j$) are white noises with variance $\upsilon^{2}>0$, and we recall the
notations $\xi^{\varepsilon,\zeta}$ and
$\widehat{H}^{\varepsilon,\zeta}=H+\xi^{\varepsilon,\zeta}$ introduced in
Definition 5.12 and (5.22).

###  8.1. Scalar Real-Valued Reduction

The results proved in this section can all be reduced to similar results from
[26] in the scalar setting with $\mathbb{F}=\mathbb{R}$. This is because, by
linearity/bilinearity, the form $\xi^{\varepsilon,\zeta}(f,g)$ evaluated in
any functions $f,g:\mathcal{A}\to\mathbb{F}$ can be written as a linear
combination of white noises in $\mathbb{R}$ evaluated in functions from $I$ to
$\mathbb{R}$. In this section, we make this observation more precise.

To see this, we first recall that we can write

(8.1)
$\displaystyle\xi^{\varepsilon,\zeta}(f,g)=\sum_{i=1}^{r}\xi^{\varepsilon}_{i,i}\big{(}f(i,\cdot),g(i,\cdot)\big{)}+\sum_{i\neq
j}\xi^{\zeta}_{i,j}\big{(}f(i,\cdot),g(j,\cdot)\big{)},\qquad\varepsilon,\zeta\geq
0;$

moreover, letting $(W_{i,j})_{i\leq i<j\leq r}$ be the Brownian motions
introduced in Definition 5.12, we have that

(8.2)
$\displaystyle\xi^{\varepsilon}_{i,i}\big{(}f(i,\cdot),g(i,\cdot)\big{)}=\sigma(W_{i,i}\star\overline{\varrho}_{\varepsilon})^{\prime}\big{(}f(i,\cdot),g(i,\cdot)\big{)}\qquad\text{for
all }1\leq i\leq r,$

and

(8.3)
$\displaystyle\xi^{\zeta}_{i,j}\big{(}f(i,\cdot),g(j,\cdot)\big{)}=\upsilon(W_{i,j}\star\overline{\varrho}_{\zeta})^{\prime}\big{(}f(i,\cdot),g(j,\cdot)\big{)}\qquad\text{for
all }1\leq i<j\leq r.$

We note that $W_{i,i}$ are already standard Brownian motions in $\mathbb{R}$;
for $1\leq i<j\leq r$, we recall from Definition 5.12 that we can expand (8.3)
as

(8.4)
$(W_{i,j}\star\overline{\varrho}_{\zeta})^{\prime}\big{(}f(i,\cdot),g(j,\cdot)\big{)}\\\
=\begin{cases}(W_{i,j;1}\star\overline{\varrho}_{\zeta})^{\prime}\big{(}f(i,\cdot),g(j,\cdot)\big{)}&\text{if
}\mathbb{F}=\mathbb{R},\\\
\frac{1}{\sqrt{2}}\Big{(}(W_{i,j;1}\star\overline{\varrho}_{\zeta})^{\prime}\big{(}f(i,\cdot),g(j,\cdot)\big{)}+(W_{i,j;\mathsf{i}}\star\overline{\varrho}_{\zeta})^{\prime}\big{(}f(i,\cdot),\mathsf{i}g(j,\cdot)\big{)}\Big{)}&\text{if
}\mathbb{F}=\mathbb{C},\\\
\frac{1}{2}\Big{(}\sum_{\ell\in\\{1,\mathsf{i},\mathsf{j},\mathsf{k}\\}}(W_{i,j;\ell}\star\overline{\varrho}_{\zeta})^{\prime}\big{(}f(i,\cdot),\ell
g(j,\cdot)\big{)}\Big{)}&\text{if }\mathbb{F}=\mathbb{H},\end{cases}$

where $W_{i,j;\ell}$ are i.i.d. standard Brownian motions in $\mathbb{R}$.

(8.2) and (8.4) show that $\xi^{\varepsilon,\zeta}(f,g)$ can always be written
as a linear combination of white noises in $\mathbb{R}$ evaluated in scalar-
valued functions from $I$ to $\mathbb{F}$. We now finish this introductory
section by showing that we need only consider functions from $I$ to
$\mathbb{R}$. When $\mathbb{F}=\mathbb{R}$, this observation is of course
entirely vacuous, as $f(i,x),g(j,x)\in\mathbb{R}$ for all $x\in I$. Otherwise,
pick an arbitrary basis element

$\ell\in\begin{cases}\\{1,\mathsf{i}\\}&\text{if }\mathbb{F}=\mathbb{C},\\\
\\{1,\mathsf{i},\mathsf{j},\mathsf{k}\\}&\text{if
}\mathbb{F}=\mathbb{H};\end{cases}$

define $h=\ell g$; and finally expand the components of $f$ as

$f(i,x)=\begin{cases}f_{1}(i,x)+f_{2}(i,x)\mathsf{i}&\text{if
}\mathbb{F}=\mathbb{C}\\\
f_{1}(i,x)+f_{2}(i,x)\mathsf{i}+f_{3}(i,x)\mathsf{j}+f_{4}(i,x)\mathsf{k}&\text{if
}\mathbb{F}=\mathbb{H}\end{cases}$

and similarly for $\big{(}h_{m}(i,\cdot)\big{)}_{m\geq 1}$. To simplify the
presentation, we use the notational shorthand
$\mathcal{N}_{\varepsilon}=(W\star\overline{\varrho}_{\varepsilon})^{\prime}$.
Then, by bilinearity and the complex product, one has

(8.5)
$\mathcal{N}_{\varepsilon}\big{(}f(i,\cdot),h(j,\cdot)\big{)}=\mathcal{N}_{\varepsilon}\big{(}f_{1}(i,\cdot),h_{1}(j,\cdot)\big{)}-\mathcal{N}_{\varepsilon}\big{(}f_{2}(i,\cdot),h_{2}(j,\cdot)\big{)}\\\
+\Big{(}\mathcal{N}_{\varepsilon}\big{(}f_{1}(i,\cdot),h_{2}(j,\cdot)\big{)}+\mathcal{N}_{\varepsilon}\big{(}f_{2}(i,\cdot),h_{1}(j,\cdot)\big{)}\Big{)}\mathsf{i}.$

when $\mathbb{F}=\mathbb{C}$, and by bilinearity and the definition of
quaternion product,

(8.6)
$\displaystyle\mathcal{N}_{\varepsilon}\big{(}f(i,\cdot),h(j,\cdot)\big{)}$
$\displaystyle=\mathcal{N}_{\varepsilon}\big{(}f_{1}(i,\cdot),h_{1}(j,\cdot)\big{)}-\mathcal{N}_{\varepsilon}\big{(}f_{2}(i,\cdot),h_{2}(j,\cdot)\big{)}-\mathcal{N}_{\varepsilon}\big{(}f_{3}(i,\cdot),h_{3}(j,\cdot)\big{)}-\mathcal{N}_{\varepsilon}\big{(}f_{2}(i,\cdot),h_{2}(j,\cdot)\big{)}$
$\displaystyle+\Big{(}\mathcal{N}_{\varepsilon}\big{(}f_{1}(i,\cdot),h_{2}(j,\cdot)\big{)}+\mathcal{N}_{\varepsilon}\big{(}f_{2}(i,\cdot),h_{1}(j,\cdot)\big{)}+\mathcal{N}_{\varepsilon}\big{(}f_{3}(i,\cdot),h_{4}(j,\cdot)\big{)}-\mathcal{N}_{\varepsilon}\big{(}f_{4}(i,\cdot),h_{3}(j,\cdot)\big{)}\Big{)}\mathsf{i}$
$\displaystyle+\Big{(}\mathcal{N}_{\varepsilon}\big{(}f_{1}(i,\cdot),h_{3}(j,\cdot)\big{)}-\mathcal{N}_{\varepsilon}\big{(}f_{2}(i,\cdot),h_{4}(j,\cdot)\big{)}+\mathcal{N}_{\varepsilon}\big{(}f_{3}(i,\cdot),h_{1}(j,\cdot)\big{)}+\mathcal{N}_{\varepsilon}\big{(}f_{4}(i,\cdot),h_{2}(j,\cdot)\big{)}\big{)}\mathsf{j}$
$\displaystyle+\Big{(}\mathcal{N}_{\varepsilon}\big{(}f_{1}(i,\cdot),h_{4}(j,\cdot)\big{)}+\mathcal{N}_{\varepsilon}\big{(}f_{2}(i,\cdot),h_{3}(j,\cdot)\big{)}-\mathcal{N}_{\varepsilon}\big{(}f_{3}(i,\cdot),h_{2}(j,\cdot)\big{)}+\mathcal{N}_{\varepsilon}\big{(}f_{4}(i,\cdot),h_{1}(j,\cdot)\big{)}\Big{)}\mathsf{k}$

when $\mathbb{F}=\mathbb{H}$, noting that $f_{m}(i,x),h_{m}(j,x)\in\mathbb{R}$
for all $x\in I$ and $m\geq 1$. We are now ready to embark on the proofs of
Propositions 2.12, 5.18, and 5.19.

###  8.2. Proof of Propositions 2.12 and 5.18

####  8.2.1. Proof of Proposition 2.12 Part 1: (2.8) and Equivalence

For every $1\leq i\leq r$, let us denote the norm

(8.7)
$\displaystyle\|f\|_{\star;i}^{2}=\|f^{\prime}\|_{2}^{2}+\|f\|_{2}^{2}+\|V_{+}(i,\cdot)^{1/2}f\|_{2}^{2},\qquad
L^{2}(I,\mathbb{F}).$

By [26, Proposition 3.2], there exists constants $C_{i}>0$ large enough so
that

$\|f\|_{+1;i}=\sqrt{\mathcal{E}_{i}(f,f)+(C_{i}+1)\|f\|_{2}}$

is a norm on $D(\mathcal{E}_{i})$, which is equivalent to
$\|f\|_{\star;i}^{2}$ (in [26, Proposition 3.2], this result is only stated
for real-valued functions, but it can be trivially extended to $\mathbb{C}$
and $\mathbb{F}$ by linearity and noting that $L^{2}(I,\mathbb{C})\cong
L^{2}(I,\mathbb{R})^{\oplus 2}$ and $L^{2}(I,\mathbb{H})\cong
L^{2}(I,\mathbb{R})^{\oplus 4}$, the latter with $\prec\cdot,\cdot\succ$). We
then obtain the claim regarding (2.8) in Proposition 2.12 by noting that, for
$f\in L^{2}(\mathcal{A},\mathbb{F})$, if we let $C=C_{1}+\cdots+C_{i}$, then

(8.8)
$\displaystyle\|f\|_{\star}^{2}=\sum_{i=1}^{r}\|f(i,\cdot)\|_{\star;i}^{2}\qquad\text{and}\qquad\|f\|_{+1}^{2}=\sum_{i=1}^{r}\|f(i,\cdot)\|_{+1;i}^{2}$

####  8.2.2. Proof of Proposition 2.12 Part 2: (2.9)

Suppose first that $\xi_{i,j}$ are white noises for all $1\leq i,j\leq r$. By
equivalence of $\|\cdot\|_{+1}$ and $\|\cdot\|_{\star}$, it is enough to show
that almost surely, for every $\kappa\in(0,1)$, there exists $\nu>0$ such that
$|\xi(f,f)|\leq\kappa\|f\|^{2}_{\star}+\nu\|f\|_{\mu}^{2}$ for every $f\in
C_{0}^{\infty}(\mathcal{A},\mathbb{F})$. Toward this end, by (8.2)–(8.4) (with
$\varepsilon=\zeta=0$) and the triangle inequality, it suffices to show the
following: Let $W$ be a standard Brownian motion in $\mathbb{R}$. For any
choice of $\kappa\in(0,1)$ and

$\ell\in\begin{cases}\\{1\\}&\text{if }\mathbb{F}=\mathbb{R}\\\
\\{1,\mathsf{i}\\}&\text{if }\mathbb{F}=\mathbb{C}\\\
\\{1,\mathsf{i},\mathsf{j},\mathsf{k}\\}&\text{if
}\mathbb{F}=\mathbb{H},\end{cases}$

there exists $\nu>0$ such that

(8.9) $\displaystyle\big{|}W^{\prime}\big{(}f(i,\cdot),\ell
f(j,\cdot)\big{)}\big{|}\leq\kappa\|f\|_{\star}^{2}+\nu\|f\|_{\mu}^{2}$

for every $f\in C_{0}^{\infty}(\mathcal{A},\mathbb{F})$ and $1\leq i,j\leq r$.

Suppose first that we are in Case 1 or Case 2. Consider the averaged process

$\tilde{W}(x)=\int_{x}^{x+1}W(x)~{}\mathrm{d}x,\qquad x\in\mathbb{R},$

and write $W=\tilde{W}+(W-\tilde{W})$. Then, by definition of $W^{\prime}$ in
(2.4) and an integration by parts, we have that

$W^{\prime}\big{(}f(i,\cdot),\ell
f(j,\cdot)\big{)}=f(i,0)^{*}\tilde{W}(0)\mathsf{\ell}f(j,0)+\big{\langle}f(i,\cdot),\tilde{W}^{\prime}\ell
f(j,\cdot)\big{\rangle}\\\ +\big{\langle}f^{\prime}(i,\cdot),(\tilde{W}-W)\ell
f(j,\cdot)\big{\rangle}+\big{\langle}f(i,\cdot),(\tilde{W}-W)\ell
f^{\prime}(j,\cdot)\big{\rangle}.$

By [26, Lemma 3.1], for every $\kappa^{\prime}>0$, there exists some
$\nu^{\prime}>0$ such that

(8.10)
$\displaystyle|f(i,x)|^{2}\leq\kappa^{\prime}\|f^{\prime}(i,\cdot)\|_{2}^{2}+\nu^{\prime}\|f(i,\cdot)\|_{2}^{2}\qquad\text{for
every }x\in I.$

Thus, since $\tilde{W}(0)$ is finite and $xy\leq x^{2}+y^{2}$ for $x,y\geq 0$,
the latter of which implies

(8.11) $\displaystyle|f(i,0)^{*}\tilde{W}(0)\ell
f(j,0)|\leq|f(i,0)|^{2}|\tilde{W}(0)|+|f(j,0)|^{2}|\tilde{W}(0)|,$

in order to prove (8.9), it suffices to show that almost surely, for every
$\kappa\in(0,1)$, there exists $\nu>0$ such that,

(8.12) $\Big{|}\big{\langle}f(i,\cdot),\tilde{W}^{\prime}\ell
f(j,\cdot)\big{\rangle}\Big{|}+\Big{|}\big{\langle}f^{\prime}(i,\cdot),(\tilde{W}-W)\ell
f(j,\cdot)\big{\rangle}\Big{|}+\Big{|}\big{\langle}f(i,\cdot),(\tilde{W}-W)\ell
f^{\prime}(j,\cdot)\big{\rangle}\Big{|}\\\
\leq\kappa\|f\|_{\star}^{2}+\nu\|f\|_{\mu}^{2}.$

For this purpose, by combining Jensen’s inequality with the fact that
$xy\leq\frac{\theta}{2}x^{2}+\frac{1}{2\theta}y^{2}$ for any $x,y\geq 0$ and
$\theta>0$,

(8.13) $\Big{|}\big{\langle}f(i,\cdot),\tilde{W}^{\prime}\ell
f(j,\cdot)\big{\rangle}\Big{|}\leq\int_{I}|f(i,x)||\tilde{W}^{\prime}(x)|^{1/2}|f(j,x)||\tilde{W}^{\prime}(x)|^{1/2}~{}\mathrm{d}x\\\
\leq\frac{1}{2}\int_{I}|f(i,x)|^{2}|\tilde{W}^{\prime}(x)|~{}\mathrm{d}x+\frac{1}{2}\int_{I}|f(j,x)|^{2}|\tilde{W}^{\prime}(x)|~{}\mathrm{d}x;$
(8.14) $\Big{|}\big{\langle}f^{\prime}(i,\cdot),(\tilde{W}-W)\ell
f(j,\cdot)\big{\rangle}\Big{|}\leq\int_{I}|f^{\prime}(i,x)||W(x)-\tilde{W}(x)||f(j,x)|~{}\mathrm{d}x\\\
\leq\frac{\theta}{2}\|f^{\prime}(i,\cdot)\|_{2}^{2}+\frac{1}{2\theta}\int_{I}|f(j,x)|^{2}\big{(}W(x)-\tilde{W}(x)\big{)}^{2}~{}\mathrm{d}x;$

and similarly,

(8.15) $\displaystyle\Big{|}\big{\langle}f^{\prime}(i,\cdot),(\tilde{W}-W)\ell
f(j,\cdot)\big{\rangle}\Big{|}\leq\frac{\theta}{2}\|f^{\prime}(j,\cdot)\|_{2}^{2}+\frac{1}{2\theta}\int_{I}|f(i,x)|^{2}\big{(}W(x)-\tilde{W}(x)\big{)}^{2}~{}\mathrm{d}x.$

By [26, (4.2)], there exists a finite random variable $C>0$ such that, almost
surely,

$|\tilde{W}^{\prime}(x)|,\big{(}W(x)-\tilde{W}(x)\big{)}^{2}\leq C\log(2+|x|)$

for all $x\in I$. In particular, by the growth lower bound in Assumption 2.8,
this means that almost surely, for any $\theta>0$, there exists a
$\nu_{\theta}>0$ such that for every $1\leq i\leq r$,

$|\tilde{W}^{\prime}(x)|,\big{(}W(x)-\tilde{W}(x)\big{)}^{2}\leq\theta^{2}V(i,x)_{+}+\nu_{\theta}$

for all $x\in I$. If we plug this back into (8.13)–(8.15), then we get

(8.16) $\Big{|}\big{\langle}f(i,\cdot),\tilde{W}^{\prime}\ell
f(j,\cdot)\big{\rangle}\Big{|}\\\
\leq\tfrac{\theta^{2}}{2}\big{(}\|f(i,\cdot)V(i,\cdot)_{+}^{1/2}\|_{2}^{2}+\|f(j,\cdot)V(j,\cdot)_{+}^{1/2}\|_{2}^{2}\big{)}+\tfrac{\nu_{\theta}}{2}\big{(}\|f(i,\cdot)\|_{2}^{2}+\|f(j,\cdot)\|_{2}^{2}\big{)};$
(8.17) $\displaystyle\Big{|}\big{\langle}f(i,\cdot),\tilde{W}^{\prime}\ell
f(j,\cdot)\big{\rangle}\Big{|}\leq\tfrac{\theta}{2}\|f^{\prime}(i,\cdot)\|_{2}^{2}+\tfrac{\theta}{2}\|f(j,\cdot)V(j,\cdot)_{+}^{1/2}\|_{2}^{2}+\tfrac{\nu_{\theta}}{2\theta}\|f(j,\cdot)\|_{2}^{2};$

and similarly

(8.18) $\displaystyle\Big{|}\big{\langle}f^{\prime}(i,\cdot),(\tilde{W}-W)\ell
f(j,\cdot)\big{\rangle}\Big{|}\leq\tfrac{\theta}{2}\|f^{\prime}(j,\cdot)\|_{2}^{2}+\tfrac{\theta}{2}\|f(i,\cdot)V(i,\cdot)_{+}^{1/2}\|_{2}^{2}+\tfrac{\nu_{\theta}}{2\theta}\|f(i,\cdot)\|_{2}^{2}$

Given that $\theta>0$ above can be taken arbitrarily small, and that

$\|f^{\prime}(i,\cdot)\|_{2}^{2},\|f(i,\cdot)V(i,\cdot)_{+}^{1/2}\|_{2}^{2}\leq\|f\|_{\star}^{2}\qquad\text{and}\qquad\|f(i,\cdot)\|_{2}^{2}\leq\|f\|_{\mu}^{2}$

for all $1\leq i\leq r$, we obtain (8.12) from (8.16)–(8.18).

We now finish the proof of (2.9)—and thus of Proposition 2.12 in the case of
white noises—by establishing (8.9) in Case 3. In that case, by definition of
$W^{\prime}$, we can write

$\big{|}W^{\prime}\big{(}f(i,\cdot),\ell
f(j,\cdot)\big{)}\big{|}\leq|f(i,\vartheta)||f(j,\vartheta)||W(\vartheta)|+\big{|}\big{\langle}f^{\prime}(i,\cdot),Wf(j,\cdot)\big{\rangle}\big{|}+\big{|}\big{\langle}f(i,\cdot),Wf^{\prime}(j,\cdot)\big{\rangle}\big{|}.$

With this in hand, we obtain (2.9) by controlling
$|f(i,\vartheta)||f(j,\vartheta)||W(\vartheta)|$ using (8.10) and (8.11), and
then noting that

(8.19) $\big{|}\big{\langle}f^{\prime}(i,\cdot),W\ell
f(j,\cdot)\big{\rangle}\big{|}\leq\sup_{y\in[0,\vartheta]}|W(y)|\int_{I}|f^{\prime}(i,x)||f(j,x)|~{}\mathrm{d}x\\\
\leq\sup_{y\in[0,\vartheta]}|W(y)|\big{(}\tfrac{\theta}{2}\|f^{\prime}(i,\cdot)\|_{2}^{2}+\tfrac{1}{2\theta}\|f(j,x)\|_{2}^{2}\big{)}$

for any choice of $\theta>0$ (a similar bound holds for $\langle
f(i,\cdot),W\ell f^{\prime}(j,\cdot)\rangle$).

As a final remark, we note that exactly the same argument applies if $\xi$
contain regular noises, as per Definition 2.6. For this, we simply replace $W$
in the above argument by the integral
$\Xi(x)=\int_{0}^{x}\mathcal{X}(x)~{}\mathrm{d}x$, where $\Xi$ is one of the
regular noises used to define any of the $\xi_{i,j}$. This case is also
covered by [26, (4.2)] (in the terminology of [26], the case of a regular
noise is called bounded, which comes from the fact that the noise’s covariance
function is in $L^{\infty}(\mathbb{R})$; see [26, Example 2.28]). Thus, the
proof of Proposition 2.12 is now complete.

####  8.2.3. Proof of Proposition 5.18

We now conclude this subsection by proving Proposition 5.18. The reason why we
prove this in the same subsection as Proposition 2.12 is that its proof is
very similar to (2.9). Indeed, by the variational characterization of the
eigenvalues in (2.10) and the equivalence of $\|\cdot\|_{\star}$ with the norm
induced by $H$’s quadratic form in (2.8), the statement of Proposition 5.18
can be reduced to the following: Almost surely, for every $\kappa\in(0,1)$,
there exists some $\nu>0$ such that for every $\varepsilon,\zeta\in[0,1)$ and
$f\in C_{0}^{\infty}(\mathcal{A},\mathbb{F})$, one has

(8.20)
$\displaystyle|\xi^{\varepsilon,\zeta}(f,f)|\leq\kappa\|f\|^{2}_{\star}+\nu\|f\|_{\mu}^{2}.$

In similar fashion to the reduction we obtained in (8.9), by (8.2)–(8.4), this
latter claim can be reduced to showing that

(8.21)
$\displaystyle\big{|}(W\star\overline{\varrho}_{\varepsilon})^{\prime}\big{(}f(i,\cdot),\ell
f(j,\cdot)\big{)}\big{|}\leq\kappa\|f\|_{\star}^{2}+\nu\|f\|_{\mu}^{2}$

for every $\varepsilon>0$ (here $\varepsilon$ plays the role of both
$\varepsilon$ and $\zeta$) and $f\in C_{0}^{\infty}(\mathcal{A},\mathbb{F})$.
By replicating the proof of (8.9) above verbatim, for this it suffices to show
that if we define

$\tilde{W}_{\varepsilon}(x)=\int_{x}^{x+1}W\star\overline{\varrho}_{\varepsilon}(x)~{}\mathrm{d}x,\qquad
x\in\mathbb{R},$

then there exists a finite random variable $C>0$ such that, almost surely,

$|\tilde{W}_{\varepsilon}^{\prime}(x)|,\big{(}W_{\varepsilon}(x)-\tilde{W}_{\varepsilon}(x)\big{)}^{2}\leq
C\log(2+|x|)$

for all $\varepsilon\in[0,1)$, and moreover that

$\sup_{\varepsilon\in[0,1)}\sup_{x\in[0,\vartheta]}|W\star\overline{\varrho}_{\varepsilon}(x)|<\infty.$

These are proved in [26, Lemma 5.18].

###  8.3. Proof of Proposition 5.19

####  8.3.1. Part 1: Two Lemmas

This proof relies on two technical lemmas:

######  Lemma 8.1.

If $(f_{n})_{n\in\mathbb{N}}\subset D(\mathcal{E})$ is such that
$\sup_{n}\|f_{n}\|_{\star}<\infty$, then there exists $f\in D(\mathcal{E})$
and a subsequence $(n_{i})_{i\in\mathbb{N}}$ along which

  1. (1)

$\displaystyle\lim_{i\to\infty}\|f_{n_{i}}-f\|_{\mu}=0;$

  2. (2)

$\displaystyle\lim_{i\to\infty}\langle
g,f^{\prime}_{n_{i}}\rangle_{\mu}=\langle g,f^{\prime}\rangle_{\mu}$ for every
$g\in L^{2}(\mathcal{A},\mathbb{F})$;

  3. (3)

$\displaystyle\lim_{i\to\infty}f_{n_{i}}=f$ uniformly on compact sets; and

  4. (4)

$\displaystyle\lim_{i\to\infty}\langle g,f_{n_{i}}\rangle_{\star}=\langle
g,f\rangle_{\star}$ for every $g\in D(\mathcal{E})$.

######  Lemma 8.2.

Let $(\mathcal{Q}_{\varepsilon})_{\varepsilon\in[0,1)}$ be a collection of
bilinear forms such that the following conditions hold:

  1. (1)

There exists $\kappa\in(0,1)$ and $\nu>0$ such that

$|\mathcal{Q}_{\varepsilon}(f,f)|\leq\kappa\mathcal{E}(f,f)+\nu\|f\|_{\mu}^{2}\qquad\text{for
every }\varepsilon\in[0,1)\text{ and }f\in D(\mathcal{E}).$

  2. (2)

$(\mathcal{Q}_{\varepsilon})_{\varepsilon\in[0,1)}$ have a common form core
$\mathrm{FC}$. For every $f,g\in\mathrm{FC}$, one has

(8.22) $\displaystyle\lim_{\varepsilon\to
0}\mathcal{Q}_{\varepsilon}(f,g)=\mathcal{Q}_{0}(f,g).$

  3. (3)

If $(\varepsilon_{n})_{n\in\mathbb{N}}\subset(0,1)$ is a vanishing sequence,
$\sup_{n}\|f_{n}\|_{\star}<\infty$, and $f_{n}\to f$ in the sense of Lemma 8.1
(1)–(4), then for every $g\in\mathrm{FC}$, one has

(8.23)
$\displaystyle\lim_{n\to\infty}\mathcal{Q}_{\varepsilon_{n}}(f_{n},g)=\mathcal{Q}_{0}(f,g).$

If we let $T_{\varepsilon}$ denote the unique self-adjoint operators with form
$\mathcal{E}+\mathcal{Q}_{\varepsilon}$, then every vanishing sequence
$(\varepsilon_{n})_{n\in\mathbb{N}}\subset(0,1)$ has a subsequence
$(\varepsilon_{n_{\ell}})_{\ell\in\mathbb{N}}$ along which

(8.24)
$\displaystyle\lim_{\ell\to\infty}\lambda_{k}(T_{\varepsilon_{n_{\ell}}})=\lambda_{k}(T_{0})\qquad\text{for
every }k\geq 1.$

Since the proof of these lemmas are mostly technical in nature and amount to
trivial modifications of results previously proved in [26], we omit a full
proof and opt instead to provide precise references:

On the one hand, Lemma 8.1 follows from expanding
$\|\cdot\|_{\star}^{2}=\sum_{i}\|\cdot\|_{\star;i}^{2}$ using (8.7) and (8.8),
and noting that [26, Lemma 5.15] proved the same statement as Lemma 8.2 with
$\|\cdot\|_{\star}$, $D(\mathcal{E})$, and $\langle\cdot,\cdot\rangle_{\mu}$ /
$\|\cdot\|_{\mu}$ / $L^{2}(\mathcal{A},\mathbb{F})$ respectively replaced by
$\|\cdot\|_{\star;i}$, $D(\mathcal{E}_{i})$, and $\langle\cdot,\cdot\rangle$ /
$\|\cdot\|_{2}$ / $L^{2}(I,\mathbb{F})$ (see also [50, Fact 2.2], [4, Fact
2.2], and [5, Fact 3.2]).

On the other hand, Lemma 8.2 follows from a verbatim application of the
argument presented in [26, Pages 33 and 34, starting from (5.31)]. More
specifically:

  1. (1)

Lemma 8.1 plays the role of [26, Lemma 5.15];

  2. (2)

the forms $\mathcal{Q}_{\varepsilon}$ and the operators $T_{\varepsilon}$ play
the role of $\mathcal{E}_{\varepsilon}$ and $\widehat{H}_{\varepsilon}$ in
[26];

  3. (3)

Lemma 8.2-(1) plays the role of [26, Lemmas 5.17 and 5.18 and Remark 5.19];

  4. (4)

Lemma 8.2-(2) plays the role of [26, Lemma 5.20, (5.29)]; and

  5. (5)

Lemma 8.2-(3) plays the role of [26, Lemma 5.20, (5.30)].

####  8.3.2. Part 2: Application of Lemmas

At this point, in order to prove Proposition 5.19, it suffices to show that
for every fixed $\zeta>0$, almost surely, the sequence of forms
$(\mathcal{Q}_{\varepsilon})_{\varepsilon\in[0,1)}=(\xi^{\varepsilon,\zeta})_{\varepsilon\in[0,1)}$
satisfies the conditions (1)–(3) in the statement of Lemma 8.2, and similarly
for
$(\mathcal{Q}_{\varepsilon})_{\varepsilon\in[0,1)}=(\xi^{0,\varepsilon})_{\varepsilon\in[0,1)}$
(where in this latter sequence $\varepsilon$ now plays the role of the
parameter $\zeta$ in (5.34)).

For this purpose, we first note that the claim of Lemma 8.2-(1) for both
sequences follows from (8.20) and the equivalence of the norms
$\|\cdot\|_{\star}$ and $\|\cdot\|_{+1}$ (the latter of which was established
in the proof of Proposition 2.12 above). Next, regarding items (2) and (3), we
have by Remark 5.4 that all forms have a common core
$\mathrm{FC}=\mathrm{FC}_{1}\oplus\cdots\oplus\mathrm{FC}_{r}$, where the
$\mathrm{FC}_{i}$ are defined in Remark 5.1. If we combine this with the
decompositions (8.1), (8.5), and (8.6), then the claims in (8.22) and (8.23)
can be reduced to the following: Let $W$ be a standard Brownian motion, and
let $1\leq i\leq r$ be fixed. For every real-valued $f,g\in\mathrm{FC}_{i}$,
one has

(8.25) $\displaystyle\lim_{\varepsilon\to
0}(W\star\overline{\varrho}_{\varepsilon})^{\prime}(f,g)=W^{\prime}(f,g).$

Moreover, if $(\varepsilon_{n})_{n\in\mathbb{N}}\subset(0,1)$ is a vanishing
sequence, the sequence of functions $f_{n}\in L^{2}(I,\mathbb{R})$ is such
that $\sup_{n}\|f_{n}\|_{\star;i}<\infty$, and $f_{n}\to f$ in the sense of
Lemma 8.1 (1)–(4) (with $\|\cdot\|_{\star}$ and
$\langle\cdot,\cdot\rangle_{\mu}$ / $\|\cdot\|_{\mu}$ /
$L^{2}(\mathcal{A},\mathbb{F})$ respectively replaced by $\|\cdot\|_{\star;i}$
and $\langle\cdot,\cdot\rangle$ / $\|\cdot\|_{2}$ / $L^{2}(I,\mathbb{F})$),
then for every real-valued $g\in\mathrm{FC}_{i}$, one has

(8.26)
$\displaystyle\lim_{n\to\infty}(W\star\overline{\varrho}_{\varepsilon_{n}})^{\prime}(f_{n},g)=W^{\prime}(f,g).$

These are the statements proved in [26, Lemma 5.20], in the special case
where, in the notation of [26], $\Xi=W$ and $\xi=W^{\prime}$. The proof of
Proposition 5.19 is thus complete.

##  9\. Trace Covariance Asymptotics

We now conclude the proofs of the technical results stated in our roadmap in
Section 5 by establishing Proposition 5.21.

######  Notation 9.1.

Throughout this section, we use $C,c>0$ to denote constants that are
independent of $t_{1},t_{2}$, whose exact value may change from one display to
the next.

######  Remark 9.2.

In Cases 1 and 2, we assume that there exists $\kappa,\nu>0$ such that

(9.1) $\displaystyle V(i,x)\geq\kappa|x|-\nu\qquad\text{ for all $1\leq i\leq
r$ and $x\in I$}.$

In these two cases, the fact that $I$ is infinite makes this a nontrivial
strengthening of Assumption 2.8. However, the assumption that $V$ is bounded
below and the fact that $I$ is bounded in Case 3 implies that (9.1) also holds
in that case. Thus, for convenience we assume that (9.1) holds in all cases in
this section.

###  9.1. Step 1. Covariance Formula

Our first step toward Proposition 5.21 is to use the trace moment formula in
Proposition 5.14 to provide an expression for

(9.2)
$\mathbf{Cov}\Big{[}\mathrm{Tr}\big{[}\mathrm{e}^{-t_{1}\widehat{H}^{0,\zeta}}\big{]},\mathrm{Tr}\big{[}\mathrm{e}^{-t_{2}\widehat{H}^{0,\zeta}}\big{]}\Big{]}\\\
=\mathbf{E}\Big{[}\mathrm{Tr}\big{[}\mathrm{e}^{-t_{1}\widehat{H}^{0,\zeta}}\big{]}\mathrm{Tr}\big{[}\mathrm{e}^{-t_{2}\widehat{H}^{0,\zeta}}\big{]}\Big{]}-\mathbf{E}\Big{[}\mathrm{Tr}\big{[}\mathrm{e}^{-t_{1}\widehat{H}^{0,\zeta}}\big{]}\Big{]}\mathbf{E}\Big{[}\mathrm{Tr}\big{[}\mathrm{e}^{-t_{2}\widehat{H}^{0,\zeta}}\big{]}\Big{]}$

that is amenable to computation. For the first term on the right-hand side of
(9.2), Proposition 5.14 in the case $n=2$ yields:

(9.3)
$\mathbf{E}\Big{[}\mathrm{Tr}\big{[}\mathrm{e}^{-t_{1}\widehat{H}^{0,\zeta}}\big{]}\mathrm{Tr}\big{[}\mathrm{e}^{-t_{2}\widehat{H}^{0,\zeta}}\big{]}\Big{]}=\int_{\mathcal{A}^{2}}\Pi_{A}(\boldsymbol{t};\boldsymbol{a},\boldsymbol{a})\mathbf{E}\Bigg{[}\mathfrak{m}^{\boldsymbol{\zeta}}_{\boldsymbol{t}}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\\\
\cdot\mathrm{e}^{-\int_{0}^{|\boldsymbol{t}|}V(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}}(s))~{}\mathrm{d}s+\mathfrak{B}_{|\boldsymbol{t}|}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})+\frac{\sigma^{2}}{2}\|L_{[0,t_{1})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})+L_{[t_{1},t_{1}+t_{2})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\|_{\mu}^{2}}\Bigg{]}~{}\mathrm{d}\mu^{2}(\boldsymbol{a}),$

where we denote $\boldsymbol{t}=(t_{1},t_{2})$ and
$\boldsymbol{\zeta}=(\zeta,\zeta)$. In contrast, two applications of
Proposition 5.14 in the case $n=1$ yields

(9.4)
$\mathbf{E}\Big{[}\mathrm{Tr}\big{[}\mathrm{e}^{-t_{1}\widehat{H}^{0,\zeta}}\big{]}\Big{]}\mathbf{E}\Big{[}\mathrm{Tr}\big{[}\mathrm{e}^{-t_{2}\widehat{H}^{0,\zeta}}\big{]}\Big{]}=\prod_{k=1}^{2}\int_{\mathcal{A}}\Pi_{A}(t_{k};a_{k},a_{k})\mathbf{E}\Bigg{[}\mathfrak{m}^{\zeta}_{t_{k}}(A^{a_{k},a_{k}}_{t_{k}})\\\
\cdot\mathrm{e}^{-\int_{0}^{t_{k}}V(A^{a_{k},a_{k}}_{t_{k}}(s))~{}\mathrm{d}s+\mathfrak{B}_{t_{k}}(A^{a_{k},a_{k}}_{t_{k}})+\frac{\sigma^{2}}{2}\|L_{[0,t_{k})}(A^{a_{k},a_{k}}_{t_{k}})\|_{\mu}^{2}}\Bigg{]}~{}\mathrm{d}\mu(a_{k}).$

In order to relate this to (9.4), we introduce the following couplings:

######  Notation 9.3.

For any given $\boldsymbol{a}=(a_{1},a_{2})\in\mathcal{A}^{2}$, where
$a_{k}=(i_{k},x_{k})$ for $k=1,2$, we let
$A^{a_{1},a_{1}}_{t_{1}}=(U^{i_{1},i_{1}}_{t_{1}},Z^{x_{1},x_{1}}_{t_{1}})$ be
as in Definitions 3.2 and 3.3, and let
$\overline{A}^{a_{2},a_{2}}_{t_{2}}=(\overline{U}^{i_{2},i_{2}}_{t_{2}},\overline{Z}^{x_{2},x_{2}}_{t_{2}})$
be a process that has the same distribution as $A_{t_{2}}^{a_{2},a_{2}}$, but
which we assume is independent of $A^{a_{1},a_{1}}_{t_{1}}$. Then we can
couple these two processes with the concatenated process
$A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}}$ in (9.4) in such a way
that

$A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}}(s)=\begin{cases}A^{a_{1},a_{1}}_{t_{1}}(s)&\text{if
}0\leq s<t_{1},\vspace{5pt}\\\
\overline{A}^{a_{2},a_{2}}_{t_{2}}(s-t_{1})&\text{if }t_{1}\leq s\leq
t_{1}+t_{2}.\end{cases}$

Under this coupling, the additivity of integrals and boundary local time
yields

$\int_{0}^{t_{1}}V\big{(}A^{a_{1},a_{1}}_{t_{1}}(s)\big{)}~{}\mathrm{d}s+\int_{0}^{t_{2}}V\big{(}\overline{A}^{a_{2},a_{2}}_{t_{2}}(s)\big{)}~{}\mathrm{d}s=\int_{0}^{|\boldsymbol{t}|}V\big{(}A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}}(s)\big{)}~{}\mathrm{d}s,$
$\mathfrak{B}_{t_{1}}(A^{a_{1},a_{1}}_{t_{1}})+\mathfrak{B}_{t_{2}}(\overline{A}^{a_{2},a_{2}}_{t_{2}})=\mathfrak{B}_{|\boldsymbol{t}|}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}}),$

where $\boldsymbol{a}=(a_{1},a_{2})$. Moreover,

$L_{[0,t_{1})}(A^{a_{1},a_{1}}_{t_{1}})=L_{[0,t_{1})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}});\qquad
L_{[0,t_{2})}(\overline{A}^{a_{2},a_{2}}_{t_{2}})=L_{[t_{1},t_{1}+t_{2})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}});$

and
$\prod_{k=1}^{2}\Pi_{A}(t_{k};a_{k},a_{k})=\Pi(\boldsymbol{t};\boldsymbol{a},\boldsymbol{a})$.
Therefore, an application of Fubini’s theorem yields that (9.4) is equal to

(9.5)
$\int_{\mathcal{A}^{2}}\Pi_{A}(\boldsymbol{t};\boldsymbol{a},\boldsymbol{a})\mathbf{E}\Bigg{[}\mathfrak{m}^{\zeta}_{t_{1}}(A^{a_{1},a_{1}}_{t_{1}})\mathfrak{m}^{\zeta}_{t_{2}}(\overline{A}^{a_{2},a_{2}}_{t_{2}})\\\
\cdot\mathrm{e}^{-\int_{0}^{|\boldsymbol{t}|}V(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}}(s))~{}\mathrm{d}s+\mathfrak{B}_{|\boldsymbol{t}|}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})+\frac{\sigma^{2}}{2}(\|L_{[0,t_{1})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\|_{\mu}^{2}+\|L_{[t_{1},t_{1}+t_{2})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\|_{\mu}^{2})}\Bigg{]}~{}\mathrm{d}\mu^{2}(\boldsymbol{a}).$

In summary, the only two differences between (9.3) and (9.5) are

$\mathfrak{m}^{\boldsymbol{\zeta}}_{\boldsymbol{t}}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\quad\text{vs.}\quad\mathfrak{m}^{\zeta}_{t_{1}}(A^{a_{1},a_{1}}_{t_{1}})\mathfrak{m}^{\zeta}_{t_{2}}(\overline{A}^{a_{2},a_{2}}_{t_{2}})$

and

$\tfrac{\sigma^{2}}{2}\|L_{[0,t_{1})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})+L_{[t_{1},t_{1}+t_{2})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\|_{\mu}^{2}\quad\text{vs.}\quad\tfrac{\sigma^{2}}{2}(\|L_{[0,t_{1})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\|_{\mu}^{2}+\|L_{[t_{1},t_{1}+t_{2})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\|_{\mu}^{2}).$

In order to analyze the contributions of these differences to the covariance
one at a time, we add and subtract the intermediary term

$\int_{\mathcal{A}^{2}}\Pi_{A}(\boldsymbol{t};\boldsymbol{a},\boldsymbol{a})\mathbf{E}\Bigg{[}\mathfrak{m}^{\boldsymbol{\zeta}}_{\boldsymbol{t}}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\\\
\cdot\mathrm{e}^{-\int_{0}^{|\boldsymbol{t}|}V(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}}(s))~{}\mathrm{d}s+\mathfrak{B}_{|\boldsymbol{t}|}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})+\frac{\sigma^{2}}{2}(\|L_{[0,t_{1})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\|_{\mu}^{2}+\|L_{[t_{1},t_{1}+t_{2})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\|_{\mu}^{2})}\Bigg{]}~{}\mathrm{d}\mu^{2}(\boldsymbol{a});$

in doing so, the covariance in (9.2) (which we write as the difference between
(9.3) and (9.5) with the intermediary term added and subtracted) can be
written as the sum of the following two terms:

(9.6)
$\int_{\mathcal{A}^{2}}\Pi_{A}(\boldsymbol{t};\boldsymbol{a},\boldsymbol{a})\mathbf{E}\Bigg{[}\mathfrak{m}^{\boldsymbol{\zeta}}_{\boldsymbol{t}}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\mathrm{e}^{-\int_{0}^{|\boldsymbol{t}|}V(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}}(s))~{}\mathrm{d}s+\mathfrak{B}_{|\boldsymbol{t}|}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})}\\\
\cdot\left(\mathrm{e}^{\frac{\sigma^{2}}{2}\|L_{[0,t_{1})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})+L_{[t_{1},t_{1}+t_{2})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\|_{\mu}^{2}}-\mathrm{e}^{\frac{\sigma^{2}}{2}(\|L_{[0,t_{1})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\|_{\mu}^{2}+\|L_{[t_{1},t_{1}+t_{2})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\|_{\mu}^{2})}\right)\Bigg{]}~{}\mathrm{d}\mu^{2}(\boldsymbol{a});$

and

(9.7)
$\int_{\mathcal{A}^{2}}\Pi_{A}(\boldsymbol{t};\boldsymbol{a},\boldsymbol{a})\mathbf{E}\Bigg{[}\left(\mathfrak{m}^{\boldsymbol{\zeta}}_{\boldsymbol{t}}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})-\mathfrak{m}^{\zeta}_{t_{1}}(A^{a_{1},a_{1}}_{t_{1}})\mathfrak{m}^{\zeta}_{t_{2}}(\overline{A}^{a_{2},a_{2}}_{t_{2}})\right)\\\
\cdot\mathrm{e}^{-\int_{0}^{|\boldsymbol{t}|}V(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}}(s))~{}\mathrm{d}s+\mathfrak{B}_{|\boldsymbol{t}|}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})+\frac{\sigma^{2}}{2}(\|L_{[0,t_{1})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\|_{\mu}^{2}+\|L_{[t_{1},t_{1}+t_{2})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\|_{\mu}^{2})}\Bigg{]}~{}\mathrm{d}\mu^{2}(\boldsymbol{a}).$

The proof of Proposition 5.21 now relies on showing that for any choice of
$\zeta>0$,

(9.8) $\displaystyle|\eqref{Equation: Covariance Formula Piece
1}|~{},|\eqref{Equation: Covariance Formula Piece 2}|\leq\text{right-hand side
of }\eqref{Equation: Rigidity RHS}.$

We bound these terms one at a time.

###  9.2. Step 2. Upper Bound for (9.6)

Noting that

$\|L_{[0,t_{1})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})+L_{[t_{1},t_{1}+t_{2})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\|_{\mu}^{2}\\\
=\|L_{[0,t_{1})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\|_{\mu}^{2}+\|L_{[t_{1},t_{1}+t_{2})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\|_{\mu}^{2}+2\big{\langle}L_{[0,t_{1})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}}),L_{[t_{1},t_{1}+t_{2})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\big{\rangle}_{\mu},$

and then recalling the bound
$|\mathfrak{m}^{\boldsymbol{\zeta}}_{\boldsymbol{t}}(A)|\leq\mathrm{e}^{(r-1)|\boldsymbol{t}|}\overline{\mathfrak{m}}^{\boldsymbol{\zeta}}_{\boldsymbol{t}}(A),$
where $\overline{\mathfrak{m}}^{\boldsymbol{\zeta}}_{\boldsymbol{t}}(A)$ was
defined in (7.37), for any $t_{1},t_{2}\in(0,1]$ we can bound the absolute
value of (9.6) as follows:

(9.9) $|\eqref{Equation: Covariance Formula Piece 1}|\leq
C\int_{\mathcal{A}^{2}}\Pi_{A}(\boldsymbol{t};\boldsymbol{a},\boldsymbol{a})\mathbf{E}\Bigg{[}\overline{\mathfrak{m}}^{\boldsymbol{\zeta}}_{\boldsymbol{t}}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\mathrm{e}^{-\int_{0}^{|\boldsymbol{t}|}V(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}}(s))~{}\mathrm{d}s+\mathfrak{B}_{|\boldsymbol{t}|}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})}\\\
\cdot\mathrm{e}^{\frac{\sigma^{2}}{2}(\|L_{[0,t_{1})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\|_{\mu}^{2}+\|L_{[t_{1},t_{1}+t_{2})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\|_{\mu}^{2})}\left(\mathrm{e}^{\sigma^{2}\langle
L_{[0,t_{1})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}}),L_{[t_{1},t_{1}+t_{2})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\rangle_{\mu}}-1\right)\Bigg{]}~{}\mathrm{d}\mu^{2}(\boldsymbol{a}).$

Recall from (4.4) that

$\sum_{i=1}^{r}L_{[0,t_{1})}^{(i,x)}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})=L_{t_{1}}^{x}(Z^{x_{1},x_{1}}_{t_{1}})\qquad\text{and}\qquad\sum_{i=1}^{r}L_{[t_{1},t_{1}+t_{2})}^{(i,x)}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})=L_{t_{2}}^{x}(\overline{Z}^{x_{2},x_{2}}_{t_{2}}).$

In particular, since local time is nonnegative,

$\langle
L_{[0,t_{1})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}}),L_{[t_{1},t_{1}+t_{2})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\rangle_{\mu}\leq\langle
L_{t_{1}}(Z^{x_{1},x_{1}}_{t_{1}}),L_{t_{2}}(\overline{Z}^{x_{2},x_{2}}_{t_{2}})\rangle.$

Moreover, by (7.14) applied to the case where $f$ is
$A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}}$’s local time,

$\|L_{[0,t_{1})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\|_{\mu}^{2}\leq\|L_{t_{1}}(Z^{x_{1},x_{1}}_{t_{1}})\|_{2}^{2},\quad\|L_{[t_{1},t_{1}+t_{2})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\|_{\mu}^{2}\leq\|L_{t_{2}}(\overline{Z}^{x_{2},x_{2}}_{t_{2}})\|_{2}^{2}.$

Next, by [28, Proposition A.1], there exists some $C>0$ such that

$\sup_{\boldsymbol{x}\in
I^{2}}\Pi_{Z}(\boldsymbol{t};\boldsymbol{x},\boldsymbol{x})\leq
C(t_{1}t_{2})^{-1/2},\qquad\text{for all }t_{1},t_{2}\in(0,1].$

If we combing all this with (7.8) and (9.1), we get from (9.9) that

(9.10) $|\eqref{Equation: Covariance Formula Piece 1}|\leq
C(t_{1}t_{2})^{-1/2}\int_{\mathcal{A}^{2}}\Pi_{U}(\boldsymbol{t};\boldsymbol{i},\boldsymbol{i})\mathbf{E}\Bigg{[}\overline{\mathfrak{m}}^{\boldsymbol{\zeta}}_{\boldsymbol{t}}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\mathrm{e}^{-\int_{0}^{|\boldsymbol{t}|}\kappa|Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(s)|~{}\mathrm{d}s+\overline{\mathfrak{B}_{|\boldsymbol{t}|}}(Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}})}\\\
\cdot\mathrm{e}^{\frac{\sigma^{2}}{2}(\|L_{t_{1}}(Z^{x_{1},x_{1}}_{t_{1}})\|_{2}^{2}+\|L_{t_{2}}(\overline{Z}^{x_{2},x_{2}}_{t_{2}})\|_{2}^{2})}\left(\mathrm{e}^{\sigma^{2}\langle
L_{t_{1}}(Z^{x_{1},x_{1}}_{t_{1}}),L_{t_{2}}(\overline{Z}^{x_{2},x_{2}}_{t_{2}})\rangle}-1\right)\Bigg{]}~{}\mathrm{d}\mu^{2}(\boldsymbol{a}).$

At this point, we notice that the only term that depends on $U$ in (9.10) is
$\overline{\mathfrak{m}}^{\boldsymbol{\zeta}}_{\boldsymbol{t}}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})$.
Thus, by replicating the analysis performed in (7.38)–(7.42), if we first
average over $U$ only in (9.10), then we get from (9.10) the
$\zeta$-independent bound

(9.11) $|\eqref{Equation: Covariance Formula Piece 1}|\leq
C(t_{1}t_{2})^{-1/2}\int_{I^{2}}\mathbf{E}\Bigg{[}\mathrm{e}^{-\int_{0}^{|\boldsymbol{t}|}\kappa|Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(s)|~{}\mathrm{d}s+\overline{\mathfrak{B}_{|\boldsymbol{t}|}}(Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}})}\\\
\cdot\mathrm{e}^{\frac{n(r-1)^{2}\upsilon^{2}+\sigma^{2}}{2}(\|L_{t_{1}}(Z^{x_{1},x_{1}}_{t_{1}})\|_{2}^{2}+\|L_{t_{2}}(\overline{Z}^{x_{2},x_{2}}_{t_{2}})\|_{2}^{2})}\left(\mathrm{e}^{\sigma^{2}\langle
L_{t_{1}}(Z^{x_{1},x_{1}}_{t_{1}}),L_{t_{2}}(\overline{Z}^{x_{2},x_{2}}_{t_{2}})\rangle}-1\right)\Bigg{]}~{}\mathrm{d}\boldsymbol{x}.$

Then, an application of Hölder’s inequality in (9.11) yields

(9.12) $|\eqref{Equation: Covariance Formula Piece 1}|\leq
C(t_{1}t_{2})^{-1/2}\int_{I^{2}}\mathbf{E}\left[\mathrm{e}^{-4\int_{0}^{|\boldsymbol{t}|}\kappa|Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(s)|~{}\mathrm{d}s}\right]^{1/4}\mathbf{E}\left[\mathrm{e}^{4\overline{\mathfrak{B}_{|\boldsymbol{t}|}}(Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}})}\right]^{1/4}\\\
\cdot\mathbf{E}\left[\mathrm{e}^{2(n(r-1)^{2}\upsilon^{2}+\sigma^{2})(\|L_{t_{1}}(Z^{x_{1},x_{1}}_{t_{1}})\|_{2}^{2}+\|L_{t_{2}}(\overline{Z}^{x_{2},x_{2}}_{t_{2}})\|_{2}^{2})}\right]^{1/4}\\\
\cdot\mathbf{E}\left[\left(\mathrm{e}^{\sigma^{2}\langle
L_{t_{1}}(Z^{x_{1},x_{1}}_{t_{1}}),L_{t_{2}}(\overline{Z}^{x_{2},x_{2}}_{t_{2}})\rangle}-1\right)^{4}\right]^{1/4}~{}\mathrm{d}\boldsymbol{x}.$

If we use exactly the same Brownian scaling arguments as in [28, Lemma 4.6],
then we see that there exists a constant $C>0$ such that

(9.13) $\displaystyle\sup_{\boldsymbol{t}\in(0,1]^{2},~{}\boldsymbol{x}\in
I^{2}}\mathbf{E}\left[\mathrm{e}^{4\overline{\mathfrak{B}_{|\boldsymbol{t}|}}(Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}})}\right]^{1/4}\mathbf{E}\left[\mathrm{e}^{2(n(r-1)^{2}\upsilon^{2}+\sigma^{2})(\|L_{t_{1}}(Z^{x_{1},x_{1}}_{t_{1}})\|_{2}^{2}+\|L_{t_{2}}(\overline{Z}^{x_{2},x_{2}}_{t_{2}})\|_{2}^{2})}\right]^{1/4}<C.$

Thus, we get from (9.12) that

(9.14) $|\eqref{Equation: Covariance Formula Piece 1}|\leq
C(t_{1}t_{2})^{-1/2}\int_{I^{2}}\mathbf{E}\left[\mathrm{e}^{-4\int_{0}^{|\boldsymbol{t}|}\kappa|Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(s)|~{}\mathrm{d}s}\right]^{1/4}\\\
\cdot\mathbf{E}\left[\left(\mathrm{e}^{\sigma^{2}\langle
L_{t_{1}}(Z^{x_{1},x_{1}}_{t_{1}}),L_{t_{2}}(\overline{Z}^{x_{2},x_{2}}_{t_{2}})\rangle}-1\right)^{4}\right]^{1/4}~{}\mathrm{d}\boldsymbol{x}.$

At this point, in order to prove that (9.14) is bounded by the right-hand side
of (5.41), we separate our analysis for Cases 1 and 2 versus 3.

We begin with Case 3, as it is the simplest. In this case, we notice that
since the interval $I=(0,\vartheta)$ is bounded and
$\mathrm{e}^{-4\int_{0}^{|\boldsymbol{t}|}\kappa|Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(s)|~{}\mathrm{d}s}\leq
1$, (9.14) yields

(9.15) $\displaystyle|\eqref{Equation: Covariance Formula Piece 1}|\leq
C(t_{1}t_{2})^{-1/2}\sup_{\boldsymbol{x}\in
I^{2}}\mathbf{E}\left[\left(\mathrm{e}^{\sigma^{2}\langle
L_{t_{1}}(Z^{x_{1},x_{1}}_{t_{1}}),L_{t_{2}}(\overline{Z}^{x_{2},x_{2}}_{t_{2}})\rangle}-1\right)^{4}\right]^{1/4}.$

Then, if we apply the inequality
$(\mathrm{e}^{x}-1)^{4}\leq(x\mathrm{e}^{x})^{4}$, followed by Hölder’s
inequality, we get the upper bound

(9.16) $\displaystyle|\eqref{Equation: Covariance Formula Piece 1}|\leq
C(t_{1}t_{2})^{-1/2}\sup_{\boldsymbol{x}\in
I^{2}}\prod_{k=1}^{2}\mathbf{E}\left[\|L_{t_{k}}(Z^{x_{k},x_{k}}_{t_{k}})\|_{2}^{8}\right]^{1/8}\mathbf{E}\left[\mathrm{e}^{8\sigma^{2}\|L_{t_{k}}(Z^{x_{k},x_{k}}_{t_{k}})\|_{2}}\right]^{1/8},$

where the expectation factors into a product over $k=1,2$ because of the
independence of $Z^{x_{1},x_{1}}_{t_{1}}$ and
$\overline{Z}^{x_{2},x_{2}}_{t_{2}}$. Thanks to (9.13), this further reduces
to

(9.17) $\displaystyle|\eqref{Equation: Covariance Formula Piece 1}|\leq
C(t_{1}t_{2})^{-1/2}\sup_{\boldsymbol{x}\in
I^{2}}\prod_{k=1}^{2}\mathbf{E}\left[\|L_{t_{k}}(Z^{x_{k},x_{k}}_{t_{k}})\|_{2}^{8}\right]^{1/8}.$

By [28, (4.31)] (in the notation of [28], the parameter $\mathfrak{d}$ is
equal to $3/2$ in the case of white noise; see [28, (2.16)]), we have the
following estimate:

######  Lemma 9.4.

In Cases 1, 2, and 3, for any $k=1,2$ and $\theta\geq 1$, there exists a
constant $C$ such that

$\sup_{x_{k}\in
I}\mathbf{E}\left[\|L_{t_{k}}(Z^{x_{k},x_{k}}_{t_{k}})\|_{2}^{\theta}\right]^{1/\theta}\leq
Ct_{k}^{3/4}\qquad\text{for all $t_{k}\in(0,1]$.}$

The combination of this with (9.17) implies that (9.8) holds for (9.6) in Case
3.

Consider now Cases 1 and 2. If we replicate the bound (6.12) with
$\overline{\mathfrak{a}}=1$ and apply (6.18), then we get from (9.14) that

(9.18) $\displaystyle|\eqref{Equation: Covariance Formula Piece 1}|\leq
C(t_{1}t_{2})^{-1/2}\int_{I^{2}}\mathrm{e}^{-t_{1}|x_{1}|-t_{2}|x_{2}|}\mathbf{E}\left[\left(\mathrm{e}^{\sigma^{2}\langle
L_{t_{1}}(Z^{x_{1},x_{1}}_{t_{1}}),L_{t_{2}}(\overline{Z}^{x_{2},x_{2}}_{t_{2}})\rangle}-1\right)^{4}\right]^{1/4}~{}\mathrm{d}\boldsymbol{x}.$

In order for the random variable inside the expectation in (9.18) to be
nonzero, it is necessary that the supports of the independent Brownian bridges
$Z^{x_{1},x_{1}}_{t_{1}}$ and $\overline{Z}^{x_{2},x_{2}}_{t_{2}}$ intersect.
Thus, by introducing the indicator of this event in the expectation and
applying Hölder’s inequality, we get from (9.18) that

(9.19) $|\eqref{Equation: Covariance Formula Piece 1}|\leq
C(t_{1}t_{2})^{-1/2}\int_{I^{2}}\mathrm{e}^{-t_{1}|x_{1}|-t_{2}|x_{2}|}\mathbf{P}\left[Z^{x_{1},x_{1}}_{t_{1}}\text{
and }\overline{Z}^{x_{2},x_{2}}_{t_{2}}\text{ intersect}\right]^{1/8}\\\
\cdot\mathbf{E}\left[\left(\mathrm{e}^{\sigma^{2}\langle
L_{t_{1}}(Z^{x_{1},x_{1}}_{t_{1}}),L_{t_{2}}(\overline{Z}^{x_{2},x_{2}}_{t_{2}})\rangle}-1\right)^{8}\right]^{1/8}~{}\mathrm{d}\boldsymbol{x}.$

By replicating the analysis in (9.15)–(9.17) and Lemma 9.4, (9.19) implies
that

(9.20) $\displaystyle|\eqref{Equation: Covariance Formula Piece 1}|\leq
C(t_{1}t_{2})^{1/4}\int_{I^{2}}\mathrm{e}^{-t_{1}|x_{1}|-t_{2}|x_{2}|}\mathbf{P}\left[Z^{x_{1},x_{1}}_{t_{1}}\text{
and }\overline{Z}^{x_{2},x_{2}}_{t_{2}}\text{
intersect}\right]^{1/8}~{}\mathrm{d}\boldsymbol{x}.$

By [27, (4.14) and the two following displays], there exists $C,c>0$ such that

$\mathbf{P}\left[Z^{x_{1},x_{1}}_{t_{1}}\text{ and
}\overline{Z}^{x_{2},x_{2}}_{t_{2}}\text{ intersect}\right]^{1/8}\leq
C\left(\mathrm{e}^{-c(x_{1}-x_{2})^{2}/t_{1}}+\mathrm{e}^{-c(x_{1}-x_{2})^{2}/t_{2}}\right)$

for all $t_{1},t_{2}\in(0,1]$; hence (9.20) yields

(9.21) $\displaystyle|\eqref{Equation: Covariance Formula Piece 1}|\leq
C(t_{1}t_{2})^{1/4}\int_{\mathbb{R}^{2}}\mathrm{e}^{-t_{1}|x_{1}|-t_{2}|x_{2}|}\left(\mathrm{e}^{-c(x_{1}-x_{2})^{2}/t_{1}}+\mathrm{e}^{-c(x_{1}-x_{2})^{2}/t_{2}}\right)~{}\mathrm{d}\boldsymbol{x},$

where we have replaced $I$ by $\mathbb{R}$ in Case 2, which only increases the
size of the integral. Denote $t_{*}=\max\\{t_{1},t_{2}\\}$. Then, (9.21)
implies that

(9.22) $\displaystyle|\eqref{Equation: Covariance Formula Piece 1}|\leq
C(t_{1}t_{2})^{1/4}\int_{\mathbb{R}^{2}}\mathrm{e}^{-t_{*}|x_{1}|}\mathrm{e}^{-c(x_{1}-x_{2})^{2}/t_{*}}~{}\mathrm{d}\boldsymbol{x}=C(t_{1}t_{2})^{1/4}\cdot\frac{2\sqrt{\pi}}{\sqrt{ct_{*}}};$

hence (9.8) holds for (9.6) in Case 1 and 2 as well.

###  9.3. Step 3. Upper Bound for (9.7)

Recall the coupling introduced in Definition 9.3. Given that
$\boldsymbol{\zeta}=(\zeta,\zeta)$, we have from (5.27) that the difference
$\mathfrak{m}^{\boldsymbol{\zeta}}_{\boldsymbol{t}}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})-\mathfrak{m}^{\zeta}_{t_{1}}(A^{a_{1},a_{1}}_{t_{1}})\mathfrak{m}^{\zeta}_{t_{2}}(A^{a_{2},a_{2}}_{t_{2}})$
can be written as

(9.23)
$\mathrm{e}^{(r-1)|\boldsymbol{t}|}\Bigg{(}\mathbf{1}_{\\{N(|\boldsymbol{t}|)\text{
is
even}\\}}\sum_{p\in\mathcal{P}_{N(|\boldsymbol{t}|)}}\mathfrak{C}_{|\boldsymbol{t}|}(p,U^{\boldsymbol{i},\boldsymbol{i}}_{\boldsymbol{t}})\prod_{\\{\ell_{1},\ell_{2}\\}\in
p}\upsilon^{2}\varrho_{\zeta}\big{(}Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(\tau_{\ell_{1}})-Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(\tau_{\ell_{2}})\big{)}\\\
-\mathbf{1}_{\\{N(t_{1}),N(|\boldsymbol{t}|)-N(t_{1})\text{ are
even}\\}}\sum_{p_{1}\in\mathcal{P}_{N(t_{1})},~{}p_{2}\in\mathcal{P}_{N(|\boldsymbol{t}|)-N(t_{1})}}\Bigg{\\{}\cdots\\\
\cdots\mathfrak{C}_{t_{1}}(p_{1},U^{i_{1},i_{1}}_{t_{1}})\mathfrak{C}_{t_{2}}(p_{2},\overline{U}^{i_{2},i_{2}}_{t_{2}})\prod_{\\{\ell_{1},\ell_{2}\\}\in
p_{1}\cup
p_{2}}\upsilon^{2}\varrho_{\zeta}\big{(}Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(\tau_{\ell_{1}})-Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(\tau_{\ell_{2}})\big{)}\Bigg{\\}}\Bigg{)}.$

Suppose that $N(t_{1})$ and $N(|\boldsymbol{t}|)-N(t_{1})$ are even, and let
$p\in\mathcal{P}_{N(|\boldsymbol{t}|)}$ be such that we can write $p=p_{1}\cup
p_{2}$ for some $p_{1}\in\mathcal{P}_{N(t_{1})}$ and
$p_{2}\in\mathcal{P}_{N(|\boldsymbol{t}|)-N(t_{1})}$. We claim that in such a
case, one has

(9.24)
$\displaystyle\mathfrak{C}_{|\boldsymbol{t}|}(p,U^{\boldsymbol{i},\boldsymbol{i}}_{\boldsymbol{t}})=\mathfrak{C}_{|\boldsymbol{t}|}(p_{1}\cup
p_{2},U^{\boldsymbol{i},\boldsymbol{i}}_{\boldsymbol{t}})=\mathfrak{C}_{t_{1}}(p_{1},U^{i_{1},i_{1}}_{t_{1}})\mathfrak{C}_{t_{2}}(p_{2},\overline{U}^{i_{2},i_{2}}_{t_{2}}).$

When $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$, this follows directly by
Definition 4.2-(1) and -(2): The matching $p=p_{1}\cup p_{2}$ only contains
pairs $\\{\ell_{1},\ell_{2}\\}$ such that $J_{\ell_{1}}=J_{\ell_{2}}$ and/or
$J_{\ell_{1}}=J_{\ell_{2}}^{*}$ if and only if the same is true for both
$p_{1}$ and $p_{2}$. When $\mathbb{F}=\mathbb{H}$, by Definition 4.2-(3) we
obtain (9.24) if we additionally ensure that

$\displaystyle\mathfrak{D}_{|\boldsymbol{t}|}(p,U^{\boldsymbol{i},\boldsymbol{i}}_{\boldsymbol{t}})=\mathfrak{D}_{|\boldsymbol{t}|}(p_{1}\cup
p_{2},U^{\boldsymbol{i},\boldsymbol{i}}_{\boldsymbol{t}})=\mathfrak{D}_{t_{1}}(p_{1},U^{i_{1},i_{1}}_{t_{1}})\mathfrak{D}_{t_{2}}(p_{2},\overline{U}^{i_{2},i_{2}}_{t_{2}})$

whenever $J_{\ell_{1}}=J_{\ell_{2}}$ or $J_{\ell_{1}}=J_{\ell_{2}}^{*}$ for
all $\\{\ell_{1},\ell_{2}\\}\in p$. By (4.1), this reduces to

(9.25) $\sum_{m\in\mathcal{B}^{0,0}_{N(|\boldsymbol{t}|)},~{}m\text{ respects
}(p,J)}(-1)^{f(m,p,J)}=\sum_{m\in\mathcal{B}^{0,0}_{N(|\boldsymbol{t}|)},~{}m\text{
respects }(p_{1}\cup p_{2},J)}(-1)^{f(m,p_{1}\cup p_{2},J)}\\\
=\left(\sum_{m_{1}\in\mathcal{B}^{0,0}_{N(t_{1})},~{}m_{1}\text{ resp.
}(p_{1},J_{1})}(-1)^{f(m_{1},p_{1},J_{1})}\right)\left(\sum_{m_{2}\in\mathcal{B}^{0,0}_{N(|\boldsymbol{t}|)-N(t_{1})},~{}m_{2}\text{
resp.
}(p_{2},\overline{J}_{2})}(-1)^{f(m_{2},p_{2},\overline{J}_{2})}\right),$

where $J_{1}$ denotes the jumps in the path $U^{i_{1},i_{1}}_{t_{1}}$, and
$\overline{J}_{2}$ the jumps in the path $\overline{U}^{i_{2},i_{2}}_{t_{2}}$.
If $m\in\mathcal{B}^{0,0}_{N(|\boldsymbol{t}|)}$ can be written as the
concatenation of $m_{1}\in\mathcal{B}^{0,0}_{N(t_{1})}$ and
$m_{2}\in\mathcal{B}^{0,0}_{N(|\boldsymbol{t}|)-N(t_{1})}$, then clearly
$f(m_{1},p_{1},J_{1})+f(m_{2},p_{2},\overline{J}_{2})=f(m,p,J)$ since flips
can only occur in pairs matched by $p=p_{1}\cup p_{2}$; moreover, the fact
that $m_{1}$ respects $(p_{1},J_{1})$ and $m_{2}$ respects
$(p_{2},\overline{J}_{2})$ implies that $m$ respects $(p,J)$. Thus, (9.25)
follows if we show that every $m\in\mathcal{B}^{0,0}_{N(|\boldsymbol{t}|)}$
that respects $(p_{1}\cup p_{2},J)$ can be written as a concatenation of
$m_{1}\in\mathcal{B}^{0,0}_{N(t_{1})}$ which respects $(p_{1},J_{1})$ and
$m_{2}\in\mathcal{B}^{0,0}_{N(|\boldsymbol{t}|)-N(t_{1})}$ which respects
$(p_{2},\overline{J}_{2})$. For this, we recall from Definition 4.2-(3.1) and
-(3.2) that in order for $m$ to respect $(p_{1}\cup p_{2},J)$, the steps in
$m$ that are paired up by $p$ must be taken from the following set of pairs of
steps:

$\Big{\\{}\big{\\{}(0,0),(1,1)\big{\\}},\big{\\{}(0,1),(1,0)\big{\\}},\big{\\{}(0,0),(0,0)\big{\\}},\big{\\{}(1,1),(1,1)\big{\\}}\Big{\\}}.$

In the above, the only steps that move from $0$ to $1$ or vice versa are
$(0,1)$ and $(1,0)$, and these must be paired by $p$. In particular, any
uninterrupted segment of $m$ that only contains pairs of steps matched by $p$
must start and end and the same element. Consequently, the first $N(t_{1})$
steps in $m$, which are matched by $p_{1}$, must form a path in
$\mathcal{B}^{0,0}_{N(t_{1})}$ that respects $(p_{1},J_{1})$, and the
following $N(|\boldsymbol{t}|)-N(t_{1})$ steps in $m$, which are matched by
$p_{2}$, must form a path in
$\mathcal{B}^{0,0}_{N(|\boldsymbol{t}|)-N(t_{1})}$ that respects
$(p_{2},\overline{J}_{2})$. This then concludes the proof of (9.24).

Thanks to (9.24), we now see that every
$p\in\mathcal{P}_{N(|\boldsymbol{t}|)}$ that we can write $p=p_{1}\cup p_{2}$
for some $p_{1}\in\mathcal{P}_{N(t_{1})}$ and
$p_{2}\in\mathcal{P}_{N(|\boldsymbol{t}|)-N(t_{1})}$ is cancelled out by the
difference in (9.23). Thus, if we let $\mathcal{R}_{m_{1},m_{2}}$ denote the
set of $p\in\mathcal{P}_{m_{1}+m_{2}}$ that cannot be written as $p=p_{1}\cup
p_{2}$ for some $p_{1}\in\mathcal{P}_{m_{1}}$ and
$p_{2}\in\mathcal{P}_{m_{2}}$, then (9.23) simplifies to

$\mathfrak{m}^{\boldsymbol{\zeta}}_{\boldsymbol{t}}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})-\mathfrak{m}^{\zeta}_{t_{1}}(A^{a_{1},a_{1}}_{t_{1}})\mathfrak{m}^{\zeta}_{t_{2}}(A^{a_{2},a_{2}}_{t_{2}})\\\
=\mathrm{e}^{(r-1)|\boldsymbol{t}|}\mathbf{1}_{\\{N(|\boldsymbol{t}|)\text{ is
even}\\}}\sum_{p\in\mathcal{R}_{N(t_{1}),N(|\boldsymbol{t}|)-N(t_{1})}}\mathfrak{C}_{|\boldsymbol{t}|}(p,U^{\boldsymbol{i},\boldsymbol{i}}_{\boldsymbol{t}})\prod_{\\{\ell_{1},\ell_{2}\\}\in
p}\upsilon^{2}\varrho_{\zeta}\big{(}Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(\tau_{\ell_{1}})-Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(\tau_{\ell_{2}})\big{)}.$

At this point, if we combine (4.2) with the fact that $\mathcal{R}_{0,0}$ is
empty (as $\varnothing=\varnothing\cup\varnothing$), then we obtain the bound

(9.26)
$\big{|}\mathfrak{m}^{\boldsymbol{\zeta}}_{\boldsymbol{t}}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})-\mathfrak{m}^{\zeta}_{t_{1}}(A^{a_{1},a_{1}}_{t_{1}})\mathfrak{m}^{\zeta}_{t_{2}}(A^{a_{2},a_{2}}_{t_{2}})\big{|}\\\
\leq\mathrm{e}^{(r-1)|\boldsymbol{t}|}\mathbf{1}_{\\{N(|\boldsymbol{t}|)\geq
2\text{ is
even}\\}}\sum_{p\in\mathcal{R}_{N(t_{1}),N(|\boldsymbol{t}|)-N(t_{1})}}\prod_{\\{\ell_{1},\ell_{2}\\}\in
p}\upsilon^{2}\varrho_{\zeta}\big{(}Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(\tau_{\ell_{1}})-Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(\tau_{\ell_{2}})\big{)}.$

Thus, plugging this back into (9.7) yields

(9.27) $|\eqref{Equation: Covariance Formula Piece
2}|\leq\int_{\mathcal{A}^{2}}\Pi_{A}(\boldsymbol{t};\boldsymbol{a},\boldsymbol{a})\mathbf{E}\Bigg{[}\mathrm{e}^{(r-1)|\boldsymbol{t}|}\mathbf{1}_{\\{N(|\boldsymbol{t}|)\geq
2\text{ is even}\\}}\\\
\cdot\sum_{p\in\mathcal{R}_{N(t_{1}),N(|\boldsymbol{t}|)-N(t_{1})}}\prod_{\\{\ell_{1},\ell_{2}\\}\in
p}\upsilon^{2}\varrho_{\zeta}\big{(}Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(\tau_{\ell_{1}})-Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(\tau_{\ell_{2}})\big{)}\\\
\cdot\mathrm{e}^{-\int_{0}^{|\boldsymbol{t}|}V(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}}(s))~{}\mathrm{d}s+\mathfrak{B}_{|\boldsymbol{t}|}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})+\frac{\sigma^{2}}{2}(\|L_{[0,t_{1})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\|_{\mu}^{2}+\|L_{[t_{1},t_{1}+t_{2})}(A^{\boldsymbol{a},\boldsymbol{a}}_{\boldsymbol{t}})\|_{\mu}^{2})}\Bigg{]}~{}\mathrm{d}\mu^{2}(\boldsymbol{a}).$

Then, if we replicate the bounds on $V$, $\Pi_{Z}$, $\mathfrak{B}_{t}(A)$, and
the local time $L_{[u,v)}(A)$ used to arrive to (9.10)—together with an
application of the tower property in the expectation with respect to $U$—we
obtain from (9.27) that

(9.28) $|\eqref{Equation: Covariance Formula Piece 2}|\leq
C(t_{1}t_{2})^{-1/2}\int_{\mathcal{A}^{2}}\mathbf{E}\Bigg{[}\cdots\\\
\cdot\Pi_{U}(\boldsymbol{t};\boldsymbol{i},\boldsymbol{i})\mathbf{E}_{U^{\boldsymbol{i},\boldsymbol{i}}_{\boldsymbol{t}}}\Bigg{[}\mathbf{1}_{\\{N(|\boldsymbol{t}|)\geq
2\text{ is
even}\\}}\sum_{p\in\mathcal{R}_{N(t_{1}),N(|\boldsymbol{t}|)-N(t_{1})}}\prod_{\\{\ell_{1},\ell_{2}\\}\in
p}\upsilon^{2}\varrho_{\zeta}\big{(}Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(\tau_{\ell_{1}})-Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(\tau_{\ell_{2}})\big{)}\Bigg{]}\\\
\cdot\mathrm{e}^{-\int_{0}^{|\boldsymbol{t}|}\kappa|Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(s)|~{}\mathrm{d}s+\overline{\mathfrak{B}_{|\boldsymbol{t}|}}(Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}})+\frac{\sigma^{2}}{2}(\|L_{t_{1}}(Z^{x_{1},x_{1}}_{t_{1}})\|_{2}^{2}+\|L_{t_{2}}(\overline{Z}^{x_{2},x_{2}}_{t_{2}})\|_{2}^{2})}\Bigg{]}~{}\mathrm{d}\mu^{2}(\boldsymbol{a}).$

Using the fact that

$\Pi_{U}(\boldsymbol{t};\boldsymbol{i},\boldsymbol{i})\mathbf{E}\big{[}F(U^{\boldsymbol{i},\boldsymbol{i}}_{\boldsymbol{t}})\big{]}=\mathbf{E}\big{[}F(U^{\boldsymbol{i}}_{\boldsymbol{t}})\mathbf{1}_{\\{\forall
k:~{}U^{\boldsymbol{i}}_{\boldsymbol{t}}(s)\to i_{k}\text{ as
}s\to(t_{0}+\cdots+t_{k})^{+}\\}}\big{]}\leq\mathbf{E}\big{[}F(U^{\boldsymbol{i}}_{\boldsymbol{t}})\big{]}$

for any nonnegative functional $F$, we note that

$\displaystyle\Pi_{U}(\boldsymbol{t};\boldsymbol{i},\boldsymbol{i})\mathbf{E}_{U^{\boldsymbol{i},\boldsymbol{i}}_{\boldsymbol{t}}}\Bigg{[}\mathbf{1}_{\\{N(|\boldsymbol{t}|)\geq
2\text{ is
even}\\}}\sum_{p\in\mathcal{R}_{N(t_{1}),N(|\boldsymbol{t}|)-N(t_{1})}}\prod_{\\{\ell_{1},\ell_{2}\\}\in
p}\upsilon^{2}\varrho_{\zeta}\big{(}Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(\tau_{\ell_{1}})-Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(\tau_{\ell_{2}})\big{)}\Bigg{]}$
(9.29)
$\displaystyle\leq\mathbf{E}_{U^{\boldsymbol{i}}_{\boldsymbol{t}}}\Bigg{[}\mathbf{1}_{\\{N(|\boldsymbol{t}|)\geq
2\text{ is
even}\\}}\sum_{p\in\mathcal{R}_{N(t_{1}),N(|\boldsymbol{t}|)-N(t_{1})}}\prod_{\\{\ell_{1},\ell_{2}\\}\in
p}\upsilon^{2}\varrho_{\zeta}\big{(}Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(\tau_{\ell_{1}})-Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(\tau_{\ell_{2}})\big{)}\Bigg{]}.$

Then, using the law of total expectation, and the facts that
$N(|\boldsymbol{t}|)$ is a Poisson random variable with parameter
$(r-1)|\boldsymbol{t}|$ and that conditional on $N(t_{1})=2m$, the jump times
$(\tau_{1},\ldots,\tau_{2m})$ are the order statistics of i.i.d. uniform
random points on $[0,|\boldsymbol{t}|)^{2m}$, we can bound (9.29) by:

(9.30) $\eqref{Equation: Covariance Formula Piece 2 -
7}\leq\sum_{m=1}^{\infty}\Bigg{(}\int_{[0,|\boldsymbol{t}|)^{2m}_{<}}\sum_{p\in\mathcal{R}_{m_{1}(\boldsymbol{s}),m_{2}(\boldsymbol{s})}}\left\\{\prod_{\\{\ell_{1},\ell_{2}\\}\in
p}\upsilon^{2}\varrho_{\zeta}\big{(}Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(s_{\ell_{1}})-Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(s_{\ell_{2}})\big{)}\right\\}\\\
\cdot\frac{(2m)!}{|\boldsymbol{t}|^{2m}}~{}\mathrm{d}\boldsymbol{s}\Bigg{)}\frac{\big{(}(r-1)|\boldsymbol{t}|\big{)}^{2m}}{(2m)!}\mathrm{e}^{-(r-1)|\boldsymbol{t}|},$

where we recall that $[0,|\boldsymbol{t}|)_{<}^{2m}$ denotes the tuples
$\boldsymbol{s}=(s_{1},\ldots,s_{2m})\in[0,|\boldsymbol{t}|)^{2m}$ such that
$s_{1}<s_{2}<\cdots<s_{2m}$, and we use $m_{1}(\boldsymbol{s})$ and
$m_{2}(\boldsymbol{s})$ respectively to denote the number of components of
$\boldsymbol{s}$ that land in $[0,t_{1})$ and $[t_{1},t_{1}+t_{2})$ (i.e.,
$m_{1}(\boldsymbol{s})$ is the largest integer $\ell$ such that
$s_{\ell}<t_{1}$, and $2m=m_{1}(\boldsymbol{s})+m_{2}(\boldsymbol{s})$). If
$p\in\mathcal{R}_{m_{1}(\boldsymbol{s}),m_{2}(\boldsymbol{s})}$, then this
means that there exists at least one pair of indices
$1\leq\ell_{1}<\ell_{2}\leq 2m$ such that $s_{\ell_{1}}\in[0,t_{1})$,
$s_{\ell_{2}}\in[t_{1},|\boldsymbol{t}|)$, and $\\{\ell_{1},\ell_{2}\\}\in p$.
The indicator function that the tuple $\boldsymbol{s}$ satisfies this is
obviously bounded by the sum of indicators

$\mathcal{I}^{t_{1},t_{2}}(\boldsymbol{s})=\sum_{u,v=1}^{2m}\mathbf{1}_{\\{s_{u}\in[0,t_{1})\\}}\mathbf{1}_{\\{s_{v}\in[t_{1},t_{1}+t_{2})\\}}.$

If we combine this upper bound in (9.30) with the replacement of the sum over
$p\in\mathcal{R}_{m_{1}(\boldsymbol{s}),m_{2}(\boldsymbol{s})}$ to the sum
over $p\in\mathcal{P}_{2m}$, then we get the further upper bound

(9.31) $\eqref{Equation: Covariance Formula Piece 2 -
7}\leq\sum_{m=1}^{\infty}\Bigg{(}\int_{[0,|\boldsymbol{t}|)^{2m}_{<}}\sum_{p\in\mathcal{P}_{2m}}\mathcal{I}^{t_{1},t_{2}}(\boldsymbol{s})\left\\{\prod_{\\{\ell_{1},\ell_{2}\\}\in
p}\upsilon^{2}\varrho_{\zeta}\big{(}Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(s_{\ell_{1}})-Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(s_{\ell_{2}})\big{)}\right\\}\\\
\cdot\frac{(2m)!}{|\boldsymbol{t}|^{2m}}~{}\mathrm{d}\boldsymbol{s}\Bigg{)}\frac{\big{(}(r-1)|\boldsymbol{t}|\big{)}^{2m}}{(2m)!}\mathrm{e}^{-(r-1)|\boldsymbol{t}|},$

Now, we note that the function being integrated in (9.31) is invariant with
respect to permuting the entries of $\boldsymbol{s}$, hence we can rewrite
(9.31) by removing the condition that the $s_{i}$’s be ordered as follows:

$\eqref{Equation: Covariance Formula Piece 2 -
7}\leq\sum_{m=1}^{\infty}\Bigg{(}\int_{[0,|\boldsymbol{t}|)^{2m}}\sum_{p\in\mathcal{P}_{2m}}\mathcal{I}^{t_{1},t_{2}}(\boldsymbol{s})\left\\{\prod_{\\{\ell_{1},\ell_{2}\\}\in
p}\upsilon^{2}\varrho_{\zeta}\big{(}Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(s_{\ell_{1}})-Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(s_{\ell_{2}})\big{)}\right\\}\\\
\cdot\frac{1}{|\boldsymbol{t}|^{2m}}~{}\mathrm{d}\boldsymbol{s}\Bigg{)}\frac{\big{(}(r-1)|\boldsymbol{t}|\big{)}^{2m}}{(2m)!}\mathrm{e}^{-(r-1)|\boldsymbol{t}|}.$

Now, if we cancel out both appearances of $|\boldsymbol{t}|^{2m}$, pull out
the sums over $p$ and with in $\mathcal{I}_{t_{1},t_{2}}(\boldsymbol{s})$, and
then apply a change of variables such that the variables $u$ and $v$ in
$\mathbf{1}_{\\{s_{u}\in[0,t_{1})\\}}\mathbf{1}_{\\{s_{v}\in[t_{1},t_{1}+t_{2})\\}}$
are always $u=1$ and $v=2$, then Tonelli’s theorem yields

$\eqref{Equation: Covariance Formula Piece 2 -
7}\leq\sum_{m=1}^{\infty}(2m-1)!!(2m)(2m-1)\Bigg{(}\int_{[0,t_{1}]\times[t_{1},t_{1}+t_{2}]}\upsilon^{2}\varrho_{\zeta}\big{(}Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(s_{1})-Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(s_{2})\big{)}~{}\mathrm{d}\boldsymbol{s}\Bigg{)}\\\
\Bigg{(}\int_{[0,|\boldsymbol{t}|)^{2}}\upsilon^{2}\varrho_{\zeta}\big{(}Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(s_{1})-Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(s_{2})\big{)}~{}\mathrm{d}\boldsymbol{s}\Bigg{)}^{m-1}\frac{\big{(}(r-1)\big{)}^{2m}}{(2m)!}\mathrm{e}^{-(r-1)|\boldsymbol{t}|},$

where the prefactors in front of the integral come from the fact that
$|\mathcal{P}_{2m}|=(2m-1)!!$ and that there are $(2m)(2m-1)$ summands in
$\mathcal{I}_{t_{1},t_{2}}(\boldsymbol{s})$. Replicating the application of
Young’s convolution inequality in (7.41), we get that for intervals
$\mathcal{K}_{1},\mathcal{K}_{2}\subset[0,\infty)$,

$\int_{\mathcal{K}_{1}\times\mathcal{K}_{2}}\upsilon^{2}\varrho_{\zeta}\big{(}Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(s_{1})-Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(s_{2})\big{)}~{}\mathrm{d}\boldsymbol{s}\leq\upsilon^{2}\big{\langle}L_{\mathcal{K}_{1}}(Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}),L_{\mathcal{K}_{2}}(Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}})\big{\rangle}.$

Therefore, we get that (9.29)$\cdot\mathrm{e}^{(r-1)|\boldsymbol{t}|}$ is
bounded above by

$\frac{\big{\langle}L_{t_{1}}(Z^{x_{1},x_{1}}_{t_{1}}),L_{t_{2}}(\overline{Z}^{x_{2},x_{2}}_{t_{2}})\big{\rangle}}{\|L_{|\boldsymbol{t}|}(Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}})\|_{2}^{2}}\sum_{m=1}^{\infty}\frac{(2m-1)!!(2m)(2m-1)}{(2m)!}\left(\frac{(r-1)^{2}\upsilon^{2}\|L_{|\boldsymbol{t}|}(Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}})\|_{2}^{2}}{2}\right)^{m}\\\
=(r-1)^{2}\upsilon^{2}\big{\langle}L_{t_{1}}(Z^{x_{1},x_{1}}_{t_{1}}),L_{t_{2}}(\overline{Z}^{x_{2},x_{2}}_{t_{2}})\big{\rangle}\left(\|L_{|\boldsymbol{t}|}(Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}})\|_{2}^{2}(r-1)^{2}\upsilon^{2}+1\right)\mathrm{e}^{\frac{1}{2}(r-1)^{2}\upsilon^{2}\|L_{|\boldsymbol{t}|}(Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}})\|_{2}^{2}},$

where the second line follows from direct calculation.

If we now combine this with (9.28) and (9.29), then we obtain

$|\eqref{Equation: Covariance Formula Piece 2}|\leq
C(t_{1}t_{2})^{-1/2}\int_{I^{2}}\mathbf{E}\Bigg{[}\cdots\\\
\cdot\big{\langle}L_{t_{1}}(Z^{x_{1},x_{1}}_{t_{1}}),L_{t_{2}}(\overline{Z}^{x_{2},x_{2}}_{t_{2}})\big{\rangle}\left(\|L_{|\boldsymbol{t}|}(Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}})\|_{2}^{2}(r-1)^{2}\upsilon^{2}+1\right)\mathrm{e}^{\frac{1}{2}(r-1)^{2}\upsilon^{2}\|L_{|\boldsymbol{t}|}(Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}})\|_{2}^{2}}\\\
\cdot\mathrm{e}^{-\int_{0}^{|\boldsymbol{t}|}\kappa|Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}}(s)|~{}\mathrm{d}s+\overline{\mathfrak{B}_{|\boldsymbol{t}|}}(Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}})+\frac{\sigma^{2}}{2}(\|L_{t_{1}}(Z^{x_{1},x_{1}}_{t_{1}})\|_{2}^{2}+\|L_{t_{2}}(\overline{Z}^{x_{2},x_{2}}_{t_{2}})\|_{2}^{2})}\Bigg{]}~{}\mathrm{d}\boldsymbol{x}.$

At this point, we obtain (9.8) for (9.7) by replicating essentially the same
analysis that we performed in (9.15): The only difference is that we replace

$\mathrm{e}^{\sigma^{2}\langle
L_{t_{1}}(Z^{x_{1},x_{1}}_{t_{1}}),L_{t_{2}}(\overline{Z}^{x_{2},x_{2}}_{t_{2}})\rangle}-1$

by

$\big{\langle}L_{t_{1}}(Z^{x_{1},x_{1}}_{t_{1}}),L_{t_{2}}(\overline{Z}^{x_{2},x_{2}}_{t_{2}})\big{\rangle}\left(\|L_{|\boldsymbol{t}|}(Z^{\boldsymbol{x},\boldsymbol{x}}_{\boldsymbol{t}})\|_{2}^{2}(r-1)^{2}\upsilon^{2}+1\right),$

but both of these terms are controlled in the same way. That is, we introduce
an indicator that the paths of $Z^{x_{1},x_{1}}_{t_{1}}$ and
$\overline{Z}^{x_{2},x_{2}}_{t_{2}}$ must intersect, and then use Cauchy-
Schwarz with Lemma 9.4. With this in hand, the proof of Theorem 4.12, and thus
of all of our results, is complete.

## References

  * [1] Romain Allez and Khalil Chouk.  The continuous anderson hamiltonian in dimension two.  ArXiv Preprint; 1511.02718, 2015.
  * [2] Jinho Baik, Gérard Ben Arous, and Sandrine Péché.  Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices.  Ann. Probab., 33(5):1643–1697, 2005.
  * [3] Patrick Billingsley.  Convergence of probability measures.  Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons, Inc., New York, second edition, 1999.  A Wiley-Interscience Publication.
  * [4] Alex Bloemendal and Bálint Virág.  Limits of spiked random matrices I.  Probab. Theory Related Fields, 156(3-4):795–825, 2013.
  * [5] Alex Bloemendal and Bálint Virág.  Limits of spiked random matrices II.  Ann. Probab., 44(4):2726–2769, 2016.
  * [6] Alexander I. Bufetov.  Rigidity of determinantal point processes with the Airy, the Bessel and the gamma kernel.  Bull. Math. Sci., 6(1):163–172, 2016.
  * [7] X. Chen.  Random walk intersections, volume 157 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, RI, 2010.  Large deviations and related topics.
  * [8] Reda Chhaibi and Joseph Najnudel.  Rigidity of the ${\rm Sine}_{\beta}$ process.  Electron. Commun. Probab., 23:Paper No. 94, 8, 2018.
  * [9] Khalil Chouk and Willem van Zuijlen.  Asymptotics of the eigenvalues of the Anderson Hamiltonian with white noise potential in two dimensions.  Ann. Probab., 49(4):1917–1964, 2021.
  * [10] Kai Lai Chung and Zhong Xin Zhao.  From Brownian motion to Schrödinger’s equation, volume 312 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, Berlin, 1995.
  * [11] Robert C. Dalang and Carl Mueller.  Intermittency properties in a hyperbolic Anderson problem.  Ann. Inst. Henri Poincaré Probab. Stat., 45(4):1150–1164, 2009\.
  * [12] Robert C. Dalang, Carl Mueller, and Roger Tribe.  A Feynman-Kac-type formula for the deterministic and stochastic wave equations and other P.D.E.’s.  Trans. Amer. Math. Soc., 360(9):4681–4703, 2008.
  * [13] G. F. De Angelis and G. Jona-Lasinio.  A stochastic description of a spin-${1\over 2}$ particle in a magnetic field.  J. Phys. A, 15(7):2053–2061, 1982.
  * [14] G. F. De Angelis, G. Jona-Lasinio, and M. Sirugue.  Probabilistic solution of Pauli type equations.  J. Phys. A, 16(11):2433–2444, 1983.
  * [15] D. Dereudre, A. Hardy, T. Leblé, and M. Maïda.  DLR equations and rigidity for the sine-beta process.  Comm. Pure Appl. Math., 74(1):172–222, 2021.
  * [16] John D. Dollard and Charles N. Friedman.  Product integration with applications to differential equations, volume 10 of Encyclopedia of Mathematics and its Applications.  Addison-Wesley Publishing Co., Reading, MA, 1979.  With a foreword by Felix E. Browder, With an appendix by P. R. Masani.
  * [17] Laure Dumaz and Cyril Labbé.  Localization of the continuous Anderson Hamiltonian in 1-D.  Probab. Theory Related Fields, 176(1-2):353–419, 2020.
  * [18] Laure Dumaz and Cyril Labbé.  The delocalized phase of the Anderson Hamiltonian in 1-D.  Ann. Probab., 51(3):805–839, 2023.
  * [19] Laure Dumaz and Cyril Labbé.  Localization crossover for the continuous anderson hamiltonian in 1-d.  Inventiones mathematicae, 2023.
  * [20] Laure Dumaz and Cyril Labbé.  Anderson localization for the 1-d Schrödinger operator with white noise potential.  J. Funct. Anal., 286(1):Paper No. 110191, 45, 2024.
  * [21] K.-J. Engel and R. Nagel.  One-parameter semigroups for linear evolution equations, volume 194 of Graduate Texts in Mathematics.  Springer-Verlag, New York, 2000.
  * [22] L. Erdős.  Gaussian decay of the magnetic eigenfunctions.  Geom. Funct. Anal., 6(2):231–248, 1996.
  * [23] László Erdős.  Magnetic Lieb-Thirring inequalities.  Comm. Math. Phys., 170(3):629–668, 1995.
  * [24] Douglas R. Farenick and Barbara A. F. Pidkowich.  The spectral theorem in quaternions.  Linear Algebra Appl., 371:75–102, 2003.
  * [25] Irene Fonseca and Giovanni Leoni.  Modern methods in the calculus of variations: $L^{p}$ spaces.  Springer Monographs in Mathematics. Springer, New York, 2007.
  * [26] Pierre Yves Gaudreau Lamarre.  Semigroups for one-dimensional Schrödinger operators with multiplicative Gaussian noise.  Electron. J. Probab., 26:Paper No. 107, 47, 2021.
  * [27] Pierre Yves Gaudreau Lamarre, Promit Ghosal, Wenxuan Li, and Yuchen Liao.  Rigidity of the Stochastic Airy Operator.  Int. Math. Res. Not. IMRN, 2022.  Online Version: https://doi.org/10.1093/imrn/rnac265.
  * [28] Pierre Yves Gaudreau Lamarre, Promit Ghosal, and Yuchen Liao.  Spectral rigidity of random Schrödinger operators via Feynman-Kac formulas.  Ann. Henri Poincaré, 21(7):2259–2299, 2020.
  * [29] Pierre Yves Gaudreau Lamarre, Promit Ghosal, and Yuchen Liao.  On spatial conditioning of the spectrum of discrete random Schrödinger operators.  J. Spectr. Theory, 12(3):1109–1153, 2022.
  * [30] Pierre Yves Gaudreau Lamarre and Mykhaylo Shkolnikov.  Edge of spiked beta ensembles, stochastic Airy semigroups and reflected Brownian motions.  Ann. Inst. Henri Poincaré Probab. Stat., 55(3):1402–1438, 2019\.
  * [31] Subhroshekhar Ghosh.  Determinantal processes and completeness of random exponentials: the critical case.  Probab. Theory Related Fields, 163(3-4):643–665, 2015.
  * [32] Subhroshekhar Ghosh and Yuval Peres.  Rigidity and tolerance in point processes: Gaussian zeros and Ginibre eigenvalues.  Duke Math. J., 166(10):1789–1858, 2017.
  * [33] Richard D. Gill and Søren Johansen.  A survey of product-integration with a view toward application in survival analysis.  Ann. Statist., 18(4):1501–1555, 1990.
  * [34] Vadim Gorin and Mykhaylo Shkolnikov.  Stochastic Airy semigroup through tridiagonal matrices.  Ann. Probab., 46(4):2287–2344, 2018.
  * [35] J.-C. Gruet and Z. Shi.  The occupation time of Brownian motion in a ball.  J. Theoret. Probab., 9(2):429–445, 1996.
  * [36] Batu Güneysu.  Multiplicative matrix-valued functionals and the continuity properties of semigroups corresponding to partial differential operators with matrix-valued coefficients.  J. Math. Anal. Appl., 380(2):709–725, 2011.
  * [37] Batu Güneysu.  Covariant Schrödinger semigroups on Riemannian manifolds, volume 264 of Operator Theory: Advances and Applications.  Birkhäuser/Springer, Cham, 2017.
  * [38] Alexander E. Holroyd and Terry Soo.  Insertion and deletion tolerance of point processes.  Electron. J. Probab., 18:no. 74, 24, 2013.
  * [39] Yueh-Sheng Hsu and Cyril Labbé.  Asymptotic of the smallest eigenvalues of the continuous Anderson Hamiltonian in $d\leq 3$.  Stoch. Partial Differ. Equ. Anal. Comput., 11(3):1089–1122, 2023\.
  * [40] L. Isserlis.  On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables.  Biometrika, 12(1/2):134–139, 1918.
  * [41] Rajeeva L. Karandikar.  On pathwise stochastic integration.  Stochastic Process. Appl., 57(1):11–18, 1995.
  * [42] Claude Kipnis and Claudio Landim.  Scaling limits of interacting particle systems, volume 320 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, Berlin, 1999.
  * [43] Cyril Labbé.  The continuous Anderson Hamiltonian in $d\leq 3$.  J. Funct. Anal., 277(9):3187–3235, 2019.
  * [44] Tomoyu Matsuda and Willem van Zuijlen.  Anderson hamiltonians with singular potentials.  ArXiv Preprint; 2211.01199, 2023.
  * [45] Toyomu Matsuda.  Integrated density of states of the Anderson Hamiltonian with two-dimensional white noise.  Stochastic Process. Appl., 153:91–127, 2022.
  * [46] James A. Mingo and Roland Speicher.  Free probability and random matrices, volume 35 of Fields Institute Monographs.  Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto, ON, 2017.
  * [47] M. Y. Mo.  Rank 1 real Wishart spiked model.  Comm. Pure Appl. Math., 65(11):1528–1638, 2012.
  * [48] Vassilis G. Papanicolaou.  The probabilistic solution of the third boundary value problem for second order elliptic equations.  Probab. Theory Related Fields, 87(1):27–77, 1990.
  * [49] S. Péché.  The largest eigenvalue of small rank perturbations of Hermitian random matrices.  Probab. Theory Related Fields, 134(1):127–173, 2006.
  * [50] José A. Ramírez, Brian Rider, and Bálint Virág.  Beta ensembles, stochastic Airy spectrum, and a diffusion.  J. Amer. Math. Soc., 24(4):919–944, 2011.
  * [51] M. Reed and B. Simon.  Methods of modern mathematical physics. IV. Analysis of operators.  Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.
  * [52] Michael Reed and Barry Simon.  Methods of modern mathematical physics. II. Fourier analysis, self-adjointness.  Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975.
  * [53] Michael Reed and Barry Simon.  Methods of modern mathematical physics. I.  Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, second edition, 1980.  Functional analysis.
  * [54] Daniel Revuz and Marc Yor.  Continuous martingales and Brownian motion, volume 293 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, Berlin, third edition, 1999.
  * [55] B. Simon.  Schrödinger semigroups.  Bull. Amer. Math. Soc. (N.S.), 7(3):447–526, 1982.
  * [56] Barry Simon.  Operator theory.  A Comprehensive Course in Analysis, Part 4. American Mathematical Society, Providence, RI, 2015.
  * [57] Antonín Slavík.  Product integration, its history and applications, volume 1 of Nec̆as Center for Mathematical Modeling.  Matfyzpress, Prague, 2007.  Corrected translation of the Czech original, Dějiny Matematiky/History of Mathematics, 29.
  * [58] Dong Wang.  The largest sample eigenvalue distribution in the rank 1 quaternionic spiked model of Wishart ensemble.  Ann. Probab., 37(4):1273–1328, 2009.

##  Appendix A Illustrations

Figure 1. This figure shows a realization $A=(U,Z)$. The $y$-axis represents
the output of $Z$, and the colors represent the value of the process $U$ (the
jump times $\tau_{k}$ therein are represented by the symbol $\otimes$). Figure
2. This figure shows a realization of the càdlàg concatenated path
$A^{\boldsymbol{a},\boldsymbol{b}}_{\boldsymbol{t}}$ with
$\boldsymbol{a}=\big{(}(1,1),(3,0)\big{)}$,
$\boldsymbol{b}=\big{(}(1,1),(3,0)\big{)}$, and $\boldsymbol{t}=(1,1)$ (i.e.,
the first path goes from $(i,x)=(1,1)$ to itself in time $1$, and the second
path goes from $(i,x)=(3,0)$ to itself in time $1$). Figure 3. In this figure,
the arrows represent the first six jumps of $U$ by time $t$, namely,
$(1,2),(2,3),(3,1),(1,2),(2,3),(3,1)$. The partition above the arrows serve to
highlight which jumps were paired by the perfect matching
$p=\big{\\{}\\{1,4\\},\\{2,5\\},\\{3,6\\}\big{\\}}$. With this in hand, we
conclude that this particular combination of $(p,U)$ is such that
$\mathfrak{C}_{t}(p,U)\neq 0$ when $\mathbb{F}=\mathbb{R},\mathbb{H}$, since
$J_{1}=J_{4}$, $J_{2}=J_{5}$, and $J_{3}=J_{6}$. However,
$\mathfrak{C}_{t}(p,U)=0$ when $\mathbb{F}=\mathbb{C}$ because paired jumps
are not reversed. Figure 4. In this figure, the six jumps by time $t$ are
$(1,2),(2,1),(1,3),(3,1),(1,2),(2,1)$, and
$p=\big{\\{}\\{1,6\\},\\{2,5\\},\\{3,4\\}\big{\\}}$. In this case,
$\mathfrak{C}_{t}(p,U)\neq 0$ for all
$\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H}$, since every paired jumps are
reversed (i.e., $J_{1}=J_{6}^{*}$, $J_{2}=J_{5}^{*}$, and $J_{3}=J_{4}^{*}$).
Figure 5. In this figure, the black arrows are the steps $J_{k}$ until $N(t)$,
the blue arrows are the steps of a binary sequence
$m\in\mathcal{B}^{0,0}_{N(t)}$, and the partition above all that represents
the pairs given by $p$. In order to help decipher whether this choice of $m$
contributes to $\mathfrak{D}_{t}(p,U)$, the matched pairs of steps such that
$J_{\ell_{1}}=J_{\ell_{2}}$ are colored in green, and the matched pairs such
that $J_{\ell_{1}}=J_{\ell_{2}}^{*}$ are colored in red. In this case, $m$
does not respect $(p,J)$, and thus it does not contribute to
$\mathfrak{D}_{t}(p,U)$. (E.g., $J_{4}$ and $J_{5}$ are matched and in
opposite directions, but the corresponding steps in the binary sequence,
$\big{(}(0,0),(0,1)\big{)}$, are not among those listed in Definition
4.2-(3.2)) Figure 6. In this figure, $m$ contributes to
$\mathfrak{D}_{t}(p,U)$, since every matched pairs of jumps
$J_{\ell_{1}},J_{\ell_{2}}$ are accompanied by steps in the binary sequence
that satisfy Definition 4.2-(3.1) and -(3.2). Moreover, there is one flip that
occurs in this sequence: $J_{1}=J_{6}$ and
$\big{(}(m_{0},m_{1}),(m_{5},m_{6})\big{)}=\big{(}(0,1),(1,0)\big{)}$; hence
$f(m,p,J)=1$. Figure 7. This is another example where $m$ contributes to
$\mathfrak{D}_{t}(p,U)$. Moreover, in this case we have $f(m,p,J)=2$; the two
flips occur at the pairs $J_{1},J_{5}$ and $J_{2},J_{6}$. Figure 8. In this
figure, the colored path represents a realization of $(\widehat{U},Z)$ with
$Z(0)=1=Z(1)$. The symbols $\otimes$ along the path represent the jump times
$\widehat{\tau}_{k}$, and the different colors represent the values of the
process $\widehat{U}(t)$. The jumps $J_{k}$ and the random matching
$\widehat{\pi}(P)$ (the latter of which is represented by the partition above
$Z$’s path) are the same as $J_{k}$ and $p$ in Figure 4. In this illustration,
we observe that $Z(\widehat{\tau}_{\ell_{1}})=Z(\widehat{\tau}_{\ell_{2}})$
for every matched pair $\\{\ell_{1},\ell_{2}\\}\in\widehat{\pi}(P)$, which is
consistent with the fact that the pairs $\widehat{\tau}_{\ell_{1}}$ and
$\widehat{\tau}_{\ell_{2}}$ were sampled according to $Z$’s self-intersection
measure.