Analytic conventions and the two-color localization setup

Global conventions¶

All measures are Borel probability measures on $\R^n$ unless otherwise stated. A log-concave measure has density $e^{-V}$ on its convex support, with $V$ convex. For a measure $\nu$, write

The Hilbert–Schmidt inner product on symmetric matrices is

For positive semidefinite matrices, $B\preceq A$ denotes the Loewner order.

For log-concave $\nu$, the isoperimetric profile $I_\nu$ is concave in the usual generalized sense; this is part of the Bobkov theory of log-concave isoperimetry Bobkov, 1999, and is also closely related to the convexity principles emphasized in Milman, 2009. This concavity allows the Cheeger problem to be tested on balanced volumes up to universal constants; Lemma 33.1 below sharpens this to the exact identity $h_\nu=2I_\nu(1/2)$.

Stochastic localization¶

Starting from an isotropic log-concave measure $\mu$, Eldan stochastic localization produces random measures

where $(c_t)$ is adapted to a Brownian motion $(W_t)$ in $\R^n$. The posterior $\mu_t$ is $t$-uniformly log-concave because of the Gaussian factor. Put

The basic martingale identity is

for every bounded measurable $\varphi$. In particular,

The covariance process satisfies

where

This is the standard covariance SDE in stochastic localization Eldan, 2013Lee & Vempala, 2024. The pointwise density process $F_t(x)=\dd\mu_t/\dd\mu(x)$ is a nonnegative local martingale satisfying

a true martingale on bounded intervals under compact support; this fact drives the perimeter martingale of Lemma 32.3.

Two consequences of $t$-uniform log-concavity are used repeatedly. The Brascamp–Lieb inequality applied to linear functions gives the pathwise cap

and the Bakry–Emery mechanism gives the dimension-free Cheeger bound at scale $\sqrt t$ used in Theorem 30.1 Brascamp & Lieb, 1976Bakry et al., 2014.

Two-color notation¶

Fix a measurable set $E$ and write $F=E^c$. Define

The conditional means and covariances are

Set

The whitened color Hessian is denoted

using the inverse on the support of $\mu_t$ if $A_t$ is singular. This notation deliberately differs from the weighted mean curvature $H_\nu$ used later.

The coarse balanced stopping time is

For the boundary approach, it is useful to work with a tighter window. For $0<\eta<1/2$ set

All stopped statements are understood with nested windows: for the coarse window start with $p_0\in[2/5,3/5]$, and for the tight window start with $p_0=1/2$ or $\abs{p_0-1/2}\le\eta/2$. This avoids the irrelevant degeneracy $\tau=0$ at the boundary of a stopping interval.

The excess processes attached to $E$ are

the second being the Cheeger excess of Definition 33.1; at $p_0=1/2$ the two coincide at time zero (Lemma 33.1).

Boundary notation¶

When $\nu=e^{-V}dx$ is smooth and log-concave on a convex support $K$, its generator is

For a smooth finite-perimeter set $E$, write

The weighted mean curvature is

where $n$ is a chosen unit normal. The weighted Jacobi potential is

The second variation form is

where $\calB_{\partial K}$ denotes the support or free-boundary contribution. The full constant-mode curvature is

For full smooth support, $\mathfrak K_\Sigma=\int_\Sigma\qJac_\Sigma\dd\sigma_\nu$.

Regularity convention¶

Each stochastic identity below is used only in the regularity class stated at its label or in its full proof. In particular, no blanket rough-set approximation is used to identify lower outer Minkowski perimeter with weighted reduced-boundary area.

For every fixed Borel set $E$ with finite initial lower outer Minkowski perimeter, Lemma 32.3 proves the conditional supermartingale inequality and hence

If the initial density is smooth and compactly supported and $E$ has a $C^2$ boundary with a tubular neighborhood over that support, the same lower Minkowski perimeter is weighted surface area and (23.9) upgrades the inequality to the driftless surface martingale identity. No equality is asserted here for an arbitrary rough set merely from compact support.