\rho-\pi\|_{L^1}\leq \sqrt{2\mathcal H_\pi(\rho)}$ to conclude. The key point is really here that the $\lambda$-convexity of the potential $V$ (or in other words the $\lambda$-displacement convexity of the relative entropy $\mathcal H_\pi(\cdot)$) is somehow equivalent to the Log-Sobolev inequality \eqref{LSI}. Of course this is very sketchy and there are various subtle points here, but let me remain formal for the sake of exposition.

It is well-known from the celebrated paper [JKO] that \eqref{FP} is the Wasserstein gradient flow of the relative entropy $$ \mathcal H_{\pi}(\rho)=\int_\Omega\frac{\rho(x)}{\pi(x)}\log\left(\frac{\rho(x)}{\pi(x)}\right) \pi(x) dx. $$ Here the Gibbs distribution $$ \pi(x)=\frac{1}{Z}e^{-V(x)} $$ is the unique stationary solution of \eqref{FP} ($Z>0$ is a normalizing factor so that $\int \pi=1$). This gradient flow structure can be formalized and made completely rigorous in the context of metric gradient flows and curves of maximal slope, see [AGS]. It is moreover known that, if $Hess V\geq \lambda$ for some $\lambda>0$, then the relative entropy $\rho\mapsto\mathcal H_\pi(\rho)$ is $\lambda$-geodesically convex (aka $\lambda$-displacement convex as introduced by R. McCann in [M]). This in turn can be exploited to prove long-time convergence $$ W_2(\rho_t,\pi) \leq C e^{-\lambda t} \qquad\mbox{and}\qquad \|\rho_t-\pi\|_{L^1}\leq C e^{-\lambda t}. $$ (here $W_2(\mu,\nu)$ is the quadratic Wasserstein distance between probability measures, and the various constants only depend on the intial entropy $\mathcal H_\pi(\rho_0)$ and $\lambda$) Roughly speaking, the proof is as follows: the entropy is dissipated along the evolution by the relative Fisher-information $$ \frac{d}{dt}\mathcal H_\pi(\rho_t)=-\mathcal I_\pi(\rho_t)\overset{def}{=}- \int_\Omega \left|\nabla\log\left(\frac{\rho_t(x)}{\pi(x)}\right)\right|^2 \rho_t(x) dx $$ The $\lambda$-convexity of $V$ guarantees next the Logarithmic-Sobolev Inequality \begin{equation} \label{LSI} \mathcal I_\pi(\rho)\geq 2\lambda\mathcal H_\pi(\rho), \tag{LSI} \end{equation} which by Grönwall's lemma immediately gives entropic decay $$ \mathcal H_\pi(\rho_t)\leq e^{-2\lambda t}\mathcal H_\pi(\rho_0). $$ On can then use a Talagrand inequality $W_2(\rho,\pi)\leq \sqrt{\frac{2\mathcal H_\pi(\rho)}{\lambda}}$ or a Csiszár-Kullback-Pinsker inequality $\|\rho-\pi\|_{L^1}\leq \sqrt{2\mathcal H_\pi(\rho)}$ to conclude. The key point is really here that the $\lambda$-convexity of the potential $V$ (or in other words the $\lambda$-displacement convexity of the relative entropy $\mathcal H_\pi(\cdot)$) is somehow equivalent to the Log-Sobolev inequality \eqref{LSI}. Of course this is very sketchy and there are various subtle points here, but let me remain formal for the sake of exposition.