The excess and the perimeter martingale - The KLS theorem and its methods

Part of the fixed-cut archive, Chapters The fixed cut: approach and lessons and The fixed cut: the near-Cheeger variant; the reading order is on the full proofs page.

Overview. This dossier proves Lemma 32.3, Proposition 32.2, Lemma 32.4, Lemma 32.5 and Proposition 28.1. Under stochastic localization, the lower Minkowski perimeter of a fixed cut is a supermartingale in general and a true martingale in the compact-support smooth class. The resulting excess bounds are then used to audit what an unweighted Stein-trace estimate would consume.

Lemma D4.1: countable infima of bounded boundary-layer mass martingales increase to $P_t(E)$ , which gives (D4.5). In the compact smooth class, the surface representation (D4.14), the $L^2$ density bound (D4.16) and stochastic Fubini give the martingale SDE (D4.6).
Proposition D4.1: $e_t\le P_t$ and Step 1 give (D4.20). Here $I_\mu(1/2)<1$ follows from Bobkov’s one-dimensional estimates.
Lemma D4.2: in the compact class only, the martingale property gives (D4.22).
Lemma D4.3: the infimum over a fixed countable family of cuts is a supermartingale. The comparison (D4.24) is only pointwise, and no supermartingale property is claimed for the random windowed infimum.
Proposition D4.2: (i) is Step 2. (ii) Step 2 and Lemma 32.1 turn the unweighted estimate (D4.27) into the absorptive Carleson bound (D4.29). Corollary 30.1 and Lemma 33.1 then give KLS. (iii) Proposition 25.2 rules out proving the unweighted estimate slice by slice.

Scope and regularity. Let $(\mu_t)_{t\geq0}$ be Eldan’s stochastic-localization posterior started from a log-concave probability measure $\mu$ on $\mathbb R^n$ . We use

F_t(x)=\frac{\dd\mu_t}{\dd\mu}(x),\qquad \dd F_t(x)=F_t(x)(x-a_t)\mathbin\cdot\dd W_t, \qquad a_t=\int x\,\dd\mu_t(x).

(D4.1)

For a Borel set $E$ and $r>0$ , put

E_r=\{x\in\mathbb R^n:\operatorname{dist}(x,E)<r\},

(D4.2)

and use the lower outer Minkowski convention of the manuscript,

P_t(E)=\mu_t^+(E) :=\liminf_{r\downarrow0}\frac{\mu_t(E_r)-\mu_t(E)}r.

(D4.3)

We also write

p_t=\mu_t(E),\qquad e_t(E)=P_t(E)-I_{\mu_t}(p_t).

(D4.4)

The general fixed-cut result below assumes $P_0(E)<\infty$ and uses only bounded-test mass martingales. The stronger martingale identity is stated separately in the compact-support smooth regularity class: the initial density is smooth and compactly supported and $E$ has a $C^2$ boundary with a tubular neighborhood over the support of the density, so the one-sided tube formula identifies (D4.3) with weighted surface area. No rough-set approximation or identification of inequivalent perimeter conventions is used.

1. Perimeter under localization¶

Lemma D4.1 (Perimeter martingale; Lemma 32.3)

For every Borel set $E$ with $P_0(E)<\infty$ , the lower Minkowski perimeter process is an integrable nonnegative supermartingale:

\mathbb E[P_t(E)\mid\mathcal F_s]\leq P_s(E),\qquad 0\leq s\leq t.

(D4.5)

In the compact-support smooth regularity class specified above, $P_t(E)$ is instead a true martingale and

\dd P_t(E) =\left(\int_{\partial^*E}(x-a_t)\,\dd\sigma_t(x)\right)\mathbin\cdot\dd W_t,

(D4.6)

where $\sigma_t$ is the weighted surface measure of $\mu_t$ on $\partial^*E$ . Thus its drift vanishes identically and $\mathbb E P_t(E)=P_0(E)$ .

Proof D4.1

We first prove the general lower-Minkowski assertion directly. For $q\in\mathbb Q_{>0}$ define

X_{q,t}=\frac{\mu_t(E_q)-\mu_t(E)}q =\frac{\mu_t(E_q\setminus E)}q.

(D4.7)

For each fixed $q$ , this is $q^{-1}$ times the mass of a fixed Borel set. The bounded-test localization identity (23.5) therefore makes $(X_{q,t})_{t\geq0}$ a bounded, nonnegative true martingale. For $n\geq1$ let

Y_{n,t}=\inf\{X_{q,t}:q\in\mathbb Q,\ 0<q<1/n\}.

(D4.8)

The infimum is over a nonempty countable fixed family. For every admissible $q$ and $0\leq s\leq t$ ,

\mathbb E[Y_{n,t}\mid\mathcal F_s] \leq\mathbb E[X_{q,t}\mid\mathcal F_s]=X_{q,s}.

(D4.9)

Because there are only countably many $q$ , the inequalities may be realized on one common full-probability set. Infimizing there gives

\mathbb E[Y_{n,t}\mid\mathcal F_s]\leq Y_{n,s}.

(D4.10)

Also $0\leq Y_{n,t}\leq X_{q_n,t}$ for any one admissible rational $q_n$ , so $Y_{n,t}$ is integrable.

For any probability measure $\nu$ , the map $r\mapsto\nu(E_r\setminus E)$ is continuous from below: if $q_k\uparrow r$ , then $E_{q_k}\setminus E\uparrow E_r\setminus E$ . Approximating every real $r>0$ from below by rationals consequently shows that the infimum over rational $q\in(0,1/n)$ equals the infimum over all real $r\in(0,1/n)$ . Hence, pathwise,

Y_{n,t}\uparrow \liminf_{r\downarrow0}\frac{\mu_t(E_r)-\mu_t(E)}r=P_t(E).

(D4.11)

Ordinary monotone convergence and (D4.10) with $s=0$ give

\mathbb E P_t(E) =\lim_n\mathbb E Y_{n,t} \leq\lim_nY_{n,0}=P_0(E)<\infty.

(D4.12)

Thus $P_t(E)$ is integrable. Conditional monotone convergence in (D4.10) now yields

\mathbb E[P_t(E)\mid\mathcal F_s] =\lim_n\mathbb E[Y_{n,t}\mid\mathcal F_s] \leq\lim_nY_{n,s}=P_s(E),

(D4.13)

which proves (D4.5) for arbitrary support and without a rough-set approximation.

It remains to prove the stronger statement in the compact-support smooth class, using the same Minkowski convention. Let $w_0$ be the smooth compactly supported density of $\mu$ and put $\dd\sigma_0=w_0\,\dd\mathcal H^{n-1}|_{\partial^*E}$ . The one-sided tubular-neighborhood formula for a smooth boundary and the continuous density $w_t=F_tw_0$ gives, for every fixed $t$ and sample path,

P_t(E)=\int_{\partial^*E}F_t(x)\,\dd\sigma_0(x), \qquad \dd\sigma_t(x)=F_t(x)\,\dd\sigma_0(x).

(D4.14)

Indeed, in outward normal coordinates the Jacobian is $1+O(r)$ and $w_t(x+r\mathbf n_x)\to w_t(x)$ , so division of the boundary-layer integral by its thickness and dominated convergence gives the displayed surface integral. Thus (D4.14) is precisely the lower Minkowski perimeter (D4.3), not a change of convention. Its initial surface measure is finite.

Let $K$ contain the support of $\mu$ and set $D=\operatorname{diam}K$ . Both $x$ and $a_u$ belong to the convex hull of $K$ for $\sigma_0$ -almost every $x$ , so $|x-a_u|\leq D$ . For fixed $x$ , let $\rho_m\uparrow\infty$ localize the stochastic integral in the pointwise SDE and apply Itô’s formula to $F_{u\wedge\rho_m}(x)^2$ . Taking expectations gives

\mathbb E F_{u\wedge\rho_m}(x)^2 \leq 1+D^2\int_0^u\mathbb E F_{v\wedge\rho_m}(x)^2\,\dd v.

(D4.15)

Gronwall and then Fatou as $m\uparrow\infty$ give

\mathbb E F_u(x)^2\leq \exp(D^2u).

(D4.16)

Consequently, for every finite $T$ ,

\begin{aligned} \mathbb E\int_0^T\int_{\partial^*E} |F_u(x)(x-a_u)|^2\,\dd\sigma_0(x)\,\dd u &\leq D^2\sigma_0(\partial^*E)\int_0^T e^{D^2u}\,\dd u<\infty. \end{aligned}

(D4.17)

Since $\sigma_0$ is finite, Cauchy–Schwarz also gives

\mathbb E\int_0^T \left|\int_{\partial^*E}F_u(x)(x-a_u)\,\dd\sigma_0(x)\right|^2\dd u<\infty.

(D4.18)

These are pre-interchange stochastic-Fubini bounds. Stochastic Fubini applied to the pointwise density SDE therefore proves (D4.6); $F_t\,\dd\sigma_0=\dd\sigma_t$ gives its displayed coefficient.

Finally, the same bound (or Novikov directly) makes every fixed- $x$ process $F_t(x)$ a true martingale on bounded intervals. Conditional Tonelli in (D4.14) gives

\mathbb E[P_t(E)\mid\mathcal F_s] =\int_{\partial^*E}\mathbb E[F_t(x)\mid\mathcal F_s]\,\dd\sigma_0(x) =P_s(E).

(D4.19)

Thus the smooth compact-support lower Minkowski perimeter is a true martingale with the stated SDE.

2. The unconditional excess bound and the exact identity¶

Proof D4.2

The profile is nonnegative, so $0\leq e_t(E)\leq P_t(E)$ pathwise. Consequently, Tonelli and (D4.5) give

\mathbb E\int_0^{T\wedge\tau}e_t(E)\,\dd t =\int_0^T\mathbb E[\mathbf 1_{\{t<\tau\}}e_t(E)]\,\dd t \leq\int_0^T\mathbb E P_t(E)\,\dd t \leq T P_0(E).

(D4.21)

This calculation uses neither optional stopping nor independence of $\tau$ ; only the deterministic-time supermartingale bound is used.

To bound $P_0(E)$ , take a coordinate marginal of $\mu$ . It is a one-dimensional log-concave law $\nu$ of variance one. If $f$ is its density and $m$ a median, the coordinate halfspace $H=\{x:x_1\leq m\}$ has $\mu(H)=1/2$ and $\mu^+(H)=f(m)$ . Bobkov’s one-dimensional isoperimetric identity and variance estimate Bobkov, 1999, Proposition 4.1 and (4.2) give $\operatorname{Is}(\nu)=2f(m)$ and $\operatorname{Is}(\nu)^2\leq2/\operatorname{Var}_\nu(X)$ , hence $I_\mu(1/2)\leq f(m)\leq1/\sqrt2<1$ . Therefore $P_0(E)=I_\mu(1/2)+e_0\leq1+e_0$ , proving (D4.20).

3. What a countable infimum of fixed-cut perimeter supermartingales implies¶

Lemma D4.3 (Fixed competitor families; Lemma 32.5)

Let $\mathfrak S$ be a fixed nonempty countable family of Borel sets of finite initial lower Minkowski perimeter and define $J_t=\inf_{S\in\mathfrak S}P_t(S)$ . Then $J$ is a supermartingale. The same conclusion holds when a prescribed uncountable family has a fixed nonempty countable determining subfamily $\mathfrak S_0$ satisfying $\inf_{S\in\mathfrak S}P_t(S)=\inf_{S\in\mathfrak S_0}P_t(S)$ almost surely at every time under consideration; in that case $J$ means the latter countable infimum. On $\{t<\tau_\eta\}$ one has the pointwise comparison

I_{\mu_t}(p_t)\geq J_t(\eta), \qquad J_t(\eta)=\inf\{P_t(S):\mu_t(S)\in[1/2-\eta,1/2+\eta]\}.

(D4.24)

The family in the last infimum depends on $(t,\omega)$ ; the first assertion therefore does not say that $J_t(\eta)$ or $I_{\mu_t}(p_t)$ is a supermartingale.

Proof D4.4

Fix $0\leq s\leq t$ . For every $S\in\mathfrak S$ , $J_t\leq P_t(S)$ , hence by monotonicity of conditional expectation and the noncompact part of Lemma D4.1,

\mathbb E[J_t\mid\mathcal F_s] \leq\mathbb E[P_t(S)\mid\mathcal F_s] \leq P_s(S).

(D4.25)

Taking the infimum in $S$ on the right gives $\mathbb E[J_t\mid\mathcal F_s]\leq J_s$ . This argument needs only that each fixed-cut perimeter is a supermartingale; in the compact-support class the second inequality happens to be an equality. Moreover, choosing one $S_\star\in\mathfrak S$ gives $0\leq J_t\leq P_t(S_\star)$ and $\mathbb E P_t(S_\star)\leq P_0(S_\star)<\infty$ , so $J_t$ is integrable. The determining-subfamily variant is exactly the same countable argument. No conclusion is claimed for a raw uncountable pointwise infimum merely assumed measurable.

On $\{t<\tau_\eta\}$ , $p_t\in[1/2-\eta,1/2+\eta]$ . Thus the exact-mass competitor class $\{S:\mu_t(S)=p_t\}$ is contained in the windowed class in (D4.24). The infimum over the smaller class is at least the infimum over the larger one, proving the pointwise comparison. But eligibility for the windowed class is the random condition $\mu_t(S)\in[1/2-\eta,1/2+\eta]$ ; it is not a fixed family to which the conditional-expectation argument can be applied.

4. Consumption audit¶

Proposition D4.2 (Consumption audit; Proposition 28.1)

The following three assertions hold.

For every isotropic log-concave $\mu$ , every Borel $E$ with $\mu(E)=1/2$ and $P_0(E)<\infty$ , every $T>0$ , and every stopping time $\tau$ ,
$\mathbb E\int_0^{T\wedge\tau}e_t(E)\,\dd t\leq(1+e_0)T.$
(D4.26)
Replace the covariance weight in the stable Stein-trace estimate by 1, keeping its fixed tight window and absorption margin. Then this unweighted estimate, without any separate excess-propagation hypothesis, implies KLS.
There are log-concave pairs $(\nu,E)$ with $\nu(E)=1/2$ , $r=D=0$ , $e(E)\to0$ , and $\mathcal S_\nu(E)/s\to\infty$ . Hence the unweighted estimate cannot be proved by a universal one-time-slice inequality of the asserted form.

Proof D4.5

The first assertion is Proposition D4.1.

For the second, spell out exactly what is consumed. Fix a universal $\eta\in(0,1/6]$ and constants $C_0,C_1,C_2$ , $\beta\geq0$ satisfying $2\beta+64\eta^2<1$ . The unweighted stable Stein-trace estimate is

\begin{split} \mathbb E\int_0^{T\wedge\tau_\eta} \frac{\mathcal S_{\mu_t}(E)}{s_t}\,\dd t \leq{}&C_0T+C_1\mathbb E\int_0^{T\wedge\tau_\eta}r_t\,\dd t +\beta\mathbb E\int_0^{T\wedge\tau_\eta}D_t\,\dd t\\ &+C_2\mathbb E\int_0^{T\wedge\tau_\eta}e_t(E)\,\dd t . \end{split}

(D4.27)

The exact tight-window Stein/source conversion (Lemma 32.1) is

\mathbb E\int_0^{T\wedge\tau_\eta}S_t\,\dd t \leq2\mathbb E\int_0^{T\wedge\tau_\eta} \frac{\mathcal S_{\mu_t}(E)}{s_t}\,\dd t +64\eta^2\mathbb E\int_0^{T\wedge\tau_\eta}D_t\,\dd t.

(D4.28)

For the near-Cheeger cuts used in the contradiction argument, take $e_0\leq1$ . Applying the first assertion to (D4.27) gives

\begin{split} \mathbb E\int_0^{T\wedge\tau_\eta}S_t\,\dd t \leq{}&(2C_0+4C_2)T +2C_1\mathbb E\int_0^{T\wedge\tau_\eta}r_t\,\dd t\\ &+(2\beta+64\eta^2) \mathbb E\int_0^{T\wedge\tau_\eta}D_t\,\dd t. \end{split}

(D4.29)

This is precisely an absorptive tight-window two-color Carleson estimate, with damping coefficient strictly below one. Corollary 30.1 then gives a positive universal boundary lower bound for every balanced near-Cheeger cut to which (D4.27) applies.

For completeness, if KLS failed, there would be isotropic log-concave $\mu_k$ with $h_{\mu_k}\to0$ . The balanced characterization from Lemma 33.1, $h_\nu=2I_\nu(1/2)$ , permits choices of sets $E_k$ of mass $1/2$ with $P_0(E_k)\leq I_{\mu_k}(1/2)+o(1)$ . Thus $P_0(E_k)\to0$ and $e_0(E_k)\leq1$ for all large $k$ , whereas (D4.29) and tight-window consumption give a uniform positive lower bound for the same perimeters, a contradiction. No weighted excess estimate was used.

The third assertion is exactly the certified anisotropic two-tail construction of Proposition 25.2: it has $r=D=0$ , vanishing absolute excess, and divergent $\mathcal S/s$ . A slice-wise version of (D4.27) would have a uniformly bounded right-hand side on that family and a divergent left-hand side. This disproves that slice-wise route while making no claim against nonlocal-in-time proofs or the covariance-weighted formulation.

Obstructions respected. No node in this dossier has a ledger bounded_by edge. Nevertheless, Lemma D4.2 and Lemma D4.3 explicitly preserve the content of rem:profile-circularity: the exact identity is restricted to compact support, and no supermartingale property is inferred for the random time-dependent balanced-profile infimum. Proposition D4.2 uses the separate two-tail obstruction only to refute a slice-wise unweighted estimate; it does not promote numerical or directional evidence to proof.

References¶

Bobkov, S. G. (1999). Isoperimetric and Analytic Inequalities for Log-Concave Probability Measures. Annals of Probability, 27(4), 1903–1921. 10.1214/aop/1022874820