BKL: inverse-covariance cumulant dynamics - The KLS theorem and its methods

Part of the Bizeul–Klartag–Lehec proof, Chapter Bizeul–Klartag–Lehec: cumulants and suspension; the reading order is on the full proofs page.

Overview. This dossier reconstructs Section 4 of Bizeul et al., 2026, version 1, equations (67)–(79). It derives the inverse-covariance localization equations and proves that their solution is global for compactly supported initial laws. A separate, deliberately dimension-dependent moment estimate justifies the expectation arguments used later. It is not an input to the dimension-free induction.

Author. plan_framework researcher, unknown, 2026-10-06.

Dependencies. The conventions are Definition 8.1. We use Proposition 26.1 together with Theorem 25.1, discharging the former’s antecedent, to obtain the third-cumulant estimate. We use ordinary finite-dimensional Itô calculus, local existence and uniqueness for locally Lipschitz SDE coefficients, and preservation of log-concavity under linear images. No KLS estimate, Riccati covariance process, higher-cumulant bound or Song–Zhang criterion is used.

Theorem 132.1 (Dynamics and finite-time integrability)

Let $\mu$ be a compactly supported isotropic log-concave probability measure on $\mathbb R^n$ . There is a unique global strong solution, starting from $c_0=Q_0=0$ , of

\frac{d\mu_t}{d\mu}(x)=Z_t^{-1} e^{\langle c_t,x\rangle-\langle Q_tx,x\rangle/2},\qquad dc_t=A_t^{-1}a_t\,dt+A_t^{-1/2}\,dB_t,\qquad dQ_t=A_t^{-1}\,dt,

(132.1)

where $a_t=\int x\,d\mu_t$ and $A_t=\operatorname{Cov}(\mu_t)$ . The covariance is positive definite at every finite time. Put $\kappa_m(t)=\kappa_m^{\mu_t}$ and $H_{i,t}=\kappa_3(t)(\cdot,\cdot,A_t^{-1/2}e_i)$ . Then

da_t=A_t^{1/2}dB_t,\qquad dA_t=\sum_iH_{i,t}\,dB_{t,i}-A_t\,dt,\qquad \mathbb EA_t=e^{-t}I.

(132.2)

For $m\ge3$ , define

L_m(t)(h_1,\ldots,h_m)= \sum_{\substack{J\subset[m],\ 1\in J\\2\le |J|\le m-2}} \left\langle\kappa_{|J|+1}(t)(h_J,\cdot), A_t^{-1}\kappa_{m-|J|+1}(t)(h_{J^c},\cdot)\right\rangle.

(132.3)

The sum is zero for $m=3$ . The cumulants satisfy

d\kappa_m(t)=\kappa_{m+1}(t)(\cdot,\ldots,\cdot,A_t^{-1/2}dB_t) -(m\kappa_m(t)+L_m(t))\,dt.

(132.4)

For a tensor of order $r$ set $|T|_t^2=\langle T,(A_t^{-1})^{\otimes r}T\rangle$ . For every deterministic unit vector $u$ , let $\mathcal E_m(t)=|\kappa_m(t)(u,\cdot,\ldots,\cdot)|_t^2$ . Then

\sum_iH_{i,t}A_t^{-1}H_{i,t}\le8A_t,\qquad \mathcal E_2(t)=\langle A_tu,u\rangle,\qquad \mathcal E_3(t)\le8\langle A_tu,u\rangle.

(132.5)

For every fixed $n,m$ the processes $\mathcal E_m$ and $|L_m(t)(u)|_t^2$ have deterministic bounds depending on $n,m$ and the initial support radius. The Itô local martingale term of each $\mathcal E_m$ is a true square-integrable martingale on every bounded time interval.

Proof 132.1

Local construction and the third-order input. Suppose the support is contained in a ball of radius $R$ centered at the origin. For every finite pair $(c,Q)$ , the tilted law has the same support as $\mu$ . Its covariance is positive definite because isotropy makes that support full dimensional. Compact support permits differentiation of the partition function to every order. Thus its mean and covariance, the inverse covariance and its positive square root are smooth locally in $(c,Q)$ . The SDE consequently has a unique strong solution up to its explosion time. Along this solution $Q_t$ is positive semidefinite, so $\mu_t$ is log-concave.

The stated Letwin inputs give, for any isotropic log-concave $\nu$ and any $v$ ,

|\kappa_3^\nu(v)|^2\le8|v|^2.

(132.6)

Indeed, for symmetric $M$ , the covariance of $\langle v,X\rangle$ with $X^TMX$ is $\langle\kappa_3^\nu(v),M\rangle$ ; Cauchy–Schwarz and the quadratic-chaos variance bound give its square at most $8|v|^2|M|_{\rm HS}^2$ . Take the supremum over symmetric matrices of Hilbert–Schmidt norm one.

The density and covariance equations. All calculations initially take place before an exit time from a compact subset of the finite $(c,Q)$ parameter space. For $g_t(x)=\exp(\langle c_t,x\rangle-\langle Q_tx,x\rangle/2)$ , Itô’s formula gives

\frac{dg_t(x)}{g_t(x)}=\langle x,A_t^{-1}a_t\rangle dt +\langle x,A_t^{-1/2}dB_t\rangle.

(132.7)

Integrating gives $dZ_t/Z_t=\langle a_t,A_t^{-1}a_t\rangle dt+ \langle a_t,A_t^{-1/2}dB_t\rangle$ . The quotient rule, including its cross variation, cancels the drift and yields

dp_t(x)=p_t(x)\langle A_t^{-1/2}(x-a_t),dB_t\rangle, \qquad p_t=g_t/Z_t.

(132.8)

Consequently, for any bounded measurable $f$ on the support,

d\int f\,d\mu_t= \left\langle A_t^{-1/2}\int f(x)(x-a_t)\,d\mu_t(x),dB_t\right\rangle.

(132.9)

Apply this identity to $x$ and $xx^T$ , and subtract $d(a_ta_t^T)$ . The barycenter noise is $A_t^{1/2}dB_t$ , whose quadratic variation is $A_tdt$ . The centered third moment is the remaining covariance noise. This proves the two claimed SDEs, initially up to explosion.

Let $\nu_t$ be the image of $\mu_t$ under $x\mapsto A_t^{-1/2}(x-a_t)$ . For $m\ge2$ , affine covariance of the derivatives of the logarithmic Laplace transform gives

|\kappa_m(t)(v)|_t^2 =|\kappa_m^{\nu_t}(A_t^{1/2}v)|^2.

(132.10)

In particular, the third-order estimate gives $\mathcal E_3\le8\langle A_tu,u\rangle$ . The quadratic form of $\sum_iH_iA_t^{-1}H_i$ at $v$ is exactly $|\kappa_3(t)(v)|_t^2$ . This proves the matrix inequality.

No finite-time explosion. Set $G_i=A_t^{-1/2}H_iA_t^{-1/2}$ . The preceding inequality implies $\sum_iG_i^2\le8I$ . Applying Itô to $\log\det A_t$ gives

d\log\det A_t=\sum_i\operatorname{Tr}(G_i)dB_{t,i} -\left(n+\frac12\sum_i\operatorname{Tr}(G_i^2)\right)dt.

(132.11)

Its drift lies between $-5n$ and $-n$ , and the bracket of its local martingale is at most $8n^2dt$ , since $(\operatorname{Tr}G_i)^2\le n\operatorname{Tr}(G_i^2)$ . These bounds hold uniformly under stopping. On any deterministic interval $[0,T]$ the stopped stochastic integral extends continuously to the explosion time: extend its integrand by zero after that time, and use its bounded quadratic variation. Hence $\log\det A_t$ is bounded below before $T$ on each path. Compact support gives $A_t\le R^2I$ . The determinant lower bound therefore bounds $A_t^{-1}$ on that path before $T$ . It also bounds the coefficients of $c_t,Q_t$ , because $|a_t|\le R$ . Their drift integrals and stochastic integrals have finite limits. Local existence at these finite limiting parameters extends the solution, contradicting a finite explosion time. The solution is therefore global.

Expectation of the covariance. The process $e^tA_t$ is a local martingale by its SDE, and all its entries are bounded by $e^TR^2$ on $[0,T]$ . It is a true martingale. Since $A_0=I$ , its expectation is $I$ . This proves $\mathbb EA_t=e^{-t}I$ and, in particular, $\mathbb E\mathcal E_2(t)=e^{-t}$ and $\mathbb E\mathcal E_3(t)\le8e^{-t}$ .

The cumulant equation. Write $M_t(z)=\int e^{\langle z,x\rangle}\,d\mu_t(x)$ , $\Lambda_t=\log M_t$ , and $v_t(z)=\nabla\Lambda_t(z)-a_t$ . The bounded support and stopped parameter coefficients justify differentiation and stochastic integration in either order, on compact sets of $z$ . The density equation and scalar Itô formula give

d\Lambda_t(z)=\langle A_t^{-1/2}v_t(z),dB_t\rangle -\tfrac12\langle v_t(z),A_t^{-1}v_t(z)\rangle dt.

(132.12)

At zero, $v_t(0)=0$ and its $r$ th derivative, contracted against a further vector, is $\kappa_{r+1}(t)$ . The $m$ th derivative of the noise is the asserted $(m+1)$ st cumulant. Leibniz’s rule for the quadratic drift is the sum over nonempty proper subsets $J\subset[m]$ . Its $2m$ singleton or co-singleton terms each equal $\kappa_m(t)$ , because one factor is $A_th_j$ and cancels $A_t^{-1}$ . The remaining terms occur in equal complementary pairs. Selecting the member containing the first index and accounting for the prefactor $1/2$ gives precisely $L_m(t)$ . This proves the equation, including $L_3=0$ .

Crude moment bounds, independent of the induction. For completeness there are finite constants $D_{n,m}$ such that every isotropic log-concave $\nu$ on $\mathbb R^n$ satisfies $|\kappa_m^\nu(v)|^2\le D_{n,m}|v|^2$ . No dimension-free claim is made here. To see this, a centered variance-one one-dimensional log-concave marginal $Y$ has a log-concave survival function $S(t)$ . This last fact follows either by integration of a log-concave density or directly from its increasing hazard: for density $f$ , the ratio $S(t)/f(t)=\int_0^\infty f(t+s)/f(t)\,ds$ is nonincreasing, by concavity of $\log f$ . Endpoints follow by limits. Chebyshev gives $S(2)\le1/4$ and $S(-2)\ge3/4$ . Concavity of $\log S$ therefore gives, for $t\ge2$ ,

S(t)\le\tfrac14\exp\bigl(-(\log3)(t-2)/4\bigr).

(132.13)

If $S(2)=0$ , its upper tail vanishes and the displayed bound is immediate. The same argument applies to $-Y$ . Integration of the two tails gives a finite universal upper bound for $\mathbb E|Y|^p$ at every fixed $p$ .

For unit vectors $v_1,\ldots,v_m$ , Hölder bounds every mixed moment $\mathbb E\prod_{j\in J}\langle v_j,X\rangle$ in terms of $|J|$ alone. The moment-cumulant formula

\kappa_m(v_1,\ldots,v_m)= \sum_{\pi\in\mathcal P([m])}(-1)^{|\pi|-1}(|\pi|-1)! \prod_{J\in\pi}\mathbb E\prod_{j\in J}\langle v_j,X\rangle

(132.14)

then bounds each mixed cumulant by a finite $C_m$ . Summing its square over the $n^{m-1}$ coordinate choices gives $D_{n,m}=n^{m-1}C_m^2$ . Whitening the posterior now gives

\mathcal E_m(t)\le D_{n,m}\langle A_tu,u\rangle\le D_{n,m}R^2.

(132.15)

For a summand of $L_m$ with $|J|=a$ , sum first over the whitened coordinates in $J^c$ , apply the same bound for $\kappa_{m-a+1}$ , and then sum over those in $J\setminus\{1\}$ . Its squared weighted norm is at most $D_{n,m-a+1}\mathcal E_{a+1}(t)$ : the contracted vector is $A_t^{-1}\kappa_{a+1}(t)(u,h_{J\setminus\{1\}},\cdot)$ , whose $A_t$ -norm is exactly the sum over the final whitened contraction index. The finite sum over $J$ therefore bounds $|L_m(t)(u)|_t^2$ deterministically as well.

Finally write $r=m-1$ , $T=\kappa_m(t)(u)$ , $W_i=\kappa_{m+1}(t)(u,\cdot,\ldots,\cdot,A_t^{-1/2}e_i)$ , $\widetilde T=(A_t^{-1/2})^{\otimes r}T$ , and $\widetilde W_i=(A_t^{-1/2})^{\otimes r}W_i$ . The noise coefficient of $\mathcal E_m$ is

\beta_i=2\langle\widetilde T,\widetilde W_i\rangle -\left\langle\widetilde T,\sum_{s=1}^rG_i^{(s)}\widetilde T\right\rangle,

(132.16)

where $G_i^{(s)}$ acts on tensor slot $s$ . Since $\sum_iG_i^2\le8I$ ,

\sum_i\beta_i^2\le8\mathcal E_m\mathcal E_{m+1} +16r^2\mathcal E_m^2.

(132.17)

This is bounded deterministically. The drift is integrable on finite intervals as well: the second derivative of the inverse metric consists of finitely many terms bounded by $\mathcal E_m$ , $\mathcal E_{m+1}$ and $\sqrt{\mathcal E_m}|L_m(u)|_t$ , using the same matrix inequality. Thus localization may be removed in expectations. These observations establish the final integrability assertion and supply the precise justification for the energy argument of Lemma 8.4. All assertions concerning a fixed vector extend from unit $u$ to arbitrary $u$ by homogeneity; each energy and squared norm acquires a factor $|u|^2$ . In particular, $\int_0^\infty\mathbb E\mathcal E_2(t)\,dt=|u|^2$ .

Fences respected. There is no assigned bounded_by fence. All stochastic expectations here are for compactly supported initial laws; passage to noncompact laws is made only for the final static cumulant estimate. The covariance drift is $-A_t$ , not the $-A_t^2$ drift of another localization clock.

References¶

Bizeul, P., Klartag, B., & Lehec, J. (2026). Presenting a Proof of the Kannan–Lovasz–Simonovits Conjecture. https://arxiv.org/abs/2610.05474v1