Family 5: transport maps and the entropic barrier

KLS is now proved, by three different arguments compared in Chapter The proofs of KLS compared. None of them uses the transport maps of this chapter, although all three start from Letwin’s quadratic estimate, which rests on the moment map of Chapter Family 4: moment maps, Monge–Ampère, and Stein kernels. This chapter describes what transport controls on its own and what it does not give.

Object followed. A map pushing a reference measure — usually a Gaussian, or Wiener measure — onto $\mu$, together with the derivative of that map.

What it buys. A reformulation in which KLS becomes a regularity statement about a single transport map rather than a spectral statement about a diffusion. The reformulation is exact and appealing; the difficulty is that the three natural instances each fail in an instructive and different way.

Caffarelli¶

If the target is uniformly more log-concave than a Gaussian, Caffarelli’s contraction theorem makes the Brenier map from that Gaussian to $\mu$ 1-Lipschitz. The Gaussian Poincaré inequality then transfers directly, with no loss.

This settles the strongly log-concave case, the same case already settled by Bakry–Émery for structured families. It cannot cover arbitrary log-concave measures, whose curvature may vanish identically — the uniform measure on a convex body being the extreme instance.

Brownian (Föllmer) transport¶

The Brownian transport map sends Wiener measure to a target $\mu$ Mikulincer & Shenfeld, 2024. A dimension-free estimate on its Malliavin derivative,

would give $\Var_\mu f\le C\Lip(f)^2$ for every Lipschitz $f$; Milman’s equivalence between Lipschitz concentration and a spectral gap for log-concave measures Milman, 2009 would then upgrade that to KLS.

This is the reason the family has not yet produced an independent bound: current estimates on (5.1) are polylogarithmic and largely inherit existing KLS-scale input rather than supplying it. The averaged operator derivative is the plausible target.

The entropic barrier¶

For a convex body $K$ set

Then

where $p_\theta\propto e^{\inner\theta x}\one_K$, and $D^3\Lambda(\theta)$ is the third-moment tensor of the same exponential tilt Bubeck & Eldan, 2014. The entropic barrier therefore packages exactly the covariance and third-moment geometry that stochastic localization manipulates — (5.3) and (2.5) are two descriptions of one object, one indexed by a tilt parameter and one by a time.

The gap is a matter of which contractions are controlled. Ordinary self-concordance controls scalar contractions of $D^3\Lambda$, of the form $D^3\Lambda[u,u,u]$. Localization needs matrix or Hilbert–Schmidt contractions — precisely the parameter $\kappa_n$ of (0.13). Letwin’s theorem now supplies the latter (Proposition 26.1), which closes this particular mismatch; that estimate alone does not supply the all-functions spectral step, which the three proofs obtain through polynomials of every degree (Chapter The proofs of KLS compared).

What it does not reach alone. The dimension-free averaged bound (5.1) on the expected operator norm of the transport derivative. No pathwise dimension-free Lipschitz map exists rules out the pathwise approach, so any argument must work in expectation, and an expected operator norm is not accessible by the pathwise convexity techniques that make Caffarelli’s theorem work. Available derivative bounds either lose the alignment between the derivative and the test function, or reuse KLS-scale input and so cannot improve it.

Where this family meets the alternative mechanisms. It enters none of them directly. The nearest open question is the coupling one of the next family, Chapter Family 6: parallel coupling of exponential tilts, whose extension beyond linear tilts is described in Section A further direction: parallel coupling beyond linear tilts; no labelled statement of this manuscript formulates it. Transport enters the alternative mechanisms only through the Brascamp–Lieb cap that stochastic localization uses.