Full matrix dissipation - The KLS theorem and its methods

Part of the shared technical foundations, Chapter The two-color Riccati identities; the reading order is on the full proofs page.

Overview. This dossier proves a refined form of Corollary 24.2: under Eldan localization of an isotropic log-concave $\mu$ with a nontrivial cut $E$ , the full matrix dissipation $\E\int_0^\infty(R_t^2+s_tG_t^2)\dd t$ is bounded by $R_0\preceq I_n$ in Loewner order, with no balance or stopping hypothesis. The proof feeds the certified matrix Riccati identity into a general positive-drift lemma for matrix semimartingales.

Lemma D2.1: a positive semidefinite process with local-martingale part and positive drift $Q$ satisfies (D2.6) and (D2.7). The proof scalarizes along each direction, then uses localization, optional sampling, Fatou and monotone convergence. Positive semidefiniteness gives integrability of the off-diagonal entries.
The certified identity Lemma 24.1 gives (D2.15) with $Q_t=R_t^2+s_tG_t^2\succeq0$ . Step 1 then yields (D2.3) and (D2.4), and isotropy gives $R_0\preceq I_n$ .
Testing against a direction gives (D2.17), which recovers and strengthens Corollary 24.1. A Gaussian halfspace example shows that the extra $R_t^2$ term is not vacuous.
The trace (D2.19) is only bounded by $n$ . The result gives no dimension-free operator-to-trace upgrade.

Scope and notation. Let $\mu$ be an isotropic log-concave probability measure on $\R^n$ , let $E$ be a fixed measurable set with $0<\mu(E)<1$ , and run Eldan stochastic localization. We use the standing two-color notation

p_t=\mu_t(E),\qquad q_t=1-p_t,\qquad s_t=p_tq_t,

(D2.1)

B_t=s_t\delta_t\delta_t^T, \qquad R_t=A_t-B_t=p_t\Sigma_t^E+q_t\Sigma_t^{E^c}\succeq0, \qquad G_t=\Sigma_t^E-\Sigma_t^{E^c}.

(D2.2)

The finite-time localization density is strictly positive relative to $\mu$ , so the two conditional laws, and hence these matrices, are defined at every finite time. All matrix integrals below are understood entrywise; the proof shows their absolute integrability rather than assuming it.

We first isolate the only stochastic expectation argument that is needed.

Lemma D2.1 (Positive matrix drift survives localization)

Let $(X_t)_{t\geq0}$ be a continuous adapted process with values in the positive semidefinite symmetric matrices, with deterministic initial matrix $X_0$ . Suppose that, entrywise,

X_t=X_0+N_t-\int_0^tQ_u\dd u,

(D2.5)

where $N_0=0$ is a matrix local martingale and $Q_t\succeq0$ is progressively measurable and locally integrable. Then the matrices in the following display are integrable and, for every finite $T$ ,

\E X_T+\E\int_0^TQ_t\dd t\preceq X_0.

(D2.6)

Moreover the improper integral $\int_0^\infty Q_t\dd t$ exists entrywise and absolutely almost surely, is integrable, and

\E\int_0^\infty Q_t\dd t\preceq X_0.

(D2.7)

Proof D2.1

Fix a deterministic vector $\theta\in\R^n$ and set

x_t^\theta=\theta^TX_t\theta, \qquad M_t^\theta=\theta^TN_t\theta, \qquad a_t^\theta=\int_0^t\theta^TQ_u\theta\dd u.

(D2.8)

Thus $x^\theta,a^\theta\geq0$ , $a^\theta$ is increasing, and

x_t^\theta+a_t^\theta=x_0^\theta+M_t^\theta.

(D2.9)

Choose increasing localizing times $\sigma_k\uparrow\infty$ almost surely such that $(M^\theta)^{\sigma_k}$ is a true martingale. At the bounded time $T\wedge\sigma_k$ , (D2.9) and optional sampling give

\E\bigl[x_{T\wedge\sigma_k}^\theta+a_{T\wedge\sigma_k}^\theta\bigr] =x_0^\theta.

(D2.10)

There is no integrability assumption hidden here: the right-hand side of (D2.9) is integrable after stopping, while its left-hand side is nonnegative, so both terms on that side are integrable.

Continuity of $X$ and local integrability of $Q$ imply convergence of the stopped left-hand side to $x_T^\theta+a_T^\theta$ . Fatou’s lemma, applied to that nonnegative sum, now yields

\E x_T^\theta+\E a_T^\theta\leq x_0^\theta.

(D2.11)

In particular this estimate does not require uniform integrability of the unstopped local martingale or of the terminal matrix process.

For completeness, the scalar quadratic-form estimates really do reconstruct ordinary integrable matrices. Apply (D2.11) to the coordinate vectors $e_i$ . It gives integrability of the diagonal entries of $X_T$ and $H_T:=\int_0^TQ_t\dd t$ . Positivity gives, pathwise,

\abs{(X_T)_{ij}}\leq\frac{(X_T)_{ii}+(X_T)_{jj}}2, \qquad \int_0^T\abs{(Q_t)_{ij}}\dd t \leq\frac{(H_T)_{ii}+(H_T)_{jj}}2.

(D2.12)

Hence all entries are absolutely integrable. Polarization, or simply testing (D2.11) for every deterministic $\theta$ , gives

\theta^T\left(\E X_T+\E H_T\right)\theta \leq\theta^TX_0\theta \qquad(\theta\in\R^n),

(D2.13)

which is exactly the Loewner inequality (D2.6).

Finally $a_T^\theta\uparrow a_\infty^\theta$ as $T\uparrow\infty$ . Monotone convergence in (D2.11) gives

\E a_\infty^\theta\leq x_0^\theta.

(D2.14)

For $\theta=e_i$ , this says that every diagonal integral is finite almost surely and in $L^1$ . The positive-semidefinite entry bound just used, now on $[0,\infty)$ , makes every off-diagonal integral absolutely convergent and integrable. Polarization reconstructs the matrix expectation, and the preceding inequality for all $\theta$ proves (D2.7).

Relation to the per-direction estimate and the trace gate. Testing (D2.4) against any deterministic $\theta$ gives

\E\int_0^\infty \left(\abs{R_t\theta}^2+s_t\abs{G_t\theta}^2\right)\dd t \leq\theta^TR_0\theta.

(D2.17)

Restricting the integral to $[0,T]$ recovers the first display of Corollary 24.1; discarding the nonnegative $R_t^2$ term recovers its source-only Loewner display. Thus the new conclusion is the infinite-horizon matrix form of the full directional dissipation and retains strictly more information than the source-only bound.

The extra information is non-vacuous. For the standard Gaussian measure and the halfspace $E=\{x_1\leq0\}$ , every localization posterior remains a product across the coordinate axes and has covariance $(1+t)^{-1}I_n$ . For each spectator direction $e_j$ , $j\geq2$ , conditioning on $E$ does not change that coordinate, and hence

G_te_j=0, \qquad R_te_j=(1+t)^{-1}e_j, \qquad \int_0^\infty\abs{R_te_j}^2\dd t=1.

(D2.18)

The source-only estimate sees zero in those directions, whereas the full dissipation estimate uses their entire unit budgets.

Taking a trace in (D2.4) gives only

\E\int_0^\infty \left(\norm{R_t}_{\HS}^2+s_t\norm{G_t}_{\HS}^2\right)\dd t \leq\Tr R_0\leq n.

(D2.19)

Indeed, the Gaussian halfspace example already contributes $n-1$ from its spectator $R_t^2$ directions. The result therefore supplies a genuine matrix coercivity budget but no dimension-free operator-to-trace upgrade.

Hypotheses and fences. The only mathematical input beyond the standing two-color setup is the already certified matrix Riccati identity. Isotropy is used solely for $R_0\preceq I_n$ ; the estimate with right side $R_0$ holds for any initial law for which that identity and the two-color covariances are defined. Nontriviality of the cut is exactly what makes the conditional covariances meaningful. No balance, smooth-boundary, compact-support, stopping-time, fourth-moment, or endpoint uniform- integrability hypothesis is used. The ledger node has no bounded_by edge. The dimension-dependent trace consequence (D2.19) explicitly respects the operator-to-trace obstruction emphasized immediately after the manuscript corollary.