\chapter{The unnormalized process and its moments}\label{sec:process}

This section introduces the stochastic representation of the Langevin dynamics on which our moment bounds rest, and proves those bounds.
For real numbers $\rho_1,\dots,\rho_N$, the unnormalized process is the solution $Y$ in $\R^N$ of the stochastic differential equation $dY_i=-2\rho_iY_i\,dt+\sqrt{2/N}\,|Y|\,dB_i$.
We show in \cref{lem:proc-normalization} that it never vanishes and that its direction $\sqrt N\,Y/|Y|$ is the diffusion on $S_N$ with potential $\sum_i\rho_ix_i^2$, with the same time parameter and with noise coefficient $\sqrt2$.
If $Y(0)$ is a standard Gaussian vector, this diffusion starts from $\nu_N$.
The coordinates of $Y$ have linear drifts, and when $\E|Y(0)|^2<\infty$, the expectations of the squares $Y_i^2$ solve the linear equation with matrix $-4\diag(\rho)+(2/N)\ind\ind^\T$ studied in \cref{sec:heat}.
The only nonlinear feature is the random noise amplitude $|Y|$, and the estimates of this section bound the squared norm $|Y_t|^2$ from above and from below by solutions of the linear equation.

\Cref{sec:proc-normalization} proves the representation (\cref{lem:proc-normalization}), together with a representation $Y=\Phi X_0$, where $\Phi$ is a matrix-valued process independent of the initial point $X_0$.
\Cref{sec:proc-martingales} proves a criterion for positive martingales (\cref{pr-positive-martingale}): a uniform bound on conditional relative variances implies bounds on conditional moments of the maximum and of the inverse of the minimum, and on exponential moments of the quadratic variation of the logarithm.
In \cref{sec:proc-backward}, we bound conditional moments of $h(u)^\T Z_t$, where $Z_i=Y_i^2$ and $h$ is the solution of the linear equation started from $\ind$, by the corresponding solution of the linear equation (\cref{lem:proc-backward-moments}). We also bound moments of single coordinates by moments of the norm (\cref{lem:proc-coordinate-moments}).
\Cref{sec:proc-critical} applies these results in the critical window (\cref{cl2-horizon-second,cl2-direct-second,cd-adjoint-variance,cd-lift-moments}).
\Cref{sec:proc-hot,sec:proc-initial,sec:proc-maximal} treat precision sequences whose smallest element $d$ is small but satisfies $Nd^{3/2}\gg1$, which is the situation at temperatures above and near the critical temperature, on time intervals of length of order $d^{-1}\log(Nd^{3/2})$.
In \cref{sec:proc-hot}, we prove conditional moment bounds (\cref{iw-whole-horizon-positive}), inverse moment bounds (\cref{iw-w-inverse,iw-backward-inverse}), and an exponential moment bound for the time integral of $\Gamma(\log\varphi)$ along the path, where $\varphi$ is the quadratic form defined in \eqref{eq:proc-F} (\cref{iw-chemical-occupation}).
\Cref{sec:proc-initial} compares the initial state with all future solutions of the linear equation at once when the initial law is a Bingham comparison law (\cref{iw-early-pin-simultaneous-centers}).
\Cref{sec:proc-maximal} upgrades the fixed-time inverse moments of $|Y_t|^2$ to a bound for the supremum over the whole time interval (\cref{hmr-main}).
Finally, \cref{sec:proc-fk} proves a Feynman--Kac bound for a weighted gradient of the semigroup (\cref{hfk-pointwise-fk}).

We use the model and its stochastic differential equation from \cref{sec:model}, the conditioned Gaussian laws of \cref{sec:conditioning} (\cref{prop:cond-bingham,gp-growing-head}), the estimates for the linear equation from \cref{sec:heat} (\cref{cd-heat-profiles,iw-all-time-heat,hmr-heat-derivatives}), and the weighted Bochner identity \eqref{sgl-canonical-weighted-identity} from \cref{sec:inequalities}.
The representation is used in \cref{sec:cold-fixed,sec:cold-meso,sec:crit-process,sec:crit-profile}.
The estimates of \cref{sec:proc-critical} are used in \cref{sec:crit-process,sec:crit-profile}.
The estimates of \cref{sec:proc-hot,sec:proc-initial,sec:proc-maximal,sec:proc-fk} are used in \cref{sec:hot-meso}, and through it in \cref{sec:outer-hot}.

All statements in this section are deterministic in the disorder: the numbers $\rho_i$ and the precisions $p_i$ below are fixed, and probabilities and expectations refer only to the initial state and the Brownian motion.
We work on a filtered probability space $(\Omega,\mathcal F,(\mathcal F_t)_{t\ge0},\PP)$ satisfying the usual conditions and carrying an $(\mathcal F_t)$-Brownian motion $B=(B_1,\dots,B_N)$.
Stopping times are taken with respect to $(\mathcal F_t)$.
For a random variable $\xi$ and $r>0$, we write $\|\xi\|_r=(\E|\xi|^r)^{1/r}$; for $r\ge1$, this is the usual $L^r$ norm.
Since $B$ is an $(\mathcal F_t)$-Brownian motion, it is independent of $\mathcal F_0$; in particular, every $\mathcal F_0$-measurable initial value below is independent of $B$.
The estimates below are moment bounds for the process itself, obtained from It\^o's formula and martingale inequalities.
None of them uses a Gaussian approximation for the law of $Y_t$ or for the conditional law of a coordinate given the past.

\section{The unnormalized process}\label{sec:proc-normalization}

For $\rho=(\rho_1,\dots,\rho_N)\in\R^N$, we call the numbers $\rho_i$ the quadratic potential coefficients; they are the coefficients of the potential $\sum_i\rho_ix_i^2$.
We write $D_\rho=\diag(\rho_1,\dots,\rho_N)$ and
\begin{equation}\label{eq:proc-generator}
L_\rho=\Delta_{S_N}-\nabla_{S_N}\Big(\sum_{i=1}^N\rho_ix_i^2\Big)\cdot\nabla_{S_N}.
\end{equation}
In the notation of \eqref{eq:model-general}, we have $L_\rho=L_A$ with $A=-2D_\rho$, and by \eqref{eq:model-shift}, $L_{\rho+\tau\ind}=L_\rho$ for every $\tau\in\R$.

\begin{definition}[The unnormalized process]\label{def:proc-unnormalized}
Let $\rho\in\R^N$, and let $Y(0)$ be an $\mathcal F_0$-measurable random vector in $\R^N$ with $Y(0)\ne0$ almost surely.
The unnormalized process with quadratic potential coefficients $\rho$ and initial value $Y(0)$ is the solution of
\begin{equation}\label{eq:proc-sde}
dY_i(t)=-2\rho_iY_i(t)\,dt+\sqrt{2/N}\,|Y(t)|\,dB_i(t),\qquad 1\le i\le N,\quad t\ge0.
\end{equation}
We write
\begin{equation}\label{eq:proc-notation}
Z_i(t)=Y_i(t)^2,\qquad \mathcal A_t=|Y(t)|^2=\sum_{i=1}^NZ_i(t),\qquad X(t)=\frac{\sqrt N\,Y(t)}{|Y(t)|},
\end{equation}
and we call $\mathcal A_t$ the squared norm.
\end{definition}

We use the letter $\mathcal A$ for the squared norm to distinguish it from the matrix $A$ of \eqref{eq:model-general}.
When convenient, we also write $Y_t$, $Z_t$, and $X_t$ for $Y(t)$, $Z(t)$, and $X(t)$.

\begin{lemma}[The unnormalized process]\label{lem:proc-normalization}\label{cd-lift}\label{cq-ce-source-valid-unshifted-matrix-lift}
Let $N\ge2$ and $\rho\in\R^N$.
\begin{enumerate}
\item For every $\mathcal F_0$-measurable $Y(0)$ with $Y(0)\ne0$ almost surely, the equation \eqref{eq:proc-sde} has a pathwise unique strong solution.
Conditionally on $Y(0)=y$, the process $Y$ has the law of the solution started at $y$.
Almost surely, $Y(t)\ne0$ for all $t\ge0$, and
\begin{equation}\label{eq:proc-log-norm}
d\log\mathcal A_t=\Big(2-4\bar\rho_t-\frac4N\Big)\,dt+\sqrt{8/N}\,d\hat B_t,\qquad \bar\rho_t=\frac{1}{\mathcal A_t}\sum_{i=1}^N\rho_iZ_i(t),
\end{equation}
where $\hat B$ is a standard one-dimensional Brownian motion.
\item The process $X$ is the diffusion on $S_N$ generated by $L_\rho$, with noise coefficient $\sqrt2$ and initial point $X(0)=\sqrt N\,Y(0)/|Y(0)|$; no change of time is involved.
If $Y(0)$ is independent of $B$ and $X(0)$ has law $\mu$, then $X$ is this diffusion with initial law $\mu$.
In particular, if $Y(0)$ is independent of $B$ and is either a standard Gaussian vector or a random vector with law $\nu_N$, then $X$ is the diffusion generated by $L_\rho$ with initial law $\nu_N$.
\item Let $\lambda>0$ be an $\mathcal F_0$-measurable random variable.
Then $\lambda Y$ is the unnormalized process with coefficients $\rho$ and initial value $\lambda Y(0)$, driven by the same $B$.
For $\tau\in\R$, the process $(e^{-2\tau t}Y(t))_{t\ge0}$ is the unnormalized process with coefficients $\rho+\tau\ind$ and initial value $Y(0)$, driven by the same $B$.
In both cases $X$ is unchanged.
\item The squares satisfy
\begin{equation}\label{eq:proc-squares}
\begin{gathered}
dZ_i=\Big(-4\rho_iZ_i+\frac2N\,\mathcal A\Big)\,dt+dM_i,\qquad
dM_i=2\sqrt{2/N}\,Y_i\,|Y|\,dB_i,\\
d[M_i,M_j]_t=\delta_{ij}\,\frac8N\,\mathcal A_tZ_i(t)\,dt.
\end{gathered}
\end{equation}
\item Let $\mathcal B=(\mathcal B_{ij})_{i,j=1}^N$ be a matrix of independent standard Brownian motions, let $X_0$ be a random vector independent of $\mathcal B$ with $X_0\ne0$ almost surely, and let $\Phi$ be the solution of the linear matrix equation
\begin{equation}\label{eq:proc-matrix-flow}
d\Phi(t)=-2D_\rho\Phi(t)\,dt+\sqrt{2/N}\,d\mathcal B(t)\,\Phi(t),\qquad \Phi(0)=I.
\end{equation}
Then almost surely, for all $t\ge0$,
\begin{equation}\label{eq:proc-determinant}
\det\Phi(t)=\exp\Big\{-\Big(1+2\sum_{i=1}^N\rho_i\Big)t+\sqrt{2/N}\,\Tr\mathcal B(t)\Big\}>0 .
\end{equation}
The process $\Phi$ is independent of $X_0$.
The process $Y=\Phi X_0$ is the unnormalized process with coefficients $\rho$ and initial value $X_0$, with respect to the filtration generated by $X_0$ and $\mathcal B$ and driven by the Brownian motion $B(t)=\int_0^t d\mathcal B(s)\,Y(s)/|Y(s)|$, which is independent of $X_0$.
In particular, conditionally on $X_0=y$, the process $Y$ has the law of the unnormalized process started at $y$.
\end{enumerate}
\end{lemma}

\begin{proof}
We begin with the first part.
The drift $y\mapsto-2D_\rho y$ is linear, and the diffusion coefficient $y\mapsto\sqrt{2/N}\,|y|\,I$ is globally Lipschitz:
\[
\big\|\sqrt{2/N}\,\big(|y|-|y'|\big)I\big\|_{\mathrm{HS}}=\sqrt2\,\big||y|-|y'|\big|\le\sqrt2\,|y-y'| .
\]
By \cite[Theorem~IX.2.1, p.~375]{RevuzYor1999}, the equation \eqref{eq:proc-sde} has a pathwise unique strong solution for every deterministic initial value. The same Picard construction, localized on $\{|Y(0)|\le m\}$ and followed by $m\to\infty$, applies to every $\mathcal F_0$-measurable initial value.
For the conditional law, we use a stability estimate.
If $Y$ and $Y'$ are two solutions driven by the same $B$, with $\mathcal F_0$-measurable initial values, then $\Delta=Y-Y'$ satisfies $d\Delta=-2D_\rho\Delta\,dt+\sqrt{2/N}\,(|Y|-|Y'|)\,dB$.
Doob's $L^2$ inequality \cite[Theorem~II.1.7]{RevuzYor1999}, the Lipschitz bound above, and Gronwall's lemma, applied after localization and on events in $\mathcal F_0$, give
\begin{equation}\label{eq:proc-stability}
\E\Big[\sup_{s\le t}|\Delta_s|^2\,\Big|\,\mathcal F_0\Big]\le C_t\,|\Delta_0|^2,\qquad t\ge0,
\end{equation}
where $C_t$ depends only on $t$ and $\max_i|\rho_i|$.
For deterministic $y$, let $Y^y$ denote the solution started at $y$; it is adapted to the filtration generated by $B$.
Fix $t>0$ and a bounded continuous function $\Phi$ on $C([0,t],\R^N)$, and let $\psi(y)=\E\Phi(Y^y)$; by \eqref{eq:proc-stability}, $\psi$ is continuous.
For $n\ge1$, let $Y_n(0)$ be a measurable function of $Y(0)$ that takes countably many values $y_{n,k}$ and satisfies $|Y_n(0)-Y(0)|\le1/n$.
Since stochastic integrals commute with multiplication by indicators of events in $\mathcal F_0$, the process $Y^{(n)}=\sum_k\ind_{\{Y_n(0)=y_{n,k}\}}Y^{y_{n,k}}$ solves \eqref{eq:proc-sde} with initial value $Y_n(0)$.
As $Y(0)$ is independent of $B$, we have $\E[\Phi(Y^{(n)})g(Y(0))]=\E[\psi(Y_n(0))g(Y(0))]$ for every bounded measurable $g$.
By \eqref{eq:proc-stability}, $Y^{(n)}\to Y$ uniformly on $[0,t]$ in probability, and $\psi(Y_n(0))\to\psi(Y(0))$.
We conclude that $\E[\Phi(Y)g(Y(0))]=\E[\psi(Y(0))g(Y(0))]$, which is the second assertion of the first part.

Let $\zeta=\inf\{t\ge0:\mathcal A_t=0\}$.
For $t<\zeta$, It\^o's formula and the identity $\sum_id[Y_i]_t=2\mathcal A_t\,dt$ give
\[
d\mathcal A_t=(2-4\bar\rho_t)\,\mathcal A_t\,dt+2\sqrt{2/N}\,|Y_t|\,Y_t\cdot dB_t .
\]
Let $\hat B_t=\int_0^t\ind_{\{s<\zeta\}}|Y_s|^{-1}Y_s\cdot dB_s+\int_0^t\ind_{\{s\ge\zeta\}}\,dB_1(s)$.
Its integrands have unit length. Thus $\hat B$ is a continuous local martingale with $[\hat B]_t=t$, and by L\'evy's characterization \cite[Theorem~IV.3.6]{RevuzYor1999}, it is a standard Brownian motion.
For $t<\zeta$, we obtain \eqref{eq:proc-log-norm} from It\^o's formula for $\log\mathcal A_t$; the term $-4/N$ is $-\frac12\cdot\frac8N$.
Since $|\bar\rho_t|\le\max_i|\rho_i|$, we have
\[
\log\mathcal A_t\ge\log\mathcal A_0-\Big(2+4\max_i|\rho_i|+\frac4N\Big)t-\sqrt{8/N}\,\sup_{s\le t}|\hat B_s|,\qquad t<\zeta .
\]
On the event $\{\zeta<\infty\}$, the left side tends to $-\infty$ as $t\uparrow\zeta$, by continuity of $\mathcal A$, while the right side stays bounded below.
Hence $\zeta=\infty$ almost surely, and \eqref{eq:proc-log-norm} holds for all $t\ge0$.

For the second part, the map $f(y)=\sqrt N\,y/|y|$ is smooth on $\R^N\setminus\{0\}$, where $Y$ stays by the first part.
With $\Pi(x)=I-xx^\T/N$ as in \eqref{eq:model-sde}, its derivatives are
\[
\partial_kf_i(y)=\frac{\sqrt N}{|y|}\,\Pi\big(f(y)\big)_{ik},\qquad
\sum_{k=1}^N\partial_k^2f_i(y)=-\frac{(N-1)\sqrt N\,y_i}{|y|^3}.
\]
The martingale parts of the coordinates of $Y$ have covariations $(2/N)|Y|^2\delta_{kl}\,dt$, so the It\^o correction in $dX_i$ is $\frac12\cdot\frac2N|Y|^2\sum_k\partial_k^2f_i(Y)\,dt=-\frac{N-1}N\,X_i\,dt$.
It\^o's formula gives
\begin{equation}\label{eq:proc-normalized-sde}
dX_t=\sqrt2\,\Pi(X_t)\,dB_t+\Big(\Pi(X_t)(-2D_\rho)X_t-\frac{N-1}N\,X_t\Big)\,dt,
\end{equation}
where we used $\sqrt N\,Y/|Y|=X$ in the drift and $(\sqrt N/|Y|)\cdot\sqrt{2/N}\,|Y|=\sqrt2$ in the noise.
In coordinates, the drift of $X_i$ is $\big(-2\rho_i+\frac2N\sum_j\rho_jX_j^2-\frac{N-1}N\big)X_i$.
Equation \eqref{eq:proc-normalized-sde} is \eqref{eq:model-sde} with $cW_N$ replaced by $A=-2D_\rho$.
As shown in \cref{sec:model-model}, the solutions of this equation started on $S_N$ remain on $S_N$ and form the diffusion generated by $L_A=L_\rho$.
Its coefficients are smooth; the solution is therefore pathwise unique, and its law is determined by the law of its initial point.
If $Y(0)$ is independent of $B$, then so is $X(0)$. This gives the assertion about the initial law $\mu$.
If $Y(0)$ is a standard Gaussian vector, then $X(0)$ has law $\nu_N$ by the first part of \cref{prop:cond-bingham}; if $Y(0)$ has law $\nu_N$, then $X(0)=Y(0)$.

For the third part, since $\lambda$ is $\mathcal F_0$-measurable and $\lambda|Y|=|\lambda Y|$, multiplying \eqref{eq:proc-sde} by $\lambda$ shows that $\lambda Y$ solves \eqref{eq:proc-sde} with the same $B$.
The product rule shows that $\tilde Y(t)=e^{-2\tau t}Y(t)$ satisfies
\[
d\tilde Y_i=-2\tau\tilde Y_i\,dt+e^{-2\tau t}\big(-2\rho_iY_i\,dt+\sqrt{2/N}\,|Y|\,dB_i\big)=-2(\rho_i+\tau)\tilde Y_i\,dt+\sqrt{2/N}\,|\tilde Y|\,dB_i .
\]
In both cases $Y$ is multiplied by a positive scalar process, which leaves $X$ unchanged.
The fourth part follows from $dZ_i=2Y_i\,dY_i+d[Y_i]$ and $d[Y_i,Y_j]_t=\delta_{ij}(2/N)\mathcal A_t\,dt$.

For the last part, the linear equation \eqref{eq:proc-matrix-flow} has a unique strong solution, which is a measurable function of $\mathcal B$ and is independent of $X_0$.
The determinant is a polynomial in the entries of $\Phi$ that is affine in each row of $\Phi$.
The martingale part of $d\Phi_{ij}$ is $\sqrt{2/N}\sum_k\Phi_{kj}\,d\mathcal B_{ik}$, by \eqref{eq:proc-matrix-flow}. In particular, two entries in different rows have zero covariation.
Thus the second-order terms in It\^o's formula for $\det\Phi$ vanish, and we have
\[
\begin{aligned}
d\det\Phi&=\Tr\big(\operatorname{adj}(\Phi)\,d\Phi\big)
=\Tr\Big(\Phi\operatorname{adj}(\Phi)\Big(-2D_\rho\,dt+\sqrt{2/N}\,d\mathcal B\Big)\Big)\\
&=\det\Phi\,\Big(-2\sum_{i}\rho_i\,dt+\sqrt{2/N}\,d\Tr\mathcal B\Big),
\end{aligned}
\]
where $\operatorname{adj}(\Phi)$ is the adjugate matrix, which satisfies $\Phi\operatorname{adj}(\Phi)=\det(\Phi)I$.
The process $\Tr\mathcal B$ is a Brownian motion with variance $N$ per unit time.
The linear equation $dD=D\,dS$, $D_0=1$, with $S_t=-2t\sum_i\rho_i+\sqrt{2/N}\,\Tr\mathcal B(t)$, has the unique solution $D_t=\exp(S_t-[S]_t/2)$, and $[S]_t=2t$.
This proves \eqref{eq:proc-determinant}.
In particular, $\Phi(t)$ is invertible for all $t$, and $Y=\Phi X_0$ never vanishes.
Let $(\mathcal G_t)$ be the usual augmentation of the filtration generated by $X_0$ and $\mathcal B$; by independence, $\mathcal B$ is a $(\mathcal G_t)$-Brownian motion.
The process $B_i(t)=\sum_k\int_0^tY_k(s)|Y(s)|^{-1}\,d\mathcal B_{ik}(s)$ satisfies $d[B_i,B_j]_t=\delta_{ij}\,dt$. L\'evy's characterization implies that $B$ is a $(\mathcal G_t)$-Brownian motion, and it is independent of $\mathcal G_0\ni X_0$.
Multiplying \eqref{eq:proc-matrix-flow} by $X_0$ on the right, we obtain
\[
dY_i=-2\rho_iY_i\,dt+\sqrt{2/N}\sum_kY_k\,d\mathcal B_{ik}=-2\rho_iY_i\,dt+\sqrt{2/N}\,|Y|\,dB_i ,
\]
so $Y$ solves \eqref{eq:proc-sde} with initial value $X_0$ on the filtered space with filtration $(\mathcal G_t)$, and the last assertion follows from the first part.
\end{proof}

\begin{remark}[The two choices of coefficients]\label{rem:proc-choices}
Let $A$ be a real symmetric matrix with Lagrange multiplier $z$ and precisions $p_i=z-\alpha_i$ (\cref{def:model-multiplier}), and work in eigencoordinates of $A$.
On $S_N$ we have $-\frac12\sum_ip_ix_i^2=\frac12x^\T Ax-\frac{zN}2$, so $L_\rho=L_A$ for the half-precisions $\rho_i=r_i=p_i/2$.
The unnormalized process with these coefficients has drift rates $2r_i=p_i$ and solves
\begin{equation}\label{cd-lift-equation}
dY_i=-2r_iY_i\,dt+\sqrt{2/N}\,|Y|\,dB_i=-p_iY_i\,dt+\sqrt{2/N}\,|Y|\,dB_i .
\end{equation}
This choice is used in the critical window and above the critical temperature, and it is the equation \eqref{eq:intro-unnormalized} of the introduction.
For the model at low temperature, we use instead the scaled eigenvalue gaps $\rho_i=a_i=c(\lambda_1-\lambda_i)/2$ of \eqref{eq:model-raw} as coefficients, for which $L_\rho$ is the generator with potential $F=\sum_{i\ge2}a_ix_i^2$; the drift rates are $2a_i$, and the equation reads
\begin{equation}\label{eq:proc-sde-cold}
dY_i=-2a_iY_i\,dt+\sqrt{2\mathcal A_t/N}\,dB_i,\qquad \mathcal A_t=|Y(t)|^2 .
\end{equation}
Since $a_1=0$, the process $Y_1(t)-Y_1(0)$ is a continuous local martingale in this case; if $\E|Y(0)|^2<\infty$, then $Y_1$ is a square-integrable martingale on every bounded time interval, because $e^{-2s}\mathcal A_s$ is a nonnegative local supermartingale by \eqref{eq:proc-squares}, so that $\E\mathcal A_s\le e^{2s}\E\mathcal A_0$.
For $A=cW_N$, we have $r_i=a_i+d_N/2$, where $d_N=z_N-c\lambda_1$.
Using the third part of \cref{lem:proc-normalization}, we see that the two processes with the same initial value and the same Brownian motion are related by $Y^{(r)}(t)=e^{-d_Nt}\,Y^{(a)}(t)$, and that they have the same direction $X$.
\end{remark}

In \cite{CugliandoloDean1995}, Cugliandolo and Dean diagonalized the interaction of this model and, in the limit $N\to\infty$, reduced its dynamics to linear equations for the eigencoordinates and a scalar Volterra equation for the Lagrange multiplier \cite[Section~II, (II.2)--(II.7)]{CugliandoloDean1995}.
The unnormalized process is an exact analogue at finite $N$: its coordinates have linear drifts, and when $\E|Y(0)|^2<\infty$, the expectations of their squares solve the closed linear equation of \cref{sec:heat} (see \eqref{eq:proc-mean-squares} below), while its noise amplitude $|Y|$ is random.

\section{Positive martingales}\label{sec:proc-martingales}\label{pr-sec:positive-martingales}

The following criterion converts a bound on conditional relative variances of a positive martingale into bounds on its maximum, on the inverse of its minimum, and on the quadratic variation of its logarithm.
It is used in \cref{sec:proc-critical,sec:proc-hot} for martingales built from the squares $Z_i$, and its hypothesis is verified separately in each case.

\begin{lemma}[Maximum, minimum, and the quadratic variation of $\log V$]\label{pr-positive-martingale}
Let $T>0$, let $V=(V_t)_{0\le t\le T}$ be a strictly positive continuous square-integrable martingale, and let $\varepsilon\ge0$.
Suppose that for every stopping time $\tau\le T$,
\begin{equation}\label{pr-eq:conditional-relative-variance}
\E\big[(V_T-V_\tau)^2\,\big|\,\mathcal F_\tau\big]\le\varepsilon V_\tau^2 .
\end{equation}
\begin{enumerate}
\item For every $p>0$ with $2^p\cdot4\varepsilon<1$ and every stopping time $\tau\le T$,
\begin{equation}\label{pr-eq:inverse-max}
\E\Big[\sup_{\tau\le s\le T}\Big(\frac{V_\tau}{V_s}\Big)^p\,\Big|\,\mathcal F_\tau\Big]\le\frac{2^p}{1-2^p\cdot4\varepsilon} .
\end{equation}
\item For every $p>0$ with $2^p\varepsilon<1$ and every stopping time $\tau\le T$,
\begin{equation}\label{eq:proc-positive-max}
\E\Big[\sup_{\tau\le s\le T}\Big(\frac{V_s}{V_\tau}\Big)^p\,\Big|\,\mathcal F_\tau\Big]\le\frac{2^p}{1-2^p\varepsilon} .
\end{equation}
\item Suppose in addition that $\varepsilon\le1/32$, and that the quadratic variation of the logarithm of $V$ has the form
\[
Q_t=\int_0^tV_s^{-2}\,d[V]_s=\int_0^tq_s\,ds,\qquad 0\le t\le T,
\]
where $q\ge0$ is progressively measurable and $Q_T\le\bar Q$ for a deterministic constant $\bar Q$.
Then for every stopping time $\tau\le T$,
\begin{equation}\label{pr-eq:conditional-bracket}
\E\big[Q_T-Q_\tau\,\big|\,\mathcal F_\tau\big]\le K,\qquad K=6\sqrt{2\varepsilon},
\end{equation}
and for every $\lambda\ge0$ with $\lambda K<1$,
\begin{equation}\label{pr-eq:occupation}
\E\big[e^{\lambda(Q_T-Q_\tau)}\,\big|\,\mathcal F_\tau\big]\le\frac1{1-\lambda K} .
\end{equation}
\end{enumerate}
The constants in \eqref{pr-eq:inverse-max}--\eqref{pr-eq:occupation} depend only on $p$, $\varepsilon$, and $\lambda$; they do not depend on $T$ or $\bar Q$.
\end{lemma}

The bound $\bar Q$ is used only to show that the stochastic logarithm of $V$ is a square-integrable martingale.
The successive hitting-time argument in Steps~1--3 follows the classical martingale John--Nirenberg proof pattern; see \cite[Chapter~II, Exercise~(3.16), part~5, pp.~75--76]{RevuzYor1999}. We prove the conditional ordered-integral estimate \eqref{pr-eq:occupation} in Step~4 below; its Markov-process antecedent is \cite[Lemma~3, pp.~337--338]{Khasminskii1959}.

\begin{proof}
The proof has four steps: a bound for the probability that $V$ moves by a fixed fraction of its current value, its iteration at successive hitting times, the resulting moment bounds, and the two bounds on $Q$.

\paragraph{Step 1: one step.}
Fix a stopping time $\tau\le T$ and $\theta>0$, and let
\[
\sigma=\inf\{s\ge\tau:|V_s-V_\tau|\ge\theta V_\tau\}\wedge T,\qquad
G_\theta=\Big\{\sup_{\tau\le s\le T}|V_s-V_\tau|\ge\theta V_\tau\Big\}.
\]
By optional sampling, $\E[V_T-V_\tau\mid\mathcal F_\sigma]=V_\sigma-V_\tau$, and by the conditional Jensen inequality, $\E[(V_T-V_\tau)^2\mid\mathcal F_\sigma]\ge(V_\sigma-V_\tau)^2$.
On $G_\theta$, we have $|V_\sigma-V_\tau|=\theta V_\tau$ by continuity of $V$.
Taking conditional expectations given $\mathcal F_\tau$ and using \eqref{pr-eq:conditional-relative-variance}, we obtain
\[
\varepsilon V_\tau^2\ge\E\big[(V_\sigma-V_\tau)^2\mid\mathcal F_\tau\big]\ge\theta^2V_\tau^2\,\PP(G_\theta\mid\mathcal F_\tau),
\]
so $\PP(G_\theta\mid\mathcal F_\tau)\le\varepsilon/\theta^2$.
The events $\{\inf_{\tau\le s\le T}V_s\le V_\tau/2\}$ and $\{\sup_{\tau\le s\le T}V_s\ge2V_\tau\}$ are contained in $G_{1/2}$ and $G_1$, respectively.
We conclude that
\begin{equation}\label{eq:proc-one-step}
\PP\Big(\inf_{\tau\le s\le T}V_s\le\frac{V_\tau}2\,\Big|\,\mathcal F_\tau\Big)\le4\varepsilon,\qquad
\PP\Big(\sup_{\tau\le s\le T}V_s\ge2V_\tau\,\Big|\,\mathcal F_\tau\Big)\le\varepsilon .
\end{equation}

\paragraph{Step 2: iteration.}
Fix $\tau$, and for $j\ge0$ let $\sigma_j=\inf\{s\ge\tau:V_s\le2^{-j}V_\tau\}$, with $\inf\emptyset=\infty$, and $E_j=\{\sigma_j\le T\}$.
Then $\sigma_0=\tau$, $E_j\subset E_{j-1}$, and $E_{j-1}\in\mathcal F_{\sigma_{j-1}\wedge T}$.
On $E_{j-1}$, continuity gives $V_{\sigma_{j-1}}=2^{1-j}V_\tau$, and $V$ stays above $2^{-j}V_\tau$ before $\sigma_{j-1}$, so
\[
E_j=E_{j-1}\cap\Big\{\inf_{\sigma_{j-1}\wedge T\le s\le T}V_s\le\frac{V_{\sigma_{j-1}\wedge T}}2\Big\}.
\]
Applying \eqref{eq:proc-one-step} at the stopping time $\sigma_{j-1}\wedge T$ and conditioning on $\mathcal F_\tau$, we find $\PP(E_j\mid\mathcal F_\tau)\le4\varepsilon\,\PP(E_{j-1}\mid\mathcal F_\tau)$.
By induction,
\[
\PP\Big(\sup_{\tau\le s\le T}\frac{V_\tau}{V_s}\ge2^j\,\Big|\,\mathcal F_\tau\Big)=\PP(E_j\mid\mathcal F_\tau)\le(4\varepsilon)^j,\qquad j\ge0 .
\]
The same argument with the successive first hitting times of the levels $2^jV_\tau$ and the second bound in \eqref{eq:proc-one-step} yields
\[
\PP\Big(\sup_{\tau\le s\le T}\frac{V_s}{V_\tau}\ge2^j\,\Big|\,\mathcal F_\tau\Big)\le\varepsilon^j,\qquad j\ge0 .
\]

\paragraph{Step 3: moments.}
Let $R=\sup_{\tau\le s\le T}V_\tau/V_s$, which is finite and at least one.
Splitting according to the dyadic interval $[2^j,2^{j+1})$ that contains $R$, we have
\[
\E[R^p\mid\mathcal F_\tau]\le\sum_{j\ge0}2^{p(j+1)}\,\PP(R\ge2^j\mid\mathcal F_\tau)\le2^p\sum_{j\ge0}(2^p\cdot4\varepsilon)^j=\frac{2^p}{1-2^p\cdot4\varepsilon},
\]
which is \eqref{pr-eq:inverse-max}.
With the second bound of Step~2 in place of the first, the same computation gives \eqref{eq:proc-positive-max}.

\paragraph{Step 4: the quadratic variation of the logarithm.}
Let $\varepsilon\le1/32$, fix a stopping time $\tau\le T$, and let $R'=V_T/V_\tau$.
The hypothesis \eqref{pr-eq:conditional-relative-variance} states that $\E[(R'-1)^2\mid\mathcal F_\tau]\le\varepsilon$.
By \eqref{pr-eq:inverse-max} with $p=2$, for which $2^2\cdot4\varepsilon\le1/2$, we have $\E[R'^{-2}\mid\mathcal F_\tau]\le8$, and then $\E[(1+R'^{-1})^2\mid\mathcal F_\tau]\le2(1+8)=18$.
For every $r>0$, we have $|\log r|\le|r-1|(1+r^{-1})$: for $r\ge1$ this follows from $\log r\le r-1$, and for $r<1$ from $\log(1/r)\le1/r-1=(1-r)/r$.
Using the conditional Cauchy--Schwarz inequality, we find
\[
\E\big[|\log R'|\,\big|\,\mathcal F_\tau\big]\le\sqrt{\varepsilon}\cdot\sqrt{18}=3\sqrt{2\varepsilon}.
\]
Since $[\mathcal L]_T=Q_T\le\bar Q$, the stochastic logarithm $\mathcal L_t=\int_0^tV_s^{-1}\,dV_s$ is a square-integrable martingale on $[0,T]$, and by It\^o's formula, $\log V_t=\log V_0+\mathcal L_t-\frac12Q_t$.
Taking conditional expectations, we have
\[
\E[Q_T-Q_\tau\mid\mathcal F_\tau]=-2\,\E[\log R'\mid\mathcal F_\tau]\le6\sqrt{2\varepsilon},
\]
which is \eqref{pr-eq:conditional-bracket}.

For \eqref{pr-eq:occupation}, we show by induction that $\E[(Q_T-Q_\tau)^j\mid\mathcal F_\tau]\le j!\,K^j$ for every stopping time $\tau\le T$ and every integer $j\ge1$.
The case $j=1$ is \eqref{pr-eq:conditional-bracket}.
Since $s\mapsto(Q_T-Q_s)^j$ is absolutely continuous with derivative $-jq_s(Q_T-Q_s)^{j-1}$, we have
\[
(Q_T-Q_\tau)^j=j\int_0^T\ind_{\{s>\tau\}}\,q_s\,(Q_T-Q_s)^{j-1}\,ds .
\]
The factor $\ind_{\{s>\tau\}}q_s$ is $\mathcal F_s$-measurable.
By the conditional Tonelli theorem, the tower property, and the induction hypothesis at the deterministic time $s$, we obtain
\[
\begin{aligned}
\E\big[(Q_T-Q_\tau)^j\mid\mathcal F_\tau\big]
&=j\int_0^T\E\Big[\ind_{\{s>\tau\}}q_s\,\E\big[(Q_T-Q_s)^{j-1}\mid\mathcal F_s\big]\,\Big|\,\mathcal F_\tau\Big]\,ds\\
&\le j!\,K^{j-1}\,\E[Q_T-Q_\tau\mid\mathcal F_\tau]\le j!\,K^j .
\end{aligned}
\]
Summing the exponential series by monotone convergence, we find $\E[e^{\lambda(Q_T-Q_\tau)}\mid\mathcal F_\tau]\le\sum_{j\ge0}(\lambda K)^j=(1-\lambda K)^{-1}$.
\end{proof}

\section{Moment bounds with backward semigroup weights}\label{sec:proc-backward}

In this section, $\rho\in[0,\infty)^N$.
We write
\begin{equation}\label{eq:proc-heat}
H_\rho=-4D_\rho+\frac2N\,\ind\ind^\T,\qquad h(t)=e^{tH_\rho}\ind,\qquad \bar h(t)=\frac1N\ind^\T h(t),\qquad m(t)=\max_ih_i(t),
\end{equation}
and, for $y\in[0,\infty)^N$, we set $\mathcal H_y(t)=h(t)^\T y$.
When $\rho_i=p_i/2$ for precisions $p_i$, the matrix $H_\rho=-2\diag(p)+(2/N)\ind\ind^\T$ is the matrix $H$ of \cref{sec:heat}.
The matrix $H_\rho$ is symmetric and its off-diagonal entries are nonnegative.
For $c_\rho=4\max_i\rho_i$, the matrix $H_\rho+c_\rho I$ has nonnegative entries, so $e^{tH_\rho}=e^{-c_\rho t}e^{t(H_\rho+c_\rho I)}$ has nonnegative entries; this is the positivity used in the first part of \cref{lem:heat-algebra}.
By the variation of constants formula for the coordinates of the equation $h'=H_\rho h$, we have
\begin{equation}\label{eq:proc-coordinate-duhamel}
h_i(t)=e^{-4\rho_it}+\frac2N\int_0^te^{-4\rho_i(t-s)}\,N\bar h(s)\,ds\ge e^{-4\rho_it}>0 .
\end{equation}
In particular, $\mathcal H_y(t)>0$ for every nonzero $y\in[0,\infty)^N$.
We have $\ind^\T e^{tH_\rho}y=\mathcal H_y(t)$ by symmetry of $H_\rho$, and the same argument applied to $e^{tH_\rho}y$ yields
\begin{equation}\label{eq:proc-duhamel-y}
(e^{tH_\rho}y)_i=e^{-4\rho_it}y_i+\frac2N\int_0^te^{-4\rho_i(t-s)}\,\mathcal H_y(s)\,ds .
\end{equation}
For the unnormalized process with coefficients $\rho\in[0,\infty)^N$ and $\E|Y(0)|^2<\infty$, taking expectations in \eqref{eq:proc-squares} gives
\begin{equation}\label{eq:proc-mean-squares}
\E Z_t=e^{tH_\rho}\,\E Z_0,\qquad t\ge0 .
\end{equation}
Indeed, for a deterministic initial value, the local martingales $M_i$ in \eqref{eq:proc-squares} are square integrable on bounded intervals by the moment bound \eqref{eq:proc-crude} proved below, so $\E Z_t$ solves the linear equation with matrix $H_\rho$; the general case follows by conditioning on $Y(0)$, using the first part of \cref{lem:proc-normalization}.

The squares of the unnormalized process satisfy the following equations with $\epsilon\equiv0$, by \eqref{eq:proc-squares}.
The estimates below also allow a nonnegative perturbation $\epsilon$ of the drift, which may depend on the past of the process.
We say that a continuous adapted process $Z$ with values in $[0,\infty)^N$ and with $\mathcal A_t=\sum_iZ_i(t)>0$ for all $t$ satisfies \eqref{cl2-direct-lift} with perturbation bound $E_*\ge0$ if
\begin{equation}\label{cl2-direct-lift}
dZ_i=\Big[-4\rho_iZ_i+\big(2+\epsilon_i(t)\big)\frac{\mathcal A}N\Big]\,dt+dM_i,\qquad
d[M_i,M_j]_t=\delta_{ij}\,\frac8N\,\mathcal A_tZ_i(t)\,dt,
\end{equation}
where the $M_i$ are continuous local martingales and the $\epsilon_i$ are progressively measurable with $\epsilon_i\ge0$ and $\sum_i\epsilon_i(t)\le E_*$ for all $t$.

The following lemma bounds moments of $h(u)^\T Z_t$ by the corresponding power of $\mathcal H_{Z(0)}(u+t)$.
When $\epsilon\equiv0$, the process $s\mapsto h(u+t-s)^\T Z_s$, $0\le s\le u+t$, is a local martingale with initial value $\mathcal H_{Z(0)}(u+t)$, which we call a martingale with backward semigroup weights, and the lemma bounds its second and higher moments by powers of its initial value.
The size of the bound is governed by the quantity
\begin{equation}\label{eq:proc-Lambda}
\Lambda(\ell,\Theta)=\sup\bigg\{\int_0^tm(w-s)\,\frac{\mathcal H_y(s)}{\mathcal H_y(w)}\,ds:\ y\in[0,\infty)^N\setminus\{0\},\ 0\le t\le\ell,\ t\le w\le\Theta\bigg\},
\end{equation}
which depends only on $\rho$, $\ell$, and $\Theta$.

\begin{lemma}[Moment bounds with backward semigroup weights]\label{lem:proc-backward-moments}
Let $\rho\in[0,\infty)^N$, let $Z$ satisfy \eqref{cl2-direct-lift} with perturbation bound $E_*$, let $k\ge2$ be an integer, and let $0<\ell\le\Theta<\infty$.
Set
\begin{equation}\label{eq:proc-alpha}
\alpha=\frac{k\,(E_*+4k-4)}N\,\Lambda(\ell,\Theta),
\end{equation}
and suppose that $\alpha<1$.
Then for every bounded stopping time $\tau$, and all $\mathcal F_\tau$-measurable random variables $u\ge0$ and $t\in[0,\ell]$ with $u+t\le\Theta$,
\begin{equation}\label{eq:proc-backward-moments}
\E\big[(h(u)^\T Z_{\tau+t})^k\,\big|\,\mathcal F_\tau\big]\le\frac{\mathcal H_{Z_\tau}(u+t)^k}{1-\alpha}\qquad\text{almost surely}.
\end{equation}
In particular, if $Z(0)=y$ is deterministic, then $\E[(h(u)^\T Z_t)^k]\le\mathcal H_y(u+t)^k/(1-\alpha)$ for all $t\in[0,\ell]$ and $u\ge0$ with $u+t\le\Theta$.
\end{lemma}

The conditional statement allows the drift perturbation $\epsilon$ after time $\tau$ to depend on the whole past; no Markov property is used.

\begin{proof}
The proof has three steps.
We first prove a crude bound on moments of $\mathcal A$, which shows that a normalized supremum $\mathcal B$ of the left side of \eqref{eq:proc-backward-moments} is finite.
We then compute the evolution of $h(w-s)^\T Z_{\tau+s}$ with It\^o's formula, and finally we show that $\mathcal B\le1+\alpha\mathcal B$.

\paragraph{Step 1: a crude moment bound.}
Let $\mathcal M=\sum_iM_i$; as the $M_i$ have zero mutual covariation, $d[\mathcal M]_t=(8/N)\mathcal A_t^2\,dt$.
Summing \eqref{cl2-direct-lift} and using $\rho\ge0$ and $\sum_i\epsilon_i\le E_*$, we see that $\mathcal A$ has drift at most $(2+E_*/N)\mathcal A$.
For an integer $j\ge1$, It\^o's formula shows that the drift of $\mathcal A^j$ is at most $c_j\mathcal A^j$, with
\[
c_j=2j+\frac{jE_*+4j(j-1)}N .
\]
Fix a bounded stopping time $\sigma$.
Relative to the filtration $(\mathcal F_{\sigma+s})_{s\ge0}$, It\^o's formula shows that
\[
R_s=e^{-c_js}\Big(\frac{\mathcal A_{\sigma+s}}{\mathcal A_\sigma}\Big)^j
\]
is a nonnegative local supermartingale with $R_0=1$; by Fatou's lemma, it is a supermartingale, without any moment assumption on $\mathcal A_\sigma$.
If $\sigma\le\sigma'\le\sigma+\ell$ for a stopping time $\sigma'$, then $\sigma'-\sigma$ is a stopping time for the shifted filtration, and optional sampling gives $\E[e^{-c_j(\sigma'-\sigma)}(\mathcal A_{\sigma'}/\mathcal A_\sigma)^j\mid\mathcal F_\sigma]\le1$.
Since $c_j\ge0$ and $\mathcal A_\sigma$ is $\mathcal F_\sigma$-measurable, we obtain
\begin{equation}\label{eq:proc-crude}
\E\big[\mathcal A_{\sigma'}^j\,\big|\,\mathcal F_\sigma\big]\le e^{c_j\ell}\,\mathcal A_\sigma^j .
\end{equation}
Let $m_\Theta=\max_{s\le\Theta}m(s)$ and $h_\Theta=\min_{s\le\Theta}\min_ih_i(s)$, which is positive by \eqref{eq:proc-coordinate-duhamel}.
Call a pair $(u,t)$ of $\mathcal F_\tau$-measurable random variables admissible if $u\ge0$, $t\in[0,\ell]$, and $u+t\le\Theta$.
For an admissible pair, $\tau+t$ is a bounded stopping time. We set
\[
\beta(u,t)=\frac{\E[(h(u)^\T Z_{\tau+t})^k\mid\mathcal F_\tau]}{\mathcal H_{Z_\tau}(u+t)^k} .
\]
We have $h(u)^\T Z_{\tau+t}\le m_\Theta\mathcal A_{\tau+t}$ and $\mathcal H_{Z_\tau}(u+t)\ge h_\Theta\mathcal A_\tau$, and \eqref{eq:proc-crude} implies that $\beta(u,t)\le(m_\Theta/h_\Theta)^ke^{c_k\ell}$.
Let $\mathcal B$ be the essential supremum of $\beta(u,t)$ over all admissible pairs.
It is an $\mathcal F_\tau$-measurable random variable, $\beta(u,t)\le\mathcal B$ almost surely for every admissible pair, and
\begin{equation}\label{eq:proc-B-finite}
\mathcal B\le(m_\Theta/h_\Theta)^k\,e^{c_k\ell}<\infty .
\end{equation}

\paragraph{Step 2: the backward semigroup weights.}
Fix an admissible pair $(u,t)$, let $w=u+t$, and set $X_s=h(w-s)^\T Z_{\tau+s}$ for $0\le s\le t$.
Since $h'=H_\rho h$ and $H_\rho$ is symmetric, the terms of \eqref{cl2-direct-lift} that involve $H_\rho$ cancel, and we have
\[
dX_s=D_s\,ds+\sum_ih_i(w-s)\,dM_i(\tau+s),\qquad
D_s=\frac{\mathcal A_{\tau+s}}N\sum_i\epsilon_i(\tau+s)\,h_i(w-s).
\]
The drift satisfies $0\le D_s\le(E_*/N)\,m(w-s)\,\mathcal A_{\tau+s}$, and since $\sum_ih_i^2Z_i\le m\sum_ih_iZ_i$, the quadratic variation satisfies
\[
d[X]_s=\frac8N\,\mathcal A_{\tau+s}\sum_ih_i(w-s)^2Z_i(\tau+s)\,ds\le\frac8N\,m(w-s)\,\mathcal A_{\tau+s}X_s\,ds .
\]
Applying It\^o's formula to $X^k$, we see that the drift of $X^k$ is at most
\[
kX_s^{k-1}D_s+\frac{k(k-1)}2X_s^{k-2}\,\frac{d[X]_s}{ds}\le\frac{k(E_*+4k-4)}N\,m(w-s)\,\mathcal A_{\tau+s}X_s^{k-1}.
\]
We stop at the first time that $\mathcal A_{\tau+s}$ exceeds $n$, take conditional expectations given $\mathcal F_\tau$, and let $n\to\infty$, using conditional Fatou on the left and monotone convergence on the right.
Since $X_0=\mathcal H_{Z_\tau}(w)$ and $X_t=h(u)^\T Z_{\tau+t}$, this gives
\begin{equation}\label{eq:proc-ito-backward}
\begin{split}
\E\big[(h(u)^\T Z_{\tau+t})^k\,\big|\,\mathcal F_\tau\big]&\le\mathcal H_{Z_\tau}(w)^k\\
&\quad+\frac{k(E_*+4k-4)}N\int_0^\ell\ind_{\{s<t\}}\,m\big(w-(s\wedge t)\big)\,\E\big[\mathcal A_{\tau+s}X_{s\wedge t}^{k-1}\,\big|\,\mathcal F_\tau\big]\,ds .
\end{split}
\end{equation}
\aliaslabel{cl2-direct-ito-bound}{eq:proc-ito-backward}%
Here we used that $\ind_{\{s<t\}}$ and $m(w-(s\wedge t))$ are $\mathcal F_\tau$-measurable, which allows us to take them out of the conditional expectation; the argument $w-(s\wedge t)$ is nonnegative, and on $\{s<t\}$ it equals $w-s$.

\paragraph{Step 3: absorption.}
For $s\le\ell$, both $(0,s)$ and $(w-s\wedge t,s\wedge t)$ are admissible pairs, and $\mathcal A_{\tau+s}=h(0)^\T Z_{\tau+s}$.
By the conditional H\"older inequality and the definition of $\mathcal B$, on the event $\{s<t\}$,
\[
\E\big[\mathcal A_{\tau+s}X_s^{k-1}\,\big|\,\mathcal F_\tau\big]
\le\E\big[\mathcal A_{\tau+s}^k\,\big|\,\mathcal F_\tau\big]^{1/k}\,\E\big[X_s^k\,\big|\,\mathcal F_\tau\big]^{(k-1)/k}
\le\mathcal B\,\mathcal H_{Z_\tau}(s)\,\mathcal H_{Z_\tau}(w)^{k-1}.
\]
Inserting this into \eqref{eq:proc-ito-backward} and dividing by $\mathcal H_{Z_\tau}(w)^k$, we obtain
\[
\beta(u,t)\le1+\frac{k(E_*+4k-4)}N\,\mathcal B\int_0^tm(w-s)\,\frac{\mathcal H_{Z_\tau}(s)}{\mathcal H_{Z_\tau}(w)}\,ds\le1+\alpha\mathcal B,
\]
where in the last step we used the definition \eqref{eq:proc-Lambda} with $y=Z_\tau$, which is a nonzero vector in $[0,\infty)^N$.
This holds almost surely for every admissible pair, and we conclude that $\mathcal B\le1+\alpha\mathcal B$ almost surely.
Since $\mathcal B$ is finite by \eqref{eq:proc-B-finite} and $\alpha<1$, we conclude that $\mathcal B\le(1-\alpha)^{-1}$, which is \eqref{eq:proc-backward-moments}.
\end{proof}

The next lemma bounds the moments of a single square by moments of the squared norm.

\begin{lemma}[Moments of single coordinates]\label{lem:proc-coordinate-moments}
Let $\rho\in[0,\infty)^N$, let $k\ge1$ be an integer, and let $Y$ be the unnormalized process with coefficients $\rho$ and an $\mathcal F_0$-measurable initial value with $\E|Y(0)|^{4k}<\infty$.
Then for every $i$ and $t\ge0$,
\begin{equation}\label{eq:proc-coordinate-moments}
\|Z_i(t)\|_k\le e^{-4\rho_it}\,\|Z_i(0)\|_k+\frac{2(2k-1)}N\int_0^te^{-4\rho_i(t-s)}\,\|\mathcal A_s\|_k\,ds .
\end{equation}
\end{lemma}

\begin{proof}
All moments used below are finite and locally bounded in time, by \eqref{eq:proc-crude} with $\sigma=0$ and $j=2k$.
Using \eqref{eq:proc-squares} and It\^o's formula, we find
\[
d(Z_i^k)=\Big[-4k\rho_iZ_i^k+\frac{2k}N\,\mathcal AZ_i^{k-1}+\frac{k(k-1)}2\cdot\frac8N\,\mathcal AZ_i^{k-1}\Big]dt+kZ_i^{k-1}\,dM_i,
\]
and $2k+4k(k-1)=2k(2k-1)$.
The martingale term has integrable quadratic variation, since $Z_i^{2k-2}\mathcal AZ_i\le\mathcal A^{2k}$.
Taking expectations, we see that $\phi(t)=\E Z_i(t)^k$ is absolutely continuous, and by H\"older's inequality,
\[
\begin{aligned}
\phi'(t)&=-4k\rho_i\phi(t)+\frac{2k(2k-1)}N\,\E\big[\mathcal A_tZ_i(t)^{k-1}\big]\\
&\le-4k\rho_i\phi(t)+\frac{2k(2k-1)}N\,\|\mathcal A_t\|_k\,\phi(t)^{(k-1)/k}.
\end{aligned}
\]
For $\eta>0$, let $\psi_\eta=(\phi+\eta)^{1/k}$.
Using $\phi(\phi+\eta)^{1/k-1}\ge(\phi+\eta)^{1/k}-\eta^{1/k}$ and $\phi^{(k-1)/k}(\phi+\eta)^{1/k-1}\le1$, we find
\[
\psi_\eta'(t)\le-4\rho_i\psi_\eta(t)+4\rho_i\eta^{1/k}+\frac{2(2k-1)}N\,\|\mathcal A_t\|_k .
\]
Integrating this linear differential inequality and letting $\eta\downarrow0$, we obtain \eqref{eq:proc-coordinate-moments}.
\end{proof}

\section{The critical window}\label{sec:proc-critical}\label{cl2-direct-moments-section}

In this section, we specialize \cref{lem:proc-backward-moments,lem:proc-coordinate-moments} to the half-precisions of the critical window.
Let $0<r_1\le\dots\le r_N$ be real numbers, and suppose that for fixed constants $0<c_0\le C_0<\infty$,
\begin{equation}\label{eq:proc-critical-weyl}
\sum_{i=1}^N\frac1{2r_i}=N,\qquad c_0\Big(\frac iN\Big)^{2/3}\le r_i\le C_0\Big(\frac iN\Big)^{2/3}\quad(1\le i\le N).
\end{equation}
These are the hypotheses \eqref{eq:heat-weyl} of \cref{cd-heat-profiles}, and they are the first two of the critical spectral conditions \eqref{cg-weyl-hypotheses}; the bound on the first gap in \eqref{cg-weyl-hypotheses} is not used in this section.
For the model, fix $\sigma_0>0$ and $\eta\in(0,1)$; the second part of \cref{cg-canonical-weyl} supplies constants depending only on $\sigma_0$ and $\eta$, and events $\mathcal E_N$ with $\liminf_{N\to\infty}\PP(\mathcal E_N)\ge1-\eta$, on which \eqref{cg-weyl-hypotheses} holds simultaneously for the half-precisions of $(1+\sigma N^{-1/3})W_N$, $|\sigma|\le\sigma_0$, when $N>(2\sigma_0)^3$.
Here the probability is over the disorder $W_N$, and the constants are fixed before $N\to\infty$.
Note that $r_i\le C_0$ for all $i$.
We take $\rho=r$ in \eqref{eq:proc-heat}; then $H_r$ is the matrix $H$ of \cref{sec:heat} with $p=2r$.
By \cref{cd-heat-profiles}, for every fixed $K<\infty$, uniformly in $0\le t\le KN^{2/3}$,
\begin{equation}\label{eq:proc-critical-heat}
N\bar h(t)\asymp\frac{N}{\sqrt{1+t}},\qquad h_i(t)\asymp\frac{\sqrt{1+t}}{1+r_it},
\end{equation}
with constants depending only on $c_0$, $C_0$, and $K$.

The first two statements of this section assume only a two-sided bound on the coordinates of $h$ on a general time interval $[0,\Theta]$; they do not use \eqref{eq:proc-critical-weyl}.

\begin{corollary}[Second-moment bounds with backward semigroup weights]\label{cl2-horizon-second}
Let $N\ge1$, let $0<r_i\le C_0$ for $1\le i\le N$, and let $h(t)=e^{tH_r}\ind$ with $H_r=-4\diag(r)+(2/N)\ind\ind^\T$.
Fix $\Theta>0$, and assume that for $0\le t\le\Theta$,
\begin{equation}\label{cl2-direct-profile}
c_*\,\frac{\sqrt{1+t}}{1+r_it}\le h_i(t)\le C_*\,\frac{\sqrt{1+t}}{1+r_it},\qquad 1\le i\le N .
\end{equation}
Let $Z$ satisfy \eqref{cl2-direct-lift} with $\rho=r$ and perturbation bound $E_*$.
There exists a constant $C$ depending only on $c_*$, $C_*$, $C_0$, and $E_*$ such that the following holds.
For $0<\ell\le\Theta$, let $\alpha=C(1+\Theta)\sqrt\ell/N$.
If $\alpha<1$, then every deterministic nonzero initial state $Z(0)=y\in[0,\infty)^N$ satisfies
\begin{equation}\label{cl2-horizon-second-bound}
\E\big[(h(u)^\T Z_t)^2\big]\le\frac{[h(u+t)^\T y]^2}{1-\alpha},\qquad 0\le t\le\ell,\quad u\ge0,\quad u+t\le\Theta .
\end{equation}
The same bound holds conditionally on $\mathcal F_\tau$, for every bounded stopping time $\tau$, with $y$ replaced by $Z_\tau$ and with $\mathcal F_\tau$-measurable $u$ and $t$ in the same ranges.
\end{corollary}

\begin{proof}
We bound $\Lambda(\ell,\Theta)$ and apply \cref{lem:proc-backward-moments} with $k=2$.
Let $0\le s\le w\le\Theta$ and $y\in[0,\infty)^N\setminus\{0\}$.
The ratio $\mathcal H_y(s)/\mathcal H_y(w)$ is at most $\max_ih_i(s)/h_i(w)$, because $y\ge0$.
The function $x\mapsto(1+xw)/(1+xs)$ is nondecreasing, and for $r_i\le1$ it is at most $(1+w)/(1+s)$.
For $1\le r_i\le C_0$, we have $(1+r_iw)/(1+r_is)\le r_i(1+w)/(1+s)$.
Thus
\[
\frac{1+r_iw}{1+r_is}\le\max(1,C_0)\,\frac{1+w}{1+s},
\]
and by \eqref{cl2-direct-profile}, we have
\[
\frac{h_i(s)}{h_i(w)}\le\frac{C_*}{c_*}\sqrt{\frac{1+s}{1+w}}\cdot\frac{1+r_iw}{1+r_is}\le\frac{C_*}{c_*}\max(1,C_0)\sqrt{\frac{1+w}{1+s}} .
\]
Together with $m(w-s)\le C_*\sqrt{1+w}$, this yields
\begin{equation}\label{cl2-direct-heat-ratio}
m(w-s)\,\frac{\mathcal H_y(s)}{\mathcal H_y(w)}\le C_1\,\frac{1+w}{\sqrt{1+s}},\qquad C_1=\frac{C_*^2}{c_*}\max(1,C_0).
\end{equation}
Since $\int_0^t(1+s)^{-1/2}\,ds\le2\sqrt t$, we have $\Lambda(\ell,\Theta)\le2C_1(1+\Theta)\sqrt\ell$.
With $k=2$, the quantity \eqref{eq:proc-alpha} is at most $(2E_*+8)\cdot2C_1(1+\Theta)\sqrt\ell/N$.
Thus \cref{lem:proc-backward-moments} applies with $C=2(2E_*+8)C_1$ and gives \eqref{cl2-horizon-second-bound} and its conditional form.
\end{proof}

\begin{corollary}[Second moments on short intervals in the critical window]\label{cl2-direct-second}
Fix $K>0$, and suppose that the hypotheses of \cref{cl2-horizon-second} hold with $\Theta=KN^{2/3}$ and with constants $c_*$, $C_*$, $C_0$, $E_*$.
For every $\eta>0$, there exists $\delta>0$, depending only on $\eta$, $K$, $c_*$, $C_*$, $C_0$, and $E_*$, such that every deterministic nonzero initial state $Z(0)=y$ satisfies
\begin{equation}\label{cl2-direct-short-second}
\E\big[(h(u)^\T Z_t)^2\big]\le(1+\eta)\,[h(u+t)^\T y]^2,\qquad 0\le t\le\delta N^{2/3},\quad u\ge0,\quad u+t\le KN^{2/3}.
\end{equation}
The same bound holds conditionally after every bounded stopping time, as in \cref{cl2-horizon-second}.
\end{corollary}

\begin{proof}
Let $C$ be the constant of \cref{cl2-horizon-second}, and choose
\[
\delta=\min\Big\{K,\ \Big[\frac{\eta}{(1+\eta)\,C\,(K+1)}\Big]^2\Big\}.
\]
Let $\ell=\delta N^{2/3}\le\Theta$ in \cref{cl2-horizon-second}.
Since $N\ge1$, we have $1+KN^{2/3}\le(1+K)N^{2/3}$, and therefore
\[
\alpha=\frac{C(1+KN^{2/3})\sqrt{\delta N^{2/3}}}N\le C(1+K)\sqrt\delta\le\frac\eta{1+\eta}.
\]
Then $(1-\alpha)^{-1}\le1+\eta$, and \eqref{cl2-horizon-second-bound} implies \eqref{cl2-direct-short-second}.
\end{proof}

For the rest of this section, $Y$ is the unnormalized process with coefficients $r$, so that by \cref{rem:proc-choices} its direction is the diffusion generated by $L_\rho$ with $\rho=r$, and its squares satisfy \eqref{cl2-direct-lift} with $\epsilon\equiv0$.
We see from \eqref{eq:proc-critical-heat} that the hypotheses of \cref{cl2-direct-second} hold with $\Theta=KN^{2/3}$, $E_*=0$, and constants $c_*,C_*$ depending only on $c_0$, $C_0$, and $K$.

\begin{lemma}[A martingale with small conditional variance]\label{cd-adjoint-variance}
Assume \eqref{eq:proc-critical-weyl}, fix $K<\infty$ and a terminal time $T\le KN^{2/3}$, and let $\mathsf M_s=h(T-s)^\T Z(s)$ for $0\le s\le T$, where $Y$ has a deterministic nonzero initial value.
Then $\mathsf M$ is a positive continuous square-integrable martingale with $\mathsf M_T=\mathcal A_T$.
For every $\epsilon_0>0$ there exists $\delta>0$, depending only on $c_0$, $C_0$, $K$, and $\epsilon_0$, such that for every interval $[s_1,s_2]\subset[0,T]$ of length at most $\delta N^{2/3}$ and every stopping time $\tau$ with $s_1\le\tau\le s_2$,
\begin{equation}\label{cd-cond-var}
\E\big[(\mathsf M_{s_2}-\mathsf M_\tau)^2\,\big|\,\mathcal F_\tau\big]\le\epsilon_0\,\mathsf M_\tau^2 .
\end{equation}
This holds uniformly over all deterministic nonzero initial values.
\end{lemma}

\begin{proof}
The process $\mathsf M$ is a positive continuous local martingale, by Step~2 of the proof of \cref{lem:proc-backward-moments} with $\epsilon\equiv0$.
Its quadratic variation satisfies $[\mathsf M]_T\le C\int_0^T\mathcal A_s^2\,ds$, with $C=(8/N)\max_{v\le T}m(v)^2$, and the right side is integrable by \eqref{eq:proc-crude}. Thus $\mathsf M$ is a square-integrable martingale.
We have $\mathsf M_T=\mathcal A_T$, because $h(0)=\ind$.
Let $\delta$ be given by \cref{cl2-direct-second} with $\eta=\epsilon_0$ and $E_*=0$.
For a stopping time $\tau\in[s_1,s_2]$, we apply the conditional form of \cref{cl2-direct-second} with $u=T-s_2$ and $t=s_2-\tau$, which is $\mathcal F_\tau$-measurable and at most $\delta N^{2/3}$; note that $u+t=T-\tau\le KN^{2/3}$.
It gives
\[
\E\big[\mathsf M_{s_2}^2\,\big|\,\mathcal F_\tau\big]=\E\big[(h(T-s_2)^\T Z_{\tau+t})^2\,\big|\,\mathcal F_\tau\big]\le(1+\epsilon_0)\,\big(h(T-\tau)^\T Z_\tau\big)^2=(1+\epsilon_0)\,\mathsf M_\tau^2 .
\]
By optional sampling, $\E[\mathsf M_{s_2}\mid\mathcal F_\tau]=\mathsf M_\tau$, and \eqref{cd-cond-var} follows.
\end{proof}

\begin{lemma}[Two-sided moments in the critical window]\label{cd-lift-moments}
Assume \eqref{eq:proc-critical-weyl}, and fix $p>0$ and $K<\infty$.
There exists $C_{K,p}$, depending only on $c_0$, $C_0$, $K$, and $p$, such that the following holds for $0\le T\le KN^{2/3}$.
\begin{enumerate}
\item For every deterministic initial value $Y(0)=y\ne0$, the two inequalities
\begin{equation}\label{cd-anisotropic-moments}
\E\,\mathcal A_T^{\,p}\le C_{K,p}\Big(\sum_ih_i(T)y_i^2\Big)^{p},\qquad
\E\,\mathcal A_T^{-p}\le C_{K,p}\Big(\sum_ih_i(T)y_i^2\Big)^{-p}
\end{equation}
hold.
\item Let $Y(0)=g$ be a standard Gaussian vector independent of $B$.
There exists $N_0$, depending only on $c_0$, $C_0$, $K$, and $p$, such that for $N\ge N_0$,
\begin{equation}\label{cd-two-sided-gaussian}
\E\Big[\Big(\frac{\mathcal A_T}{N\bar h(T)}\Big)^p\Big]+\E\Big[\Big(\frac{N\bar h(T)}{\mathcal A_T}\Big)^p\Big]\le C_{K,p},
\end{equation}
and, if in addition $p\ge1$,
\begin{equation}\label{cd-coordinate-moments}
\|Y_i(T)^2\|_p\le C_{K,p}\,h_i(T),\qquad 1\le i\le N .
\end{equation}
\end{enumerate}
\end{lemma}

In the notation of \cref{sec:heat}, $N\bar h(T)=\ind^\T h(T)$ is the total mass of the solution of the linear equation started from $\ind$; by \eqref{eq:proc-critical-heat}, it is of order $N/\sqrt{1+T}$.

\begin{proof}
For the first part, let $\epsilon_0>0$ satisfy $2^p\cdot4\epsilon_0\le1/2$, and let $\delta$ be given by \cref{cd-adjoint-variance}.
Partition $[0,T]$ into $J\le\lceil K/\delta\rceil$ intervals $[s_{j-1},s_j]$ of length at most $\delta N^{2/3}$.
On each interval, the martingale $(\mathsf M_s)_{s_{j-1}\le s\le s_j}$, with respect to $(\mathcal F_s)$, satisfies the hypothesis \eqref{pr-eq:conditional-relative-variance} of \cref{pr-positive-martingale} with terminal time $s_j$ and $\varepsilon=\epsilon_0$, by \eqref{cd-cond-var}.
Then \eqref{pr-eq:inverse-max} and \eqref{eq:proc-positive-max} at $\tau=s_{j-1}$ imply that
\[
\E\Big[\Big(\frac{\mathsf M_{s_{j-1}}}{\mathsf M_{s_j}}\Big)^p\,\Big|\,\mathcal F_{s_{j-1}}\Big]\le2^{p+1},\qquad
\E\Big[\Big(\frac{\mathsf M_{s_j}}{\mathsf M_{s_{j-1}}}\Big)^p\,\Big|\,\mathcal F_{s_{j-1}}\Big]\le2^{p+1},
\]
since $2^p\epsilon_0\le2^p\cdot4\epsilon_0\le1/2$.
Writing $\mathsf M_T/\mathsf M_0$ as a product of the $J$ ratios and taking iterated conditional expectations, we have $\E[(\mathsf M_T/\mathsf M_0)^{\pm p}]\le2^{(p+1)J}$.
Here $\mathsf M_0=h(T)^\T Z(0)=\sum_ih_i(T)y_i^2$ is deterministic and $\mathsf M_T=\mathcal A_T$, which proves \eqref{cd-anisotropic-moments} with $C_{K,p}=2^{(p+1)\lceil K/\delta\rceil}$.

For the second part, let $S=\sum_ih_i(T)g_i^2$, $h_{\max}=\max_ih_i(T)$, and $D=N\bar h(T)/(2h_{\max})$.
Conditionally on $g$, the process $Y$ is the solution started at $g$, by the first part of \cref{lem:proc-normalization}. Then \eqref{cd-anisotropic-moments} implies that $\E[\mathcal A_T^{\pm p}\mid g]\le C_{K,p}S^{\pm p}$.
It remains to bound $\E S^{\pm p}$ by a constant times $(N\bar h(T))^{\pm p}$.
For the positive moments, we have $\E S^p\le C_p(N\bar h(T))^p$ by Minkowski's inequality when $p\ge1$ and by Jensen's inequality when $p<1$, since $\E S=N\bar h(T)$.
For the inverse moments, note that the function $x\mapsto\log(1+x)$ is concave and vanishes at zero, which implies that $\log(1+2\lambda h_i(T))\ge(h_i(T)/h_{\max})\log(1+2\lambda h_{\max})$ for $\lambda\ge0$.
This gives
\[
\E e^{-\lambda S}=\prod_i\big(1+2\lambda h_i(T)\big)^{-1/2}\le(1+2\lambda h_{\max})^{-D}.
\]
Let $\kappa_p=\int_0^\infty x^{p-1}e^{-x}\,dx$.
Using $s^{-p}=\kappa_p^{-1}\int_0^\infty\lambda^{p-1}e^{-\lambda s}\,d\lambda$ and substituting $x=2\lambda h_{\max}$, we find
\[
\E S^{-p}\le\frac{(2h_{\max})^{-p}}{\kappa_p}\int_0^\infty x^{p-1}(1+x)^{-D}\,dx .
\]
If $D\ge2p+2$, then the last integral is at most $C_pD^{-p}$: on $[0,1]$ we use $(1+x)^{-D}\le e^{-Dx/2}$, and on $[1,\infty)$ we use $(1+x)^{-D}\le2^{-D/2}x^{-D/2}$.
We conclude that $\E S^{-p}\le C_p(2h_{\max}D)^{-p}=C_p(N\bar h(T))^{-p}$.
Recall from \eqref{eq:proc-critical-heat} that $N\bar h(T)\ge c_1N/\sqrt{1+T}$ and $h_{\max}\le C\sqrt{1+T}$, with $c_1>0$ and $C$ depending only on $c_0$, $C_0$, and $K$. It follows that $D\ge(c_1/2C)N/(1+T)\ge c_KN^{1/3}$, which is at least $2p+2$ for $N\ge N_0$.
This proves \eqref{cd-two-sided-gaussian}.

For \eqref{cd-coordinate-moments}, let first $p$ be an integer.
Then \eqref{cd-two-sided-gaussian} implies that $\|\mathcal A_s\|_p\le C_{K,p}N\bar h(s)$ for $0\le s\le T$.
Using \cref{lem:proc-coordinate-moments} with $k=p$ and $\|g_i^2\|_p\le C_p$, we obtain
\[
\|Z_i(T)\|_p\le C_pe^{-4r_iT}+\frac{2(2p-1)C_{K,p}}N\int_0^Te^{-4r_i(T-s)}\,N\bar h(s)\,ds\le C'_{K,p}\,h_i(T),
\]
where in the last step we compared with \eqref{eq:proc-coordinate-duhamel}.
For noninteger $p$, note that $\|\cdot\|_p\le\|\cdot\|_{\lceil p\rceil}$.
\end{proof}

\section{Above the critical temperature: moments and occupation}\label{sec:proc-hot}

In \cref{sec:proc-hot,sec:proc-initial,sec:proc-maximal}, we fix $K\ge1$ and a sequence of precisions $0<p_1\le p_2\le\dots\le p_N$ that satisfies the hypothesis \eqref{hi3-indexed-geometry} of \cref{sec:heat}, that is,
\[
\begin{gathered}
0<d\le1,\qquad p_1=d,\qquad \frac1K\Big(\frac{i-1}N\Big)^{2/3}\le p_i-d\le K\Big(\frac{i-1}N\Big)^{2/3}\quad(2\le i\le N),\\
\sum_{i=1}^N\frac1{p_i}=N,\qquad Nd^{3/2}\ge1 .
\end{gathered}
\]
We write $q_i=1/p_i$ and $M=Nd^{3/2}$.
For a real symmetric matrix $A$ with eigenvalues $\alpha_1\ge\dots\ge\alpha_N$ and Lagrange multiplier $z$, the precisions $p_i=z-\alpha_i$ are nondecreasing in $i$; for a general sequence, the increasing rearrangement satisfies \eqref{hi3-indexed-geometry} with the same $K$ (\cref{rem:heat-hot-hypothesis}).
For the model with $c_N=1-\delta_N$, $\delta_N\to0$, and $N\delta_N^3\to\infty$, and for every $\eta\in(0,1)$, there exist $K=K_\eta$ and events $\mathcal E_N$ with $\liminf_{N\to\infty}\PP(\mathcal E_N)\ge1-\eta$ on which the precisions satisfy \eqref{hi3-indexed-geometry}; the probability is over the disorder, and $K_\eta$ is fixed before $N\to\infty$. This is checked in \cref{sec:hot-meso}.
Note that $p_N\le d+K\le1+K$ and $q_i\in[(1+K)^{-1},d^{-1}]$.

Throughout these three sections, $Y$ is the unnormalized process with coefficients $\rho_i=p_i/2$, which solves \eqref{cd-lift-equation}.
Its direction $X$ is the diffusion generated by $L_A$ when $p_i=z-\alpha_i$ (\cref{rem:proc-choices}).
The heat matrix \eqref{eq:proc-heat} is $H=-2\diag(p)+(2/N)\ind\ind^\T$, as in \eqref{eq:heat-objects}. We use the notation $h$, $\bar h$, $m$, $\mathcal H_y$ of \eqref{eq:proc-heat}.
As in \cref{sec:heat,sec:inequalities}, we set
\begin{equation}\label{eq:proc-F}
\mathsf a(t)=\sqrt d+(1+t)^{-1/2},\qquad \vartheta=\frac{N}{\sum_iq_i^2},\qquad \varphi(x)=\frac1N\sum_iq_ix_i^2,
\end{equation}
so that $\varphi(x)=x^\T(zI-A)^{-1}x/N$ in eigencoordinates of $A$.
Along the unnormalized process, we write
\begin{equation}\label{eq:proc-W}
\mathcal W_t=\sum_iq_iZ_i(t),\qquad V_2(t)=\sum_iq_i^2Z_i(t),\qquad \varphi(X_t)=\frac{\mathcal W_t}{\mathcal A_t}.
\end{equation}
The last identity holds since $X_i(t)^2=NZ_i(t)/\mathcal A_t$.
Since $q_i\le1/d$ and $1=p_iq_i\le p_Nq_i$, we have
\begin{equation}\label{eq:proc-W-bounds}
\mathcal A_t\le p_N\mathcal W_t,\qquad V_2(t)\le d^{-1}\mathcal W_t .
\end{equation}

We use the following properties of the heat vector.
Recall from \cref{lem:heat-algebra} that $H$ is symmetric and negative semidefinite, its kernel is spanned by $q$, the function $\bar h$ decreases to $\vartheta$, and $\sum_iq_ih_i(t)=N$.
The vector $h(t)$ converges to the orthogonal projection of $\ind$ onto the kernel, which is $(q^\T\ind/|q|^2)\,q=\vartheta q$, and by self-adjointness, $\ind^\T e^{tH}e^{uH}\ind=\ind^\T e^{(t+u)H}\ind$.
We obtain
\begin{equation}\label{eq:proc-hot-identities}
\lim_{t\to\infty}h(t)=\vartheta q,\qquad \bar h(t)\ge\vartheta,\qquad \sum_ih_i(t)h_i(u)=N\bar h(t+u),\qquad \sum_iq_ih_i(t)=N .
\end{equation}
We see from \eqref{eq:heat-duhamel} that $h_i(t)$ is a nonincreasing function of $p_i$; in particular, $h_1(t)\ge h_2(t)\ge\dots\ge h_N(t)$.
By \cref{iw-all-time-heat}, there exists $C_{\mathrm h}\ge1$, depending only on $K$, such that for all $t\ge0$,
\begin{equation}\label{eq:proc-hot-heat}
\begin{gathered}
C_{\mathrm h}^{-1}\mathsf a(t)\le \bar h(t)\le C_{\mathrm h}\mathsf a(t),\qquad
C_{\mathrm h}^{-1}\mathsf a(t)\frac{1+t}{1+p_it}\le h_i(t)\le C_{\mathrm h}\mathsf a(t)\frac{1+t}{1+p_it},\qquad
m(t)\le\frac{C_{\mathrm h}}{\sqrt d},
\end{gathered}
\end{equation}
and, by \eqref{iw-backward-ratio}, for every nonzero $y\in[0,\infty)^N$,
\begin{equation}\label{eq:proc-hot-ratio}
m(w-s)\,\frac{\mathcal H_y(s)}{\mathcal H_y(w)}\le C_{\mathrm h}\,\frac{\mathsf a(s)}d,\qquad 0\le s\le w .
\end{equation}

The statements below hold for all sufficiently large $M$, uniformly in $N$, $d$, the sequence $(p_i)$, and, in \cref{sec:proc-initial,sec:proc-maximal}, the time parameter $u\ge0$ of the initial law.
\Cref{tab:proc-parameters} lists the parameters and the order in which they are chosen.

\begin{table}[ht]
\centering
\caption{Parameters in \cref{sec:proc-hot,sec:proc-initial,sec:proc-maximal}, in the order in which they are fixed.}\label{tab:proc-parameters}
\begin{tabular}{@{}lll@{}}
\toprule
Parameter & Role & Depends on \\
\midrule
$K$ & constant in \eqref{hi3-indexed-geometry} & fixed first \\
$C_T$ & time range $[0,T_*]$, $T_*=C_Td^{-1}\log M$ & fixed \\
$k$, $r$, $\gamma$, $\lambda$ & moment orders and exponential tilt & fixed \\
$C$, $C_r$, \dots & constants in the conclusions & $K$, $C_T$, orders \\
$M_1$, $M_2$, \dots & lower bounds for $M=Nd^{3/2}$ & $K$, $C_T$, orders \\
$N$, $d$, $(p_i)$, $u$ & precisions, initial law & arbitrary, with $M$ large \\
\bottomrule
\end{tabular}
\end{table}

\begin{lemma}[Conditional moments on long time intervals]\label{iw-whole-horizon-positive}
Assume \eqref{hi3-indexed-geometry}.
For every integer $k\ge2$, let $C_k=8k(k-1)C_{\mathrm h}$ and
\begin{equation}\label{eq:proc-alpha-hot}
\alpha_k(T)=\frac{C_k}{Nd}\big(\sqrt T+\sqrt d\,T\big),\qquad T>0 .
\end{equation}
If $\alpha_k(T)<1$, then for every deterministic nonzero initial value $Y(0)$, with $y=Z(0)$,
\begin{equation}\label{iw-positive-bound}
\E\big[(h(u)^\T Z_t)^k\big]\le\frac{\mathcal H_y(u+t)^k}{1-\alpha_k(T)},\qquad 0\le t\le T,\quad u\ge0 .
\end{equation}
The same bound holds conditionally on $\mathcal F_\tau$, for every $\mathcal F_0$-measurable initial value and every bounded stopping time $\tau$, with $y$ replaced by $Z_\tau$ and with $\mathcal F_\tau$-measurable $u\ge0$ and $t\in[0,T]$.
If $\alpha_k(T)\le1/2$, then for every $\omega\in[0,\infty)^N$,
\begin{equation}\label{iw-positive-quadratic}
\E\big[(\omega^\T Z_t)^k\big]\le C'_k\,\big(\omega^\T e^{tH}y\big)^k,\qquad 0\le t\le T,\qquad C'_k=\big(2^{1/k}(2k-1)\big)^k .
\end{equation}
\end{lemma}

\begin{proof}
Using \eqref{eq:proc-hot-ratio}, we find that for every $\Theta\ge T$,
\[
\Lambda(T,\Theta)\le\frac{C_{\mathrm h}}d\int_0^T\mathsf a(s)\,ds\le\frac{C_{\mathrm h}}d\big(\sqrt d\,T+2\sqrt T\big),
\]
since $\int_0^T(1+s)^{-1/2}\,ds\le2\sqrt T$.
The squares satisfy \eqref{cl2-direct-lift} with $\epsilon\equiv0$, and the quantity \eqref{eq:proc-alpha} with $E_*=0$ and $\ell=T$ is at most $4k(k-1)\Lambda(T,\Theta)/N\le\alpha_k(T)$, for every $\Theta\ge T$.
Let $u\ge0$ and $t\in[0,T]$ be $\mathcal F_\tau$-measurable.
For $n\ge1$, \cref{lem:proc-backward-moments} with $\Theta=n+T$ applies to the pair $(u\wedge n,t)$, and on the $\mathcal F_\tau$-measurable event $\{u\le n\}$ it yields the conditional form of \eqref{iw-positive-bound}.
Letting $n\to\infty$ proves the conditional form, and the case $\tau=0$ is \eqref{iw-positive-bound}.

Now let $\alpha_k(T)\le1/2$.
Then \eqref{iw-positive-bound} with $u=0$ implies that $\|\mathcal A_s\|_k\le2^{1/k}\mathcal H_y(s)$ for $s\le T$.
Using \cref{lem:proc-coordinate-moments} with $\rho_i=p_i/2$, and then \eqref{eq:proc-duhamel-y}, we find
\[
\|Z_i(t)\|_k\le e^{-2p_it}y_i+\frac{2(2k-1)2^{1/k}}N\int_0^te^{-2p_i(t-s)}\,\mathcal H_y(s)\,ds\le2^{1/k}(2k-1)\,(e^{tH}y)_i .
\]
Since $\|\omega^\T Z_t\|_k\le\sum_i\omega_i\|Z_i(t)\|_k$ by Minkowski's inequality, \eqref{iw-positive-quadratic} follows.
\end{proof}

For $T\le C_Td^{-1}\log M$, we have $\sqrt T/(Nd)\le\sqrt{C_T\log M}/M$ and $\sqrt d\,T/(Nd)\le C_T\log M/M$, so
\begin{equation}\label{eq:proc-alpha-small}
\alpha_k(T)\le C_k\,\frac{\sqrt{C_T\log M}+C_T\log M}{M},\qquad T\le C_Td^{-1}\log M .
\end{equation}
This tends to zero as $M\to\infty$, for fixed $k$ and $C_T$.

The vector $q$ is a stationary solution of the linear equation, and weighting the squares by $q$ produces a martingale.

\begin{lemma}[The martingale $\mathcal W_t=\sum_iq_iZ_i(t)$]\label{lem:proc-zero-mode}
Assume \eqref{hi3-indexed-geometry}, and let $Y(0)$ be $\mathcal F_0$-measurable with $\E|Y(0)|^4<\infty$.
Then $\mathcal W$ is a strictly positive continuous square-integrable martingale on every bounded interval, with
\begin{equation}\label{eq:proc-W-bracket}
d\mathcal W_t=\sum_iq_i\,dM_i(t),\qquad d[\mathcal W]_t=\frac8N\,\mathcal A_tV_2(t)\,dt .
\end{equation}
Let $k\ge2$ be an integer and let $T>0$ satisfy $\alpha_k(T)<1$.
For every bounded stopping time $\tau$ and every $\mathcal F_\tau$-measurable $t\in[0,T]$,
\begin{equation}\label{iw-w-positive}
\E\big[\mathcal W_{\tau+t}^k\,\big|\,\mathcal F_\tau\big]\le\frac{\mathcal W_\tau^k}{1-\alpha_k(T)} .
\end{equation}
In particular, if $\alpha_2(T)<1$, then for all stopping times $\tau\le D\le T$,
\begin{equation}\label{iw-w-conditional-variance}
\E\big[(\mathcal W_D-\mathcal W_\tau)^2\,\big|\,\mathcal F_\tau\big]\le\varepsilon_T\,\mathcal W_\tau^2,\qquad \varepsilon_T=\frac{\alpha_2(T)}{1-\alpha_2(T)} .
\end{equation}
\end{lemma}

\begin{proof}
Using \eqref{eq:proc-squares} and $Hq=0$, we find that the drift of $\mathcal W=q^\T Z$ is $q^\T HZ=(Hq)^\T Z=0$, and \eqref{eq:proc-W-bracket} holds because the $M_i$ have zero mutual covariation.
We have $\mathcal A_sV_2(s)\le d^{-2}\mathcal A_s^2$ by \eqref{eq:proc-W-bounds}, and the right side is integrable on bounded intervals by \eqref{eq:proc-crude}. Thus $\mathcal W$ is a square-integrable martingale; it is positive because $\mathcal W\ge\mathcal A/p_N>0$.
For \eqref{iw-w-positive}, apply the conditional form of \eqref{iw-positive-bound} with lag $u$ and let $u\to\infty$.
By \eqref{eq:proc-hot-identities}, we have $h(u)^\T Z_{\tau+t}\to\vartheta\mathcal W_{\tau+t}$ and $\mathcal H_{Z_\tau}(u+t)\to\vartheta\mathcal W_\tau$. The conditional Fatou lemma then implies \eqref{iw-w-positive}, after division by $\vartheta^k$.

For \eqref{iw-w-conditional-variance}, note that $D\le T\le\tau+T$. By optional sampling, $\mathcal W_D=\E[\mathcal W_{\tau+T}\mid\mathcal F_D]$, and by the conditional Jensen inequality and \eqref{iw-w-positive} with $k=2$ and $t=T$,
\[
\E\big[\mathcal W_D^2\,\big|\,\mathcal F_\tau\big]\le\E\big[\mathcal W_{\tau+T}^2\,\big|\,\mathcal F_\tau\big]\le\frac{\mathcal W_\tau^2}{1-\alpha_2(T)} .
\]
Since $\E[\mathcal W_D\mid\mathcal F_\tau]=\mathcal W_\tau$, the conditional variance is at most $(1-\alpha_2(T))^{-1}\mathcal W_\tau^2-\mathcal W_\tau^2=\varepsilon_T\mathcal W_\tau^2$.
\end{proof}

\begin{lemma}[Moments of the maximum and inverse minimum of $\mathcal W$]\label{iw-w-inverse}
In the setting of \cref{lem:proc-zero-mode}, let $p>0$, and suppose that $\alpha_2(T)<1$ and $2^p\cdot4\varepsilon_T\le1/2$.
Then for all stopping times $\tau\le D\le T$,
\begin{equation}\label{iw-inverse-bound}
\E\Big[\sup_{\tau\le s\le D}\Big(\frac{\mathcal W_\tau}{\mathcal W_s}\Big)^p\,\Big|\,\mathcal F_\tau\Big]\le2^{p+1}.
\end{equation}
In particular, if $\mathcal W_0=N$, then $\E[\sup_{s\le T}(N/\mathcal W_s)^p]\le2^{p+1}$.
Further, for every integer $k\ge2$ with $\alpha_k(T)<1$,
\begin{equation}\label{eq:proc-W-max}
\E\Big[\sup_{\tau\le s\le D}\Big(\frac{\mathcal W_s}{\mathcal W_\tau}\Big)^k\,\Big|\,\mathcal F_\tau\Big]\le\Big(\frac k{k-1}\Big)^k\frac1{1-\alpha_k(T)} .
\end{equation}
\end{lemma}

\begin{proof}
We apply \cref{pr-positive-martingale} on $[0,T]$ to the stopped martingale $\mathcal W^D_s=\mathcal W_{s\wedge D}$, with $\varepsilon=\varepsilon_T$.
To verify \eqref{pr-eq:conditional-relative-variance}, let $\sigma\le T$ be a stopping time and $G\in\mathcal F_\sigma$.
Then $G\cap\{\sigma<D\}\in\mathcal F_{\sigma\wedge D}$, and $\mathcal W^D_T-\mathcal W^D_\sigma$ vanishes outside $\{\sigma<D\}$.
By \eqref{iw-w-conditional-variance} at the stopping times $\sigma\wedge D\le D$, we obtain
\[
\E\big[\ind_G(\mathcal W_D-\mathcal W_{\sigma\wedge D})^2\big]=\E\big[\ind_{G\cap\{\sigma<D\}}(\mathcal W_D-\mathcal W_{\sigma\wedge D})^2\big]\le\varepsilon_T\,\E\big[\ind_G\,\mathcal W_{\sigma\wedge D}^2\big],
\]
which is \eqref{pr-eq:conditional-relative-variance} for $\mathcal W^D$.
Then \eqref{pr-eq:inverse-max} implies \eqref{iw-inverse-bound}, since $2^p/(1-2^p\cdot4\varepsilon_T)\le2^{p+1}$.
For \eqref{eq:proc-W-max}, Doob's $L^k$ inequality \cite[Theorem~II.1.7]{RevuzYor1999}, applied conditionally on $\mathcal F_\tau$ to the martingale $(\mathcal W_{(\tau+s)\wedge D})_{s\ge0}$, yields
\[
\E\Big[\sup_{\tau\le s\le D}\mathcal W_s^k\,\Big|\,\mathcal F_\tau\Big]\le\Big(\frac k{k-1}\Big)^k\E\big[\mathcal W_D^k\,\big|\,\mathcal F_\tau\big].
\]
As in the proof of \eqref{iw-w-conditional-variance}, optional sampling, the conditional Jensen inequality, and \eqref{iw-w-positive} give $\E[\mathcal W_D^k\mid\mathcal F_\tau]\le\E[\mathcal W_{\tau+T}^k\mid\mathcal F_\tau]\le\mathcal W_\tau^k/(1-\alpha_k(T))$.
\end{proof}

\begin{lemma}[Inverse moments of the squared norm]\label{iw-backward-inverse}
Assume \eqref{hi3-indexed-geometry}, let $p>0$, and suppose that $\alpha_2(T)<1$ and $2^p\cdot4\varepsilon_T\le1/2$, with $\varepsilon_T$ as in \eqref{iw-w-conditional-variance}.
Fix a deterministic $D\in(0,T]$ and $u\ge0$, and let $V_s=h(u+D-s)^\T Z_s$ for $0\le s\le D$.
For every deterministic nonzero initial value, with $y=Z(0)$, and every stopping time $\tau\le D$,
\[
\E\Big[\sup_{\tau\le s\le D}\Big(\frac{V_\tau}{V_s}\Big)^p\,\Big|\,\mathcal F_\tau\Big]\le2^{p+1}.
\]
In particular,
\begin{equation}\label{iw-terminal-norm-inverse}
\E\,\mathcal A_D^{-p}\le2^{p+1}\,\mathcal H_y(D)^{-p},\qquad 0<D\le T .
\end{equation}
If $Y(0)$ is random and independent of $B$, then $\E[\mathcal A_D^{-p}\mid Y(0)]\le2^{p+1}\mathcal H_{Z(0)}(D)^{-p}$.
\end{lemma}

\begin{proof}
The process $V$ is a positive continuous local martingale by Step~2 of the proof of \cref{lem:proc-backward-moments}, and it is square integrable, as in the proof of \cref{cd-adjoint-variance}.
Let $\sigma\le D$ be a stopping time.
By the conditional form of \eqref{iw-positive-bound} with $k=2$, lag $u$, and duration $D-\sigma$, which is $\mathcal F_\sigma$-measurable and at most $T$, we have
\[
\E\big[V_D^2\,\big|\,\mathcal F_\sigma\big]=\E\big[(h(u)^\T Z_{\sigma+(D-\sigma)})^2\,\big|\,\mathcal F_\sigma\big]\le\frac{\mathcal H_{Z_\sigma}(u+D-\sigma)^2}{1-\alpha_2(T)}=\frac{V_\sigma^2}{1-\alpha_2(T)} .
\]
Since $\E[V_D\mid\mathcal F_\sigma]=V_\sigma$, the hypothesis \eqref{pr-eq:conditional-relative-variance} holds on $[0,D]$ with $\varepsilon=\varepsilon_T$, and the first assertion follows from \eqref{pr-eq:inverse-max}.
For $u=0$, we have $V_D=h(0)^\T Z_D=\mathcal A_D$ and $V_0=\mathcal H_y(D)$, and the first assertion at $\tau=0$ implies \eqref{iw-terminal-norm-inverse}.
The last assertion follows from the first part of \cref{lem:proc-normalization}.
\end{proof}

We now turn to the occupation bound.
For $x\in S_N$, we compute the tangential gradient of $\varphi$ and find
\begin{equation}\label{eq:proc-gamma-logF}
\begin{gathered}
\Gamma(\log\varphi)(x)=\frac{|\nabla_{S_N}\varphi(x)|^2}{\varphi(x)^2}=\frac4N\bigg(\frac{N\sum_iq_i^2x_i^2}{\big(\sum_iq_ix_i^2\big)^2}-1\bigg),\\
\Gamma(\log\varphi)(X_t)=\frac4N\Big(\frac{\mathcal A_tV_2(t)}{\mathcal W_t^2}-1\Big).
\end{gathered}
\end{equation}
Indeed, the tangential gradient is $\nabla_{S_N}\varphi(x)=(2/N)\big((q_i-\varphi(x))x_i\big)_i$.
Using $\sum_iq_ix_i^2=N\varphi(x)$ and $|x|^2=N$, we find $|\nabla_{S_N}\varphi(x)|^2=(4/N^2)\big(\sum_iq_i^2x_i^2-N\varphi(x)^2\big)$, which is the first formula after division by $\varphi(x)^2$.
The second follows from $X_i(t)^2=NZ_i(t)/\mathcal A_t$.

\begin{theorem}[Exponential moments of $\int_0^T\Gamma(\log\varphi)(X_s)\,ds$]\label{iw-chemical-occupation}
Fix $C_T>0$ and $\lambda>0$.
There exist $M_1$, depending only on $K$, $C_T$, and $\lambda$, and $C$, depending only on $K$ and $C_T$, such that every sequence satisfying \eqref{hi3-indexed-geometry} with $M\ge M_1$ and every $T\le C_Td^{-1}\log M$ satisfy
\begin{equation}\label{iw-occupation-bound}
\sup_{Y(0)\ne0}\E\exp\Big[\lambda\int_0^T\Gamma(\log\varphi)(X_s)\,ds\Big]\le\frac1{1-3\sqrt2\,\lambda\sqrt{\varepsilon_T}}\le1+C\lambda\sqrt{\frac{\log M}M},
\end{equation}
where the supremum is over deterministic initial values.
The same bound holds for every initial value $Y(0)$ independent of $B$, and hence for the diffusion generated by $L_A$ with an arbitrary initial law.
\end{theorem}

\begin{proof}
Using \eqref{eq:proc-alpha-small}, we choose $M_1$ such that $M\ge M_1$ implies $\alpha_2(T)\le1/64$, $\varepsilon_T\le1/32$, and $3\sqrt2\,\lambda\sqrt{\varepsilon_T}\le1/2$.
Let $Y(0)\ne0$ be deterministic.
The quadratic variation of $\log\mathcal W$ is, by \eqref{eq:proc-W-bracket},
\[
Q_t=\int_0^t\frac{d[\mathcal W]_s}{\mathcal W_s^2}=\int_0^t\frac8N\,\frac{\mathcal A_sV_2(s)}{\mathcal W_s^2}\,ds,
\]
and by \eqref{eq:proc-W-bounds} its density is at most $8p_N/(Nd)$; in particular, $Q_T\le8p_NT/(Nd)$ deterministically.
The process $\mathcal W$ satisfies \eqref{pr-eq:conditional-relative-variance} on $[0,T]$ with $\varepsilon=\varepsilon_T$, by \cref{lem:proc-zero-mode}.
Then \eqref{pr-eq:conditional-bracket} and \eqref{pr-eq:occupation} apply with $K=6\sqrt{2\varepsilon_T}$, and we obtain
\begin{equation}\label{iw-bracket-conditional-mean}
\E[Q_T-Q_\tau\mid\mathcal F_\tau]\le6\sqrt{2\varepsilon_T},\qquad
\E e^{\lambda Q_T/2}\le\frac1{1-\lambda K/2}=\frac1{1-3\sqrt2\,\lambda\sqrt{\varepsilon_T}} .
\end{equation}
Using \eqref{eq:proc-gamma-logF}, we obtain the exact identity
\[
\int_0^T\Gamma(\log\varphi)(X_s)\,ds=\frac{Q_T}2-\frac{4T}N,
\]
and dropping the negative term proves the first inequality in \eqref{iw-occupation-bound}.
For the second, we use $(1-x)^{-1}\le1+2x$ for $x\le1/2$ and $\sqrt{\varepsilon_T}\le\sqrt{2\alpha_2(T)}\le C\sqrt{\log M/M}$, by \eqref{eq:proc-alpha-small}.
The bound does not depend on the deterministic initial value, and the first part of \cref{lem:proc-normalization} extends it to every initial value independent of $B$.
Every initial law of $X$ arises in this way, by the second part of that lemma; for instance, one may take $Y(0)=X(0)$.
\end{proof}

\section{Initial laws equal to Bingham comparison laws}\label{sec:proc-initial}

We keep the setting of \cref{sec:proc-hot}.
For $u\ge0$, let $v_i(u)=h_i(u)/\bar h(u)$, so that $\sum_iv_i(u)=N$, and let $\mu_u=\mu_{v(u)}$ be the Gaussian law with variances $v(u)$ conditioned on $S_N$ (\cref{def:cond-finite}).
By the second part of \cref{prop:cond-bingham}, $\mu_u$ is the Bingham comparison law at time $u$ of \cref{sec:heat}, and $\mu_0=\nu_N$.
We consider the unnormalized process with the initial value
\begin{equation}\label{eq:proc-hot-initial}
Y(0)=\frac{X(0)}{\sqrt{\varphi(X(0))}},\qquad X(0)\sim\mu_u\ \text{independent of }B .
\end{equation}
Then $\mathcal W_0=\sum_iq_iX_i(0)^2/\varphi(X(0))=N$, and $\sqrt N\,Y(0)/|Y(0)|=X(0)$ since $|X(0)|^2=N$.
We see from the second part of \cref{lem:proc-normalization} that $X$ is the diffusion generated by $L_A$ with initial law $\mu_u$.
We write
\begin{equation}\label{eq:proc-center}
\mathcal H(t)=\mathcal H_{Z(0)}(t)=\sum_ih_i(t)\,Y_i(0)^2 .
\end{equation}
The martingale with backward semigroup weights $s\mapsto h(t-s)^\T Z_s$, $0\le s\le t$, is a martingale (see the proof of \cref{iw-backward-inverse}) with terminal value $\mathcal A_t$, and we have $\E[\mathcal A_t\mid Y(0)]=\mathcal H(t)$; that is, $\mathcal H(t)$ is the conditional mean of the squared norm given the initial state.
The following lemma compares $\mathcal H(t)$ with its deterministic counterpart $N\bar h(u+t)$ simultaneously for all $t\ge0$, with one random constant.

\begin{lemma}[Uniform-in-time comparison of the initial quadratic forms]\label{iw-early-pin-simultaneous-centers}\label{hmr-all-pin-centers}
For every $r>0$ there exist $M_2$ and $C_r$, depending only on $r$ and $K$, such that the following holds for every sequence satisfying \eqref{hi3-indexed-geometry} with $M\ge M_2$ and every $u\ge0$.
With the initial value \eqref{eq:proc-hot-initial}, there exists a random variable $C_{\mathrm{init}}\ge1$, measurable with respect to $X(0)$, with $\E C_{\mathrm{init}}^r\le C_r$ and such that simultaneously for all $t\ge0$,
\begin{equation}\label{iw-early-pin-heat-center-bound}
C_{\mathrm{init}}^{-1}\le\frac{\mathcal H(t)}{N\bar h(u+t)}\le C_{\mathrm{init}} .
\end{equation}
\aliaslabel{hmr-all-pin-center-bound}{iw-early-pin-heat-center-bound}%
\end{lemma}

The point of the lemma is that one random constant serves for all future times.
The proof uses one comparison of the initial law with independent Gaussian variables on each of two sets of coordinates, together with Abel summation; it involves no union bound over times.

\begin{proof}
The proof has four steps.

\paragraph{Step 1: the weights carry no macroscopic fraction.}
Let $\zeta_i=X_i(0)/\sqrt{v_i(u)}$ be the standardized coordinates of $X(0)$.
For $t\ge0$, define the weight sequences
\[
w_i(t)=h_i(t)\,v_i(u),\qquad w_i^q=q_i\,v_i(u),\qquad 1\le i\le N .
\]
Each of them is a positive nonincreasing function of $i$, since $h_i(t)$, $v_i(u)$, and $q_i$ are.
Using \eqref{eq:proc-hot-identities}, we compute
\begin{equation}\label{eq:proc-row-sums}
\sum_iw_i(t)=\frac{N\bar h(u+t)}{\bar h(u)},\qquad\sum_iw_i^q=\frac N{\bar h(u)},\qquad\sum_iv_i(u)^2=\frac{N\bar h(2u)}{\bar h(u)^2} .
\end{equation}
By \eqref{eq:proc-hot-heat} with $p_1=d$, we have $h_1(t)\le C_{\mathrm h}\mathsf a(t)(1+t)/(1+dt)\le4C_{\mathrm h}\mathsf a(t)^{-1}$, where we used that $\mathsf a(t)^2(1+t)\le2(d(1+t)+1)\le4(1+dt)$ for $d\le1$.
Using $\mathsf a\ge\sqrt d$ and $\bar h\ge C_{\mathrm h}^{-1}\mathsf a$, we find
\[
\frac{w_1(t)}{\sum_iw_i(t)}=\frac{h_1(t)h_1(u)}{N\bar h(u+t)}\le\frac{C}{N\mathsf a(t)\mathsf a(u)\mathsf a(u+t)}\le\frac CM,\qquad
\frac{w^q_1}{\sum_iw^q_i}=\frac{h_1(u)}{Nd}\le\frac{C}{Nd\mathsf a(u)}\le\frac CM,
\]
and similarly $v_1(u)^2/\sum_iv_i(u)^2=h_1(u)^2/(N\bar h(2u))\le C/M$.
Fix $C_*\ge1$, depending only on $K$, such that each of these three fractions is at most $C_*/M$, for all $t,u\ge0$.
In the notation \eqref{gp-standardization} of \cref{sec:conditioning}, the last bound gives $\eta_{v(u)}^2\le C_*/(2M)$.

\paragraph{Step 2: two blocks of coordinates.}
Let $I_1$ and $I_2$ be the sets of odd and of even indices.
For a positive nonincreasing weight sequence $x$, the alternating sum $x_1-x_2+x_3-\cdots$ lies in $[0,x_1]$, so $\sum_{I_1}x$ and $\sum_{I_2}x$ both lie in $[\frac12(\sum_ix_i-x_1),\frac12(\sum_ix_i+x_1)]$.
Applied to the three weight sequences of Step~1, this shows that for $M\ge3C_*$, each block carries between $1/3$ and $2/3$ of the sum of each of them, and that the maximal entry of $w(t)$ or $w^q$ on a block is at most $3C_*/M$ times the sum of that weight sequence over the block.
In particular, the fraction $h_{I_b}=\sum_{i\in I_b}v_i(u)^2/\sum_iv_i(u)^2$ of the variance of the radial constraint carried by the block $I_b$ is at most $2/3$, for $b=1,2$.
By the second part of \cref{gp-growing-head} with $\epsilon=1/3$, for $M$ large enough that $\eta_{v(u)}\le\eta(1/3)$, the law of $(\zeta_i)_{i\in I_b}$ under $\mu_u$ has a density with respect to the standard Gaussian measure on $\R^{I_b}$ that is bounded by a universal constant $C_{1/3}$.

\paragraph{Step 3: prefix sums.}
Fix $b\in\{1,2\}$, enumerate $I_b$ in increasing order as $j_1<j_2<\cdots<j_n$, and let $\Sigma_k=\sum_{l\le k}\zeta_{j_l}^2$.
Let $m_0=\lfloor M/(24C_*)\rfloor$; since $M\le N$, we have $m_0\le n$.
We define
\[
\underline s_b=\min_{m_0\le k\le n}\frac{\Sigma_k}k,\qquad \overline s_b=\max_{m_0\le k\le n}\frac{\Sigma_k}k .
\]
Under independent standard Gaussian variables, $\Sigma_k$ has the chi-squared distribution with $k$ degrees of freedom, and Chernoff's bound yields $\PP(\Sigma_k\le kx)\le(xe^{1-x})^{k/2}\le(ex)^{k/2}$ for $x\le1$, and $\PP(\Sigma_k\ge kx)\le e^{-kx/4}2^{k/2}\le e^{-kx/8}$ for $x\ge8$.
Summing over $k\ge m_0$, we obtain
\[
\PP(\underline s_b<x)\le2(ex)^{m_0/2}\quad\Big(0<x\le\frac1{4e}\Big),\qquad
\PP(\overline s_b>x)\le2e^{-m_0x/8}\quad(x\ge8).
\]
Integrating these tails, we find that for every $r>0$ there exists $C_r$ such that $\E\underline s_b^{-r}+\E\overline s_b^{\,r}\le C_r$ whenever $m_0\ge2r+2$.
Using the density bound of Step~2, we find that the same bounds hold under $\mu_u$, with $C_r$ replaced by $C_{1/3}C_r$.

Let $x$ be one of the weight sequences $w(t)$ or $w^q$, restricted to $I_b$ and written as $x_{j_1}\ge\dots\ge x_{j_n}$, and set $x_{j_{n+1}}=0$.
Abel summation gives
\[
\sum_{l=1}^nx_{j_l}\zeta_{j_l}^2=\sum_{k=1}^n(x_{j_k}-x_{j_{k+1}})\,\Sigma_k .
\]
The coefficients $x_{j_k}-x_{j_{k+1}}$ are nonnegative. We also have $\sum_k(x_{j_k}-x_{j_{k+1}})k=\sum_lx_{j_l}$ and $\sum_{k<m_0}(x_{j_k}-x_{j_{k+1}})k\le m_0\max_lx_{j_l}$.
For the lower bound, we use $\Sigma_k\ge0$ for $k<m_0$ and $\Sigma_k\ge k\underline s_b$ for $k\ge m_0$, which gives at least $\underline s_b(\sum_lx_{j_l}-m_0\max_lx_{j_l})$.
For the upper bound, we use $\Sigma_k\le\Sigma_{m_0}\le m_0\overline s_b$ for $k<m_0$ and $\Sigma_k\le k\overline s_b$ for $k\ge m_0$, which gives at most $\overline s_b(\sum_lx_{j_l}+m_0\max_lx_{j_l})$.
Since $m_0\max_lx_{j_l}\le\frac{M}{24C_*}\cdot\frac{3C_*}M\sum_lx_{j_l}=\frac18\sum_lx_{j_l}$ by Step~2, we conclude that
\begin{equation}\label{eq:proc-abel}
\frac78\,\underline s_b\sum_{i\in I_b}x_i\le\sum_{i\in I_b}x_i\zeta_i^2\le\frac98\,\overline s_b\sum_{i\in I_b}x_i .
\end{equation}

\paragraph{Step 4: conclusion.}
Let $\mathcal S=1+\underline s_1^{-1}+\overline s_1+\overline s_2$.
Using the lower bound in \eqref{eq:proc-abel} on $I_1$, the upper bounds on both blocks, and the fact that $I_1$ carries at least a third of each weight sequence, we find that simultaneously for every weight sequence $x$ among $w(t)$, $t\ge0$, and $w^q$,
\[
\frac7{24}\,\mathcal S^{-1}\sum_ix_i\le\sum_ix_i\zeta_i^2\le\frac98\,\mathcal S\sum_ix_i .
\]
No independence between the two blocks is needed.
Since $X_i(0)^2=v_i(u)\zeta_i^2$ and $\varphi(X(0))=N^{-1}\sum_iw^q_i\zeta_i^2$, we have exactly
\[
\mathcal H(t)=\frac{\sum_ih_i(t)X_i(0)^2}{\varphi(X(0))}=N\,\frac{\sum_iw_i(t)\zeta_i^2}{\sum_iw^q_i\zeta_i^2},
\]
and by \eqref{eq:proc-row-sums}, $\sum_iw_i(t)/\sum_iw_i^q=\bar h(u+t)$.
This implies \eqref{iw-early-pin-heat-center-bound} with $C_{\mathrm{init}}=\frac{27}7\mathcal S^2$.
By Step~3 and the elementary inequality $(\sum_{l=1}^4x_l)^{2r}\le4^{2r}\sum_lx_l^{2r}$, we have $\E C_{\mathrm{init}}^r\le C_r$ for $M\ge M_2(r)$.
All constants depend only on $r$ and $K$, and not on $u$, $N$, or $d$.
\end{proof}

\begin{corollary}[Fixed-time moments for Bingham comparison initial laws]\label{iw-early-pin-actual-fixed-time}
For every $r>0$ and $C_T>0$, there exist $M_3$ and $C_r$, depending only on $r$, $C_T$, and $K$, such that the following holds for every sequence satisfying \eqref{hi3-indexed-geometry} with $M\ge M_3$, every $u\ge0$, and every $0\le t\le C_Td^{-1}\log M$.
With the initial value \eqref{eq:proc-hot-initial},
\begin{equation}\label{iw-source-fixed-radius-moments}
\E\Big(\frac{\mathcal A_t}{N\bar h(u+t)}\Big)^r+\E\Big(\frac{N\bar h(u+t)}{\mathcal A_t}\Big)^r\le C_r .
\end{equation}
For the diffusion $X$ with initial law $\mu_u$,
\begin{equation}\label{iw-source-fixed-chemical-moments}
\E\big[\big(\varphi(X_t)\bar h(u+t)\big)^r\big]+\E\big[\big(\varphi(X_t)\bar h(u+t)\big)^{-r}\big]\le C_r,\qquad
\E\big[\big(\vartheta\varphi(X_t)\big)^r\big]\le C_r .
\end{equation}
These are bounds at each fixed time, uniform in the time; a bound for the expectation of the supremum over $t$ is proved in \cref{hmr-main}.
\end{corollary}

\begin{proof}
Let $T=C_Td^{-1}\log M$.
By H\"older's inequality, it suffices to treat integer $r\ge2$.
Using \eqref{eq:proc-alpha-small}, we choose $M_3\ge M_2(2r)$ so that $\alpha_{2r}(T)\le1/2$ and $2^{2r}\cdot4\varepsilon_T\le1/2$; note that $\alpha_k(T)$ increases with $k$.
We condition on $Y(0)$, which is independent of $B$.
By the conditional form of \eqref{iw-positive-bound} at $\tau=0$ with $u=0$, and by the last assertion of \cref{iw-backward-inverse}, we have
\[
\E\big[\mathcal A_t^r\,\big|\,Y(0)\big]\le2\,\mathcal H(t)^r,\qquad\E\big[\mathcal A_t^{-r}\,\big|\,Y(0)\big]\le2^{r+1}\mathcal H(t)^{-r}.
\]
By \eqref{iw-early-pin-heat-center-bound}, $\mathcal H(t)^{\pm r}\le C_{\mathrm{init}}^r(N\bar h(u+t))^{\pm r}$, and taking expectations, we obtain \eqref{iw-source-fixed-radius-moments}.
Next, we have $\varphi(X_t)\bar h(u+t)=(\mathcal W_t/N)\cdot N\bar h(u+t)/\mathcal A_t$ by \eqref{eq:proc-W}.
Since $\mathcal W_0=N$, \cref{iw-w-inverse} and \eqref{iw-w-positive} bound $\E(\mathcal W_t/N)^{\pm2r}$, and the first bound in \eqref{iw-source-fixed-chemical-moments} follows from the Cauchy--Schwarz inequality and \eqref{iw-source-fixed-radius-moments} at order $2r$.
Finally, $\bar h(u+t)\ge\vartheta$ by \eqref{eq:proc-hot-identities}. Thus $\vartheta\varphi(X_t)\le\varphi(X_t)\bar h(u+t)$ pointwise, and the last bound follows.
\end{proof}

\section{The maximal inverse of the squared norm}\label{sec:proc-maximal}\label{hmr-section}

We keep the setting of \cref{sec:proc-initial}: the precisions satisfy \eqref{hi3-indexed-geometry}, $u\ge0$, and the initial value is \eqref{eq:proc-hot-initial}.
Fix $C_T>0$ and let $T_*=C_Td^{-1}\log M$.
In this section, constants depend only on $K$, $C_T$, and the indicated moment orders, and the lower bounds for $M$ depend on the same quantities.

\begin{theorem}[Maximal inverse moments of the squared norm]\label{hmr-main}
For every $r>0$, there exist $M_4$ and $C_r$, depending only on $r$, $C_T$, and $K$, such that for every sequence satisfying \eqref{hi3-indexed-geometry} with $M\ge M_4$ and every $u\ge0$,
\begin{equation}\label{hmr-maximal-inverse}
\E\Big[\sup_{0\le t\le T_*}\Big(\frac{\mathcal H(t)}{\mathcal A_t}\Big)^r\Big]\le C_r,\qquad
\E\Big[\sup_{0\le t\le T_*}\Big(\frac{N\bar h(u+t)}{\mathcal A_t}\Big)^r\Big]\le C_r .
\end{equation}
In particular, for the diffusion $X$ with initial law $\mu_u$,
\begin{equation}\label{hmr-source-cap}
\E\Big[\sup_{0\le t\le T_*}\big(\vartheta\varphi(X_t)\big)^r\Big]\le C_r .
\end{equation}
\end{theorem}

The proof occupies the rest of this section.
Its outline is as follows.
We write $\mathcal A_t=\mathcal H(t)+S_t$, where $S_t$ is a stochastic convolution, and bound the moments of the increments of $J_t=S_t/\mathcal H(t)$ on the blocks $[0,1]$ and $[L,2L]$, $1\le L\le T_*$ (\cref{hmr-block-holder}).
A chaining argument turns these into bounds for the H\"older norm of $J$ on each block.
Combined with the fixed-time inverse moments, this yields small-ball bounds for $\inf\mathcal A_t/\mathcal H(t)$ on each block (\cref{hmr-block-smallball}), with a factor that is summable over dyadic blocks.
A union bound over the blocks then proves \cref{hmr-main}.
Both the H\"older bound and the fixed-time inverse moments are needed.

\paragraph{Inputs.}
We collect the estimates of \cref{sec:proc-hot,sec:proc-initial} that are used below, on the doubled interval $[0,2T_*]$; doubling the interval only replaces $C_T$ by $2C_T$.
For every $r>0$, there exist $C_r$ and a lower bound for $M$ such that, for all $t\ge0$ in the first two lines and all $s,t\in[0,2T_*]$ in the last two,
\begin{align}
\bar h(t)&\asymp \mathsf a(t),\qquad h_i(t)\asymp \mathsf a(t)\,\frac{1+t}{1+p_it},\label{hmr-heat-input}\\
C_{\mathrm{init}}^{-1}&\le\frac{\mathcal H(t)}{N\bar h(u+t)}\le C_{\mathrm{init}},\label{hmr-center-input}\\
\E C_{\mathrm{init}}^r&\le C_r,\qquad\|\mathcal A_s\|_r\le C_rN\bar h(u+s),\qquad\Big\|\sup_{s\le2T_*}\frac{\mathcal W_s}N\Big\|_r\le C_r,\label{hmr-positive-input}\\
\E\Big[\Big(\frac{\mathcal H(t)}{\mathcal A_t}\Big)^r\Big]&\le C_r .\label{hmr-point-inverse-input}
\end{align}
Here \eqref{hmr-heat-input} is \eqref{eq:proc-hot-heat}, \eqref{hmr-center-input} and the first bound of \eqref{hmr-positive-input} are \cref{iw-early-pin-simultaneous-centers}, the second bound of \eqref{hmr-positive-input} is \eqref{iw-source-fixed-radius-moments}, and the third follows from \eqref{eq:proc-W-max} with $\tau=0$ and $\mathcal W_0=N$.
For \eqref{hmr-point-inverse-input}, we condition on $Y(0)$ and apply \eqref{iw-terminal-norm-inverse} at the deterministic initial state $Y(0)$; no Gaussian approximation of $\mathcal A_t$ is involved.
We also use the following bounds on the derivatives of $h$, which hold for all $t,z\ge0$ by \cref{hmr-heat-derivatives}:
\begin{equation}\label{eq:proc-heat-derivatives}
\begin{gathered}
\max_ip_ih_i(t)^2\le C(1+dt),\qquad
\max_i\sqrt{p_i}\,|h_i'(t)|\le\frac C{1+t},\\
\max_ip_i\big[h_i(t+z)-h_i(t)\big]^2\le C\log^2\Big(1+\frac z{1+t}\Big),
\end{gathered}
\end{equation}
and $|\mathcal H'(t)|\le C\mathcal H(t)/(1+t)$, which follows by summing $|h_i'(t)|\le Ch_i(t)/(1+t)$ against $Y_i(0)^2\ge0$.
Finally, we use two elementary facts about $\mathsf a$: for all $u,t\ge0$,
\begin{equation}\label{eq:proc-a-facts}
\int_0^t\mathsf a(u+v)\,dv\le2t\,\mathsf a(u+t),\qquad \mathsf a(u+v)\le\sqrt2\,\mathsf a(u+t)\quad\text{for }t/2\le v\le t .
\end{equation}
The first follows from $\int_0^t(1+u+v)^{-1/2}\,dv=2t/(\sqrt{1+u+t}+\sqrt{1+u})\le2t/\sqrt{1+u+t}$, and the second from $1+u+t\le2(1+u+v)$.

\paragraph{The stochastic convolution.}
For fixed $t$, the martingale with backward semigroup weights $s\mapsto h(t-s)^\T Z_s$, $0\le s\le t$, has initial value $\mathcal H(t)$ and terminal value $\mathcal A_t$.
It follows that
\begin{equation}\label{hmr-volterra}
\mathcal A_t=\mathcal H(t)+S_t,\qquad S_t=\sum_i\int_0^th_i(t-s)\,dM_i(s),\qquad J_t=\frac{S_t}{\mathcal H(t)},
\end{equation}
so that $\mathcal A_t/\mathcal H(t)=1+J_t$.
The random variable $\mathcal H(t)$ is $\mathcal F_0$-measurable.

\begin{lemma}[Increments of the normalized stochastic convolution]\label{hmr-block-holder}
For every $r\ge2$ there exist $C_r$ and a lower bound for $M$ such that the following holds.
Let $1\le L\le T_*$, and let $L\le s\le t\le2L$.
Then
\begin{equation}\label{hmr-block-increments}
\|J_L\|_r\le C_r\sigma_L,\qquad
\|J_t-J_s\|_r\le C_r\sigma_L\sqrt{\frac{t-s}L},\qquad
\sigma_L^2=\frac{L(1+dL)}{N\mathsf a(u+L)}\le\frac{dL+(dL)^2}M .
\end{equation}
On the initial block, for $0\le s\le t\le1$, we have $J_0=0$ and $\|J_t-J_s\|_r\le C_r\sigma_*\sqrt{t-s}$, with $\sigma_*^2=[N\mathsf a(u+1)]^{-1}\le d/M$.
After the linear change of time that maps the block to $[0,1]$, let $\mathcal K$ be the sum of the supremum of $|J|$ and the $1/4$-H\"older seminorm of $J$ on $[0,1]$.
For every $r>4$,
\begin{equation}\label{hmr-holder-moment}
\|\mathcal K\|_r\le C_r\sigma_L\quad\text{on }[L,2L],\qquad \|\mathcal K\|_r\le C_r\sigma_*\quad\text{on }[0,1].
\end{equation}
\end{lemma}

The restriction $L\le T_*$ is necessary: the moment inputs above hold only on $[0,2T_*]$, and the bounds \eqref{hmr-block-increments} fail for large $L$ at fixed $N$ and $d$.
Indeed, we have $\mathcal A_tV_2(t)\ge\mathcal W_t^2$ by the Cauchy--Schwarz inequality, and \eqref{eq:proc-W-bracket} implies that $\E\mathcal W_t^2\ge N^2e^{8t/N}$; thus $\|J_t\|_2$ grows exponentially in $t/N$. Moreover, since $\log\mathcal W$ has quadratic variation at least $8t/N$, the martingale $\mathcal W_t$ tends to zero almost surely, so that $\mathcal A_t/\mathcal H(t)\to0$.

\begin{proof}
We begin with a general bound.
Let $\omega_i(v)$ be deterministic functions, and let $\mathcal N_t=\sum_i\int_0^t\omega_i(v)\,dM_i(v)$.
Its quadratic variation is $\frac8N\int_0^t\mathcal A_v\sum_i\omega_i(v)^2Z_i(v)\,dv$, and $\sum_i\omega_i^2Z_i=\sum_i(p_i\omega_i^2)q_iZ_i\le(\max_ip_i\omega_i^2)\,\mathcal W$.
By the Burkholder--Davis--Gundy inequality \cite[Theorem~IV.4.1]{RevuzYor1999} and Minkowski's inequality in $L^{r/2}$, for $r\ge2$,
\begin{equation}\label{eq:proc-bdg}
\|\mathcal N_t\|_r^2\le\frac{C_r}N\int_0^t\max_ip_i\omega_i(v)^2\,\|\mathcal A_v\mathcal W_v\|_{r/2}\,dv\le C_rN\int_0^t\max_ip_i\omega_i(v)^2\,\mathsf a(u+v)\,dv,
\end{equation}
where in the last step we used $\|\mathcal A_v\mathcal W_v\|_{r/2}\le\|\mathcal A_v\|_r\|\mathcal W_v\|_r\le C_rN^2\mathsf a(u+v)$, by \eqref{hmr-heat-input} and \eqref{hmr-positive-input}.

\paragraph{Pointwise bound.}
With $\omega_i(v)=h_i(t-v)$, the first bound in \eqref{eq:proc-heat-derivatives} and \eqref{eq:proc-a-facts} give
\begin{equation}\label{hmr-noise-point}
\|S_t\|_r^2\le C_rN\int_0^t\mathsf a(u+v)\big(1+d(t-v)\big)\,dv\le C_rN\,\mathsf a(u+t)\,t(1+dt),\qquad 0\le t\le2T_* .
\end{equation}

\paragraph{Increments on $[L,2L]$.}
Let $L\le s\le t\le2L$ and $z=t-s$, so that $z\le L\le s$.
We split $S_t-S_s$ into an old part, the integral over $[0,s]$ with $\omega_i(v)=h_i(t-v)-h_i(s-v)$, and a fresh part, the integral over $[s,t]$ with $\omega_i(v)=h_i(t-v)$.
Using \eqref{eq:proc-bdg} and \eqref{eq:proc-heat-derivatives}, we find that the squared $L^r$ norm of $S_t-S_s$ is at most a constant times
\[
N\int_0^s\mathsf a(u+v)\log^2\Big(1+\frac z{1+s-v}\Big)\,dv+N\int_s^t\mathsf a(u+v)\big(1+d(t-v)\big)\,dv .
\]
We bound the three integrals separately.
For $v\le s/2$, the logarithm is at most $z/(1+s-v)\le2z/(1+s)$, and by \eqref{eq:proc-a-facts}, $\int_0^{s/2}\mathsf a(u+v)\,dv\le s\,\mathsf a(u+s/2)\le\sqrt2\,s\,\mathsf a(u+s)$; hence the integral over $[0,s/2]$ is at most $C\mathsf a(u+s)z^2/(1+s)\le C\mathsf a(u+s)z$, where we used $z\le s$.
For $s/2\le v\le s$, we use $\mathsf a(u+v)\le\sqrt2\mathsf a(u+s)$ and
\begin{equation}\label{eq:proc-log-kernel}
\int_0^\infty\log^2\Big(1+\frac z{1+w}\Big)\,dw\le4z .
\end{equation}
To prove \eqref{eq:proc-log-kernel}, write the logarithm as $\int_0^z(1+w+x)^{-1}\,dx$; Minkowski's inequality in $L^2(dw)$ bounds its norm by $\int_0^z(1+x)^{-1/2}\,dx\le2\sqrt z$.
For the fresh part, $\mathsf a(u+v)\le \mathsf a(u+s)$ and $\int_s^t(1+d(t-v))\,dv\le z+dz^2$.
Altogether,
\begin{equation}\label{hmr-noise-increment}
\|S_t-S_s\|_r^2\le C_rN\,\mathsf a(u+s)\,(z+dz^2) .
\end{equation}
Next, we divide by $\mathcal H$.
We write
\[
J_t-J_s=\frac{S_t-S_s}{\mathcal H(t)}+J_s\Big(\frac{\mathcal H(s)}{\mathcal H(t)}-1\Big).
\]
Using H\"older's inequality and \eqref{hmr-center-input}, we find $\|(S_t-S_s)/\mathcal H(t)\|_r\le\|C_{\mathrm{init}}\|_{2r}\|S_t-S_s\|_{2r}/(N\bar h(u+t))$.
On the block, $\mathsf a(u+L)/\sqrt2\le \mathsf a(u+t)\le \mathsf a(u+s)\le \mathsf a(u+L)$, and $z+dz^2\le z(1+dL)$.
Together with \eqref{hmr-noise-increment} and \eqref{hmr-heat-input}, this gives
\[
\Big\|\frac{S_t-S_s}{\mathcal H(t)}\Big\|_r\le C_r\Big(\frac{z(1+dL)}{N\mathsf a(u+L)}\Big)^{1/2}=C_r\sigma_L\sqrt{\frac zL} .
\]
In the same way, we find from \eqref{hmr-noise-point} that $\|J_s\|_r\le C_r(s(1+ds)/(N\mathsf a(u+s)))^{1/2}\le C_r\sigma_L$ for $s\in[L,2L]$, which contains the bound on $\|J_L\|_r$.
The bound $|\mathcal H'|\le C\mathcal H/(1+t)$ gives $|\log\mathcal H(t)-\log\mathcal H(s)|\le Cz/L\le C$, and hence $|\mathcal H(s)/\mathcal H(t)-1|\le C'z/L\le C'\sqrt{z/L}$.
This proves \eqref{hmr-block-increments}.
The bound on $\sigma_L^2$ follows from $\mathsf a(u+L)\ge\sqrt d$ and $N\sqrt d=M/d$.

\paragraph{The initial block.}
For $0\le s\le t\le1$ and $z=t-s$, the second bound in \eqref{eq:proc-heat-derivatives} implies that $p_i[h_i(t-v)-h_i(s-v)]^2\le Cz^2$, and the first gives $p_ih_i(t-v)^2\le C$.
For $v\le1$, we have $\mathsf a(u+v)\le \mathsf a(u)\le\sqrt2\mathsf a(u+1)$, and \eqref{eq:proc-bdg} yields $\|S_t-S_s\|_r^2\le C_rN\mathsf a(u+1)(z^2+z)$ and $\|S_t\|_r^2\le C_rN\mathsf a(u+1)$.
Dividing by $\mathcal H(t)\ge C_{\mathrm{init}}^{-1}N\bar h(u+t)\ge c_1C_{\mathrm{init}}^{-1}N\mathsf a(u+1)$, with $c_1>0$ depending only on $K$, as before, and using $|\mathcal H'|\le C\mathcal H$, we have $\|J_t-J_s\|_r\le C_r\sigma_*\sqrt z$.
Also $J_0=0$, since $S_0=0$, and $\sigma_*^2\le(N\sqrt d)^{-1}=d/M$.

\paragraph{H\"older norms.}
We use the dyadic proof of Kolmogorov's continuity criterion, keeping its moment bound; see \cite[Theorem~I.2.1 and its proof, pp.~26--28]{RevuzYor1999}.
Let $\tilde J$ be $J$ on one of the blocks after the linear change of time to $[0,1]$, and let $\sigma$ be the corresponding $\sigma_L$ or $\sigma_*$.
The bounds above give $\|\tilde J(\theta')-\tilde J(\theta)\|_r\le C_r\sigma|\theta'-\theta|^{1/2}$ and $\|\tilde J(0)\|_r\le C_r\sigma$.
For $n\ge0$, let $\Delta_n=\max_{0\le k<2^n}|\tilde J((k+1)2^{-n})-\tilde J(k2^{-n})|$.
Bounding the maximum by the $\ell^r$ sum of its $2^n$ terms, we find $\|\Delta_n\|_r\le C_r\sigma2^{n(1/r-1/2)}$. For $r>4$, it follows that
\[
\Big\|\sum_{n\ge0}2^{n/4}\Delta_n\Big\|_r\le C_r\sigma\sum_{n\ge0}2^{n(1/r-1/4)}\le C_r\sigma .
\]
Let $\theta<\theta'$ be dyadic points in $[0,1]$ with $2^{-n-1}<\theta'-\theta\le2^{-n}$.
Approximating $\theta$ and $\theta'$ by their dyadic expansions, as in the proof of Kolmogorov's continuity criterion, we have $|\tilde J(\theta')-\tilde J(\theta)|\le2\sum_{k\ge n}\Delta_k\le2^{5/4}(\theta'-\theta)^{1/4}\sum_{k\ge0}2^{k/4}\Delta_k$.
By continuity of $\tilde J$, this bound extends to all $\theta,\theta'\in[0,1]$, and the $1/4$-H\"older seminorm of $\tilde J$ is at most $2^{5/4}\sum_k2^{k/4}\Delta_k$.
The supremum of $|\tilde J|$ is at most $|\tilde J(0)|$ plus this seminorm, which proves \eqref{hmr-holder-moment}.
\end{proof}

\begin{lemma}[Small values of the normalized squared norm on a block]\label{hmr-block-smallball}
Let $1\le L\le T_*$, let $\iota_L=\inf_{L\le t\le2L}\mathcal A_t/\mathcal H(t)$, and let $\iota_*=\inf_{0\le t\le1}\mathcal A_t/\mathcal H(t)$.
For every $\gamma>0$ and $r>4$, there exist constants $C_\gamma$, $C_{\gamma,r}$ and a lower bound for $M$ such that for $0<x\le1/2$,
\begin{equation}\label{hmr-block-smallball-bound}
\PP(\iota_L\le x)\le C_\gamma x^\gamma,\qquad
\PP(\iota_L\le x)\le C_{\gamma,r}\,x^{\gamma/2}\sigma_L^{r/2},
\end{equation}
and the same bounds hold for $\iota_*$ with $\sigma_*$ in place of $\sigma_L$.
\end{lemma}

\begin{proof}
Let $\mathsf b_t=\mathcal A_t/\mathcal H(t)=1+J_t$ on the block, after the change of time to $[0,1]$.
For $L\le T_*$, we have $\sigma_L^2\le(C_T\log M+C_T^2\log^2M)/M$ by \eqref{hmr-block-increments}, and for $M$ large every $\sigma_L$ and $\sigma_*$ is at most one.
By \eqref{hmr-holder-moment}, every moment of order greater than $4$ of the $1/4$-H\"older seminorm of $\mathsf b$, which equals that of $J$, is bounded by a constant.
We choose deterministic grid points in $[0,1]$ with spacing at most $x^8$; there are at most $2x^{-8}$ of them.
If the seminorm is at most $x^{-1}$, every point of $[0,1]$ is within $x^{-1}(x^8)^{1/4}=x$ of the value at a grid point.
The event $\{\iota_L\le x\}$ is contained in the union of the event that the seminorm exceeds $x^{-1}$ and the events that $\mathsf b\le2x$ at one of the grid points.
By Markov's inequality, the first event has probability at most $C_{\gamma'}x^{\gamma'}$ for every $\gamma'>4$. We take $\gamma'=\max(\gamma,5)$.
By \eqref{hmr-point-inverse-input} with order $\gamma+9$, each grid event has probability at most $C(2x)^{\gamma+9}$, and the union over the grid contributes at most $2x^{-8}C(2x)^{\gamma+9}\le C'x^{\gamma}$.
This proves the first bound.
For the second, we have $\{\iota_L\le1/2\}\subset\{\sup|J|\ge1/2\}$, since $\mathsf b=1+J$, and \eqref{hmr-holder-moment} and Markov's inequality imply that $\PP(\iota_L\le1/2)\le2^r\E\mathcal K^r\le C_r\sigma_L^r$.
For $x\le1/2$, the probability $\PP(\iota_L\le x)$ is at most the minimum of $C_\gamma x^\gamma$ and $C_r\sigma_L^r$, which is at most their geometric mean.
The initial block is treated in the same way, with $\mathsf b_0=1$.
\end{proof}

\begin{proof}[Proof of \cref{hmr-main}]
We take $M$ large enough that $T_*\ge1$ and that all the preceding lemmas apply with the finitely many moment orders used below.
Cover $[0,T_*]$ by $[0,1]$ and the dyadic blocks $[2^j,2^{j+1}]$, $0\le j\le j_*$, where $2^{j_*}\le T_*<2^{j_*+1}$.
Every block lies in $[0,2T_*]$, and every $L=2^j$ satisfies $1\le L\le T_*$; thus \cref{hmr-block-holder,hmr-block-smallball} apply.
For $r>4$, the noise scales satisfy
\begin{equation}\label{hmr-summable-scales}
\sigma_*^{r/2}+\sum_{j=0}^{j_*}\sigma_{2^j}^{r/2}\le C_r\Big(\frac{1+(dT_*)^2}M\Big)^{r/4}\le C_r\Big(\frac{1+\log^2M}M\Big)^{r/4} .
\end{equation}
Indeed, by \eqref{hmr-block-increments}, $\sigma_{2^j}^{r/2}\le C_r[(d2^j)^{r/4}+(d2^j)^{r/2}]M^{-r/4}$, and the two geometric sums over $j\le j_*$ are bounded by constants times their last terms, which are at most $C_r(dT_*)^{r/4}$ and $C_r(dT_*)^{r/2}$; the initial term is at most $(d/M)^{r/4}$, and $dT_*=C_T\log M$.
In particular, the number of blocks does not enter, and no factor depending on $\log(1/d)$ appears.

Let $\iota=\inf_{0\le t\le T_*}\mathcal A_t/\mathcal H(t)$.
Fix $r'>0$, and apply the second bound of \eqref{hmr-block-smallball-bound} on every block with $\gamma=2r'+2$ and $r=8$.
By \eqref{hmr-summable-scales} and a union bound, for $0<x\le1/2$,
\[
\PP(\iota\le x)\le C\,x^{r'+1}\Big(\frac{1+\log^2M}M\Big)^2 .
\]
Integrating this tail bound, we find
\[
\E\iota^{-r'}\le2^{r'}+\int_2^\infty r'y^{r'-1}\,\PP(\iota\le1/y)\,dy\le2^{r'}+C\Big(\frac{1+\log^2M}M\Big)^2\le C_{r'} ,
\]
which is the first bound in \eqref{hmr-maximal-inverse}.
For the second, we have $\sup_{t\le T_*}N\bar h(u+t)/\mathcal A_t\le C_{\mathrm{init}}\,\iota^{-1}$ by \eqref{hmr-center-input}, and the claim follows from the Cauchy--Schwarz inequality, the first bound at order $2r'$, and \eqref{hmr-positive-input}.

Finally, by \eqref{eq:proc-W}, we have the exact identity
\[
\vartheta\varphi(X_t)=\frac{\mathcal W_t}N\cdot\frac{N\bar h(u+t)}{\mathcal H(t)}\cdot\frac{\mathcal H(t)}{\mathcal A_t}\cdot\frac\vartheta{\bar h(u+t)} .
\]
The last factor is at most one, since $\bar h(u+t)\ge\vartheta$ by \eqref{eq:proc-hot-identities}.
Hence $\sup_{t\le T_*}\vartheta\varphi(X_t)\le\big(\sup_{t\le T_*}\mathcal W_t/N\big)\,C_{\mathrm{init}}\,\iota^{-1}$, and H\"older's inequality with three factors, together with \eqref{hmr-positive-input} and the first bound in \eqref{hmr-maximal-inverse}, proves \eqref{hmr-source-cap}.
The laws and times in this argument are those of the diffusion with initial law $\mu_u$, for every $u\ge0$, including $u=0$, where $\mu_0=\nu_N$.
\end{proof}

\section{A Feynman--Kac bound for a weighted gradient}\label{sec:proc-fk}

Let $A$ be a real symmetric matrix with eigenvalues $\alpha_1\ge\dots\ge\alpha_N$, Lagrange multiplier $z$, and smallest precision $d=z-\alpha_1$ (\cref{def:model-multiplier}).
As in \cref{sec:inequalities}, let $\varphi(x)=x^\T(zI-A)^{-1}x/N$; in eigencoordinates, this is the quadratic form \eqref{eq:proc-F}.
We write $\Gamma(f_1,f_2)=\nabla_{S_N}f_1\cdot\nabla_{S_N}f_2$, $\Gamma(f)=\Gamma(f,f)$, and
\[
\Gamma_{\varphi,2}(f)=\frac12\Big\{L_A\big(\varphi\Gamma(f)\big)-2\varphi\,\Gamma(f,L_Af)\Big\}.
\]
This is a weighted version of the iterated carr\'e du champ; note that $\varphi\Gamma$ is not, in general, the carr\'e du champ of $L_A$.
Let $g=\nabla_{S_N}f$, let $a=\nabla_{S_N}\log\varphi$, and let $\nabla_S^2f$ be the tangential Hessian of $f$, as in \cref{sgl-ambient-bochner}.
By \eqref{sgl-canonical-weighted-identity}, for every smooth $f$ on $S_N$,
\begin{equation}\label{eq:proc-weighted-bochner}
\Gamma_{\varphi,2}(f)=\varphi\Big\{\|\nabla_S^2f\|_{\mathrm{HS}}^2+2\,\nabla_S^2f(a,g)+g^\T\Big[(zI-A)-\frac2NI\Big]g\Big\},
\end{equation}
where $\nabla_S^2f(a,g)=a^\T(\nabla_S^2f)g$.

\begin{lemma}[A Feynman--Kac bound for the weighted gradient]\label{hfk-pointwise-fk}
Let $\bar r=d-2/N$.
For every smooth function $f$ on $S_N$, every $x\in S_N$, and every $t\ge0$,
\begin{equation}\label{hfk-fk-gradient}
\varphi(x)\,\Gamma(P_t^Af)(x)\le e^{-2\bar rt}\,\E_x\Big[\exp\Big(2\int_0^t\Gamma(\log\varphi)(X_s)\,ds\Big)\varphi(X_t)\,\Gamma(f)(X_t)\Big],
\end{equation}
where $X$ is the diffusion generated by $L_A$ started at $x$.
\end{lemma}

The inequality holds whatever the sign of $\bar r$.
In \cref{sec:hot-meso}, the exponential factor in \eqref{hfk-fk-gradient} is controlled by \cref{iw-chemical-occupation}.

\begin{proof}
We begin with a pointwise lower bound for $\Gamma_{\varphi,2}$.
For a symmetric matrix $\mathsf H$ and vectors $b,v$, the Cauchy--Schwarz inequality gives $2|\mathsf H(b,v)|\le2\|\mathsf H\|_{\mathrm{HS}}|b||v|\le\|\mathsf H\|_{\mathrm{HS}}^2+|b|^2|v|^2$.
With $\mathsf H=\nabla_S^2f$, $b=a$, and $v=g$, we have $|b|^2=\Gamma(\log\varphi)$ and $|v|^2=\Gamma(f)$.
The eigenvalues of $zI-A$ are at least $d$, and by \eqref{eq:proc-weighted-bochner}, we obtain
\begin{equation}\label{eq:proc-gamma2-lower}
\Gamma_{\varphi,2}(f)\ge\big(\bar r-\Gamma(\log\varphi)\big)\,\varphi\,\Gamma(f).
\end{equation}

Since $L_A$ is an elliptic operator with smooth coefficients on the compact manifold $S_N$, the function $(t,x)\mapsto P^A_tf(x)$ is smooth on $[0,\infty)\times S_N$ for smooth $f$, and so is $\phi(t,x)=\varphi(x)\Gamma(P^A_tf)(x)$.
Using $\partial_tP^A_tf=L_AP^A_tf$, the definition of $\Gamma_{\varphi,2}$, and \eqref{eq:proc-gamma2-lower}, we find
\[
(\partial_t-L_A)\phi=2\varphi\,\Gamma(P^A_tf,L_AP^A_tf)-L_A\big(\varphi\Gamma(P^A_tf)\big)=-2\Gamma_{\varphi,2}(P^A_tf)\le\mathcal V\phi,
\]
where $\mathcal V=2\Gamma(\log\varphi)-2\bar r$ is smooth and bounded on $S_N$.
Fix $t>0$ and $x\in S_N$, let $X$ be the diffusion generated by $L_A$ started at $x$, and let
\[
\Upsilon_s=\exp\Big(\int_0^s\mathcal V(X_v)\,dv\Big)\,\phi(t-s,X_s),\qquad 0\le s\le t .
\]
By It\^o's formula for \eqref{eq:model-sde} with $cW_N$ replaced by $A$, whose generator is $L_A$ by the calculation in \cref{sec:model-model}, the drift of $\Upsilon$ is
\[
\exp\Big(\int_0^s\mathcal V(X_v)\,dv\Big)\big(\mathcal V\phi-\partial_t\phi+L_A\phi\big)(t-s,X_s)\ge0,
\]
and its martingale part is a stochastic integral with a bounded integrand.
Thus $\Upsilon$ is a submartingale, and $\phi(t,x)=\Upsilon_0\le\E\Upsilon_t$, which is \eqref{hfk-fk-gradient}.
This is the Feynman--Kac argument; see \cite[Proposition~VIII.3.10, pp.~358--359]{RevuzYor1999} for the generator identity underlying it, after a constant shift of the bounded potential.
\end{proof}