The time-weighted source budget - The KLS theorem and its methods

Part of the fixed-eigenfunction mechanism, Chapter The fixed eigenfunction: following one eigenfunction through localization; the reading order is on the full proofs page.

Overview. This dossier proves Lemma 21.2 as Theorem D18.1. The setting is a smooth log-concave law with $\nabla^2V\succeq\kappa I_n$ , $\kappa\ge0$ , an arbitrary fixed test $f\in L^2(\mu)$ and the planted channel. The time-weighted source $\E\int_0^T(\kappa+t)\norm{H_t}_{\HS}^2\dd t$ , plus a nonnegative remainder, is bounded by $\Var_\mu(f)-\kappa\abs{g_0}^2$ ((D18.6)); for $\kappa=0$ this gives (D18.7). The proof applies Itô’s formula to $(\kappa+t)\abs{g_t}^2$ and pays the terms with the posterior variance budget. It then closes from $C_c^\infty$ tests to $L^2(\mu)$ . The weight is kept, and no unweighted initial-layer claim is made.

The posterior equations and the Brascamp–Lieb cap (D18.11) make the remainder nonnegative ((D18.12)) and give the terminal cap (D18.14).
For $f\in C_c^\infty$ , the vector $g_t$ solves (D18.17), and Itô’s formula gives (D18.18).
Localizing, taking expectations, and using step 1 with the variance identity (D18.22) gives (D18.23). Monotone convergence then removes the stopping.
Closure to $L^2(\mu)$ : $g^j\to g$ strongly ((D18.24)) and $H^j\to H$ in $L^1$ ((D18.28)). Weak lower semicontinuity then passes the step-3 bound to the limit.
Setting $\kappa=0$ , dropping the remainder and letting $T\uparrow\infty$ gives (D18.7).

Refined statement and admissible class. The following theorem makes the square-integrable admissible class and the posterior coupling in Lemma 21.2 explicit. No centering, isotropy, or eigenfunction hypothesis is imposed.

Theorem D18.1 (= Lemma 21.2)

Let

\dd\mu(x)=Z^{-1}e^{-V(x)}\dd x

(D18.1)

be a smooth log-concave probability measure on $\R^n$ , where $V\in C^\infty(\R^n)$ and $\nabla^2V\succeq\kappa I_n$ for some $\kappa\ge0$ . Take $X\sim\mu$ and an independent standard Brownian motion $B^{\mathrm{obs}}$ , and set

c_t=tX+B_t^{\mathrm{obs}}, \qquad \mathcal F_t^{\mathrm{obs}}=\sigma(c_s:0\le s\le t), \qquad \mu_t=\mathcal L(X\mid\mathcal F_t^{\mathrm{obs}}).

(D18.2)

For an arbitrary fixed $f\in L^2(\mu)$ , write $\E_t$ for integration against $\mu_t$ and put

m_t=\E_tf, \qquad a_t=\E_tX, \qquad v_t=\Var_{\mu_t}(f), \qquad A_t=\Cov_{\mu_t}(X),

(D18.3)

g_t=\E_t[(f-m_t)(X-a_t)], \qquad H_t=\E_t[(f-m_t)(X-a_t)^{\otimes2}].

(D18.4)

The tensor $H$ belongs on every finite horizon to

L^2\bigl(\Omega\times(0,T),(\kappa+t)\,\dd\Prob\,\dd t; \operatorname{Sym}_n\bigr),

(D18.5)

and, for every $T>0$ ,

\begin{aligned} &\E\int_0^T(\kappa+t)\norm{H_t}_{\HS}^2\dd t +2\E\int_0^T\left(\abs{g_t}^2-(\kappa+t)g_t^TA_tg_t\right)\dd t \\ &\hspace{7em}\le \Var_\mu(f)-\kappa\abs{g_0}^2. \end{aligned}

(D18.6)

Both integrands on the left are nonnegative. In particular, when $\kappa=0$ ,

\E\int_0^\infty t\norm{H_t}_{\HS}^2\dd t\le\Var_\mu(f).

(D18.7)

The conclusion retains one power of time and makes no unweighted initial-layer assertion.

Proof D18.1

Posterior equations and curvature bounds. Conditional on $X=x$ , the observation likelihood up to time $t$ is proportional to $\exp(c_t\cdot x-t\abs{x}^2/2)$ . Hence

\dd\mu_t(x) =\frac{\exp(c_t\cdot x-t\abs{x}^2/2)} {\int\exp(c_t\cdot y-t\abs{y}^2/2)\dd\mu(y)}\,\dd\mu(x).

(D18.8)

The innovation process

W_t=c_t-\int_0^ta_s\dd s

(D18.9)

is an $n$ -dimensional Brownian motion in the observation filtration, and the filtering identity for a fixed integrable test $\phi$ is

\dd\E_t\phi=\Cov_{\mu_t}(\phi,X)\cdot\dd W_t,

(D18.10)

first for bounded tests and then, in the square-integrable cases used below, by $L^2$ closure.

The posterior potential in (D18.8) has Hessian $\nabla^2V+tI_n\succeq(\kappa+t)I_n$ . Brascamp–Lieb applied to linear functions therefore gives, with $w_t=\kappa+t$ ,

w_tA_t\preceq I_n.

(D18.11)

When $w_t=0$ , this is read as the trivial positive-semidefinite inequality. Consequently

\mathcal R_t:=\abs{g_t}^2-w_tg_t^TA_tg_t =g_t^T(I_n-w_tA_t)g_t\ge0.

(D18.12)

There is also a terminal covariance bound. If $g_t\ne0$ , set $u_t=g_t/\abs{g_t}$ . Conditional Cauchy–Schwarz and (D18.11) yield

\abs{g_t}^2 =\Cov_{\mu_t}(f,u_t\cdot X)^2 \le v_t\,u_t^TA_tu_t \le\frac{v_t}{w_t}

(D18.13)

when $w_t>0$ ; the case $g_t=0$ is immediate. Thus, also with the evident convention at $w_t=0$ ,

w_t\abs{g_t}^2\le v_t.

(D18.14)

Compact smooth core and the fixed-function SDE. Assume temporarily that $f\in C_c^\infty(\R^n)$ . Applying (D18.10) to $f$ , to the coordinates, and to the components of $fX$ gives

\dd m_t=g_t\cdot\dd W_t, \qquad \dd a_t=A_t\dd W_t.

(D18.15)

Moreover, with matrices oriented by $\Cov_t(fX,X)_{ij}=\Cov_t(fX_i,X_j)$ ,

\Cov_t(fX,X)=H_t+m_tA_t+a_t\otimes g_t.

(D18.16)

Since $g_t=\E_t(fX)-m_ta_t$ , Itô’s product rule, including $\dd[m,a]_t=A_tg_t\dd t$ , now gives the exact equation

\dd g_t=H_t\dd W_t-A_tg_t\dd t.

(D18.17)

Itô’s formula and multiplication by $w_t$ give

\begin{aligned} \dd\bigl(w_t\abs{g_t}^2\bigr) &=2w_tg_t^TH_t\dd W_t \\ &\quad+ \left(w_t\norm{H_t}_{\HS}^2+2\mathcal R_t-\abs{g_t}^2\right)\dd t. \end{aligned}

(D18.18)

We justify taking expectations in this identity on a finite horizon. Fix $T>0$ and let

L_t=\int_0^t2w_sg_s^TH_s\dd W_s.

(D18.19)

The compact support of $f$ and the Gaussian factor in (D18.8) make all posterior coefficients finite and continuous on compact time intervals. In particular $L$ is a continuous local martingale with finite quadratic variation on $[0,T]$ . Choose increasing stopping times $\tau_N$ that simultaneously localize $L$ and all finite-variation terms in (D18.18), so that $L^{\tau_N}$ is a square-integrable martingale and $\tau_N\uparrow\infty$ almost surely, and write $\theta_N=T\wedge\tau_N$ . Integrating (D18.18) to $\theta_N$ gives

\E\int_0^{\theta_N} \left(w_t\norm{H_t}_{\HS}^2+2\mathcal R_t\right)\dd t =\E\bigl[w_{\theta_N}\abs{g_{\theta_N}}^2\bigr] -\kappa\abs{g_0}^2 +\E\int_0^{\theta_N}\abs{g_t}^2\dd t.

(D18.20)

The stopped posterior variance pays exactly for the last two terms. Indeed, $m$ is a bounded martingale for the present compactly supported $f$ , so the stopped Itô isometry and the conditional-law property at $\theta_N$ imply

\E m_{\theta_N}^2-m_0^2 =\E\int_0^{\theta_N}\abs{g_t}^2\dd t, \qquad \E v_{\theta_N}=\E_\mu f^2-\E m_{\theta_N}^2.

(D18.21)

Therefore

\E v_{\theta_N} +\E\int_0^{\theta_N}\abs{g_t}^2\dd t =\Var_\mu(f).

(D18.22)

Using (D18.14) in (D18.20), followed by (D18.22), yields

\E\int_0^{\theta_N} \left(w_t\norm{H_t}_{\HS}^2+2\mathcal R_t\right)\dd t \le\Var_\mu(f)-\kappa\abs{g_0}^2.

(D18.23)

Both terms in the integrand are nonnegative by (D18.12). Since $\theta_N\uparrow T$ , monotone convergence removes the stopping and proves (D18.6) for $f\in C_c^\infty(\R^n)$ .

Closure from $C_c^\infty$ to $L^2(\mu)$ . Let now $f\in L^2(\mu)$ be arbitrary, and choose $f_j\in C_c^\infty(\R^n)$ with $f_j\to f$ in $L^2(\mu)$ . Denote by $m^j,g^j,H^j$ the corresponding posterior quantities. For $\delta_j=f_j-f$ , put $M_t^j=\E_t\delta_j$ . The $L^2$ closure of (D18.10) and the martingale isometry give

\E\int_0^T\abs{g_t^j-g_t}^2\dd t =\E\left|M_T^j-M_0^j\right|^2 \le\norm{\delta_j}_{L^2(\mu)}^2.

(D18.24)

Thus $g^j\to g$ strongly in $L^2(\Omega\times(0,T))$ . Because $0\preceq I_n-w_tA_t\preceq I_n$ , this also gives

\E\int_0^T\mathcal R_t(f_j)\dd t \longrightarrow \E\int_0^T\mathcal R_t(f)\dd t.

(D18.25)

We next identify the tensor limit without assuming any weighted moment of $f$ . A smooth log-concave probability has finite fourth moment. Conditional Jensen gives, uniformly in $t\ge0$ ,

\E\abs{X-a_t}^4 \le8\left(\E\abs X^4+\E\abs{a_t}^4\right) \le16\E\abs X^4.

(D18.26)

Linearity of the centered posterior covariance shows that

H_t^j-H_t =\E_t\left[(\delta_j-\E_t\delta_j)(X-a_t)^{\otimes2}\right].

(D18.27)

Conditional Jensen, Cauchy–Schwarz, the tower property, and (D18.26) therefore imply, for a finite constant depending only on the fourth moment of this fixed $\mu$ ,

\sup_{t\ge0}\E\norm{H_t^j-H_t}_{\HS} \le C_\mu\norm{\delta_j}_{L^2(\mu)}\longrightarrow0.

(D18.28)

For completeness, the two terms bounded here are

\E\!\left[\abs{\delta_j(X)}\abs{X-a_t}^2\right] \quad\text{and}\quad \E\!\left[\abs{\E_t\delta_j}\,\E_t\abs{X-a_t}^2\right];

(D18.29)

each is at most a fixed multiple of $\norm{\delta_j}_{L^2(\mu)}(\E\abs X^4)^{1/2}$ .

The already proved estimate and nonnegativity of $\mathcal R_t(f_j)$ show that $H^j$ is bounded in the weighted Hilbert space appearing in the theorem. Extract a weakly convergent subsequence. The strong $L^1(\Omega\times(0,T))$ convergence in (D18.28) identifies its weak limit with the displayed posterior tensor $H$ . Weak lower semicontinuity, together with (D18.25), gives

\begin{aligned} &\E\int_0^Tw_t\norm{H_t}_{\HS}^2\dd t +2\E\int_0^T\mathcal R_t(f)\dd t\\ &\qquad\le \liminf_{j\to\infty}\left( \E\int_0^Tw_t\norm{H_t^j}_{\HS}^2\dd t +2\E\int_0^T\mathcal R_t(f_j)\dd t\right)\\ &\qquad\le \lim_{j\to\infty}\left(\Var_\mu(f_j)-\kappa\abs{g_0^j}^2\right) =\Var_\mu(f)-\kappa\abs{g_0}^2. \end{aligned}

(D18.30)

Here $g_0^j\to g_0$ because $X\in L^2(\mu)$ . This proves the finite-horizon assertion for every $f\in L^2(\mu)$ and, at the same time, proves the asserted weighted $L^2$ membership of $H$ .

Finally set $\kappa=0$ . Dropping the nonnegative remainder term from the finite-horizon bound gives

\E\int_0^Tt\norm{H_t}_{\HS}^2\dd t\le\Var_\mu(f)

(D18.31)

for every $T$ . Monotone convergence as $T\uparrow\infty$ proves (D18.7).

Obstructions respected. The ledger node has no formal bounded_by edge. The proof nevertheless stays on the permitted side of all recorded KLS fences: it contains no slice or cut estimate (rem:two-tail-slice-bounds, rem:profile-circularity, and rem:single-coordinate-cuts), no radial or projection reduction (rem:projection-ceiling), and no covariance occupation bootstrap (rem:crude-insufficient and rem:relative-ceiling). It uses the posterior covariance cap only to keep the explicitly weighted remainder nonnegative; it neither unweights the tensor nor claims dimension-free control of the initial layer.