Fokker-Planck: equivalence between linear spectral gap and nonlinear displacement convexity?

Question

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 $\int_\Omega\rho_0(x)dx=1$.

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Fact 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 $\text{Hess} V\geq \lambda$ 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.

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Fact 2

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 $t\to+\infty$.

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Question:

Is the optimal transportation rate $\lambda>0$ (related to displacement convexity) always equal to the spectral gap $\lambda_1$? If not, are they always ordered in any way? In words: as far as long time convergence is concerned, does the highly nonlinear optimal-transport point of view give better or worse predictions than linear spectral theory? (or neither...) Is there by any chance an explanation along the usual rule of thumbs that "for parabolic equations one can trade-off space regularity for time regularity"? By this I mean that measuring the deviation of the solution $\rho_t$ to its limit $\pi$ in weaker or stronger senses (Wasserstein distance/$L^1$ norm/higher order Sobolev norms) may lead to faster/slower convergence rates?

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"

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[AGS] Ambrosio, L., Gigli, N., & Savaré, G. (2005). Gradient flows: in metric spaces and in the space of probability measures. Springer Science & Business Media.

[JKO] Jordan, R., Kinderlehrer, D., & Otto, F. (1998). The variational formulation of the Fokker--Planck equation. SIAM journal on mathematical analysis, 29(1), 1-17.

[M] McCann, R. J. (1997). A convexity principle for interacting gases. Advances in mathematics, 128(1), 153-179.

Answer 1

One has the ordering of the three $\lambda$'s, i.e. $$ \lambda_{\text{convex}} \leq \lambda_{\text{LSI}} \leq \lambda_{\text{SG}}, $$ where $\lambda_{\text{convex}}$ is the one from convexity, $\lambda_{\text{LSI}}$ is the one from the inverse Log-Sobolev constant (both as in your question) and $\lambda_{\text{SG}}$ the spectral gap, characterized via the smallest constant in the weighted Poincaré inequality $$ \Vert f -1 \Vert_{L^2(\pi)}^2 \leq (2\lambda)^{-1} \int |\nabla f |^2 d\pi , \qquad \forall f : \int f \, d\pi = 1. $$ here $f$ plays the role of $d\rho/ d\pi$. The first inequality is the HWI-inequality after Otto-Villani and the second is linearization of the LSI.

The equivalence $\lambda_{\text{convex}} = \lambda_{\text{LSI}} = \lambda_{\text{SG}}$ holds for $V$ being a positive quadratic form, i.e. $V(x) = x \cdot H x$ for a fixed symmetric positive matrix $H$. Then, the eigenvalues and eigenvectors of the according Ornstein-Uhlenbeck process are explicit (products of Hermite-polynomials) and obtained in dependence of the eigenvalues of $H$. The smallest non-negative eigenvalue is then the smallest eigenvalue of $H$, in accordance to the $\lambda$-convexity. The LSI is sandwiched inbetween anyways.

Except for this particular case, I expect that the equivalence breaks down for generic $V$ (non-quadratic). This is easiest observed in the non-convex case. So let, $V$ be a double-well, with local max in $0$ and two minima in $~\pm 1$ and have convex quadratic growth outside a bounded region. Then, the Fokker-Planck evolution is $\lambda$-convex for a $\lambda<0$ being the lowest bound on the Hessian again, determined by the non-convexity of $V$ around the local maximum. However, the Log-Sobolev and spectral gap constants are still positive finite and can be obtained by combining the Bakry-Emery criterion with the Holley-Stroock perturbation principle, since $V$ is a bounded perturbation of a convex potential.

This becomes even more apparent, by considering the vanishing diffusion limit, i.e. consider for $\varepsilon>0$ the Fokker-Planck equation $$ \partial_t \rho = \varepsilon \Delta \rho + \nabla \cdot(\rho \nabla V) $$ In this case, it becomes clear that the time-scales captured by convexity and Log-Sobolev constants or spectral gaps are rather different.

From the comment of @Tobsn follows that, convexity measures local stability at every point in the space of probability measures and the setting of double-well potential it is readily checked that, one actually gets the opposite comparison of the type $$ W(\rho_t, \hat\rho_t) \geq c e^{|\lambda| t} W(\delta_{-\eta},\delta_{\eta}) \quad\text{for } t \in [0, t_0] $$ where $\eta>0$ is small and $t_0$ is also not too large. I write $|\lambda|=-\lambda$ to make clear, that the trajectories expand and do not converge. This follows, because $\rho$ and $\hat\rho$ will follow mainly the deterministic ODE $\dot X_t = - \nabla V(X_t)$, which expands at rate $|\lambda|$ close to the local maximum. In particular, this result also holds for $\varepsilon=0$.

In comparison, the log-Sobolev constant and the inverse spectral gap measure a global time-scale quantifying the ergodicity in time, showing that the diffusion converges to the measure $\pi$ in entropy or the relative density $f_t = \rho_t / \pi$ in the $L^2(\pi)$-sense. Those are global averaged quantities and hence behave in general better, as the non-convex setting shows. However, note that for the $\varepsilon$-dependent case, both $\lambda_{LSI}$ and $\lambda_{SG}$ will degenerate to $0$ exponentially like $e^{-C/\varepsilon}$, which is a statement about metastability being present in this setting (the limiting ODE dynamic is non-ergodic, since it has actually three stationary states).

There is a theory of variable Ricci bounds, which can catch a bit this different local stability and might improve the gap between those constants. One, can probably think of the Bracamp-Lieb inequality as an instance of variable curvature, since it shows that $\text{Hess}\, V$ behaves like the local Ricci-tensor for the diffusion, that is $$ \Vert f - 1 \Vert_\pi^2 \leq \int \left\langle \nabla f , \text{Hess} V\, \nabla f \right\rangle d\pi . $$

Other ways, to make the different behaviour of the constants precise, at least in the scaling regime with vanishing diffusion $\varepsilon\ll 1$, are in 1d upper and lower estimates on the Log-Sobolev constant and spectral gap with the help of the Bobkov-Götze or Muckenhoupt criterion.

I can provide more details or references on the individual mentioned observations, but I'm not aware of a general result showing that for generic V (not perfectly quadratic potentials), there are strict inequalities between all of the three $\lambda$'s, i.e. $$ \lambda_{\text{convex}} < \lambda_{\text{LSI}} < \lambda_{\text{SG}}. $$