BK: covariance-normalized localization and moving Appell variance

Part of the Balasubramanian–Kasiviswanathan proof, Chapter Balasubramanian–Kasiviswanathan: compatible integration; the reading order is on the full proofs page.

Overview. We reconstruct Sections 6 and Appendix D of Balasubramanian & Kasiviswanathan, 2026 at commit 4837c33649ba2271f43c9684e9350ecbdd725f95. Quadratic variance bounds control third moments, prevent finite-time explosion, and control ordered products of covariance matrices. A finite formal-series calculation then compares the Appell variance before and after localization.

Dependencies. The only inequality input is Theorem 25.1. The Appell normalization Definition 11.1 is stated explicitly below. Standard finite-dimensional Itô calculus and local existence for smooth SDE coefficients are used. No KLS, BKL, SZ v2, or all-degree coefficient estimate is an input.

Theorem 154.1 (Global localization and ordered covariance bounds)

Let $\mu_0$ be any compactly supported isotropic log-concave law on $\mathbb R^n$ . Write

\mu_{\theta,\Lambda}(dx)=Z(\theta,\Lambda)^{-1} e^{\theta\cdot x-x^T\Lambda x/2}\mu_0(dx),

(154.1)

and let $m(\theta,\Lambda)$ and $A(\theta,\Lambda)$ be its mean and covariance. The SDE, started at $(0,0)$ ,

d\theta_t=A_t^{-1/2}dB_t+A_t^{-1}m_tdt, \qquad d\Lambda_t=A_t^{-1}dt, \qquad \mu_t=\mu_{\theta_t,\Lambda_t},

(154.2)

has a global solution with $A_t\succ0$ . Every bounded Borel test has $\int f\,d\mu_t$ a true martingale. On the full ordered tensor spaces, for every integer $k\ge1$ , $0\le s\le t$ , every integer $d\ge1$ , and $s_1,\ldots,s_d\in[0,t]$ ,

\mathbb E[A_t^{\otimes k}\mid\mathcal F_s] \preceq e^{4k^2(t-s)}A_s^{\otimes k},\qquad \mathbb E[A_{s_1}\otimes\cdots\otimes A_{s_d}] \preceq e^{4d^2t}I.

(154.3)

For $t>0$ , $0\le q\le d$ and $\delta\ge0$ ,

\mathbb E[(A_t+\delta\Lambda_t^{-1})^{\otimes q} \otimes A_t^{\otimes(d-q)}] \preceq (1+\delta/t)^q e^{4d^2t}I.

(154.4)

This proves Lemma 11.4.

Proof 154.1

Suppose the support lies in $B_R$ . All tilt moments are smooth in the finite parameters and $A$ is positive definite: tilt densities are strictly positive relative to the original full-dimensional law. Local existence therefore holds. Put $\xi_t=A_t^{-1/2}(x-m_t)$ . Differentiating the normalization gives

d\log Z_t=m_t^TA_t^{-1/2}dB_t+\tfrac12m_t^TA_t^{-1}m_tdt, \qquad d\rho_t(x)=\rho_t(x)\xi_t(x)\cdot dB_t,

(154.5)

where $\rho_t=d\mu_t/d\mu_0$ . Consequently, initially up to localization,

dE_tf=E_t[(f-E_tf)\xi_t]\cdot dB_t.

(154.6)

These formulas follow by Itô from the smooth tilt integral also for bounded Borel $f$ , since compact support permits parameter differentiation.

For any isotropic log-concave $Y$ , set $S_i=E[Y_iYY^T]$ and $S(v)=\sum_i v_iS_i$ . By Theorem 25.1 and Cauchy--Schwarz,

|\langle S(v),H\rangle|\le |v|\sqrt{\operatorname{Var}(Y^THY)} \le\sqrt8|v|\|H\|_{\rm HS}.

(154.7)

Thus $\|S(v)\|_{\rm HS}^2\le8|v|^2$ . Symmetry of all three tensor indices identifies this squared norm with $v^T(\sum_i S_i^2)v$ . Also

0\preceq\sum_i(S_i\otimes I-I\otimes S_i)^2 =\sum_i S_i^2\otimes I+I\otimes\sum_i S_i^2-2\sum_iS_i\otimes S_i.

(154.8)

Hence $\sum_iS_i^2\preceq8I$ and $\sum_iS_i\otimes S_i\preceq8I$ . Apply these pointwise to the posterior $\xi_t$ and put $U_{i,t}=A_t^{1/2}S_{i,t}A_t^{1/2}$ . Differentiating its first two moments gives

dm_t=A_t^{1/2}dB_t,\qquad dA_t=\sum_iU_{i,t}dB_{i,t}-A_tdt,\qquad \sum_iU_{i,t}\otimes U_{i,t}\preceq8A_t\otimes A_t.

(154.9)

To prove nonexplosion, let $\zeta$ be the maximal lifetime. Matrix Itô gives

d\log\det A_t=\sum_i\operatorname{Tr}(S_{i,t})dB_{i,t} -\left(n+\tfrac12\sum_i\operatorname{Tr}(S_{i,t}^2)\right)dt.

(154.10)

The drift rate lies in $[-5n,-n]$ and the martingale quadratic variation rate is at most $8n^2$ . Extend the martingale integrand by zero beyond $\zeta$ ; it defines a continuous square-integrable martingale on every finite interval. On $\{\zeta\le T\}$ the log determinant therefore has a finite limit and is bounded below before $\zeta$ . Since $A_t\preceq R^2I$ , its least eigenvalue is at least $\det A_t/R^{2(n-1)}$ , with a positive pathwise lower bound. Consequently $A_t^{-1}$ is pathwise bounded up to $\zeta$ , as is $m_t$ . The drift integrals for $\theta,\Lambda$ have finite limits. Stopping the martingale in $\theta$ when $\|A_t^{-1}\|$ exceeds an integer gives continuous extensions with bounded quadratic variation; on each such path one of these stops is never reached. Thus $\theta,\Lambda$ have finite limits at $\zeta$ . At their limit $A$ is positive definite and local smooth-coefficient existence extends the SDE, a contradiction. Since $T$ was arbitrary, $\zeta=\infty$ . The bounded local martingale $E_tf$ is now a true martingale by bounded convergence along its localizing sequence, also conditionally.

The drift of $A_t^{\otimes k}$ has $k$ linear terms and one quadratic-variation term for each pair of slots. Tensoring the preceding two-slot bound with the positive remaining factors gives a drift at most

[-k+8\tbinom k2]A_t^{\otimes k}\preceq4k^2A_t^{\otimes k}.

(154.11)

The stochastic terms are true martingales: $A_t\preceq R^2I$ and $\sum_i\|U_{i,t}\|_{\rm HS}^2\le8nR^4$ bound their squared scalar integrands for each fixed tensor test by $8nk^2R^{4k}|z|^4$ . Multiplication by $e^{-4k^2t}$ and conditional expectation prove the first bound, initially for rational test vectors and then simultaneously by continuity.

For mixed times, permute slots into increasing time order. If the latest time $u$ occurs $k$ times and $v$ is the next earlier time (or zero), condition on $\mathcal F_v$ . All other factors form a positive $\mathcal F_v$ -measurable operator. The conditional bound replaces $A_u^{\otimes k}$ by $e^{4k^2(u-v)}A_v^{\otimes k}$ . Repeat after merging equal times. The successive intervals are disjoint and contained in $[0,t]$ , and $k\le d$ ; the accumulated factor is at most $e^{4d^2t}$ , with all final factors equal to $A_0=I$ . Undo the slot permutation.

Finally integrate the positive block matrices $\left(\begin{smallmatrix}A_s&I\\I&A_s^{-1}\end{smallmatrix}\right)$ . Their Schur complement yields $\Lambda_t^{-1}\preceq t^{-2}\int_0^t A_sds$ . Insert this comparison in each of any $j$ selected tensor slots, expand the integrals, and use the mixed-time estimate. This gives an upper bound $t^{-j}e^{4d^2t}I$ for the corresponding expectation with $\Lambda_t^{-1}$ in those slots and $A_t$ in the others. Boundedness of $A_s$ justifies all integrations entrywise. Expanding the first $q$ factors of $(A_t+\delta\Lambda_t^{-1})$ now sums to $(1+\delta/t)^q$ .

For any law with all moments define $P_d^\mu[T]=\langle T,\mathcal A_d^\mu\rangle$ by the formal identity

\frac{e^{z\cdot x}}{E_\mu e^{z\cdot X}} =\sum_{d\ge0}\frac{\langle\mathcal A_d^\mu(x),z^{\otimes d}\rangle}{d!}, \qquad c_d(\mu)=\frac1{d!}\sup_{\|T\|=1}\|P_d^\mu[T]\|_2.

(154.12)

All tensors use the ordered-index Hilbert--Schmidt norm. In particular, $D_vP_d[T]=dP_{d-1}[T(v,\cdot)]$ , and all expected derivatives of order less than $d$ vanish. If $X=m+BY$ , coefficient comparison gives $P_d^{\mathcal L(X)}[T](m+By)=P_d^{\mathcal L(Y)}[(B^T)^{\otimes d}T](y)$ .

Theorem 154.2 (Moving variance with lower-degree error)

Fix $d\ge2$ and finite constants $C_k\ge0$ such that $c_k(\nu)\le C_k$ for every isotropic log-concave law in every dimension and $1\le k<d$ . Set $\Sigma_d=\sum_{k=2}^{d-1}(d-k+1)C_kC_{d-k+1}$ , with an empty sum zero. For the localization above, fixed $T\in\operatorname{Sym}^d\mathbb R^n$ , $p_t=P_d^{\mu_t}[T]$ and $V(t)=\mathbb E E_tp_t^2$ , one has, for all $t\ge0$ ,

\sqrt{V(0)}\le e^{(d+1)t}\sqrt{V(t)} +d!\Sigma_dt\,e^{(2d^2+d+1)t}\|T\|.

(154.13)

This proves Lemma 11.5.

Proof 154.2

At a fixed time put $\Psi(z)=\log E e^{z\cdot X}$ , $\Gamma_j=D^j\Psi(0)$ , and define finite-on-polynomials constant-coefficient operators

C_i=\sum_{k\ge1}\frac1{k!}\langle A^{-1/2}e_i,\Gamma_{k+1}[D^k]\rangle, \qquad R_i=\xi_i-C_i.

(154.14)

Their symbols are $u_i(z)=[A^{-1/2}(\nabla\Psi(z)-m)]_i$ . The moment-series SDE is $dZ=Z\sum_i u_i dB_i$ . Applying Itô to $Z^{-1}$ and comparing finitely many formal coefficients gives

dp_t=-\sum_iC_ip_t,dB_i+\sum_iC_i^2p_t,dt.

(154.15)

No derivative of a time-varying operator is taken in this formula: each symbol is evaluated at that time. Combining the square differential with $d\rho=\rho\xi\cdot dB$ gives the variance drift

D_t=\sum_i\|C_ip_t\|_2^2-2\langle p_t,\sum_iR_iC_ip_t\rangle_2.

(154.16)

Differentiating the generating function in $z$ along $A^{-1/2}e_i$ gives

R_iP_j[S]=P_{j+1}[\operatorname{Sym}(A^{-1/2}e_i\otimes S)].

(154.17)

Another coefficient comparison, in $E[Xe^{z\cdot X}]/Ee^{z\cdot X}=\nabla\Psi(z)$ , gives

\Gamma_{k+1}[U]=E[(X-m)P_k[U]].

(154.18)

For an isotropic law Bessel’s inequality for its orthonormal coordinate functions implies $\|\Gamma_{k+1}:\operatorname{Sym}^k\to\mathbb R^n\|\le k!c_k$ .

Whiten at this fixed time: $y=A^{-1/2}(x-m)$ , $\widetilde T=(A^{1/2})^{\otimes d}T$ and $\bar p=P_d^{\mathcal L(\xi)}[\widetilde T]$ . The chain rule and cumulant transformation show that $C_i$ becomes $\sum_{k\ge1}(k!)^{-1}(\Gamma_{k+1}^{\xi}[D_y^k])_i$ and $R_i$ becomes $y_i-C_i$ . These are only identities for evaluating the already-computed drift. In $\sum_i R_i C_i\bar p$ , the $k=1$ contribution is $d\bar p$ : $\Gamma_2^\xi=I$ and the raising formula reconstructs each of its $d$ slots. The $k=d$ contribution is $\langle E_t[p_t\xi_t],\xi_t\rangle$ ; its pairing with $p_t$ is $|E_t[p_t\xi_t]|^2\le E_tp_t^2$ .

For $2\le k\le d-1$ , put $j=d-k+1$ and contract $k$ ordered slots of $\widetilde T$ with $\Gamma_{k+1}^\xi$ , producing $W_k\in\mathbb R^n\otimes\operatorname{Sym}^{d-k}\mathbb R^n$ . Applying the cumulant norm bound to each fixed remaining-index slice gives $\|W_k\|\le k!C_k\|\widetilde T\|$ . Differentiation and raising give precisely

\frac{d!}{k!(d-k)!}P_j^{\mathcal L(\xi)}[\operatorname{Sym}W_k].

(154.19)

Since symmetrization is an orthogonal projection, its norm is at most $d!jC_kC_j\|\widetilde T\|$ . Dropping the nonnegative square term in the drift and writing $v_t=E_tp_t^2$ therefore gives

D_t\ge-2(d+1)v_t-2d!\Sigma_d\sqrt{v_t}\|\widetilde T_t\|.

(154.20)

It remains to justify expectation without an inverse-covariance bound in mean. Every posterior moment through order $2d$ is bounded deterministically by compact support. Its diffusion coefficient satisfies $|E_t[f\xi_t]|^2\le E_tf^2\le\sup_{B_R}|f|^2$ by Bessel. The variance $v_t$ is a fixed polynomial in these finitely many moments: formal inversion uses only addition and multiplication since the constant moment is one. Its Itô drift and diffusion thus have deterministic bounds. Stopping on bounded tilt parameters and removing the stops gives convergence of stochastic integrals in $L^2$ and drifts in $L^1$ . Hence $V$ is locally absolutely continuous and $V'=\mathbb ED_t$ almost everywhere. Cauchy--Schwarz and the equal-time covariance estimate give

V'(t)\ge-2(d+1)V(t) -2d!\Sigma_d\sqrt{V(t)}e^{2d^2t}\|T\|.

(154.21)

For $T\ne0$ , the leading polynomial is nonzero and a full-dimensional log-concave law cannot be supported on its zero set; thus $V(t)>0$ . Continuity gives a positive minimum on each compact time interval. Divide by $2\sqrt V$ , multiply by $e^{(d+1)t}$ and integrate to obtain

\sqrt{V(0)}\le e^{(d+1)t}\sqrt{V(t)} +d!\Sigma_d\|T\|\int_0^t e^{(2d^2+d+1)s}ds.

(154.22)

Bounding the integral by its endpoint times $t$ proves the result. For $T=0$ it is immediate.

Fences respected. These new nodes have no assigned bounded_by edges. The brief’s projection ceiling is avoided by a full tensor estimate, not projection estimates. No adaptive cut, scalar covariance ceiling, uniform all-degree bound, sharp gate-zero claim, or open antecedent is used. Compact support is used only here; its later removal belongs to reverse transfer.

Reviewer handoff. Check the two-slot Loewner inequality on the full tensor space, the log-determinant continuation argument, mixed-time conditioning, the $k=d$ Bessel endpoint, and the moment-polynomial justification of expectation. The lower-degree bounds in the second theorem are quantified antecedents, not asserted uniform coefficient theorems. The covariance and moving-variance statements use the inputs specified above; their certifications are recorded separately.

References¶

Balasubramanian, K., & Kasiviswanathan, S. (2026). A Dimension-Free Bound on the Poincaré Constant of Isotropic Log-Concave Measures. https://arxiv.org/abs/2610.07728v1