Balanced survival and the exterior boundary - The KLS theorem and its methods

Part of the fixed-cut archive, Chapter The fixed cut: the mass martingale and the Carleson estimate; the reading order is on the full proofs page.

Overview. This standalone proof of Lemma 30.1 uses the actual lower outer Minkowski content of a fixed measurable set. Neighborhood increments and conditional expectation transfer that content back from a finite-time posterior. A curvature-preserving approximation proves the required Gaussian comparison on arbitrary convex supports. Finally, concavity of the original law’s isoperimetric profile converts balanced cuts to Cheeger isoperimetry. No Riccati or occupation estimate enters.

Conventions. For an actual set $E\subset\mathbb R^n$ , put $E^r=\{x:\operatorname{dist}(x,E)<r\}$ for $r>0$ , with $\operatorname{dist}(x,\varnothing)=+\infty$ , and set

\nu^+(E)=\liminf_{r\downarrow0}\frac{\nu(E^r)-\nu(E)}r.

(126.1)

Sets may be measurable in the completion of $\nu$ . Their neighborhoods are open, regardless of the measurability of the set itself. We retain the actual set when taking neighborhoods: changing a null subset can change this content. All posterior measures below are equivalent to the original measure, so their completions coincide. Integrals of a completed-measurable indicator are defined using a Borel representative; neighborhoods still belong to the actual set.

We first provide the two analytic ingredients, including their generality.

Proof 126.1

The posterior has the conditional-law realization obtained by sampling $X\sim\mu$ , taking independent standard Brownian motion $B$ , and observing $Y_s=sX+B_s$ . With $\mathcal F_t=\sigma(Y_s:0\le s\le t)$ , Bayes’ formula gives

\mu_t(dx)=\mathbb P(X\in dx\mid\mathcal F_t) =\frac{e^{Y_t\cdot x-t|x|^2/2}}{\int e^{Y_t\cdot z-t|z|^2/2}\mu(dz)}\mu(dx).

(126.3)

Indeed, the conditional density of $Y_t$ given $X=x$ is Gaussian with mean $tx$ and covariance $tI$ . Moreover the observation bridge $Y_s-(s/t)Y_t=B_s-(s/t)B_t$ , $s\le t$ , is independent of $(X,Y_t)$ , so the whole observation history has the same posterior as $Y_t$ . The denominator is positive and finite for every finite $Y_t$ and $t>0$ , by completing the square. This is the usual localization law: if $a_s=\mathbb E(X\mid\mathcal F_s)$ , then $W_t=Y_t-\int_0^t a_s\,ds$ is a continuous $\mathcal F_t$ martingale with quadratic covariation $tI$ , hence Brownian motion, and $dY_t=dW_t+a_t\,dt$ . The martingale assertion follows by conditioning $X-a_s$ against the past and using the independent Brownian increments; integrability holds since $\mathbb E|X|<\infty$ . These are precisely the coefficient and posterior formula of stochastic localization.

In particular, for each fixed completed-measurable $A$ , $\mu_t(A)$ is a bounded conditional-expectation martingale and $\mathbb E\mu_t(A)=\mu(A)$ . A Borel representative gives this assertion also for completed-measurable $A$ , since all finite-time likelihoods are strictly positive and finite.

For each $r>0$ apply this identity to $A=E^r\setminus E$ . The random variable

G_t(r)=\frac{\mu_t(E^r\setminus E)}r\ge0 \quad\hbox{satisfies}\quad \mathbb E G_t(r)=\frac{\mu(E^r\setminus E)}r=G_0(r).

(126.4)

The full lower limit defining $\mu_t^+(E)$ is measurable: it equals the lower limit over positive rational radii. To see this, the numerator $H(r)$ is nondecreasing and is left continuous, because $E^r=\bigcup_{s<r}E^s$ . Thus $H(q)/q\to H(r)/r$ as rational $q\uparrow r$ . Taking infima over radii in each interval $(0,\delta)$ therefore gives the same infimum over its rational radii. A countable sequence of such infima computes the lower limit.

If $\mu^+(E)=+\infty$ , the proposed inequality is automatic. Otherwise choose a deterministic sequence $r_j\downarrow0$ realizing the initial lower limit. Nonnegativity, the pointwise lower-limit inequality, and Fatou give

\mathbb E\mu_t^+(E) \le\mathbb E\liminf_jG_t(r_j) \le\liminf_j\mathbb E G_t(r_j)=\mu^+(E).

(126.5)

No boundary integral, finite-perimeter hypothesis, or exchange of an uncountable infimum with expectation is used.

Proof 126.2

The smooth input is the functional inequality

\sqrt\kappa\left[J\!\left(\int f\,d\rho\right) -\int J(f)\,d\rho\right]\le\int|\nabla f|\,d\rho, \qquad 0\le f\le1. \tag{*}

(126.7)

For a smooth potential $U$ with $\nabla^2U\succeq\kappa I$ and probability $d\rho\propto e^{-U}dx$ , this is Bakry–Ledoux, Inventiones Mathematicae 123 (1996), Corollary 2.2, equation (2.10), p. 267. Their (2.11), p. 268, uses lower outer Minkowski content. We use (*) and give the extension ourselves.

Choose an affine minorant $\ell(x)=a\cdot x+b\le W(x)$ ; such a minorant exists for a proper lower semicontinuous convex function. Define

W_j(x)=\inf_y\{W(y)+\tfrac j2|x-y|^2\},\qquad j\ge1.

(126.8)

These functions are finite and convex, are continuously differentiable with $j$ -Lipschitz gradient, and increase pointwise to $W$ . For completeness, the minimizer exists uniquely by the affine lower bound and the strictly convex quadratic term. Its optimality condition, together with monotonicity of the convex subgradient, shows that $x\mapsto x-y_j(x)$ is 1-Lipschitz; differentiating the minimum gives $\nabla W_j=j(x-y_j(x))$ . For convergence, monotonicity in $j$ is immediate. If $W_j(x)$ stays bounded along $j\to\infty$ , the affine lower bound forces $y_j(x)\to x$ , and lower semicontinuity yields $W(x)\le\liminf_j W(y_j(x))\le\liminf_j W_j(x)$ . If $W(x)$ is finite the reverse bound follows by testing $y=x$ ; if it is infinite the same argument excludes a finite limit.

Convolve $W_j$ with a nonnegative smooth even compactly supported probability mollifier of sufficiently small radius to obtain $\widetilde W_j$ with

\sup_{|x|\le j}|\widetilde W_j(x)-W_j(x)|\le j^{-1}.

(126.9)

Such a radius exists by uniform continuity on compact sets. Convexity and the $j$ -Lipschitz gradient persist under convolution, so $0\preceq\nabla^2\widetilde W_j\preceq jI$ . Also

W_j(x)\ge a\cdot x+b-\frac{|a|^2}{2j},\qquad \widetilde W_j(x)\ge a\cdot x+b-\frac{|a|^2}{2j}.

(126.10)

The second bound uses the zero mean of the mollifier. Consequently the unnormalized densities $e^{-\kappa|x|^2/2-\widetilde W_j(x)}$ converge pointwise to $e^{-\kappa|x|^2/2-W(x)}$ and are dominated by the integrable function $e^{-\kappa|x|^2/2-a\cdot x-b+|a|^2/2}$ . Their normalizations converge to $Z>0$ and the probability densities converge in $L^1$ .

For each approximating law $\nu_j$ , the potential $U_j=\kappa|x|^2/2+\widetilde W_j$ has Hessian between $\kappa I$ and $(\kappa+j)I$ . The generator $\Delta-\nabla U_j\cdot\nabla$ has globally Lipschitz drift, a nonexplosive diffusion, invariant probability $\nu_j$ , and curvature at least $\kappa$ . Thus the stated smooth theorem applies. It extends from smooth tests to every bounded Lipschitz $f:\mathbb R^n\to[0,1]$ : first multiply $f$ by smooth cutoffs equal to one on the radius- $R$ ball and zero off the radius- $2R$ ball, with gradient bounded by $C/R$ , then mollify. Mollification preserves the range and has gradients converging almost everywhere to the gradient of the Lipschitz cutoff function, with a common bound. Absolute continuity and dominated convergence pass the mollification limit in (*); then $R\to\infty$ passes the cutoff limit, since the extra gradient term has integral at most $C/R$ . This also explains explicitly the test-function domain used from the smooth theorem.

Now fix a bounded Lipschitz $f$ . The quantities $f,J(f),|\nabla f|$ are bounded measurable functions (with any representative of the almost everywhere gradient). The $L^1$ density convergence passes (*) from $\nu_j$ to $\nu$ . This is a functional limit, and does not claim continuity of set perimeters under total variation.

For an actual measurable $E$ , if $\nu(\overline E)>\nu(E)$ , then $\nu^+(E)=+\infty$ , by continuity from above of $\nu(E^r)$ as $r\downarrow0$ . Otherwise set $f_r(x)=(1-\operatorname{dist}(x,E)/r)_+$ . Empty sets are immediate. These are bounded Lipschitz functions, converge to $1_{\overline E}$ , and satisfy

\int|\nabla f_r|\,d\nu \le\frac{\nu(E^r\setminus E)}r.

(126.11)

Indeed a Lipschitz function has zero gradient almost everywhere on each of its level sets, so the gradient vanishes on $\{f_r=0\}$ and $\{f_r=1\}$ ; elsewhere it is bounded by $1/r$ . Absolute continuity transfers this Lebesgue-almost-everywhere assertion to $\nu$ . Dominated convergence gives $\nu f_r\to\nu(E)$ and $\nu J(f_r)\to0$ . Apply (*) and take the lower limit over $r\downarrow0$ to obtain $\nu^+(E)\ge\sqrt\kappa J(\nu(E))$ . Finally symmetry and concavity of $J$ , with $J(0)=0$ and $J(1/2)=1/\sqrt{2\pi}$ , give the claimed linear lower bound.

Proof 126.3 (Proof of the survival theorem)

Isotropy makes the log-concave measure full dimensional. Its density can be written $e^{-V}$ with $V$ proper lower semicontinuous convex, after modification on a Lebesgue null set if needed. This convention allows $V=+\infty$ off its convex domain. At the original time $T_0$ , each posterior has potential

V(x)-Y_{T_0}\cdot x+\tfrac{T_0}2|x|^2

(126.12)

up to an additive constant. Apply the convex-support lemma directly to this posterior with $W=V-Y_{T_0}\cdot x$ and $\kappa=T_0$ . No stochastic experiment or event is approximated. Together with the neighborhood lemma this gives

\mu^+(E)\ge\mathbb E\mu_{T_0}^+(E) \ge\sqrt{2/\pi}\sqrt{T_0}\, \mathbb E\min(p_{T_0},q_{T_0}) \ge\sqrt{2/\pi}\,c_0b_0\sqrt{T_0}.

(126.13)

For the uniform consequence put $L=\sqrt{2/\pi}\,c_0b_0\sqrt{T_0}$ and define $I_\mu(p)=\inf_{\mu(E)=p}\mu^+(E)$ . The infimum may be over Borel or completed-measurable sets: every completed-measurable $E$ contains a Borel subset of equal mass, whose neighborhoods are smaller, so these two infima agree. The balanced hypothesis implies $I_\mu(1/2)\ge L$ directly, without attainment of the infimum.

For any absolutely continuous log-concave law, $I_\mu$ is concave on $(0,1)$ by Milman (2009), Corollary 6.12, with Theorem 1.8 and its Section 6 extension. It is therefore continuous there, and Corollary 6.5 of the same source gives symmetry. That paper’s definition is the lower outer Minkowski convention used here. These results include nonsmooth densities and convex supports in every dimension, including one. For $0<\varepsilon<p<1/2$ , concavity and nonnegativity give

I_\mu(p)\ge \frac{p-\varepsilon}{1/2-\varepsilon}I_\mu(1/2) +\frac{1/2-p}{1/2-\varepsilon}I_\mu(\varepsilon) \ge\frac{p-\varepsilon}{1/2-\varepsilon}L.

(126.14)

Letting $\varepsilon\downarrow0$ and using symmetry proves $I_\mu(p)\ge2L\min(p,1-p)$ . Thus $h_\mu\ge2L$ for every isotropic log-concave law, the universal Cheeger formulation of KLS. Endpoint values may be assigned zero by the empty and full sets; no continuity at either endpoint is asserted or used.

Hypotheses and scope. The individual perimeter implication needs only a full-dimensional log-concave probability with finite first moment, a fixed deterministic measurable set, a positive finite deterministic time, and the displayed survival premise. Isotropy selects the KLS class. An impossible premise (for instance $b_0>1/2$ or $c_0>1$ ) makes the implication vacuous. The proof establishes no uniform survival premise. The only imported analytic results are the smooth Bakry–Ledoux functional comparison and Milman’s general-density profile theorem, identified above; no inaccessible book passage is required.

Fences respected. The node has no bounded_by, depends_on, or assumes edges. The survival condition is an explicit antecedent of the theorem, not an additional proved node. The argument claims neither a moving-family profile supermartingale nor a solution of any occupation or covariance gate.