Heat flow, decay of the Fisher information, and $\lambda$-displacement convexity

Question

In the whole post I will work in the flat torus $\mathbb T^d=\mathbb R^d/\mathbb Z^d$ and $\rho$ will stand for any probability measure $\mathcal P(\mathbb T^d)$. This question is strongly related to two of my previous posts, universal-decay-rate-of-the-fisher-information-along-the-heat-flow and improved-regularization-for-lambda-convex-gradient-flows.

• Fact 0: the quadratic Wasserstein distance $W_2$ induces a (formal) Riemannian structure on the space of probability measures, which gives a meaning to Wasserstein gradients $\operatorname{grad}_{W_2}F(\rho)$ of a functional $F:\mathcal P(\mathbb T^d)\to\mathbb R$ at a point $\rho$

• Fact 1: the heat flow $\partial_t\rho_t=\Delta\rho_t$ is the Wasserstein gradient flow $$ \dot\rho_t=-\operatorname{grad}_{W_2}H(\rho_t) $$ of the Boltzmann entropy $$ H(\rho)=\int_{\mathbb T^d}\rho\log\rho $$

• Fact 2: the Boltzmann entropy is $\lambda$-(displacement) convex for some $\lambda$. Its dissipation functional is the Fisher information, $$ F(\rho):=\|\operatorname{grad}_{W_2} H(\rho)\|^2_{\rho}=\int _{\mathbb T^d}|\nabla\log\rho|^2 \rho $$

• Fact 3: for abstract metric gradient flows (in the sense of [AGS]) and $\lambda$-convex functionals $\Phi:X\to\mathbb R\cup\{\infty\}$ one expects a smoothing effect for gradient flows $\dot x_t=-\operatorname{grad}\Phi(x_t)$ in the form \begin{equation} |\nabla\Phi(x_t)|^2\leq \frac{C_\lambda}{t} \Big[\Phi(x_0)-\inf_X\Phi\Big] \tag{R} \end{equation} at least for small times, where $C_\lambda$ depends only on $\lambda$ but not on $x_0$ see e.g. [AG, Proposition 3.22 (iii)].

• Fact 3': with the same notation as in Fact 3, an alternative regularization can be stated as \begin{equation} |\nabla\Phi(x_t)|^2 \leq \frac{1}{2e^{\lambda t}-1}|\nabla\Phi(y)|^2 +\frac{1}{(\int_0^te^{\lambda s}ds)^2} dist^2(x_0,y), \,\, \forall y\in X \tag{R'} \end{equation}

• Fact 4: in the Torus the Fisher information decays at a universal rate, i-e there is $C=C_d$ depending on the dimension only such that, for all $\rho_0\in \mathcal P(\mathbb T^d)$ and $t>0$, the solution $\rho_t$ of the heat flow emanating from $\rho_0$ satisfies \begin{equation} F(\rho_t)\leq \frac{C}{t} \tag{*} \end{equation} This follows from the Li-Yau inequality [LY], see this post of mine and F. Baudoin's answer.

Question: is there more to ($*$) than just the convexity of the Boltzmann functional? If the driving functional were upper-bounded $\Phi(x_0)\leq C$ (for all $x_0\in X$) in the regularization estimate (R) then we would immediately get the universal decay $|\nabla \Phi(x_t)|^2\leq \frac{C}{t}$. However, in the specific context of Facts 0-2 it is clearly not true that the Boltzmann entropy is upper-bounded. In fact there are many probability measures with infinite entropy, take e.g. any Dirac mass. Since (R) is optimal, I guess that one cannot simply deduce (*) from general $\lambda$-convexity arguments, and there is more than meets the eyes. But is there any connection? Note that both the Li-Yau inequality and the displacement convexity of the Boltzmann entropy strongly rely on the nonnegative Ricci curvature of the underlying torus.

I tried desperately to use any modified regularization estimate (e.g. R' and variants thereof instead of R), but to no avail so far. I am starting to believe that there is no direct implication, and that the work of Li-Yau is really profoundly ad-hoc (don't get me wrong, I just mean that their results cannot be generalized for abstract gradient-flows, and that their result/proof really leverages the specific structure and setting of the heat flow in Riemannian manifolds, not just any gradient flow). I would immensely appreciate any input or insight!

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[AG] Ambrosio, L., & Gigli, N. (2013). A user’s guide to optimal transport. In Modelling and optimisation of flows on networks (pp. 1-155). Springer, Berlin, Heidelberg.

[AGS] Ambrosio, L., Gigli, N., & Savaré, G. (2008). Gradient flows: in metric spaces and in the space of probability measures. Springer Science & Business Media.

[LY] Li, P., & Yau, S. T. (1986). On the parabolic kernel of the Schrödinger operator. Acta Mathematica, 156, 153-201.

Answer 1

I won't say that it is impossible, but I don't see how to obtain $(\ast)$ using only general theory. There might be a different strategy that works, but I can tell you why I don't think the Li-Yau estimate can be proven using general properties of convexity. In particular, Li-Yau relies on some careful estimates involving the Laplace-Beltrami operator (and some hard analysis), which I don't think general theory can "see".

For a detailed write up of the Li-Yau estimate, I recommend Lectures on Differential Geometry by Schoen and Yau, which was very helpful for me.  From a high level overview, the idea is to let $u$ be a non-negative solution to the heat equation, consider $\log (u + \epsilon)$ and try to bound its derivative. To do this, you consider the point which maximizes $ | \nabla \log (u + \epsilon) |^2$ and use the Bochner formula. Bochner's formula has a correction term due to the curvature, but when the manifold is Ricci positive, this has a favorable sign and we can ignore it (or use something like a barrier function to sharpen the estimate). The key insight is actually a clever use of the Cauchy-Schwarz inequality to eke out a little bit extra from the second derivative terms. It's elementary, but also a stroke of genius, and allows everything else to work.

If you read proofs of Li-Yau, the logarithm tends to appear near the end. However, it was helpful for my intuition to realize that this is not ad hoc; there was always going to be a logarithm because we are using the maximum principle applied to the function $\dfrac{|\nabla u|^2}{(u+\epsilon)^2} = | \nabla \log(u+\epsilon)|^2$.

The fact that $\nabla u$ and $u$ are raised to the same power here is crucial. When the power of $\nabla u$ is less than $u$, integrating out the resulting inequality gives a bounded function (which is significantly less useful). There's this really delicate balancing act in order for everything to work, and logs play an essential role. As a brief aside, I suspect you get different powers of $\nabla u$ and $u$ if you try the Li-Yau strategy with the porous media equation (I'm not entirely sure of this though).

So back to your question about whether this can be done using general properties of gradient flows. It might be a lack of imagination on my part, but it's hard for me to see how this would work. There's several essential steps that rely on hard analysis. For instance, you really need the Cauchy-Schwarz step to work and the resulting function that you get from integrating out should be unbounded. Furthermore, while it's possible to sharpen the estimate, the original version is already fairly sharp, in that there is not a whole lot of wiggle room. As such, while it's possible to adapt the argument to elliptic operators or to include lower order terms, it does seem like there is genuinely more here than the general theory. 

Answer 2

To complement MaoWao and André Schlichting's comment: In the general setting of your Facts 3 and 3' (plus some regularity conditions), the instantaneous regularization property

$$|\nabla \Phi(x_t)|^2 \leq \frac{n}{2t}$$

is implied by $(0,n)$-convexity of $\Phi$ (in the sense of Erbar, Kuwada and Sturm (2013)), i.e., $\mathrm{Hess} \Phi \geq \frac{1}{n} (\nabla \Phi \otimes \nabla \Phi)$.

Here is a proof of the implication in the case $X = \mathbb{R}^d$: suppose $\nabla^2 \Phi(x) \succeq \frac{1}{n} \nabla \Phi(x) \nabla \Phi(x)^\top$ for all $x$ and consider $(x_t)_{t \geq 0}$ such that $\frac{d}{dt} x_t = -\nabla \Phi(x_t)$. Consider the potential function $V(t) = \frac{1}{|\nabla \Phi(x_t)|^2}$. We have by direct computations

$$\frac{d}{dt} V(t) = \frac{2 \nabla \Phi(x_t)^\top \nabla^2 \Phi(x_t) \nabla \Phi(x_t)}{|\nabla \Phi(x_t)|^4} \geq \frac{2}{n},$$

so $V(t) \geq \frac{2}{n} t + V(0) \geq \frac{2}{n} t$ and $|\nabla \Phi(x_t)|^2 \leq \frac{n}{2t}$.

For more general $X$, I would guess that the same implication is true though I don't know how this could be proved. (Of course with enough regularity the proof above will apply.)

For $X$ being the Wasserstein space, this gives a heuristic "proof" of your Fact 4 starting from the CD(0,d) property of $\mathbb{T}^d$, that doesn't go via the Li-Yau inequality.

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Edit: just to add some context, this choice of potential function $V(t)$ is directly inspired from the proof of $O(1/t)$ convergence of gradient flow for convex objectives which poses $W(t) = \frac{1}{\Phi(x_t) - \min \Phi}$ (see e.g. this blog post)