is precisely the tilt that gets rid of the divergent term appearing in the
expectation of the left-hand side of (5.10). To be precise, we have that

$\displaystyle\mathbf{E}_{SB^{(2k)}_{N^{1/2}\nu}}\bigg{[}\prod_{i=1}^{2k}e^{-\frac{1}{2}s_{i}\sqrt{N}+N^{\frac{1}{4}}X_{s_{i}}^{i}}\phi_{i}(X_{s_{i}}^{i}-N^{\frac{1}{4}}s_{i})\bigg{]}$
(5.34)
$\displaystyle=\mathbf{E}_{SB^{(2k)}_{N^{1/2}\nu}}\bigg{[}e^{\overline{\mathcal{G}}_{N}(T)-\tfrac{1}{2}\langle\overline{\mathcal{G}}_{N}\rangle(T)}\exp\bigg{(}N^{\frac{1}{2}}\sum_{r=1}^{k}\sum_{2r-1\leq
i<j\leq
2k}\int_{t_{r-1}}^{t_{r}}\mathbf{1}_{\\{X_{u}^{i}=X_{u}^{j}\\}}du\bigg{)}\prod_{r=1}^{2k}\phi_{r}(X_{s_{r}}^{r})\bigg{]}.$

Note that $N^{1/4}\mathcal{G}_{N}^{r}$ satisfies the assumption (3.3).
Consequently, $\overline{\mathcal{G}}_{N}$ satisfies the assumption (3.3) as
well. From Theorem 3.2 (a), it is immediate that, as $N\to\infty$, the right-
hand side of (5.34) converges to the right-hand side of (5.10). This proves
(5.10) modulo (5.34). Observe that by Lemma 2.4 and Proposition 2.6, an
application of Hölder’s inequality yields that the right-hand side of (5.34)
is uniformly bounded as $\vec{t}$ varies in $\Delta_{k}(0,t)$. Thus by the
dominated convergence theorem, we have (5.12).

We now turn towards the proof of (5.34) which is done by iteratively applying
(2.5). We illustrate the proof for the $k=2$ case only; the general case
follows in the exact same manner. Let us set $k=2$ and write $\mathbf{E}$ for
the expectation with respect to ${SB^{(2k)}_{N^{1/2}\nu}}$ measure. Recall
that $s_{2i-1}=s_{2i}=t_{i}$ for $i=1,2$. Let $(\mathcal{F}_{t})_{t\leq T}$
denote the filtration of the $2k$-point sticky Brownian motion. Note that the
expression inside the first expectation in (5.34) can be written as
$\mathbf{A}\cdot\mathbf{B}$ where

$\displaystyle\mathbf{A}$
$\displaystyle:=e^{-2t_{1}\sqrt{N}+N^{1/4}\sum_{j=1}^{4}X_{t_{1}}^{j}}\prod_{i=1}^{2}\phi_{i}(X_{s_{i}}^{i}-N^{\frac{1}{4}}s_{i})$
$\displaystyle\mathbf{B}$
$\displaystyle:=e^{-(t_{2}-t_{1})\sqrt{N}+N^{1/4}(X_{t_{2}}^{3}+X_{t_{2}}^{4}-X_{t_{1}}^{3}-X_{t_{1}}^{4})}\cdot\prod_{i=3}^{4}\phi_{i}(X_{s_{i}}^{i}-N^{\frac{1}{4}}s_{i}).$

$\mathbf{A}$ is measurable with respect to $\mathcal{F}_{t_{1}}$. By the
Markov property of sticky Brownian motion
$\mathbf{E}[\mathbf{B}\mid\mathcal{F}_{t_{1}}]$ is measurable with respect to
$\sigma\big{(}X_{t_{1}}^{i})_{i=1}^{4}\big{)}$. Let us write

$\displaystyle
H\big{(}\mathbf{X}_{t_{1}}\big{)}:=\mathbf{E}[\mathbf{B}\mid\mathcal{F}_{t_{1}}].$

By the tower property of conditional expectation followed by an application of
(2.5) with $\lambda=N^{\frac{1}{4}}$, $s=0$, $t=t_{1}$, $\nu\mapsto
N^{\frac{1}{2}}\nu$, and $k=4$ we get

l.h.s. of (5.34) $\displaystyle=\mathbf{E}\left[\mathbf{A}\cdot
H\big{(}\mathbf{X}_{t_{1}}\big{)}\right]$ (5.35)
$\displaystyle=\mathbf{E}\left[e^{N^{\frac{1}{4}}\mathcal{G}_{N}^{1}(t_{1})-N^{\frac{1}{2}}\langle\mathcal{G}_{N}^{1}\rangle(t_{1})}\exp\bigg{(}N^{\frac{1}{2}}\sum_{1\leq
i<j\leq
4}\int_{t_{1}}^{t_{2}}\mathbf{1}_{\\{X_{u}^{i}=X_{u}^{j}\\}}du\bigg{)}\right.$
$\displaystyle\hskip
85.35826pt\left.\cdot\prod_{i=1}^{2}\phi_{i}(X_{s_{i}}^{i})\cdot
H\big{(}({X}_{t_{1}}^{i}+N^{\frac{1}{4}}t_{1})_{i=1}^{k}\big{)}\right].$

Observe that by the Markov property of the sticky Brownian motion, we have

$\displaystyle
H\big{(}({X}_{t_{1}}^{i}+N^{\frac{1}{4}}t_{1})_{i=1}^{k}\big{)}=\mathbf{E}\left[e^{-(t_{2}-t_{1})\sqrt{N}+N^{\frac{1}{4}}(X_{t_{2}}^{3}+X_{t_{2}}^{4}-X_{t_{1}}^{3}-X_{t_{1}}^{4})}\prod_{i=3}^{4}\phi_{i}(X_{s_{i}}^{i}-N^{\frac{1}{4}}(t_{2}-t_{1}))\mid\mathcal{F}_{t_{1}}\right]\\!.\\!$
(5.36)

We use (2.5) with $\lambda=N^{\frac{1}{4}}$, $s=t_{1}$, $t=t_{2}$, $\nu\mapsto
N^{\frac{1}{2}}\nu$, and $k=2$ to get that

r.h.s. of (5.36)
$\displaystyle=\mathbf{E}\left[e^{N^{\frac{1}{4}}\mathcal{G}_{N}^{2}(t_{1},t_{2})-N^{\frac{1}{2}}\langle\mathcal{G}_{N}^{2}\rangle(t_{1},t_{2})}\cdot\prod_{i=3}^{4}\phi_{i}(X_{s_{i}}^{i})\exp\bigg{(}N^{\frac{1}{2}}\int_{t_{1}}^{t_{2}}\mathbf{1}_{\\{X_{u}^{3}=X_{u}^{4}\\}}du\bigg{)}\mid\mathcal{F}_{t_{1}}\right],$

where $g(s,t):=g(t)-g(s)$. Inserting the above expression back in (5.35) and
again using the tower property of the conditional expectation, we derive the
identity in (5.34).

Proof of (5.13). Following the same argument as in the proof of (5.10), in the
same spirit as (5.34), one has that

$\displaystyle\mathbf{E}_{SB^{(2k)}_{N^{1/2}\nu}}\bigg{[}\prod_{i=1}^{2k}e^{-t_{i}\sqrt{N}+N^{\frac{1}{4}}(X_{t_{i}}^{2i-1}+X_{t_{i}}^{2i})}\phi_{i}(X_{t_{i}}^{2i}-N^{\frac{1}{4}}t_{i})\mathbf{1}_{\\{X_{t_{i}}^{2i-1}=X_{t_{i}}^{2i}\\}}\bigg{]}$
(5.37)
$\displaystyle=\mathbf{E}_{SB^{(2k)}_{N^{1/2}\nu}}\Bigg{[}e^{\overline{\mathcal{G}}_{N}(T)-\tfrac{1}{2}\langle\overline{\mathcal{G}}_{N}\rangle(T)}$
$\displaystyle\hskip
85.35826pt\cdot\exp\bigg{(}N^{\frac{1}{2}}\sum_{r=1}^{k}\sum_{2r-1\leq i<j\leq
2k}\int_{s_{r-1}}^{s_{r}}\mathbf{1}_{\\{X_{u}^{i}=X_{u}^{j}\\}}du\bigg{)}\prod_{r=1}^{k}\phi_{i}(X_{t_{r}}^{2r})\mathbf{1}_{\\{X_{t_{i}}^{2i-1}=X_{t_{i}}^{2i}\\}}\Bigg{]}.$

Set

$\displaystyle\mathbf{A}_{N}(\vec{t}):=e^{\overline{\mathcal{G}}_{N}(T)-\tfrac{1}{2}\langle\overline{\mathcal{G}}_{N}\rangle(T)},\quad\mathbf{B}_{N}(\vec{t}):=\exp\bigg{(}N^{\frac{1}{2}}\sum_{r=1}^{k}\sum_{2r-1\leq
i<j\leq
2k}\int_{t_{r-1}}^{t_{r}}\mathbf{1}_{\\{X_{u}^{i}=X_{u}^{j}\\}}du\bigg{)}.$
(5.38)

Recall $\gamma_{N}$ from (3.7). By Theorem 3.2 (a) and (b), any limit point of
the sequence

$(\overline{\mathcal{G}}_{N},\mathbf{X},N^{1/2}\nu([0,1])\big{(}V^{ij}\big{)}_{1\leq
i<j\leq 2k},\gamma_{N})$

is of the form

$\displaystyle\left(\overline{\mathcal{G}},\mathbf{U},\big{(}L_{0}^{U_{i}-U_{j}}\big{)}_{1\leq
i<j\leq k},\sigma^{k}\prod_{j=1}^{k}dL_{0}^{U^{2j-1}-U^{2j}}(u_{j})\right)$

for some $\overline{\mathcal{G}}$ with
$\mathbf{E}[e^{p\overline{\mathcal{G}}(T)-\frac{p}{2}\langle\overline{\mathcal{G}}\rangle(T)}]<\infty$
for all $p>0$ and satisfying (3.4). Here $U$ is standard $2k$-dimensional
Brownian motion. By continuous mapping theorem, it follows that along a
subsequence we have

$\displaystyle
N^{\frac{k}{2}}\int_{\Delta_{k}(s,t)}D_{N}(\vec{t})\cdot\prod_{i=1}^{k}\mathbf{1}_{\\{X_{t_{i}}^{2i-1}=X_{t_{i}}^{2i}\\}}dt_{i}$
(5.39) $\displaystyle\hskip 142.26378pt\stackrel{{\scriptstyle
d}}{{\to}}\sigma^{k}\int_{\Delta_{k}(s,t)}e^{\overline{\mathcal{G}}(T)-\tfrac{1}{2}\langle\overline{\mathcal{G}}\rangle(T)+\sigma\mathcal{L}_{k}(\vec{t})}\prod_{i=1}^{k}\phi_{i}(U_{t_{i}}^{2i})dL_{0}^{U^{2i-1}-U^{2i}}(t_{i}),$

where

$\displaystyle
D_{N}(\vec{t}):=\mathbf{A}_{N}(\vec{t})\cdot\mathbf{B}_{N}(\vec{t})\prod_{r=1}^{k}\phi_{i}(X_{t_{r}}^{2r}).$
(5.40)

We now upgrade (5.39) to imply convergence of first moments. To do this, we
shall show the sequence of random variables in the left-hand side of (5.39) is
uniformly integrable. Repeatedly using Proposition 2.6 and Doob’s martingale
inequality to the stochastic exponentials of each of the martingales
$\mathcal{G}_{N}^{r}$ appearing in the expression for
$\overline{\mathcal{G}}_{N}$ we find that for all $p\geq 1$, we have

$\sup_{N\geq
1}\mathbf{E}_{SB^{(2k)}_{N^{1/2}\nu}}\bigg{[}\sup_{\vec{t}\in\Delta_{k}(0,T)}\big{(}\mathbf{A}_{N}(\vec{t})\big{)}^{p}\bigg{]}<\infty,\quad\sup_{N\geq
1}\mathbf{E}_{SB^{(2k)}_{N^{1/2}\nu}}\bigg{[}\sup_{\vec{t}\in\Delta_{k}(0,T)}\big{(}\mathbf{B}_{N}(\vec{t})\big{)}^{p}\bigg{]}<\infty,$
(5.41)

and thus via the definition of $D_{N}(\vec{t})$ from (5.40) we get

$\sup_{N\geq
1}\mathbf{E}_{SB^{(2k)}_{N^{1/2}\nu}}\bigg{[}\sup_{\vec{t}\in\Delta_{k}(0,T)}\big{(}D_{N}(\vec{t})\big{)}^{p}\bigg{]}<\infty.$

In view of the above bound and (3.13), we obtain uniform integrability for the
sequence of random variables in (5.39). This implies that along the same
subsequence,

$\displaystyle\lim_{N\to\infty}N^{\frac{k}{2}}\mathbf{E}\left[\int_{\Delta_{k}(s,t)}D_{N}(\vec{t})\cdot\prod_{i=1}^{k}\mathbf{1}_{\\{X_{t_{i}}^{2i-1}=X_{t_{i}}^{2i}\\}}dt_{i}\right]$
$\displaystyle\hskip
28.45274pt=\sigma^{k}\mathbf{E}\left[\int_{\Delta_{k}(s,t)}e^{\overline{\mathcal{G}}(T)-\tfrac{1}{2}\langle\overline{\mathcal{G}}\rangle(T)+\sigma\mathcal{L}_{k}(\vec{t})}\prod_{i=1}^{k}\phi_{i}(U_{t_{i}}^{2i})dL_{0}^{U^{2i-1}-U^{2i}}(t_{i})\right]$
$\displaystyle.\hskip
28.45274pt=\sigma^{k}\mathbf{E}\left[\int_{\Delta_{k}(s,t)}e^{\sigma\mathcal{L}_{k}(\vec{t})}\mathbf{E}\left[e^{\overline{\mathcal{G}}(T)-\tfrac{1}{2}\langle\overline{\mathcal{G}}\rangle(T)}\mid\mathcal{F}_{T}(\mathbf{U})\right]\prod_{i=1}^{k}\phi_{i}(U_{t_{i}}^{2i})dL_{0}^{U^{2i-1}-U^{2i}}(t_{i})\right]$

where the last equality follows from the tower property of the conditional
expectation. By (3.4), the above inner expectation is $1$. Since the limit is
free of $\overline{\mathcal{G}}$, we have the same above limit along every
subsequence. Thus we arrive at the (5.13) formula. This completes the proof. ∎

We end this section by recording a couple of useful estimates for moments of
the increments of the quadratic martingale field.

######  Proposition 5.6 (Estimates for moments of the increments of QMF).

Fix $k\in\mathbb{N}$ and $T>0$. Then there exists a constant
$\mathrm{C}=\mathrm{C}(k,T)>0$ such that for all bounded measurable functions
$\phi$ on $\mathbb{R}$ and all $0\leq s<t\leq T$ one has that

$\displaystyle\sup_{N\geq
1}\mathbb{E}\big{[}(Q_{t}^{N}(\phi)-Q_{s}^{N}(\phi))^{k}\big{]}\leq\mathrm{C}\|\phi\|^{k}_{L^{\infty}(\mathbb{R})}(t-s)^{k/4}.$
(5.42)

Furthermore fix $p>1$ and $\varepsilon>0$. Then there exists
$\mathrm{C}=\mathrm{C}(p,\varepsilon,k,T)>0$ such that for all functions
$\phi\in L^{p}(\mathbb{R})$ and all $\varepsilon\leq s<t\leq T$ one has

$\displaystyle\lim_{N\to\infty}\mathbb{E}\big{[}(Q_{t}^{N}(\phi)-Q_{s}^{N}(\phi))^{k}\big{]}\leq\mathrm{C}\|\phi\|_{L^{p}(\mathbb{R})}^{k}(t-s)^{k}.$
(5.43)

We remark that the first bound is crude and nowhere near optimality. However,
the latter bound is more important, and it will be most powerful when $p$ is
very close to $1,$ as this will allow us to obtain optimal tightness bounds
for limit points in the next section.

###### Proof.

Proof of (5.42). Using (5.5) together with the trivial bound
$|\phi(N^{-1/2}X^{j}_{t_{j}}-N^{1/4}t_{j})|\leq\|\phi\|_{L^{\infty}},$ we
obtain that

$\displaystyle\mathbb{E}[\big{(}Q_{t}^{N}(\phi)-Q_{s}^{N}(\phi)\big{)}^{k}]$
$\displaystyle\leq
N^{k/2}k!\|\phi\|_{L^{\infty}}^{k}\int_{\Delta_{k}(s,t)}\mathbf{E}_{SB^{(2k)}_{N^{1/2}\nu}}\bigg{[}\prod_{j=1}^{k}e^{-t_{j}\sqrt{N}+N^{1/4}(X^{2j-1}_{t_{j}}+X^{2j}_{t_{j}})}\mathbf{1}_{\\{X_{t_{j}}^{2j-1}=X_{t_{j}}^{2j}\\}}\bigg{]}dt_{1}\cdots
dt_{k}$ $\displaystyle=k!\|\phi\|_{L^{\infty}}^{k}\cdot
N^{k/2}\int_{\Delta_{k}(s,t)}\mathbf{E}_{SB^{(2k)}_{N^{1/2}\nu}}\bigg{[}\mathbf{A}_{N}(\vec{t})\cdot\mathbf{B}_{N}(\vec{t})\cdot\prod_{j=1}^{k}\mathbf{1}_{\\{X_{t_{j}}^{2j-1}=X_{t_{j}}^{2j}\\}}\bigg{]}dt_{1}\cdots
dt_{k},$ (5.44)

where in above we use the same notation from (5.38):

$\displaystyle\mathbf{A}_{N}(\vec{t}):=e^{\overline{\mathcal{G}}_{N}-\tfrac{1}{2}\langle\overline{\mathcal{G}}_{N}\rangle},\quad\mathbf{B}_{N}(\vec{t}):=\exp\bigg{(}N^{\frac{1}{2}}\sum_{r=1}^{k}\sum_{2r-1\leq
i<j\leq
2k}\int_{t_{r-1}}^{t_{r}}\mathbf{1}_{\\{X_{u}^{i}=X_{u}^{j}\\}}du\bigg{)},$

and where $\overline{\mathcal{G}}_{N}=\overline{\mathcal{G}}_{N}(t)$ defined
in (5.33). The equality in (5.44) is due to (5.37). Note that by Cauchy-
Schwarz inequality and using the fact that
$\mathbf{A}^{2}\cdot\mathbf{B}^{2}\leq\mathbf{A}^{4}+\mathbf{B}^{4}$, we have

(5.44) $\displaystyle\leq
k!\|\phi\|_{L^{\infty}}^{k}\sqrt{N^{\frac{k}{2}}\int_{\Delta_{k}(s,t)}\mathbf{E}_{SB^{(2k)}_{N^{1/2}\nu}}\bigg{[}(\mathbf{A}_{N}(\vec{t})^{4}+\mathbf{B}_{N}(\vec{t})^{4})\cdot\prod_{j=1}^{k}\mathbf{1}_{\\{X_{t_{j}}^{j}=Y_{t_{j}}^{j}\\}}\bigg{]}dt_{1}\cdots
dt_{k}}$ (5.45) $\displaystyle\hskip
156.49014pt\cdot\sqrt{N^{\frac{k}{2}}\int_{\Delta_{k}(s,t)}\mathbf{E}_{SB^{(2k)}_{N^{1/2}\nu}}\bigg{[}\prod_{j=1}^{k}\mathbf{1}_{\\{X_{t_{j}}^{j}=Y_{t_{j}}^{j}\\}}\bigg{]}dt_{1}\cdots
dt_{k}}$ (5.46)

For the factor in the square root of (5.46), note that

$\displaystyle
N^{\frac{k}{2}}\int_{\Delta_{k}(s,t)}\mathbf{E}_{SB^{(2k)}_{N^{1/2}\nu}}\bigg{[}\prod_{j=1}^{k}\mathbf{1}_{\\{X_{t_{j}}^{j}=Y_{t_{j}}^{j}\\}}\bigg{]}dt_{1}\cdots
dt_{k}$
$\displaystyle\leq\mathbf{E}_{SB^{(2k)}_{N^{1/2}\nu}}\bigg{[}\prod_{j=1}^{k}N^{\frac{1}{2}}\int_{s}^{t}\mathbf{1}_{\\{X_{t}^{j}=Y_{t}^{j}\\}}dt\bigg{]}$
(5.47)
$\displaystyle\leq\prod_{j=1}^{k}\mathbf{E}_{SB^{(2k)}_{N^{1/2}\nu}}\bigg{[}\bigg{(}N^{\frac{1}{2}}\int_{s}^{t}\mathbf{1}_{\\{X_{t}^{j}=Y_{t}^{j}\\}}dt\bigg{)}^{k}\bigg{]}^{\frac{1}{k}}$
$\displaystyle\leq\mathrm{C}|t-s|^{k/2}.$

where the last inequality follows by applying Lemma 2.5 on each of the
expectations. Here the constant $\mathrm{C}>0$ depends only on $k$. We now
focus on the factor in the square root of (5.45). We shall show that this
factor can be bounded uniformly in $N\geq 1$ and $s,t\in[0,T]$. We split it
into two parts: one containing $\mathbf{A}(\vec{t})$ and
$\mathbf{B}(\vec{t})$. Recall the $\gamma_{N}$ measure from (3.7). Note that

$\displaystyle\sup_{N\geq
1}N^{k/2}\int_{\Delta_{k}(s,t)}\mathbf{E}_{SB^{(2k)}_{N^{1/2}\nu}}\bigg{[}\mathbf{A}_{N}(\vec{t})^{4}\prod_{j=1}^{k}\mathbf{1}_{\\{X_{t_{j}}^{j}=Y_{t_{j}}^{j}\\}}\bigg{]}dt_{1}\cdots
dt_{k}$ $\displaystyle\hskip 56.9055pt\leq\sup_{N\geq
1}\mathbf{E}_{SB^{(2k)}_{N^{1/2}\nu}}\bigg{[}\sup_{\vec{t}\in\Delta_{k}(0,T)}\mathbf{A}_{N}(\vec{t})^{4}\int_{\Delta_{k}(s,t)}d\gamma_{N}\bigg{]}$
$\displaystyle\hskip 56.9055pt\leq\sup_{N\geq
1}\sqrt{\mathbf{E}_{SB^{(2k)}_{N^{1/2}\nu}}\bigg{[}\sup_{\vec{t}\in\Delta_{k}(0,T)}\mathbf{A}_{N}(\vec{t})^{8}\bigg{]}\cdot\mathbf{E}_{SB^{(2k)}_{N^{1/2}\nu}}\big{[}\gamma_{N}(\Delta_{k}(0,T))^{2}\big{]}}.$

where the last inequality is due to Cauchy-Schwarz inequality. By (3.13) and
by (5.41), we find that the above expression is finite. On the other hand, for
the $\mathbf{B}(\vec{t})$ term it is clear from Theorem 3.23.2 and (b) that

$N^{k/2}\int_{\Delta_{k}(0,T)}\mathbf{E}_{SB^{(2k)}_{N^{1/2}\nu}}\bigg{[}\mathbf{B}_{N}(\vec{t})^{4}\prod_{j=1}^{k}\mathbf{1}_{\\{X_{t_{j}}^{j}=Y_{t_{j}}^{j}\\}}\bigg{]}dt_{1}\cdots
dt_{k}$

converges to some finite quantity as $N\to\infty$, which implies that it is
bounded independently of $N$. Thus the last two math displays imply that the
term in (5.45) is bounded uniformly in $N\geq 1$ and $s,t\in[0,T]$. From
(5.47), we see that (5.46) $\leq\mathrm{C}|t-s|^{k/4}$. Inserting these two
bounds back in (5.45) and (5.46), we arrive at (5.42).

Proof of (5.43). Appealing to the moment formula for the increment of QMF from
(5.5) and the convergence from (5.13) we have

$\displaystyle\lim_{N\to\infty}\mathbb{E}\bigg{[}\big{(}Q_{t}^{N}(\phi)-Q_{s}^{N}(\phi)\big{)}^{k}\bigg{]}=k!\sigma^{k}\cdot\mathbf{E}_{B^{\otimes
2k}}\bigg{[}\int_{\Delta_{k}(s,t)}e^{\sigma\mathcal{L}_{k}(\vec{t})}\prod_{j=1}^{k}\phi(U_{t_{j}}^{2j})dL_{0}^{U^{2j-1}-U^{2j}}(t_{j})\bigg{]},$
(5.48)

where $\mathcal{L}_{k}(\vec{t})$ is defined in (5.11). Lemma A.2 formalizes
the intuition that
$dL_{0}^{U^{2j-1}-U^{2j}}(t_{j})=\delta_{0}(U_{t_{j}}^{2j-1}-U^{2j}_{t_{j}})$.
Thus by appealing to that lemma, we can write the above expectation in terms
of a certain family of concatenated bridge processes whose law we will write
as $\mathbf{P}_{c}$. Specifically we have that

$\displaystyle\mbox{r.h.s.~{}of
\eqref{e.idef}}=\int_{\Delta_{k}(s,t)}\mathbf{E}_{c}\bigg{[}e^{\sigma\mathcal{L}_{k}(\vec{t})}\prod_{j=1}^{k}\phi(U^{2j}_{t_{j}})\bigg{]}\prod_{j=1}^{k}p_{2t_{j}}(0)dt_{1}\cdots
dt_{k},$ (5.49)

where the expectation is taken over a collection of paths $U^{j}$ such that

  * •

$(U^{2i-1}-U^{2i},U^{2i-1}+U^{2i})_{i=1}^{k}$ are $2k$ many independent
processes.

  * •

$U^{2i-1}+U^{2i}$ is a Brownian motion of diffusion rate $2$ for
$i=1,2,\ldots,k$.

  * •

$U^{2i-1}-U^{2i}$ is a Brownian bridge (from $0$ to $0$) of diffusion rate $2$
from $[0,t_{i}]$ and an independent Brownian motion of diffusion rate $2$ from
$[t_{i},\infty)$ for $i=1,2,\ldots,k$.

We have used a different notation for the expectation operator
$\mathbf{E}_{c}$ in (5.49) ($c$ for concatenation) just to stress that the law
is different from the standard Brownian motion. We claim that for all $p>1$ we
have

$\displaystyle\sup_{\vec{t}\in[0,T]}\mathbf{E}_{c}\bigg{[}e^{\sigma\mathcal{L}_{k}(\vec{t})}\prod_{j=1}^{k}|\phi(U^{2j}_{t_{j}})|\bigg{]}\leq\mathrm{C}\cdot\|\phi\|_{L^{p}}^{k}.$
(5.50)

where the $\mathrm{C}>0$ depends on $p,k,\varepsilon,T$. Let us assume (5.50)
for the moment. By hypothesis, $t_{j}\geq\varepsilon>0$. Thus,
$p_{2t_{j}}(0)\leq\mathrm{C}$ for some constant $\mathrm{C}>0$ depending on
$\varepsilon$. Thus, in view of (5.50), to get an upper bound for the right-
hand side of (5.49), we may take the supremum of the integrand in the right-
hand side of (5.49) and pull it outside of the integration. As the Lebesgue
measure of $\Delta_{k}(s,t)$ is $\frac{(t-s)^{k}}{k!}$, we thus have the
desired estimate in (5.43).

Let us now establish (5.50). Fix $p>1$ and take $q>1$ so that
$p^{-1}+q^{-1}=1$. Use Hölder’s inequality to write

$\displaystyle\mathbf{E}_{c}\bigg{[}e^{\sigma\mathcal{L}_{k}(\vec{t})}\prod_{j=1}^{k}|\phi(U^{2j}_{t_{j}})|\bigg{]}$
$\displaystyle\leq\mathbf{E}_{c}\bigg{[}e^{q\sigma\mathcal{L}_{k}(\vec{t})}\bigg{]}^{1/q}\mathbf{E}_{c}\bigg{[}\prod_{j=1}^{k}|\phi(U^{2j}_{t_{j}})|^{p}\bigg{]}^{1/p}.$

For the first expectation above, observe that by Lemma A.6,
$\mathbf{E}_{c}\bigg{[}e^{q\sigma\mathcal{L}_{k}(\vec{t})}\bigg{]}$ is
uniformly bounded over $\vec{t}\in\Delta_{k}(0,T)$. For the second expectation
above, note that under $\mathbf{E}_{c}$, we have

$U_{t_{j}}^{2j}=\tfrac{1}{2}(U_{t_{j}}^{2j-1}+U_{t_{j}}^{2j})-\tfrac{1}{2}(U_{t_{j}}^{2j-1}-U_{t_{j}}^{2j})=\tfrac{1}{2}(U_{t_{j}}^{2j-1}+U_{t_{j}}^{2j})-0.$

Thus under $\mathbf{E}_{c}$, $U_{t_{j}}^{2j}$ are independent Gaussian random
variables with variance $t_{j}/2$. Hence,

$\displaystyle\mathbf{E}_{c}\bigg{[}\prod_{j=1}^{k}|\phi(U^{2j}_{t_{j}})|^{p}\bigg{]}^{1/p}=\prod_{j=1}^{k}\mathbf{E}_{c}\bigg{[}|\phi(U^{2j}_{t_{j}})|^{p}\bigg{]}^{1/p}=\prod_{j=1}^{k}\bigg{(}\int_{\mathbb{R}}p_{t_{j}/2}(y)|\phi(y)|^{p}\,dy\bigg{)}^{1/p}\leq\mathrm{C}\cdot\|\phi\|_{L^{p}}^{k}.$

where the last inequality follows by again using the fact that
$t_{j}\geq\varepsilon>0$, together with the uniform bound
$\sup_{y}p_{t}(y)\leq(2\pi t)^{-1/2}$, and noting that the constant
$\mathrm{C}$ is allowed to depend on $\varepsilon$. This establishes (5.50)
completing the proof of (5.43). ∎

##  6\. Solving the martingale problem for the SHE

The goal of this section is to establish the tightness of the field
$\mathscr{X}^{N}$ defined in (1.3) and identify its limit points as the
solution of the stochastic heat equation (SHE). As the field $\mathscr{X}^{N}$
does not exist in a functional sense, we first introduce in Section 6.1
appropriate spaces and topologies that work well with the bounds derived in
the previous section. In Section 6.2 and Section 6.3 we deal with the
tightness and identification of the limit points of $\mathscr{X}^{N}$
respectively.

###  6.1. Weighted Hölder spaces and Schauder estimates

In this subsection, we introduce various natural topologies for our field
$\mathscr{X}^{N}$ and its limit points. We then discuss how the heat flow
affects these topologies and record Kolmogorov-type lemmas that will be key in
showing tightness under these topologies. We begin by recalling many familiar
and useful spaces of continuous and differentiable functions that have natural
metric structures. For $d\geq 1$, we denote by
$C_{c}^{\infty}(\mathbb{R}^{d})$ the space of all compactly supported smooth
functions on $\mathbb{R}^{d}$. For a smooth function on $\mathbb{R}^{d}$, we
define its $C^{r}$ norm as

$\displaystyle\|f\|_{C^{r}}:=\sum_{\vec{\alpha}}\sup_{\mathbf{x}\in\mathbb{R}^{d}}|D^{\vec{\alpha}}f(\mathbf{x})|$
(6.1)

where the sum is over all $\vec{\alpha}\in\mathbb{Z}_{\geq 0}^{d}$ with
$\sum\alpha_{i}\leq r$ and
$D^{\vec{\alpha}}:=\partial_{x_{1}}^{\alpha_{1}}\cdots\partial_{x_{d}}^{\alpha_{d}}$
denotes the mixed partial derivative.

We now recall the definition of weighted Hölder spaces from [HL15, Definitions
2.2 and 2.3]. For the remainder of this paper, we shall work with elliptic and
parabolic weighted Hölder spaces with polynomial weight function

$\displaystyle w(x):=(1+x^{2})^{\tau}$ (6.2)

for some fixed $\tau>1$. We introduce these weights because weighted spaces
will be more convenient to obtain tightness estimates. Since the solution of
(SHE) started from Dirac initial condition is known to be globally bounded
away from $(0,0)$, we expect that it is possible to remove the weights
throughout this section, but this would require more precise moment estimates
than the ones we derived in previous sections, which take into account spatial
decay of the fields.

######  Definition 6.1 (Elliptic Hölder spaces).

For $\alpha\in(0,1)$ we define the space $C^{\alpha,\tau}(\mathbb{R})$ to be
the completion of $C_{c}^{\infty}(\mathbb{R})$ with respect to the norm given
by

$\|f\|_{C^{\alpha,\tau}(\mathbb{R})}:=\sup_{x\in\mathbb{R}}\frac{|f(x)|}{w(x)}+\sup_{|x-y|\leq
1}\frac{|f(x)-f(y)|}{w(x)|x-y|^{\alpha}}.$

For $\alpha<0$ we let $r=-\lfloor\alpha\rfloor$ and we define
$C^{\alpha,\tau}(\mathbb{R})$ to be the closure of
$C_{c}^{\infty}(\mathbb{R})$ with respect to the norm given by

$\|f\|_{C^{\alpha,\tau}(\mathbb{R})}:=\sup_{x\in\mathbb{R}}\sup_{\lambda\in(0,1]}\sup_{\phi\in
B_{r}}\frac{(f,S^{\lambda}_{x}\phi)_{L^{2}(\mathbb{R})}}{w(x)\lambda^{\alpha}}$

where the scaling operators $S^{\lambda}_{x}$ are defined by

$\displaystyle S^{\lambda}_{x}\phi(y)=\lambda^{-1}\phi(\lambda^{-1}(x-y)),$
(6.3)

and where $B_{r}$ is the set of all smooth functions of $C^{r}$ norm less than
1 with support contained in the unit ball of $\mathbb{R}$.

######  Definition 6.2 (Function spaces).

Let $C^{\alpha,\tau}(\mathbb{R})$ be as in Definition 6.1. We define
$C([0,T],C^{\alpha,\tau}(\mathbb{R}))$ to be the space of continuous maps
$g:[0,T]\to C^{\alpha,\tau}(\mathbb{R}),$ equipped with the norm

$\|g\|_{C([0,T],C^{\alpha,\tau}(\mathbb{R}))}:=\sup_{t\in[0,T]}\|g(t)\|_{C^{\alpha,\tau}(\mathbb{R})}.$

Here and henceforth we will define $\Lambda_{[a,b]}:=[a,b]\times\mathbb{R}$
and we will define $\Lambda_{T}:=\Lambda_{[0,T]}.$

So far we have used $\phi,\psi$ for test functions on $\mathbb{R}$. To make
the distinction clear between test functions on $\mathbb{R}$ and
$\mathbb{R}^{2}$, we shall use variant Greek letters such as
$\varphi,\vartheta,\varrho$ for test functions on $\mathbb{R}^{2}$. In some
instances, we will explicitly write $(f,\varphi)_{\mathbb{R}^{2}}$ or
$(g,\phi)_{\mathbb{R}}$ when we want to be clear about the space in which we
are pairing.

######  Definition 6.3 (Parabolic Hölder spaces).

We define $C_{c}^{\infty}(\Lambda_{T})$ to be the set of functions on
$\Lambda_{T}$ that are restrictions to $\Lambda_{T}$ of some function in
$C_{c}^{\infty}(\mathbb{R}^{2})$ (in particular we do not impose that elements
of $C_{c}^{\infty}(\Lambda_{T})$ vanish at the boundaries of $\Lambda_{T}).$

For $\alpha\in(0,1)$ we define the space
$C^{\alpha,\tau}_{\mathfrak{s}}(\Lambda_{T})$ to be the completion of
$C_{c}^{\infty}(\Lambda_{T})$ with respect to the norm given by

$\|f\|_{C^{\alpha,\tau}_{\mathfrak{s}}(\Lambda_{T})}:=\sup_{(t,x)\in\Lambda_{T}}\frac{|f(t,x)|}{w(x)}+\sup_{|s-t|^{1/2}+|x-y|\leq
1}\frac{|f(t,x)-f(s,y)|}{w(x)(|t-s|^{1/2}+|x-y|)^{\alpha}}.$

For $\alpha<0$ we let $r=-\lfloor\alpha\rfloor$ and we define
$C^{\alpha,\tau}_{\mathfrak{s}}(\Lambda_{T})$ to be the closure of
$C_{c}^{\infty}(\Lambda_{T})$ with respect to the norm given by

$\|f\|_{C^{\alpha,\tau}_{\mathfrak{s}}(\Lambda_{T})}:=\sup_{(t,x)\in\Lambda_{T}}\sup_{\lambda\in(0,1]}\sup_{\varphi\in
B_{r}}\frac{(f,S^{\lambda}_{(t,x)}\varphi)_{L^{2}(\Lambda_{T})}}{w(x)\lambda^{\alpha}}$

where the scaling operators are defined by

$\displaystyle
S^{\lambda}_{(t,x)}\varphi(s,y)=\lambda^{-3}\varphi(\lambda^{-2}(t-s),\lambda^{-1}(x-y)),$
(6.4)

and where $B_{r}$ is the set of all smooth functions of $C^{r}$ norm less than
1 with support contained in the unit ball of $\mathbb{R}^{2}$.

######  Remark 6.4 (Derivatives of distributions).

By definition, any element $f\in C_{\mathfrak{s}}^{\alpha,\tau}(\Lambda_{T})$
admits an $L^{2}$-pairing with any smooth function
$\varphi:\Lambda_{T}\to\mathbb{R}$ of rapid decay, which we can write as
$(f,\varphi)_{\Lambda_{T}}.$ Consequently there is a natural embedding
$C_{\mathfrak{s}}^{\alpha,\tau}(\Lambda_{T})\hookrightarrow\mathcal{S}^{\prime}(\mathbb{R}^{2})$
which is defined by formally setting $f$ to be zero outside of
$[0,T]\times\mathbb{R}$. More rigorously, this means that the
$L^{2}(\mathbb{R}^{2})$-pairing of $f$ with any
$\varphi\in\mathcal{S}(\mathbb{R}^{2})$ is defined to be equal to
$(f,\varphi|_{\Lambda_{T}})_{\Lambda_{T}}$.

The image of this embedding consists of some specific collection of tempered
distributions that are necessarily supported on $[0,T]\times\mathbb{R}.$
Consequently we can sensibly define $\partial_{t}f$ and $\partial_{x}f$ as
elements of $\mathcal{S}^{\prime}(\mathbb{R}^{2})$ whenever $f\in
C_{\mathfrak{s}}^{\alpha,\tau}(\Lambda_{T})$. Explicitly these derivatives are
defined by the formulas

$(\partial_{t}f,\varphi)_{\Lambda_{T}}:=-(f,\partial_{t}\varphi)_{\Lambda_{T}},\;\;\;\;\;\;\;\;\;\;(\partial_{x}f,\varphi)_{\Lambda_{T}}:=-(f,\partial_{x}\varphi)_{\Lambda_{T}}$

for all smooth $\varphi:\Lambda_{T}\to\mathbb{R}$ of rapid decay. This
convention on derivatives will be useful for certain computations later. From
the definitions, when $\alpha<0$ one can check that for such $f$ one
necessarily has $\partial_{t}f\in
C_{\mathfrak{s}}^{\alpha-2,\tau}(\Lambda_{T})$ and $\partial_{x}f\in
C_{\mathfrak{s}}^{\alpha-1,\tau}(\Lambda_{T}).$

We remark that this fails for $\alpha>0$. Indeed, by our convention of
derivatives, $\partial_{t}f$ may no longer be a smooth function (or even a
function) even if $f\in C_{c}^{\infty}(\Lambda_{T})$. This is because such an
$f$ gets extended to all of $\mathbb{R}^{2}$ by setting it to be zero outside
$\Lambda_{T}$. In particular, if $f$ does not vanish on the boundary of
$\Lambda_{T}$, then it may become a discontinuous function under our
convention of extension to $\mathbb{R}^{2}$. Due to these discontinuities, the
distributional derivative $\partial_{t}f$ can be a tempered distribution with
singular parts along the boundaries (one may verify that $\partial_{t}f$ can
be at best an element of $C^{-2,\tau}_{\mathfrak{s}}(\Lambda_{T})$ for generic
$f\in C_{c}^{\infty}(\Lambda_{T})$). In our later computations, we never take
derivatives of functions in $C_{\mathfrak{s}}^{\alpha,\tau}(\Lambda_{T})$ with
$\alpha>0$.

We now discuss the smoothing effect of heat flow on these weighted Hölder
spaces.

######  Proposition 6.5 (Smoothing effect on elliptic spaces).

For $f\in C_{c}^{\infty}(\mathbb{R})$ and $t>0$ define

$P_{t}f(x):=\int_{\mathbb{R}}p_{t}(x-y)f(y)dy.$

Then for all $\alpha\leq\beta<1$, there exists
$\mathrm{C}=\mathrm{C}(\alpha,\beta,T)>0$ such that

$\|P_{t}f\|_{C^{\beta,\tau}(\mathbb{R})}\leq\mathrm{C}t^{-(\beta-\alpha)/2}\|f\|_{C^{\alpha,\tau}(\mathbb{R})}$

uniformly over $f\in C_{c}^{\infty}$ and $t\in[0,T]$. In particular, $P_{t}$
extends to a globally defined linear operator on $C^{\alpha,\tau}(\mathbb{R})$
which maps boundedly into $C^{\beta,\tau}(\mathbb{R}).$

A proof may be found in [HL15, Lemma 2.8] in the case of an exponential
weight. The proof for polynomial weights is identical.

For the parabolic Hölder spaces, the following lemma states that the heat flow
improves the regularity by a factor of $2$ and provides a Schauder-type
estimate.

######  Proposition 6.6 (Schauder estimate).

For $f\in C_{c}^{\infty}(\Lambda_{T})$ let us define

$\displaystyle Kf(t,x):=\int_{\Lambda_{T}}p_{t-s}(x-y)f(s,y)dsdy,$ (6.5)

where $p_{t}$ is the standard heat kernel for $t>0,x\in\mathbb{R}$ and
$p_{t}(x):=0$ for $t<0.$ Then for all $\alpha<-1$ with
$\alpha\notin\mathbb{Z}$ there exists $\mathrm{C}=\mathrm{C}(\alpha)>0$
independent of $f$ such that

$\|Kf\|_{C^{\alpha+2,\tau}_{\mathfrak{s}}(\Lambda_{T})}\leq\mathrm{C}\cdot\|f\|_{C^{\alpha,\tau}_{\mathfrak{s}}(\Lambda_{T})}.$

In particular $K$ extends to a globally defined linear operator on
$C^{\alpha,\tau}_{\mathfrak{s}}(\Lambda_{T})$ that maps boundedly into
$C^{\alpha+2,\tau}_{\mathfrak{s}}(\Lambda_{T}).$ Furthermore, if $f\in
C^{\alpha,\tau}_{\mathfrak{s}}(\Lambda_{T})$, then
$K(\partial_{t}-\frac{1}{2}\partial_{x}^{2})f=f$.

We remark that the last statement is only true because of our convention on
distributional derivatives that we have explained in Remark 6.4. Without that
convention, that statement would be false even for a smooth function $f$ that
does not vanish on the line $\\{t=0\\}$.

###### Proof.

Let $\varphi$ be any smooth function that equals $1$ in a ball of radius $1$
about the origin in $\mathbb{R}^{2}$ and is supported on a ball of radius $2$
about the origin in $\mathbb{R}^{2}$. For $(t,x)\in\mathbb{R}^{2}$ let us
define $P^{+}(t,x):=p_{t}(x)\varphi(t,x)$ and
$P^{-}(t,x):=p_{t}(x)-P^{+}(t,x)$ so that

$p_{t}(x)=P^{+}(t,x)+P^{-}(t,x).$

We have that $P^{-}$ is a globally smooth function on $\mathbb{R}^{2}$, and
without any loss of generality, we may assume that it is supported on
$\Lambda_{T+1}$ since values of the heat kernel outside $\Lambda_{T}$ do not
matter when convolving with distributions supported on $\Lambda_{T}$. Then we
get a corresponding decomposition $K=K^{+}+K^{-}$ of the operator defined in
the proposition statement.

Since $P^{+}$ is supported on a ball of radius 2, the proof of the weighted
Schauder estimate for $K^{+}$ is sketched in [HL15, Page 8, proof of Corollary
1.2], and we do not repeat it here. That proof is given in an elliptic
setting, but the proof of the parabolic version is no different because the
singularity of the 1+1 dimensional heat satisfies a bound of the form

$|\partial_{t}^{k_{1}}\partial_{x}^{k_{2}}p_{t}(x)|\leq\mathrm{C}(|t|^{1/2}+|x|)^{-1-2k_{1}-k_{2}},$

where $\mathrm{C}$ may depend on $k_{1},k_{2}\geq 0$ but not on
$t,x\in\mathbb{R}$.

Thus it only remains to prove the weighted Schauder estimate for $K^{-}$. As
$P^{-}$ is globally smooth on $\Lambda_{T}$ and has a Gaussian decay at
infinity, we may write

$P^{-}(t,x)=\sum_{n=1}^{\infty}\psi_{n}(t,x)$

where $\psi_{n}$ can be taken to be smooth functions supported on
$[0,T+1]\times\big{(}[n-1,n+1]\cup[-n-1,1-n]\big{)}$ and
$\|\psi_{n}\|_{L^{\infty}}\leq\mathrm{C}e^{-\frac{1}{\mathrm{C}}n^{2}}$, for
some constant $\mathrm{C}>0$ free of $n$. Consequently, if we write
$K^{-}=\sum_{n}K_{n}^{-}$ where $K_{n}^{-}$ corresponds to convolution with
$\psi_{n}$, then the infinite sum of the operator norms
$\sum_{n}\|K_{n}\|_{C^{\alpha,\tau}_{\mathfrak{s}}\to
C^{\alpha+2,\tau}_{\mathfrak{s}}}$ may be shown to be finite using the fact
that Gaussian decay of the $\psi_{n}$ dominates the growth of the polynomial
weights.

Finally, it remains to prove $K(\partial_{t}-\frac{1}{2}\partial_{x}^{2})f=f$
for $f\in C^{\alpha,\tau}_{\mathfrak{s}}(\Lambda_{T})$ with $\alpha<0$. To
prove this, we remark that $Kf$ may actually be defined for any
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{2})$ with support contained in
$[0,T]\times\mathbb{R}$, via the formula

$(Kf,\varphi):=(f,K^{s}*\varphi),$

where
$K^{s}(t,x)=\frac{1}{\sqrt{2\pi|t|}}{e^{-x^{2}/2|t|}}\mathbf{1}_{\\{t\leq
0\\}}$ and $*$ denotes convolution in both space and time. The right side is
well-defined because $f$ is supported on $[0,T]\times\mathbb{R}$, and because
one can ensure that if $\varphi\in\mathcal{S}(\mathbb{R}^{2})$ then
$K^{s}*\varphi$ agrees on that strip with an element of
$\mathcal{S}(\mathbb{R}^{2})$ thanks to the Gaussian decay properties of
$K^{s}$ in the spatial variable (this is our reason for working on
$\Lambda_{T}$ rather than all of $[0,\infty)\times\mathbb{R}$).

Then $Kf$ as defined above will be an element of
$\mathcal{S}^{\prime}(\mathbb{R}^{2})$ that is supported on
$[0,\infty)\times\mathbb{R}$, and this definition of $Kf$ agrees with the one
in the proposition statement because it agrees on smooth test functions using
integration by parts. Now we claim (more generally than stated in the
proposition) that $K(\partial_{t}-\frac{1}{2}\partial_{x}^{2})f=f$ for any
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{2})$ that is supported on
$[0,T]\times\mathbb{R}$. But this just amounts to showing that
$((\partial_{t}-\frac{1}{2}\partial_{x}^{2})f,K^{s}*\varphi)=(f,\varphi)$ for
all such $f$ and all $\varphi\in\mathcal{S}(\mathbb{R}^{2})$. By definition of
$(\partial_{t}-\frac{1}{2}\partial_{x}^{2})f$, this is equivalent to showing
$(f,(-\partial_{t}-\frac{1}{2}\partial_{x}^{2})K^{s}*\varphi)=(f,\varphi)$ for
all such $f$ and all $\varphi\in\mathcal{S}(\mathbb{R}^{2})$. As $f$ is
supported on $[0,T]\times\mathbb{R}$, this reduces to showing that
$(-\partial_{t}-\frac{1}{2}\partial_{x}^{2})K^{s}*\varphi=\varphi$ for all
$\varphi\in\mathcal{S}(\mathbb{R}^{2})$ that are supported on
$[-1,T+1]\times\mathbb{R}$ (for instance), which is clear. ∎

######  Corollary 6.7.

Define $J:C^{\alpha,\tau}(\mathbb{R})\to
C_{\mathfrak{s}}^{\alpha,\tau}(\Lambda_{T})$ by

$Jf(t,x):=\big{(}f,p_{t}(x-\cdot)\big{)}_{\mathbb{R}},$

$J$ is a bounded linear operator for any $\alpha<1$ and any $\tau>0.$ Consider
the operator $\widehat{P}_{t}:=JP_{t}$ with $P_{t}$ defined in Proposition
6.5. By the semigroup property of the heat kernel, we have that

$\displaystyle\widehat{P}_{s}f(t,x)=(Jf)(t+s,x).$ (6.6)

Furthermore, the operators satisfy

$\displaystyle\|\widehat{P}_{t}f\|_{C^{\beta,\tau}_{\mathfrak{s}}(\Lambda_{T})}\leq\mathrm{C}\cdot
t^{-(\beta-\alpha)/2}\|f\|_{C^{\alpha,\tau}(\mathbb{R})},$ (6.7)

where $\mathrm{C}=\mathrm{C}(\alpha,\beta,T)>0$ is independent of $t$ and $f$.

###### Proof.

The first part follows by using the Schauder estimate in Proposition 6.6,
noting that $Jf=K(\delta_{0}\otimes f)$ where for $f\in
C^{\alpha,\tau}(\mathbb{R})$ the latter distribution is defined by
$(\delta_{0}\otimes
f,\varphi)_{\mathbb{R}^{2}}:=(f,\varphi(0,\cdot))_{\mathbb{R}}.$ Directly from
the definitions one can check that $f\mapsto\delta_{0}\otimes f$ is bounded
from $C^{\alpha,\tau}(\mathbb{R})\to
C_{\mathfrak{s}}^{\alpha-2,\tau}(\Lambda_{T}).$ Finally, (6.7) follows from
Proposition 6.5. ∎

######  Remark 6.8.

Although we have only defined it for $0<\alpha<1$, one may actually define
$C_{\mathfrak{s}}^{\alpha,\tau}$ for all $\alpha>0$, as the closure of
$C_{c}^{\infty}(\Lambda_{T})$ with respect to

$\|f\|_{C^{\alpha,\tau}_{\mathfrak{s}}(\Lambda_{T})}:=\sup_{(t,x)\in\Lambda_{T}}\frac{|f(t,x)|}{w(x)}+\sup_{(t,x)\in\Lambda_{T}}\sup_{\lambda\in(0,1]}\sup_{\varphi\in
B_{0}}\frac{(f,S^{\lambda}_{(t,x)}\varphi)}{w(x)\lambda^{\alpha}}$

where the scaling operators are defined in (6.4) and where $B_{0}$ is the set
of all smooth functions of $C^{0}$ norm less than 1, supported on the unit
ball of $\mathbb{R}^{2}$ that are orthogonal in $L^{2}(\mathbb{R}^{2})$ to all
polynomials of parabolic degree less than or equal to $\lceil\alpha\rceil$,
see for instance equation (3.8) and (3.9) in [CW17]. One may verify that this
norm is equivalent to the one in Definition 6.3 for $\alpha\in(0,1)$. Stated
in this way, the Schauder estimate actually holds for all
$\alpha\in\mathbb{R}\backslash\mathbb{Z},$ though we will not need this.

We next define a space-time distribution that is supported on a single
temporal cross-section:

######  Definition 6.9.

Let $\alpha<0$. Given some $f\in C^{\alpha,\tau}(\mathbb{R})$ and $b\in[0,T]$
we define $\delta_{b}\otimes f\in
C_{\mathfrak{s}}^{\alpha-2,\tau}(\Lambda_{T})$ by the formula
$(\delta_{b}\otimes
f,\varphi)_{\mathbb{R}^{2}}:=(f,\varphi(b,\cdot))_{\mathbb{R}}.$

Directly from the definitions of the scaling operators in (6.3) and (6.4), it
is clear that for fixed $b\in[0,T]$, the linear map $f\mapsto\delta_{b}\otimes
f$ is bounded from $C^{\alpha,\tau}(\mathbb{R})\to
C_{\mathfrak{s}}^{\alpha-2,\tau}(\Lambda_{T})$ as long as $\alpha<0.$ It is
also clear that $\delta_{b}\otimes f$ is necessarily supported on the line
$\\{b\\}\times\mathbb{R}$. The following lemma shows that under mild
conditions, $\delta_{T}\otimes f$ vanishes upon the action of the heat flow
$K$ defined in (6.5).

######  Lemma 6.10.

Let $f\in C^{\alpha,\tau}(\mathbb{R}),$ where $\alpha>-2.$ Then viewing $K$ as
an operator from $C_{\mathfrak{s}}^{\alpha-2,\tau}(\Lambda_{T})\to
C_{\mathfrak{s}}^{\alpha,\tau}(\Lambda_{T})$, we have that
$K(\delta_{T}\otimes f)=0.$

###### Proof.

Note that $\delta_{T}\otimes f\in C_{\mathfrak{s}}^{\alpha,\tau}(\Lambda_{T})$
where $\alpha>-4.$ Thus by Proposition 6.6, we know that $K(\delta_{T}\otimes
f)\in C_{\mathfrak{s}}^{\beta,\tau}(\Lambda_{T})$ for some $\beta>-2$. Now
since $\delta_{T}\otimes f$ is supported on $\\{T\\}\times\mathbb{R}$, it
immediately follows that $K(\delta_{T}\otimes f)$ is also supported on
$\\{T\\}\times\mathbb{R}.$

Thus it suffices to show that if $g\in
C_{\mathfrak{s}}^{\beta,\tau}(\Lambda_{T})$ for some $\beta\in(-2,0),$ and if
$g$ is supported on $\\{T\\}\times\mathbb{R}$ then $g=0.$ We can prove this
directly: fix some smooth nonnegative compactly supported function $\varphi$
on $\mathbb{R}^{2}$, and define
$g^{\lambda}(t,x):=(g,S^{\lambda}_{(t,x)}\varphi),$ where the scaling
operators are given by (6.4). Using $g\in
C_{\mathfrak{s}}^{\beta,\tau}(\Lambda_{T})$ for some $\beta\in(-2,0),$ one has
that $g^{\lambda}(t,x)\leq Cw(x)|T-t|^{\frac{\beta}{2}}$ where $C$ is
independent of $\lambda,x,t,$ and where $w$ is the weight function given by
(6.2). Now if $\psi$ is any compactly supported function on $\mathbb{R}^{2}$,
then by integrability of $t\mapsto t^{\beta/2}$ on $[0,1]$ and the dominated
convergence theorem, we find that

$(g,\psi)=\lim_{\lambda\to
0}\int_{\mathbb{R}^{2}}g^{\lambda}(t,x)\psi(t,x)dtdx=\int_{\mathbb{R}^{2}}\lim_{\lambda\to
0}g^{\lambda}(t,x)\psi(t,x)dtdx=0,$

where we used the fact that $\lim_{\lambda\to 0}g^{\lambda}(t,x)=0$ at all
points $(t,x)$ outside of the measure-zero subset given by
$\\{T\\}\times\mathbb{R}.$ ∎

We remark that the last paragraph of the above proof has an intuitive analogue
in the elliptic setting as well: if $f\in C^{\alpha,\tau}(\mathbb{R})$ where
$\alpha>-1$, and if $f$ is supported on a single point, then $f=0$. The proof
proceeds in a similar manner.

We end this subsection by recording a Kolmogorov-type lemma for the three
spaces introduced at the beginning of this subsection. It will be crucial in
proving tightness in those respective spaces.

######  Lemma 6.11 (Kolmogorov lemma).

Let $L^{2}(\Omega,\mathcal{F},\mathbb{P})$ be the space of all random
variables defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$
with finite second moment. We have the following:

  1. (a)

(Elliptic Hölder Space) Let $\phi\mapsto V(\phi)$ be a bounded linear map from
$\mathcal{S}(\mathbb{R})$ into $L^{2}(\Omega,\mathcal{F},\mathbb{P})$. Recall
$S^{\lambda}_{x}$ from (6.3). Assume there exists some $p>1$ and $\alpha<0$
and $\mathrm{C}=\mathrm{C}(\alpha,p)>0$ such that one has

$\mathbf{E}[|V(S^{\lambda}_{x}\phi)|^{p}]\leq\mathrm{C}\lambda^{\alpha p},$

uniformly over all smooth functions $\phi$ on $\mathbb{R}$ supported on the
unit ball of $\mathbb{R}$ with $\|\phi\|_{L^{\infty}}\leq 1$, and uniformly
over $\lambda\in(0,1]$ and $x\in\mathbb{R}$. Then for any $\tau>1$ and any
$\beta<\alpha-1/p$ there exists a random variable $\mathscr{V}$ taking values
in $C^{\beta,\tau}(\mathbb{R})$ such that $(\mathscr{V},\phi)=V(\phi)$ almost
surely for all $\phi.$ Furthermore one has that

$\mathbf{E}[\|\mathscr{V}\|^{p}_{C^{\beta,\tau}(\mathbb{R})}]\leq\mathrm{C}^{\prime},$

where $\mathrm{C}^{\prime}$ depends on the choice of $\alpha,p,$ and the
constant $\mathrm{C}$ appearing in the moment bound above but not on
$V,\Omega,\mathcal{F},\mathbb{P}$.

  2. (b)

(Function space) Let $(t,\phi)\mapsto V(t,\phi)$ be a map from
$[0,T]\times\mathcal{S}(\mathbb{R})$ into
$L^{2}(\Omega,\mathcal{F},\mathbb{P})$ which is linear and continuous in
$\phi$. Fix a non-negative integer $r$. Assume there exists some
$\kappa>0,p>1/\kappa$ and $\alpha<-r$ and
$\mathrm{C}=\mathrm{C}(\kappa,\alpha,p,{T})>0$ such that one has

$\displaystyle\mathbf{E}[|V(t,S^{\lambda}_{x}\phi)|^{p}]$
$\displaystyle\leq\mathrm{C}\lambda^{\alpha p},$
$\displaystyle\mathbf{E}[|V(t,S^{\lambda}_{x}\phi)-V(s,S^{\lambda}_{x}\phi)|^{p}]$
$\displaystyle\leq\mathrm{C}\lambda^{(\alpha-\kappa)p}|t-s|^{\kappa p},$

uniformly over all smooth functions $\phi$ on $\mathbb{R}$ supported on the
unit ball of $\mathbb{R}$ with $\|\phi\|_{C^{r}}\leq 1$, and uniformly over
$\lambda\in(0,1]$ and $0\leq s,t\leq T$. Then for any $\tau>1$ and any
$\beta<\alpha-\kappa$ there exists a random variable
$\big{(}\mathscr{V}(t)\big{)}_{t\in[0,T]}$ taking values in
$C([0,T],C^{\beta,\tau}(\mathbb{R}))$ such that
$(\mathscr{V}(t),\phi)=V(t,\phi)$ almost surely for all $\phi$ and $t$.
Furthermore, one has that

$\mathbf{E}[\|\mathscr{V}\|^{p}_{C([0,T],C^{\beta,\tau}(\mathbb{R}))}]\leq\mathrm{C}^{\prime},$

where $\mathrm{C}^{\prime}$ depends on the choice of $\alpha,\beta,p,\kappa,$
and the constant $\mathrm{C}$ appearing in the moment bound above but not on
$V,\Omega,\mathcal{F},\mathbb{P}$.

  3. (c)

(Parabolic Hölder Space) Let $\varphi\mapsto V(\varphi)$ be a bounded linear
map from $\mathcal{S}(\mathbb{R}^{2})$ to
$L^{2}(\Omega,\mathcal{F},\mathbb{P})$. Assume $V(\varphi)=0$ for all
$\varphi$ with support contained in the complement of $\Lambda_{T}$. Recall
$S^{\lambda}_{(t,x)}$ from (6.4). Fix a non-negative integer $r$. Assume there
exists some $p>1$ and $\alpha<0$ and $\mathrm{C}=\mathrm{C}(\alpha,p)>0$ such
that one has

$\mathbf{E}[|V(S^{\lambda}_{(t,x)}\varphi)|^{p}]\leq\mathrm{C}\lambda^{\alpha
p},$

uniformly over all smooth functions $\varphi$ on $\mathbb{R}^{2}$ supported on
the unit ball of $\mathbb{R}^{2}$ with $\|\varphi\|_{C^{r}}\leq 1$, and
uniformly over $\lambda\in(0,1]$ and $(t,x)\in\Lambda_{T}$. Then for any
$\tau>1$ and any $\beta<\alpha-3/p$ there exists a random variable
$\mathscr{V}$ taking values in $C^{\beta,\tau}_{\mathfrak{s}}(\Lambda_{T})$
such that $(\mathscr{V},\varphi)=V(\varphi)$ almost surely for all $\varphi.$
Furthermore one has that

$\mathbf{E}[\|\mathscr{V}\|^{p}_{C^{\beta,\tau}_{\mathfrak{s}}(\Lambda_{T})}]\leq\mathrm{C}^{\prime},$

where $\mathrm{C}^{\prime}$ depends on the choice of $\alpha,p,$ and the
constant $\mathrm{C}$ appearing in the moment bound above but not on
$V,\Omega,\mathcal{F},\mathbb{P}$.

A proof of the above results may be adapted from the proof of Lemma 9 in
Section 5 of [MW17]. We remark that we do not actually need uniformity over a
large class of test functions as we have written above, just a single well-
chosen test function would suffice (e.g. the Littlewood-Paley blocks as used
in [MW17] or the Daubechies wavelets in [HL15]). Part (a) of the above lemma
can be stated with $C^{r}$ functions just like part (c), but we will not need
this general version.

###  6.2. Tightness

In this section, we prove tightness of the field $\mathscr{X}^{N}$ and
establish regularity estimates for the limit points.

######  Proposition 6.12.

For each $t\geq 0$, the fields $\\{\mathscr{X}_{t}^{N}\\}_{N\geq 1}$ defined
in (1.3) may be realized as random elements of the elliptic Hölder space
$C^{\alpha,\tau}(\mathbb{R})$ for any $\tau>1$ and any $\alpha<-1.$ Moreover
they are tight with respect to that topology as $N\to\infty$, in fact, we have
that
$\sup_{N}\mathbb{E}[\|\mathscr{X}_{t}^{N}\|_{C^{-\alpha,\tau}(\mathbb{R})}^{p}]<\infty$
for all $p\geq 1$. Furthermore, the $\mathscr{X}^{N}$ defined in (1.4)
themselves may be viewed as elements of the parabolic Hölder space
$C_{\mathfrak{s}}^{\alpha,\tau}(\Lambda_{T})$ for $\alpha<-1$ and $\tau>1.$
Moreover they are tight with respect to that topology as $N\to\infty$.

###### Proof.

Using Lemma 2.2 and Lemma 3.1 we obtain that

$\displaystyle\mathbb{E}[\mathscr{X}_{t}^{N}(\phi)^{2k}]$
$\displaystyle=\mathbf{E}_{D_{N^{1/4}}SB_{N^{1/2}\nu}^{(k)}}\bigg{[}\prod_{j=1}^{2k}\phi(X^{j}_{t})e^{{\frac{1}{2}}N^{1/2}\sum_{i<j}V_{t}^{ij}(\mathbf{X})}\bigg{]}$
$\displaystyle=\mathbf{E}_{SB_{N^{1/2}\nu}^{(k)}}\bigg{[}\mathcal{M}_{N}(t)\prod_{j=1}^{2k}\phi(X^{j}_{t})e^{{\frac{1}{2}}N^{1/2}\sum_{i<j}V_{t}^{ij}(\mathbf{X})}\bigg{]}$
$\displaystyle\leq\|\phi\|^{2k}_{L^{\infty}(\mathbb{R})}\mathbf{E}_{SB_{N^{1/2}\nu}^{(k)}}[\mathcal{M}_{N}(t)^{2}]^{1/2}\mathbf{E}_{SB_{N^{1/2}\nu}^{(k)}}\bigg{[}e^{N^{1/2}\sum_{i<j}V_{t}^{ij}(\mathbf{X})}\bigg{]}^{1/2},$

where $\mathcal{M}_{N}(t)$ is defined in (2.15) and the last inequality above
is due to the Cauchy-Schwarz inequality. From Remark 2.7 and Proposition 2.4
(with $\mu\mapsto N^{1/2}\nu$), we know that the above two expectations are
uniformly bounded in $N$. Therefore the last expression is bounded by some
universal constant times $\|\phi\|_{L^{\infty}}^{2k}.$ In particular we get
that

$\mathbb{E}[\mathscr{X}_{t}^{N}(S_{x}^{\lambda}\phi)^{2k}]\leq\mathrm{C}\|S_{x}^{\lambda}\phi\|_{L^{\infty}}^{2k}\leq
C\lambda^{-2k},$

uniformly over all $\phi\in C_{c}^{\infty}(\mathbb{R})$ with
$\|\phi\|_{L^{\infty}}\leq 1,$ where $S_{x}^{\lambda}$ is defined in (6.3).
Appealing to Lemma 6.11 (a) we get the desired result for fixed $t$.

To prove the second statement about tightness in the parabolic Hölder space,
we will as in (1.4) denote space-time pairings as
$(\mathscr{X}^{N},\varphi)_{L^{2}(\mathbb{R}^{2})}$. Note by Minkowski’s
inequality that

$\displaystyle\mathbb{E}[(\mathscr{X}^{N},\varphi)_{L^{2}(\mathbb{R}^{2})}^{2k}]^{1/2k}$
$\displaystyle=\mathbb{E}\bigg{[}\int_{\mathbb{R}}\mathscr{X}_{s}^{N}(\varphi(s,\cdot))ds\bigg{)}^{2k}\bigg{]}^{1/2k}\leq\int_{\mathbb{R}}\mathbb{E}[\mathscr{X}_{s}^{N}(\varphi(s,\cdot))^{2k}]^{1/2k}ds.$

Now with $S^{\lambda}_{(t,x)}$ defined in (6.4), we have

$\displaystyle\mathbb{E}[(\mathscr{X}^{N},S^{\lambda}_{(t,x)}\varphi)_{L^{2}(\mathbb{R}^{2})}^{2k}]^{1/2k}$
$\displaystyle\leq\int_{\mathbb{R}}\mathbb{E}\big{[}\mathscr{X}_{s}^{N}\big{(}(S^{\lambda}_{(t,x)}\varphi)(s,\cdot)\big{)}^{2k}\big{]}^{1/2k}ds\leq\mathrm{C}\int_{t-\lambda^{2}}^{t+\lambda^{2}}\lambda^{-3}ds\leq\mathrm{C}\lambda^{-1},$

uniformly over all $\varphi\in C_{c}^{\infty}(\mathbb{R}^{2})$ supported on
the unit ball, with $\|\varphi\|_{L^{\infty}}\leq 1.$ Here the $\lambda^{-3}$
comes from the fact that the parabolic scaling operator differs from the
elliptic one by exactly a factor of $\lambda^{-2}$. In the second bound, we
used the same bounds appearing in the proof for the elliptic case above,
noting that the constants there are independent of $t\in[0,T]$. Appealing to
Lemma 6.11 (c) we get the desired result. ∎

######  Proposition 6.13.

The fields $M^{N}$ defined in (4.4) may be realized as an element of
$C([0,T],C^{\alpha,\tau}(\mathbb{R}))$ for any $\alpha<-2$ and $\tau>1$.
Moreover, they are tight with respect to that topology. $Q^{N}$ defined in
(5.1) may be realized as an element of $C([0,T],C^{\gamma,\tau}(\mathbb{R}))$
for any $\gamma<-1$ and $\tau>1$. Moreover, they are tight with respect to
that topology. Let $(M^{\infty},Q^{\infty})$ be a joint limit point of
$(M^{N},Q^{N})$ in $C([0,T],C^{\alpha,\tau}(\mathbb{R})\times
C^{\gamma,\tau}(\mathbb{R})).$ For all $\phi\in C_{c}^{\infty}(\mathbb{R})$
the process $(M_{t}^{\infty}(\phi))_{t\in[0,T]}$ is a martingale with respect
to the canonical filtration on that space, and moreover one has

$\langle M^{\infty}(\phi)\rangle_{t}=Q_{t}^{\infty}(\phi^{2}).$ (6.8)

We remark that the canonical filtration $\mathscr{F}_{t}$ on $C([0,T],\Xi)$
for any Polish space $\Xi$ is the one generated by the random variables $X(s)$
for $s\leq t$ as $X$ varies through all elements of $C([0,T],\Xi)$.

###### Proof.

Take any $\phi\in C_{c}^{\infty}(\mathbb{R})$ with $\|\phi\|_{L^{\infty}}\leq
1.$ Recall $S_{(t,x)}^{\lambda}$ from (6.4). Using the first bound in
Proposition 5.6, we have

$\mathbb{E}[(Q_{t}^{N}(S^{\lambda}_{x}\phi)-Q_{s}^{N}(S^{\lambda}_{x}\phi))^{k}]\leq\mathrm{C}|t-s|^{k/4}\lambda^{-k},$

uniformly over $x\in\mathbb{R},\lambda\in(0,1],0\leq s,t\leq T$. Since
$Q_{0}^{N}(\phi)=0$ by definition, therefore the assumptions of Lemma 6.11 (b)
are satisfied for any $\kappa\leq 1/4$, any $p>1/\kappa$, and any
$\alpha\leq-1$, and we conclude the desired tightness for $Q^{N}$.

Now we address the tightness of the $M^{N}$. By Lemma 4.3 and the Burkholder-
Davis-Gundy inequality, we have that

$\mathbb{E}[(M_{t}^{N}(\phi)-M_{s}^{N}(\phi))^{2k}]\leq\mathrm{C}\cdot\mathbb{E}\bigg{[}\bigg{(}(Q_{t}^{N}\big{(}(\phi+N^{-1/4}\phi^{\prime})^{2}\big{)}-Q_{s}^{N}\big{(}(\phi+N^{-1/4}\phi^{\prime})^{2}\big{)}\bigg{)}^{k}\bigg{]},$

where $\mathrm{C}=\mathrm{C}(k)>0$ is free of $\phi,s,t,N.$ Using a crude
bound

$(\phi+N^{-1/4}\phi^{\prime})^{2}\leq
2\phi^{2}+2N^{-1/2}(\phi^{\prime})^{2}\leq 2\phi^{2}+2(\phi^{\prime})^{2},$

we find from the first bound in Proposition 5.6 that

$\mathbb{E}[(M_{t}^{N}(\phi)-M_{s}^{N}(\phi))^{2k}]\leq\mathrm{C}(t-s)^{k/4}(\|\phi\|_{L^{\infty}}^{2}+\|\phi^{\prime}\|_{L^{\infty}}^{2})^{k}\leq\mathrm{C}(t-s)^{k/4}\|\phi\|_{C^{1}}^{2k}.$
(6.9)

This gives

$\mathbb{E}[(M_{t}^{N}(S^{\lambda}_{x}\phi)-M_{s}^{N}(S^{\lambda}_{x}\phi))^{2k}]\leq\mathrm{C}(t-s)^{k/4}\lambda^{-4k}$

uniformly over $x\in\mathbb{R},\lambda\in(0,1],0\leq s,t\leq T$, and $\phi\in
C_{c}^{\infty}(\mathbb{R})$ with $\|\phi\|_{C^{1}}\leq 1.$ Moreover
$M_{0}(\phi)=0$ by definition, therefore the assumptions of Lemma 6.11 (b) are
satisfied for any $\kappa\leq 1/8$, any $p>1/\kappa$, and any $\alpha\leq-2$.
Hence, we conclude the desired tightness for $M^{N}$.

We next show that the limit point $M^{\infty}(\phi)$ is a martingale. Since
$M_{0}^{N}(\phi)=0$, from (6.9), we see that
$\sup_{N}\mathbb{E}[M_{t}^{N}(\phi)^{2k}]<\infty.$ Thus by uniform
integrability, $M^{\infty}(\phi)$ is a martingale. In the prelimit we know
that

$(M_{t}^{N}(\phi))^{2}-Q_{t}^{N}\big{(}(\phi+N^{-1/4}\phi^{\prime})^{2}\big{)}=(M_{t}^{N}(\phi))^{2}-Q_{t}^{N}(\phi^{2})-\mathcal{E}_{t}^{N}(\phi)$

is a martingale, where the error term $\mathcal{E}_{t}^{N}$ is defined in
(5.2). By Proposition 5.5 we know that $\mathcal{E}_{t}^{N}$ vanishes in
probability, so we conclude (again by uniform $L^{p}$ boundedness guaranteed
by Proposition 5.6) that $(M_{t}^{\infty}(\phi))^{2}-Q_{t}^{\infty}(\phi^{2})$
is a martingale. This verifies (6.8) completing the proof. ∎

######  Proposition 6.14.

For $\alpha<0$, the derivative operator
$\partial_{s}:C([0,T],C^{\alpha,\tau}(\mathbb{R}))\to
C_{\mathfrak{s}}^{\alpha-2,\tau}(\Lambda_{T})$ which is defined by sending
$(v_{t})_{t\in[0,T]}$ to the distribution

$(\partial_{s}v,\varphi)_{L^{2}(\Lambda_{T})}:=v_{T}(\varphi(T,\cdot))-\int_{0}^{T}v_{t}(\partial_{t}\varphi(t,\cdot))dt-
v_{0}(\varphi(0,\cdot)),$

whenever $\varphi\in C_{c}^{\infty}(\Lambda_{T})$ is a bounded linear map. Let
$M^{N}$ be as in Proposition 6.13, viewed as elements of
$C([0,T],C^{\alpha,\tau}(\mathbb{R}))$ for some $\alpha<-2.$ Also let
$\mathscr{X}^{N}_{t}$ and $\mathscr{X}^{N}$ and be as in Proposition 6.12,
viewed as elements of $C^{\gamma,\tau}(\mathbb{R})$ and
$C_{\mathfrak{s}}^{\gamma,\tau}(\Lambda_{T})$ respectively for some
$\gamma<-1$. Then one has that

$\displaystyle\partial_{s}M^{N}=-\delta_{(0,0)}+(\partial_{t}-\tfrac{1}{2}\partial_{x}^{2})\mathscr{X}^{N}+\delta_{T}\otimes\mathscr{X}^{N}_{T}$
(6.10)

in the sense of distributions.

###### Proof.

Recall $S^{\lambda}_{(t,x)}$ from (6.4). From the definition of the space
$C^{\alpha,\tau}(\mathbb{R})$ one verifies directly that if $v\in
C^{\alpha,\tau}(\mathbb{R})$ then

$\int_{0}^{T}|v_{t}\big{(}\partial_{t}(S^{\lambda}_{(s,y)}\varphi)(t,\cdot)\big{)}|dt\leq
w(y)\int_{s-\lambda^{2}}^{s+\lambda^{2}}\lambda^{\alpha-4}dt\leq
w(y)\lambda^{\alpha-2}$

uniformly over $(s,y)\in\Lambda_{T}$ and $\varphi\in
C_{c}^{\infty}(\mathbb{R}^{2})$ supported on the unit ball of $\mathbb{R}^{2}$
with $\|\varphi\|_{C^{r}}\leq 1$ where $r=\lceil-\alpha\rceil.$ This proves
the first part of the proposition that $\partial_{s}$ is bounded from
$C([0,T],C^{\alpha,\tau}(\mathbb{R}))\to
C_{\mathfrak{s}}^{\alpha-2,\tau}(\Lambda_{T}).$

For the second part, let us abbreviate $\mathscr{M}^{N}:=\partial_{s}M^{N}$.
We first note by definition of $\partial_{s}$ that for any smooth $\varphi$ of
compact support on $\mathbb{R}^{2}$, $\mathscr{M}^{N}(\varphi)$ is explicitly
given by

$\displaystyle\mathscr{M}^{N}(\varphi)=M_{T}^{N}(\varphi(T,\cdot))-\int_{0}^{T}M^{N}_{s}\big{(}\partial_{s}\varphi(s,\cdot)\big{)}ds,$

where we used the fact that $M_{0}^{N}\equiv 0$. Note that the above
expression is the same as $R_{T}^{N}(\varphi)$ defined in (4.5). Now observe
that using (4.4) we have

$\displaystyle\int_{0}^{T}M^{N}_{s}\big{(}\partial_{s}\varphi(s,\cdot)\big{)}ds$
$\displaystyle=\int_{0}^{T}\mathscr{X}^{N}_{s}\big{(}\partial_{s}\varphi(s,\cdot)\big{)}ds-\int_{0}^{T}\big{(}\partial_{s}\varphi(s,0)\big{)}ds$
$\displaystyle\hskip
85.35826pt-\int_{0}^{T}\frac{1}{2}\int_{0}^{s}\mathscr{X}_{u}^{N}\big{(}\partial_{xx}\partial_{s}\varphi(s,\cdot)\big{)}duds$
$\displaystyle=\int_{0}^{T}\mathscr{X}^{N}_{s}\big{(}\partial_{s}\varphi(s,\cdot)\big{)}ds+\varphi(0,0)-\varphi(T,0)$
$\displaystyle\hskip
85.35826pt-\int_{0}^{T}\frac{1}{2}\int_{0}^{s}\mathscr{X}_{u}^{N}\big{(}\partial_{xx}\partial_{s}\varphi(s,\cdot)\big{)}duds.$

The last term on the right-hand side of the above equation can be simplified
as follows.

$\displaystyle\int_{0}^{T}\frac{1}{2}\int_{0}^{s}\mathscr{X}_{u}^{N}\big{(}\partial_{xx}\partial_{s}\varphi(s,\cdot)\big{)}duds$
$\displaystyle=\int_{0}^{T}\frac{1}{2}\int_{u}^{T}\mathscr{X}_{u}^{N}\big{(}\partial_{xx}\partial_{s}\varphi(s,\cdot)\big{)}dsdu$
$\displaystyle=\frac{1}{2}\int_{0}^{T}\big{[}\mathscr{X}_{u}^{N}\big{(}\partial_{xx}\varphi(T,\cdot)\big{)}-\mathscr{X}_{u}^{N}\big{(}\partial_{xx}\varphi(u,\cdot)\big{)}\big{]}du$
$\displaystyle=-M_{T}^{N}(\varphi(T,\cdot))+\mathscr{X}_{T}^{N}(\varphi(T,\cdot))-\varphi(T,0)$
$\displaystyle\qquad-\frac{1}{2}\int_{0}^{T}\mathscr{X}_{u}^{N}(\partial_{xx}\varphi(u,\cdot)\big{)}du.$

We applied (4.4) with $\phi=\varphi(T,\cdot)$ in the last line. Combining
these expressions, we derive that

$\displaystyle\mathscr{M}^{N}(\varphi)=-\varphi(0,0)-\int_{0}^{T}\mathscr{X}_{s}^{N}(\partial_{s}\varphi(s,\cdot)ds-\frac{1}{2}\int_{0}^{T}\mathscr{X}_{s}^{N}(\partial_{xx}\varphi(s,\cdot)\big{)}ds+\mathscr{X}_{T}^{N}(\varphi(T,\cdot)).$

This verifies (6.10) completing the proof. ∎

Since we expect to obtain a Dirac initial data in the limiting SPDE, there is
an additional singularity at the origin that we have not yet taken into
account. To fix this issue, we now formulate a tightness result taking into
account this singularity, by starting our martingale-forced heat equation from
some positive time $\varepsilon$ (which should be thought of as being close to
$0$).

######  Proposition 6.15 (Tightness).

Fix $\varepsilon>0,$ and consider the fields
$\mathscr{X}^{N,\varepsilon}(t,\cdot):=\mathscr{X}_{t+\varepsilon}^{N}(\cdot),$
for $t\geq 0$. Set $\mathscr{X}^{N,\varepsilon}(t,\cdot):=0$ for $t<0$. The
fields $\mathscr{X}^{N,\varepsilon}$ may be realized as random variables
taking values in $C_{\mathfrak{s}}^{\alpha,\tau}(\Lambda_{T})$ for any
$\tau>1$ and $\alpha<-1$. Furthermore, they are tight with respect to that
topology, and any limit point necessarily lies in
$C_{\mathfrak{s}}^{-\kappa+1/2,\tau}(\Lambda_{T})$ for any $\kappa>0$.

###### Proof.

The proof that the fields $\mathscr{X}^{N,\varepsilon}$ are tight in
$C_{\mathfrak{s}}^{\alpha,\tau}(\Lambda_{T})$ for any $\tau>1$ and $\alpha<-1$
follows by using the exact same strategy followed in the proof of Proposition
6.12. Let us set

$\mathscr{M}^{N,\varepsilon}:=\big{(}\partial_{t}-\frac{1}{2}\partial_{x}^{2}\big{)}\mathscr{X}^{N,\varepsilon}+(\delta_{T}\otimes\mathscr{X}^{N}_{T+\varepsilon}-\delta_{0}\otimes\mathscr{X}^{N}_{\varepsilon}).$

We first claim that for fixed $\varepsilon>0$, the fields
$\\{\mathscr{M}^{N,\varepsilon}\\}_{N>1}$ are tight in
$C_{\mathfrak{s}}^{-3-\kappa,\tau}(\Lambda_{T})$ for any $\tau>1$ and
$\kappa>0$. As $\mathscr{X}^{N,\varepsilon}$ is tight in
$C_{\mathfrak{s}}^{-1-\kappa,\tau}(\Lambda_{T})$, the tightness of the
$(\partial_{t}-\frac{1}{2}\partial_{x}^{2})\mathscr{X}^{N,\varepsilon}$ in
$C_{\mathfrak{s}}^{-3-\kappa,\tau}(\Lambda_{T})$ is immediate since
$\partial_{t}-\frac{1}{2}\partial_{x}^{2}$ is a bounded linear map from
$C_{\mathfrak{s}}^{\alpha,\tau}(\Lambda_{T})$ into
$C_{\mathfrak{s}}^{\alpha-2,\tau}(\Lambda_{T})$ as explained in Remark 6.4.
From Proposition 6.12 we know that both $\mathscr{X}^{N}_{T+\varepsilon}$ and
$\mathscr{X}^{N}_{\varepsilon}$ are tight in $C^{-1-\kappa,\tau}(\mathbb{R})$.
As we remarked after Definition 6.9, the map $f\mapsto\delta_{a}\otimes f$ is
continuous from $C^{-1-\kappa,\tau}(\mathbb{R})\to
C_{\mathfrak{s}}^{-3-\kappa,\tau}(\Lambda_{T}),$ therefore we can immediately
conclude that
$(\delta_{T}\otimes\mathscr{X}^{N}_{T+\varepsilon}-\delta_{0}\otimes\mathscr{X}^{N}_{\varepsilon})$
are tight in $C_{\mathfrak{s}}^{-3-\kappa,\tau}(\Lambda_{T})$. Hence, the sum
given by $\mathscr{M}^{N,\varepsilon}$ is also tight in the latter space as
$N\to\infty$ (for fixed $\varepsilon$).

Next, we address the regularity of the limit points of
$\mathscr{X}^{N,\varepsilon}.$ By the last statement in Proposition 6.6, we
can apply the linear operator $K$ to both sides to obtain

$\mathscr{X}^{N,\varepsilon}=K(\delta_{0}\otimes\mathscr{X}^{N}_{\varepsilon})+K\mathscr{M}^{N,\varepsilon}-K(\delta_{T}\otimes\mathscr{X}^{N}_{T+\varepsilon}).$

By Proposition 6.12 we know that $\mathscr{X}^{N}_{T+\varepsilon}$ is tight in
$C^{\gamma,\tau}(\mathbb{R})$ for $\gamma<-1$, and from Lemma 6.10 we can
conclude that $K(\delta_{T}\otimes\mathscr{X}^{N}_{T+\varepsilon})=0$. On the
other hand,
$K(\delta_{0}\otimes\mathscr{X}^{N}_{\varepsilon})=J\mathscr{X}^{N}_{\varepsilon}$,
where $J$ is defined in Remark 6.7. We thus have the Duhamel equation

$\mathscr{X}^{N,\varepsilon}=J\mathscr{X}^{N}_{\varepsilon}+K\mathscr{M}^{N,\varepsilon}.$

For technical reasons that will be made clear below, we now replace
$\varepsilon$ by $\varepsilon/2$ and $T$ by $T+1$. If we take any joint limit
point
$(\mathscr{X}^{\infty,\varepsilon/2},\mathscr{M}^{\infty,\varepsilon/2},h)$ in
$C_{\mathfrak{s}}^{\alpha,\tau}(\Lambda_{T+1})\times
C_{\mathfrak{s}}^{\alpha+2,\tau}(\Lambda_{T+1})\times
C^{\gamma,\tau}(\mathbb{R})$ as $N\to\infty$ of
$(\mathscr{X}^{N,\varepsilon/2},\mathscr{M}^{N,\varepsilon/2},\mathscr{X}^{N}_{\varepsilon/2})$,
then (by continuity of $J$ and $K$ on the relevant spaces) the Duhamel
relation still holds in the limit, i.e.,

$\displaystyle\mathscr{X}^{\infty,\varepsilon/2}=Jh+K\mathscr{M}^{\infty,\varepsilon/2}.$
(6.11)

We now claim that

$\displaystyle\mathbb{E}[\|\mathscr{M}^{\infty,\varepsilon/2}\|_{C_{\mathfrak{s}}^{-\kappa-3/2,\tau}(\Lambda_{T+1})}^{p}]<\infty.$
(6.12)

Let us now complete the proof of regularity assuming (6.12).

  * Using (6.12) and Proposition 6.6, it follows that

$\mathbb{E}\bigg{[}\big{\|}K\mathscr{M}^{\infty,\varepsilon/2}\big{\|}_{C_{\mathfrak{s}}^{-\kappa+1/2,\tau}(\Lambda_{T+1})}^{p}\bigg{]}<\infty.$

This implies that the restriction of $K\mathscr{M}^{\infty,\varepsilon/2}$ to
$[\varepsilon/2,T+\varepsilon/2]\times\mathbb{R}$ lies in
$C_{\mathfrak{s}}^{-\kappa+1/2,\tau}(\Lambda_{[\varepsilon/2,T+\varepsilon/2]})$.

  * •

By Proposition 6.12 we have
$\mathbb{E}[\|h\|_{C^{-1-\kappa,\tau}}^{p}]<\infty$. So from (6.7) with
$\alpha:=-1-\kappa$ and $\beta:=\frac{1}{2}-\kappa$. It follows that

$\mathbb{E}[\|\widehat{P}_{\varepsilon/2}h\|_{C_{\mathfrak{s}}^{-\kappa+1/2,\tau}(\Lambda_{T+1})}^{p}]<\infty.$

Since from (6.6), we know
$Jh(t+\varepsilon/2,x)=\widehat{P}_{\varepsilon/2}h(t,x)$, we thus have that
the restriction of $Jh$ to $[\varepsilon/2,T+\varepsilon/2]\times\mathbb{R}$
lies in
$C_{\mathfrak{s}}^{-\kappa+1/2,\tau}(\Lambda_{[\varepsilon/2,T+\varepsilon/2]})$.

Thanks to the above two bullet points and the relation (6.11), we have showed
that the restriction of $\mathscr{X}^{\infty,\varepsilon/2}$ to
$[\varepsilon/2,T+\varepsilon/2]\times\mathbb{R}$ lies in
$C_{\mathfrak{s}}^{-\kappa+1/2,\tau}(\Lambda_{[\varepsilon/2,T+\varepsilon/2]})$.
This is equivalent to the fact that $\mathscr{X}^{\infty,\varepsilon}$ lies in
$C_{\mathfrak{s}}^{-\kappa+1/2,\tau}(\Lambda_{T})$. This completes the proof
modulo (6.12).

Proof of (6.12). We shall show (6.12) with $\varepsilon/2$ and $T+1$ replaced
by $\varepsilon$ and $T$ respectively. Following the same proof as in Lemma
6.14, one can check that

$\mathscr{M}^{N,\varepsilon}(\varphi)=R^{N}_{T+\varepsilon}(\varphi)-R^{N}_{\varepsilon}(\varphi),$

where $R^{N}$ are the same martingales defined in (4.5). We will denote the
right side as $R^{N,\varepsilon}(\varphi)$ and the quadratic variation up to
time $T$ as $\langle R^{N,\varepsilon}(\varphi)\rangle_{T}.$ Let us fix any
$\alpha<-3$ and take any joint limit point
$(\mathscr{M}^{\infty,\varepsilon},\mathscr{X}^{\infty,\varepsilon})$ in
$C_{\mathfrak{s}}^{\alpha,\tau}(\Lambda_{T})\times
C_{\mathfrak{s}}^{\alpha+2,\tau}(\Lambda_{T})$ of
$(\mathscr{M}^{N,\varepsilon},\mathscr{X}^{N,\varepsilon})$ as $N\to\infty$.
Let us consider any smooth function $\varphi$ supported on the unit ball of
$\mathbb{R}^{2}$ such that $\|\varphi\|_{C^{1}}\leq 1.$ Recall
$S_{(t,x)}^{\lambda}$ from (6.4). Using Skorohod’s lemma, Fatou’s lemma, and
the Burkholder-Davis-Gundy inequality, one has that

$\displaystyle\mathbb{E}[|\mathscr{M}^{\infty,\varepsilon}(S_{(t,x)}^{\lambda}\varphi)|^{p}]$
$\displaystyle\leq\liminf_{N\to\infty}\mathbb{E}[|\mathscr{M}^{N,\varepsilon}(\varphi)|^{p}]\leq\mathrm{C}\cdot\liminf_{N\to\infty}\mathbb{E}\big{[}\langle
R^{N,\varepsilon}(S_{(t,x)}^{\lambda}\varphi)\rangle_{T}^{p/2}\big{]},$

where $\mathrm{C}=\mathrm{C}(p)>0$. From Lemma 4.4, the quadratic variation is
given by

$\displaystyle\langle
R^{N,\varepsilon}(S_{(t,x)}^{\lambda}\varphi)\rangle_{T}=N^{1/2}\int_{0}^{T}\big{(}(\mathscr{X}_{s+\varepsilon}^{N})^{Sq},((1+N^{-1/4}\partial_{x})S_{(t,x)}^{\lambda}\varphi(s,\cdot))^{2}\big{)}ds.$

Using $(a+b)^{2}\leq 2(a^{2}+b^{2}),$ it is clear that
$\big{(}(1+N^{-1/4}\partial_{x})\varphi\big{)}^{2}\leq
2\varphi^{2}+2N^{-1/2}(\partial_{x}\varphi)^{2}$. Furthermore, the term with a
factor $N^{-1/2}$ will not contribute when taking the $\liminf$ as
$N\to\infty$ (by e.g. Proposition 5.5). Therefore, we can obtain that

$\liminf_{N\to\infty}\mathbb{E}[\langle
R^{N,\varepsilon}(S_{(t,x)}^{\lambda}\varphi)\rangle_{T}]\leq\liminf_{N\to\infty}N^{1/2}\mathbb{E}\bigg{[}\int_{0}^{T}\big{(}(\mathscr{X}_{s+\varepsilon}^{N})^{Sq},(S_{(t,x)}^{\lambda}\varphi)^{2}\big{)}ds\bigg{]}.$

Recall the fields $Q^{N}$ defined in (5.1). Notice that
$\big{(}S_{(t,x)}^{\lambda}\varphi\big{)}^{2}\leq\lambda^{-6}\mathbf{1}_{[t-\lambda^{2},t+\lambda^{2}]\times[x-\lambda,x+\lambda]}$.
Therefore, taking any $\delta>0$, by the above bound and by the second bound
in Proposition 5.6 we find that

$\displaystyle\mathbb{E}[|\mathscr{M}^{\infty,\varepsilon}(S_{(t,x)}^{\lambda}\varphi)|^{p}]$
$\displaystyle\leq
2\mathrm{C}\lambda^{-3p}\liminf_{N\to\infty}\mathbb{E}\big{[}\big{(}Q^{N}_{t+\varepsilon+\lambda^{2}}(\mathbf{1}_{[x-\lambda,x+\lambda]})-Q^{N}_{t+\varepsilon-\lambda^{2}}(\mathbf{1}_{[x-\lambda,x+\lambda]})\big{)}^{p/2}\big{]}$
$\displaystyle\leq
2\mathrm{C}\lambda^{-3p}(\lambda^{2})^{p/2}\|\mathbf{1}_{[x-\lambda,x+\lambda]}\|_{L^{1+\delta}(\mathbb{R})}^{p/2}=\mathrm{C}\lambda^{-\frac{3}{2}p-\big{[}\frac{\delta}{1+\delta}\big{]}p}.$

Given any $\kappa>0$, by making $\delta=\delta(\kappa)$ close to $0$ and
making $p=p(\kappa)$ large enough, we may then apply Lemma 6.11(c) to conclude
(6.12). ∎

###  6.3. Identification of the limit points

In this subsection, we identify the limit points as the solution of the
stochastic heat equation (SHE) and thereby prove our weak convergence theorem:
Theorem 1.4.

######  Theorem 6.16 (Solving the martingale problem).

Consider the triplet of processes $(\mathscr{X}^{N},M^{N},Q^{N})_{t\geq 0}$,
where $\mathscr{X}^{N}$, $M^{N}$, and $Q^{N}$ are defined in (1.3), (4.4), and
(5.1) respectively. Fix $\alpha<-1,\beta<-2,\gamma<-1,$ and $\tau>1.$ These
triples are jointly tight in the space

$C^{\alpha,\tau}_{\mathfrak{s}}(\Lambda_{T})\times
C([0,T],C^{\beta,\tau}(\mathbb{R}))\times
C([0,T],C^{\gamma,\tau}(\mathbb{R})).$

Consider any joint limit point $(X^{\infty},M^{\infty},Q^{\infty})$. Then for
any $s>0$, the process $(t,x)\mapsto X_{s+t}^{\infty}(x)$ is necessarily
supported on the space $C_{\mathfrak{s}}^{-\kappa+1/2,\tau}(\Lambda_{T})$.
Furthermore, $(M_{t}^{\infty}(\phi))_{t\geq 0}$ is a continuous martingale for
all $\phi\in C_{c}^{\infty}(\mathbb{R})$, and moreover for all $0<s\leq t<T$
one has the almost sure identities

$\displaystyle M_{t}^{\infty}(\phi)-M_{s}^{\infty}(\phi)$
$\displaystyle=\int_{\mathbb{R}}(X_{t}^{\infty}(x)-X_{s}^{\infty}(x))\phi(x)dx-\frac{1}{2}\int_{s}^{t}\int_{\mathbb{R}}X_{u}^{\infty}(x)\phi^{\prime\prime}(x)dxdu$
(6.13) $\displaystyle\langle M^{\infty}(\phi)\rangle_{t}$
$\displaystyle=Q_{t}^{\infty}(\phi^{2})$ (6.14) $\displaystyle
Q_{t}^{\infty}(\phi)-Q_{s}^{\infty}(\phi)$
$\displaystyle=\sigma\int_{\mathbb{R}}\int_{s}^{t}(X_{u}^{\infty}(x))^{2}\phi(x)\,du\,dx.$
(6.15)

Before going into the proof, we remark that with some inspection it may be
verified that all quantities make sense given the spaces they lie in, as long
as we choose $\kappa<1/2$ to ensure that $X^{\infty}$ is a continuous function
in space-time away from $t=0$.

###### Proof.

Most parts of the following theorem are already established in previous
propositions and lemmas. Note that the tightness of $\mathscr{X}^{N}$ was
proved in Proposition 6.12. The tightness of $M^{N}$ and $Q^{N}$ was shown in
Proposition 6.13. Thus the joint tightness follows from individual tightness.
Consider any limit point $(\mathscr{X}^{\infty},M^{\infty},Q^{\infty}).$ The
fact that for any $s>0$, the process $(t,x)\mapsto X^{\infty}(s+t,x)$ is
necessarily supported on the space
$C_{\mathfrak{s}}^{-\kappa+1/2,\tau}(\Lambda_{T})$ was proved in Proposition
6.15. From Proposition 6.13 we have that for all $\phi\in
C_{c}^{\infty}(\mathbb{R})$ the process $(M_{t}^{\infty}(\phi))_{t\geq 0}$ is
a martingale. (6.14) is already proven in Proposition 6.13 as (6.8). All we
are left to show is (6.13) and (6.15).

Proof of (6.13). In Lemma 6.14 we obtained that
$\partial_{s}M^{N}=-\delta_{(0,0)}+(\partial_{t}-\tfrac{1}{2}\partial_{x}^{2})\mathscr{X}^{N}+\delta_{T}\otimes\mathscr{X}^{N}_{T}$
in the sense of distributions, which by disregarding the boundary terms
implies that

$(\partial_{s}M^{N},\varphi)=((\partial_{t}-\tfrac{1}{2}\partial_{x}^{2})\mathscr{X}^{N},\varphi)$

for all $\varphi$ of compact support contained in
$\Lambda_{[\varepsilon,T-\varepsilon]}$ for some $\varepsilon>0$. Note that
this relation is still respected by any limit points and hence remains true
when $N=\infty$. We thus have

$(\partial_{s}M^{\infty},\varphi)=((\partial_{t}-\tfrac{1}{2}\partial_{x}^{2})\mathscr{X}^{\infty},\varphi)$

Since both $M^{\infty}$ and $\mathscr{X}^{\infty}$ lie in spaces with strong
enough topologies, taking $\varphi(u,x)$ to approach the function
$(u,x)\mapsto\mathbf{1}_{[s,t]}(u)\phi(x)$ in the above equation leads to
(6.13).

Proof of (6.15). Fix any $0\leq t\leq T$ and let $\mu$ be a
$\mathcal{S}^{\prime}(\mathbb{R})$ valued random variable defined as

$\displaystyle(\mu,\phi):=Q_{t}^{\infty}(\phi)-\sigma\int_{\mathbb{R}}\int_{0}^{t}(\mathscr{X}_{s}^{\infty}(x))^{2}\phi(x)\,ds\,dx.$

We claim that $(\mu,\phi)=0$ a.s. for all $\phi\in\mathcal{S}(\mathbb{R}).$
This will validate (6.15). To verify this, let us define a sequence of
function-valued random variables:

$\displaystyle\mu^{\varepsilon}(a):=Q_{t}^{\infty}(\xi_{\varepsilon}^{a})-\sigma\int_{0}^{t}(\mathscr{X}_{s}^{\infty}(\xi_{\varepsilon\sqrt{2}}^{a}))^{2}ds$

where $\xi_{\varepsilon}^{a}(x):=\varepsilon^{-1}\xi(\varepsilon^{-1}(x-a))$
with $\xi(x):=\frac{1}{\sqrt{\pi}}e^{-x^{2}}$. Observe that for any $\phi\in
C_{c}^{\infty}(\mathbb{R})$ supported on $[-S,S]$, by Cauchy-Schwarz and
Jensen’s inequality we have that

$\mathbb{E}\big{[}|(\mu^{\varepsilon},\phi)|\big{]}\leq\|\phi\|_{L^{2}(\mathbb{R})}\cdot\mathbb{E}\bigg{[}\sqrt{\int_{[-S,S]}\mu^{\varepsilon}(a)^{2}da}\bigg{]}\leq\|\phi\|_{L^{2}}\bigg{[}\int_{[-S,S]}\mathbb{E}[\mu^{\varepsilon}(a)^{2}]da\bigg{]}^{1/2}.$

We now make use of Proposition 5.3. From (5.9), we see that the integral is
uniformly bounded in $\varepsilon>0$. On the other hand, (5.8) tells us that
the integrand goes to zero as $\varepsilon\to 0$. Thus, by dominated
convergence theorem $(\mu^{\varepsilon},\phi)\to 0$ in probability. But
$(\mu^{\varepsilon},\phi)\to(\mu,\phi)$ almost surely as $\varepsilon\to 0$.
Hence, $(\mu,\phi)=0$ almost surely. This proves the claim for compactly
supported $\phi$. For general $\phi\in\mathcal{S}(\mathbb{R})$, we may find a
sequence of functions $\phi_{n}\in C_{c}^{\infty}(\mathbb{R})$ such that
$\phi_{n}\to\phi$ in the topology of $\mathcal{S}(\mathbb{R})$. Then
$0=(\mu,\phi_{n})\to(\mu,\phi)$ almost surely as $n\to\infty$. Hence,
$(\mu,\phi)=0$ almost surely for all $\phi\in\mathcal{S}(\mathbb{R})$. This
verifies the claim, completing the proof. ∎

We now complete the proof of our main theorem, Theorem 1.4.

###### Proof of Theorem 1.4(a).

We continue with the notation and setup of Theorem 6.16. We have already
established the tightness of $\mathscr{X}^{N}$ in the topology of
$C_{\mathfrak{s}}^{\alpha,\tau}(\Lambda_{T})$ in Proposition 6.12. Consider
any limit point $X^{\infty}$ of $\mathscr{X}^{N}$. From the previous theorem,
we already know that $(t,x)\mapsto X_{t+\varepsilon}^{\infty}(x)$ is a
continuous function in space and time. From the three equations (6.13),
(6.14), and (6.15) in Theorem 6.16 it follows that the martingale problem for
(SHE) is satisfied by any limit point $X^{\infty}$. We refer the reader to
[BG97, Proposition 4.11] for the characterization of the law of (SHE) as the
solution to this martingale problem.

The result there is only stated for continuous initial conditions, so what
this really shows is that for any $\varepsilon>0$ the law of the continuous
field $(t,x)\mapsto X_{t+\varepsilon}^{\infty}(x)$ is that of the solution of
(SHE) with initial condition $X_{\varepsilon}^{\infty}(\cdot).$ Thus we still
need to pin down the initial data as $\delta_{0}$, by showing that we can let
$\varepsilon\to 0$ and see that the limit of $X_{\varepsilon}^{\infty}(\cdot)$
is equal to $\delta_{0}$ in some sense.

In [Par18, Section 6] there is a general approach to do exactly this.
Specifically, it suffices to show as in Lemma 6.6 of that paper the two bounds

$\displaystyle\mathbb{E}[|X_{t}^{\infty}(x)|^{r}]^{2/r}\leq\mathrm{C}\cdot
t^{-1/2}p_{t}(x),$ (6.16)
$\displaystyle\mathbb{E}[|X_{t}^{\infty}(x)-p_{t}(x)|^{r}]^{2/r}\leq\mathrm{C}\cdot
p_{t}(x),$ (6.17)

where $r>1$ is arbitrary, $p_{t}$ is the standard heat kernel, and
$\mathrm{C}$ is independent of $t>0$ and $x\in\mathbb{R}$. Clearly, it
suffices to show this when $r$ is an even integer. The solution of (SHE) with
$\delta_{0}$ initial condition certainly satisfies this bound, and by the
moment convergence result (Proposition 1.3), we know any limit point
$X^{\infty}$ must satisfy
$\mathbb{E}[(\int_{\mathbb{R}}X_{t}^{\infty}(x)\phi(x)dx)^{k}]=\mathbf{E}[(\int_{\mathbb{R}}\mathcal{Z}_{t}(x)\phi(x)dx)^{k}]$
for all $k\in\mathbb{N}$ and $\phi\in C_{c}^{\infty}(\mathbb{R})$, where
$(t,x)\mapsto\mathcal{Z}_{t}(x)$ solves (SHE) with $\delta_{0}$ initial data
and $\sigma=\frac{1}{\nu([0,1])}$. From here we can conclude by letting
$\phi\to\delta_{x}$ that
$\mathbb{E}[X_{t}^{\infty}(x)^{k}]=\mathbf{E}[\mathcal{Z}_{t}(x)^{k}]$ for all
$k\in\mathbb{N}$ and all $x\in\mathbb{R}$. Consequently, we may immediately
deduce (6.16) and (6.17) by the corresponding bounds for $\mathcal{Z}_{t}$.
This completes the proof. ∎

Finally, we conclude this section by proving Theorem 1.4(b) which follows from
the existence of a certain bounded linear operator.

###### Proof of Theorem 1.4(b).

Fix $\alpha<0$. We claim that there exists a bounded linear map
$\overline{K\partial_{s}}:C([0,T],C^{\alpha,\tau}(\mathbb{R}))\to
C([0,T],C^{\alpha,\tau}(\mathbb{R}))$ such that for all
$v=(v_{t})_{t\in[0,T]}\in C([0,T],C^{\alpha,\tau}(\mathbb{R}))$ one has that
$\overline{K\partial_{s}}v=K(\partial_{s}(v))$ where $K$ and $\partial_{s}$
are the same operators defined in Propositions 6.6 and 6.14 respectively
(where the equality may be interpreted in the sense that both sides are viewed
as elements of $\mathcal{S}^{\prime}(\mathbb{R}^{2})$).

Note that the existence of $\overline{K\partial_{s}}$ would immediately imply
the desired result. Indeed by applying $K$ to both sides of (6.10) and using
Proposition 6.6 and Lemma 6.10 we get that

$\mathscr{X}^{N}=p+K(\partial_{s}(M^{N})),$

where $p$ is the standard heat kernel. We also know from Proposition 6.13 that
the $M^{N}$ are tight in $C([0,T],C^{\gamma,\tau}(\mathbb{R}))$ for
$\gamma<-2$, therefore tightness of $\mathscr{X}^{N}$ in
$C([0,T],C^{\gamma,\tau}(\mathbb{R}))$ for $\gamma<-2$ would be immediate by
continuity of $\overline{K\partial_{s}}$. The fact that any limit point
coincides with the law of (SHE) follows from Theorem 6.16, (6.16) and (6.17)
by taking a joint limit point with the triple appearing there, then using the
fact that $K(\partial_{s}(M^{N}))=\overline{K\partial_{s}}M^{N}$.

We now turn toward the existence of $\overline{K\partial_{s}}$. To construct
$\overline{K\partial_{s}}$ we first remark that $C_{c}^{\infty}(\Lambda_{T})$
embeds into $C([0,T],C^{\alpha,\tau}(\mathbb{R}))$ in the natural way: a
function $v\in C_{c}^{\infty}(\Lambda_{T})$ may be identified with
$(v(t))_{t\in[0,T]}$ given by
$(v(t),\phi)_{L^{2}(\mathbb{R})}:=\int_{\mathbb{R}}v(t,x)\phi(x)dx.$ Then
$\partial_{s}v=:v^{\prime}$ is given by
$(v^{\prime}(t),\phi):=\int_{\mathbb{R}}(\partial_{t}v)(t,x)\phi(x)dx.$ In
this case we may integrate by parts to obtain

$\displaystyle(K(\partial_{s}v)(t),\phi)$
$\displaystyle=\bigg{(}\int_{0}^{t}P_{t-s}v^{\prime}(s)ds\;,\;\phi\bigg{)}$
$\displaystyle=\bigg{(}v(t)-P_{t}v(0)+\int_{0}^{t}\partial_{s}P_{t-s}v(s)ds\;,\;\phi\bigg{)}$
$\displaystyle=(v(t),\phi)-(v(0),P_{t}\phi)-\int_{0}^{t}(v(s),P_{t-s}\partial_{x}^{2}\phi)ds,$

where all pairings are in $L^{2}(\mathbb{R})$ and we use that
$\partial_{s}P_{t-s}=-\partial_{x}^{2}P_{t-s},$ and $\partial_{x}^{2}$ and
$P_{t}$ are self-adjoint operators on $L^{2}(\mathbb{R})$.

Based on this calculation, we consider a general path $v\in
C([0,T],C^{\alpha,\tau}(\mathbb{R}))$ and we finally define
$\overline{K\partial_{s}}v$ suggestively by the formula

$\big{(}\big{[}\;\overline{K\partial_{s}}v\;\big{]}(t),\phi):=(v(t),\phi)-(v(0),P_{t}\phi)-\int_{0}^{t}(v(s),P_{t-s}\partial_{x}^{2}\phi)ds.$
(6.18)

First note that the pairings on the right-hand side are indeed well-defined
for all $v\in C([0,T],C^{\alpha,\tau}(\mathbb{R}))$. What remains to be seen
is that the integral is convergent and that the right-hand side is indeed an
element of $C([0,T],C^{\alpha,\tau}(\mathbb{R}))$ whose norm may be controlled
by $\|v\|_{C([0,T],C^{\alpha,\tau}(\mathbb{R}))}.$ By the definition of these
spaces, we must replace $\phi$ by $S^{\lambda}_{x}\phi$ (where the scaling
operators are given in Definition 6.1) and then study the growth as $\lambda$
becomes close to 0.

The first two terms on the right side of (6.18) are completely straightforward
to deal with: the growth is at worst $\lambda^{\alpha}(1+x^{2})^{\tau}$
uniformly over $\lambda,\phi,x,T$, since we assumed $v\in
C([0,T],C^{\alpha,\tau}(\mathbb{R})).$ To deal with the third term, we claim
that one has

$|(v(s),P_{t-s}\partial_{x}^{2}S^{\lambda}_{x}\phi)|\lesssim\big{(}\lambda^{\alpha-2}\wedge(t-s)^{\frac{\alpha}{2}-1}\big{)}(1+x^{2})^{\tau}$
(6.19)

uniformly over $s<t\in[0,T]$, as well as $\lambda,\phi,x$. Indeed the bound of
the form $\lambda^{\alpha-2}$ follows by noting that when $t-s$ is much
smaller than $\lambda$, $P_{t-s}$ is essentially the identity operator, so we
can effectively disregard the heat kernel and note that
$\partial_{x}^{2}S^{\lambda}_{x}\phi$ paired with $v(s)$ satisfies a bound of
order $\lambda^{\alpha-2}$. Likewise the bound of the form
$(t-s)^{\frac{\alpha}{2}-1}$ is obtained by noting that when $\lambda$ is very
small compared to $t-s$, $P_{t-s}S_{x}^{\lambda}\phi$ behaves like
$S_{x}^{\sqrt{t-s}}\phi$, giving a bound of order $(\sqrt{t-s})^{\alpha-2}$
after applying $\partial_{x}^{2}$ and pairing with $v(s)$. This proves the
bound (6.19).

Now the fact that $\|\overline{K\partial_{s}}v\|$ can be controlled by $\|v\|$
follows simply by noting that uniformly over $0<\lambda^{2}\leq t\leq T$ one
has

$\int_{0}^{t}\big{(}\lambda^{\alpha-2}\wedge(t-s)^{\frac{\alpha}{2}-1}\big{)}ds=\int_{0}^{t-\lambda^{2}}(t-s)^{\frac{\alpha}{2}-1}ds+\int_{t-\lambda^{2}}^{t}\lambda^{\alpha-2}ds\leq\mathrm{C}(\alpha)\cdot\lambda^{\alpha}.$

It remains to verify that $\overline{K\partial_{s}}v$ is indeed a continuous
path. So far, what we have effectively shown is that
$\overline{K\partial_{s}}$ is a bounded operator from
$L^{\infty}([0,T],C^{\alpha,\tau}(\mathbb{R}))$ to itself. Note that
$C([0,T],C^{\alpha,\tau}(\mathbb{R}))$ is precisely the closure of
$\mathcal{S}(\Lambda_{T})$ inside of
$L^{\infty}([0,T],C^{\alpha,\tau}(\mathbb{R})),$ where
$\mathcal{S}(\Lambda_{T})$ denotes restrictions to $\Lambda_{T}$ of those
functions in $\mathcal{S}(\mathbb{R}^{2})$. From here, path continuity of
$\overline{K\partial_{s}}v$ for all $v\in
C([0,T],C^{\alpha,\tau}(\mathbb{R}))$ is immediate because the bounded
operator
$\overline{K\partial_{s}}:L^{\infty}([0,T],C^{\alpha,\tau}(\mathbb{R}))\to
L^{\infty}([0,T],C^{\alpha,\tau}(\mathbb{R}))$ maps the linear subspace
$\mathcal{S}(\Lambda_{T})$ into itself.

Since $\overline{K\partial_{s}}$ is indeed a bounded operator, and since
$\overline{K\partial_{s}}v=K(\partial_{s}(v))$ for smooth paths $v\in
C_{c}^{\infty}(\Lambda_{T})$ as shown above, this equality remains true for
general $v$ by the denseness of the linear subspace
$C_{c}^{\infty}(\Lambda_{T})$ in $C([0,T],C^{\alpha,\tau}(\mathbb{R})).$ ∎

We remark that the above proof is essentially sharp, in the sense that $K$
improves parabolic regularity by $2$, while $\partial_{s}$ destroys parabolic
regularity by $2$. Hence, we expect their composition to preserve regularity,
which is what we showed. Nonetheless, the Hölder exponent appearing in Theorem
1.4(b) may not be sharp, because we do not know if we obtained tightness of
$M^{N}$ in the best possible space in Proposition 6.13.

##  7\. Quenched tail field

The goal of this section is to prove the results from Section 1.3.1, namely
Theorem 1.7 and Theorem 1.8. Recall the quenched tail field $F_{N}(t,x)$ from
Definition 1.6. We begin with a preliminary estimate for the first moment of
$F_{N}(t,x)$.

######  Lemma 7.1.

Let $B$ be a standard Brownian motion. For all $t>0,x\in\mathbb{R}$, we have

$\mathbb{E}[F_{N}(t,x)]=N^{1/4}\mathbf{E}[e^{-N^{1/4}(B_{t}-x)}\mathbf{1}_{\\{B_{t}\geq
x\\}}]\leq(2\pi t)^{-1/2}.$

###### Proof.

The proof proceeds by writing $F_{N}$ as a quenched expectation, exactly as we
did in Section 1.5.1 or Section 2. Then one takes the annealed expectation
over the quenched expectation and uses the scale invariance of Brownian motion
to obtain

$\mathbb{E}[F_{N}(t,x)]=N^{1/4}e^{\frac{t}{2}\sqrt{N}+N^{1/4}x}\mathbf{P}(B_{t}-N^{1/4}t\geq
x).$

Now applying Girsanov with the stochastic exponential of $-N^{1/4}B$ will give
the claimed equality. To prove the inequality, we note that

$\displaystyle N^{1/4}\mathbf{E}[e^{-N^{1/4}(B_{t}-x)}\mathbf{1}_{\\{B_{t}\geq
x\\}}]$ $\displaystyle=N^{1/4}\int_{x}^{\infty}e^{-N^{1/4}(u-x)}p_{t}(u)du$
$\displaystyle\leq(2\pi
t)^{-1/2}\int_{x}^{\infty}N^{1/4}e^{-N^{1/4}(u-x)}du=(2\pi t)^{-1/2}.$

∎

We first prove a variant of Theorem 1.7 where the quenched tail field is
integrated against test functions.

######  Proposition 7.2.

Fix $m\in\mathbb{N}$. For all $\phi_{1},\ldots,\phi_{m}\in
C_{c}^{\infty}(\mathbb{R})$ and $t_{1},\ldots,t_{m}>0$ we have

$\displaystyle\bigg{(}\int_{\mathbb{R}}\phi_{i}(x)F_{N}(t_{i},x)dx\bigg{)}_{i=1}^{m}\stackrel{{\scriptstyle
d}}{{\to}}\bigg{(}\int_{\mathbb{R}}\phi_{i}(x)\mathcal{Z}_{t_{i}}(x)dx\bigg{)}_{i=1}^{m}.$
(7.1)

###### Proof.

Let us formally define

$\mu_{t}^{N}(dx):=N^{1/2}e^{\frac{t}{2}\sqrt{N}}K_{0,Nt}(N^{3/4}t+N^{1/2}x)dx.$

The above definition can be interpreted rigorously as a random Borel measure
by pushing forward $K_{0,Nt}$ under a linear change of coordinates, just as we
did in (1.3). Then we notice that Definition 1.6 is equivalent to

$F_{N}(t,x):=N^{1/4}e^{N^{1/4}x}\mu_{t}^{N}[x,\infty).$

An application of Fubini’s theorem gives

$\int_{-\infty}^{x}N^{1/4}e^{N^{1/4}u}\mu_{t}^{N}[u,\infty)du=\int_{-\infty}^{x}e^{N^{1/4}u}\mu_{t}^{N}(du)+e^{N^{1/4}x}\mu_{t}^{N}[x,\infty).$

In the sense of distributions, it is clear that

$\displaystyle
F_{N}(t,x)=\partial_{x}\bigg{[}\int_{-\infty}^{x}N^{1/4}e^{N^{1/4}u}\mu_{t}^{N}[u,\infty)du\bigg{]},\quad
e^{N^{1/4}x}\mu_{t}^{N}(dx)=\partial_{x}\bigg{[}\int_{-\infty}^{x}e^{N^{1/4}u}\mu_{t}^{N}(du)\bigg{]}.$

Consequently, for each $1\leq i\leq m$, we can repeatedly integrate by parts
to obtain that

$\displaystyle\int_{\mathbb{R}}\phi_{i}(x)F_{N}(t_{i},x)dx$
$\displaystyle=-\int_{\mathbb{R}}\phi_{i}^{\prime}(x)\bigg{[}\int_{-\infty}^{x}N^{1/4}e^{N^{1/4}u}\mu_{t_{i}}^{N}[u,\infty)du\bigg{]}dx$
$\displaystyle=-\int_{\mathbb{R}}\phi_{i}^{\prime}(x)\bigg{[}\int_{-\infty}^{x}e^{N^{1/4}u}\mu_{t_{i}}^{N}(du)+e^{N^{1/4}x}\mu_{t_{i}}^{N}[x,\infty)\bigg{]}dx$
$\displaystyle=\int_{\mathbb{R}}\phi_{i}(x)e^{N^{1/4}x}\mu_{t_{i}}^{N}(dx)-\int_{\mathbb{R}}\phi_{i}^{\prime}(x)e^{N^{1/4}x}\mu_{t_{i}}^{N}[x,\infty)dx.$

Thanks to Theorem 1.4(b) (see Remark 1.5(b)) we know that as $N\to\infty$

$\bigg{(}\int_{\mathbb{R}}\phi_{i}(x)e^{N^{1/4}x}\mu_{t_{i}}^{N}(dx)\bigg{)}_{i=1}^{m}\stackrel{{\scriptstyle
d}}{{\to}}\bigg{(}\int_{\mathbb{R}}\phi_{i}(x)\mathcal{Z}_{t_{i}}(x)dx\bigg{)}_{i=1}^{m}.$

Thus, to show (7.1) it suffices to show that as $N\to\infty$,
$\int_{\mathbb{R}}\phi_{i}^{\prime}(x)e^{N^{1/4}x}\mu_{t_{i}}^{N}[x,\infty)dx$
goes to zero in $L^{1}(\mathbb{P})$ for each $1\leq i\leq m$. Indeed, observe
that

$\displaystyle\mathbb{E}\bigg{|}\int_{\mathbb{R}}\phi_{i}^{\prime}(x)e^{N^{1/4}x}\mu_{t_{i}}^{N}[x,\infty)dx\bigg{|}$
$\displaystyle\leq\int_{\mathbb{R}}|\phi_{i}^{\prime}(x)|e^{N^{1/4}x}\mathbb{E}[\mu_{t_{i}}^{N}[x,\infty)]dx.$

By Lemma 7.1 we get

$e^{N^{1/4}x}\mathbb{E}[\mu_{t_{i}}^{N}[x,\infty)]=\mathbf{E}[e^{-N^{1/4}(B_{t_{i}}-x)}\mathbf{1}_{\\{B_{t_{i}}\geq
x\\}}]\leq N^{-1/4}(2\pi t_{i})^{-1/2}$

for all $x\in\mathbb{R}$, which implies the desired $L^{1}$ convergence. This
completes the proof. ∎

Note that composition with the logarithm function is not a continuous
operation in the topology of integration against test functions, hence the
convergence result stated above cannot be directly translated to any
convergence result for the KPZ equation. Our next lemma produces a regularity
estimate two-point spatial difference of the quenched tail field. This is the
crucial result that allows us to improve Proposition 7.2 to obtain multipoint
convergence in law of $F_{N}(t,x)$ to $\mathcal{Z}_{t}(x)$ for individual
values of $(t,x)\in(0,\infty)\times\mathbb{R}$.

######  Lemma 7.3.

As $N\to\infty$, we have that

$\mathbb{E}[F_{N}(t,x)F_{N}(t,y)]\to\mathbf{E}[\mathcal{Z}_{t}(x)\mathcal{Z}_{t}(y)]$

where $\mathcal{Z}$ as in Proposition 1.3. Furthermore, the above convergence
is uniform over compact sets of $(t,x,y)\in(0,\infty)\times\mathbb{R}^{2}$.

###### Proof.

By adapting the methods in Section 2, the two-point correlation function given
by $(t,x,y)\mapsto\mathbb{E}[F_{N}(t,x)F_{N}(t,y)]$ can be written as an
expression involving only the 2-point motion associated to the kernels
$K_{0,t}$. More specifically, in the notation of Section 2, it equals

$Ne^{t\sqrt{N}}\mathbf{P}_{SB^{(2)}_{N^{1/2}\nu}}(X_{t}-N^{1/4}t\geq
x,Y_{t}-N^{1/4}t\geq y).$

Consequently the two-point correlation function only depends on the total mass
of the characteristic measure. We shall thus assume $\nu$ is a multiple of
uniform measure. Without loss of generality, we will assume $\sigma=1$ (i.e.,
$\nu$ is precisely the uniform measure).

We shall now invoke results from [BR20] which are for the uniform
characteristic measure. In [BR20], the authors consider a slightly different
field $\widehat{F}_{N}(t,x)$ which is “dual” to ours in a certain sense. More
specifically, they vary the starting point $x$ and fix the tail probability as
$[0,\infty)$, whereas we fix the starting point $0$ and vary the tail
probability as $[x,\infty)$. One may show that the distributions of both
fields are the same as space-time processes, modulo a reflection of the
characteristic measure. More precisely, we have
$F_{N}^{\nu}\stackrel{{\scriptstyle d}}{{=}}\widehat{F}_{N}^{\mu}$, where
$\mu$ is the pushforward of $\nu$ by $x\mapsto 1-x$, and the superscript
highlights the dependence of each field on the underlying characteristic
measure.111In particular, our results imply multipoint convergence of their
field $\widehat{F}_{N}$ to (SHE) as well. This is proved using an explicit
coupling of the two fields via the Brownian web. See [Yu16, Equation (1.8)]
and the subsequent discussion for a short proof, which in turn is based on a
construction of [SSS14, Section 3].

Note that for the case of uniform characteristic measure, we have $\mu=\nu$.
Proposition 1.22 in [BR20] provides exact moment formulas for the unscaled
version of $F_{N}(t,x)$ in this case. Taking the scalings into account we have
that

$\displaystyle\mathbb{E}[F_{N}(t,x)F_{N}(t,y)]$
$\displaystyle=\oint_{r_{1}+i\mathbb{R}}\oint_{r_{2}+i\mathbb{R}}\frac{N^{-1/2}(z_{2}-z_{1})}{N^{-1/2}(z_{2}-z_{1})-N^{-1/2}-N^{-3/4}(z_{1}+z_{2})-N^{-1}z_{1}z_{2}}$
$\displaystyle\hskip 28.45274pt\cdot
e^{\frac{t}{2}(z_{1}^{2}+z_{2}^{2})-(xz_{1}+yz_{2})}\frac{N^{-1/2}}{(N^{-1/4}+N^{-1/2}z_{1})(N^{-1/4}+N^{-1/2}z_{2})}\frac{dz_{1}}{2\pi
i}\frac{dz_{2}}{2\pi i},$

where $r_{1},r_{2}$ are fixed numbers not depending on $t,x,y$ and $N$ (large
enough), and such that $r_{2}>r_{1}+1.$

Now assume that we have a sequence $(t^{N},x^{N},y^{N})$ converging to
$(t,x,y)\in(0,\infty)\times\mathbb{R}^{2}$. Note that for $z_{1}\in
r_{1}+i\mathbb{R}$ and $z_{2}\in r_{2}+i\mathbb{R}$ we have that
$|e^{\frac{t^{N}}{2}(z_{1}^{2}+z_{2}^{2})-(x^{N}z_{1}+y^{N}z_{2})}|\leq\mathrm{C}e^{-\frac{1}{\mathrm{C}}(|z_{1}|^{2}+|z_{2}|^{2})}$
for some constant $\mathrm{C}>0$ which is independent of $N,z_{1},z_{2}$.
Also, note that the ratios appearing in the integral may be bounded
independently of $N$ thanks to the fact that $z_{1}\in r_{1}+i\mathbb{R}$ and
$z_{2}\in r_{2}+i\mathbb{R}$ with $r_{2}>r_{1}+1$. Therefore, by the dominated
convergence theorem (with dominating function given by
$e^{-\frac{1}{\mathrm{C}}(|z_{1}|^{2}+|z_{2}|^{2})}$) we may conclude that the
integral expression for $\mathbb{E}[F_{N}(t^{N},x^{N})F_{N}(t^{N},y^{N})]$
converges as $N\to\infty$ to

$\oint_{r_{1}+i\mathbb{R}}\oint_{r_{2}+i\mathbb{R}}\frac{z_{2}-z_{1}}{z_{2}-z_{1}-1}e^{\frac{t}{2}(z_{1}^{2}+z_{2}^{2})-(xz_{1}+yz_{2})}\frac{dz_{1}}{2\pi
i}\frac{dz_{2}}{2\pi i}.$

This is known to agree with
$\mathbf{E}[\mathcal{Z}_{t}(x)\mathcal{Z}_{t}(y)]$, see [BC14, Section 6.2],
thus proving the uniform convergence. ∎

With the above lemma in place, we now prove a stronger version of Theorem 1.7.

######  Proposition 7.4.

Suppose that the deterministic sequence of vectors
$(x^{N}_{1},\ldots,x^{N}_{m})\in\mathbb{R}^{m}$ converges as $N\to\infty$ to
$(x_{1},\ldots,x_{m})\in\mathbb{R}^{m}$. Fix $t_{1},\ldots,t_{m}>0$. Then we
have the joint convergence

$\big{(}F_{N}(t_{i},x^{N}_{i}))_{i=1}^{m}\stackrel{{\scriptstyle
d}}{{\to}}\big{(}\mathcal{Z}_{t_{i}}(x_{i})\big{)}_{i=1}^{m}.$

###### Proof.

We first prove the case when $x_{i}^{N}=x_{i}$ for $1\leq i\leq m$. We give a
proof for $m=1$ to simplify the notation, but the generalization to larger $m$
is straightforward. We will simply write $(t_{1},x_{1})$ as $(t,x)$, which
will be fixed throughout the proof. By Lemma 7.1 we get that
$\mathbb{E}[F_{N}(t,x)]\leq(2\pi t)^{-1/2}$ which is independent of $N$, thus
it follows that the $\\{F_{N}\\}_{N\geq 1}$ are tight. Consider any limit
point $\mu_{0}$ of the laws of $\\{F_{N}(t,x)\\}_{N\geq 1}$. Fix a smooth
compactly supported nonnegative function $\phi:\mathbb{R}\to\mathbb{R}$ which
integrates to $1$, and define
$\phi^{\lambda}_{x}(y)=\lambda^{-1}\phi(\lambda^{-1}(y-x)).$ By Proposition
7.2 we know that for each $\lambda,x\in\mathbb{R}$,
$\big{\\{}\int_{\mathbb{R}}\phi^{\lambda}_{x}(y)F_{N}(t,y)dy\big{\\}}_{N\geq
1}$ is a tight sequence. For $r\geq 1$, consider any measure $\mu_{r}$ on
$\mathbb{R}^{r+1}$ which is a joint limit point as $N\to\infty$ of

$\bigg{(}F_{N}(t,x)\ ,\
\big{(}\int_{\mathbb{R}}\phi^{2^{-k}}_{x}(y)F_{N}(t,y)dy\big{)}_{k=1}^{r}\bigg{)},$

such that the first marginal of $\mu_{r}$ is $\mu_{0}$. Using a
diagonalization argument, these measures $\mu_{r}$ may be chosen so that they
form a projective family, and therefore by the Kolmogorov extension theorem,
we may consider any projective limit $\mu$ which will be a measure on the
space of sequences $(a_{k})_{k\geq 0}\in\mathbb{R}^{\mathbb{N}_{0}}$, equipped
with the $\sigma$-algebra generated by the projection maps. By Proposition
7.2, we find that for such a measure $\mu$ the marginal distribution of
$(a_{k})_{k\geq 1}$ is simply equal to the law of
$\big{(}\int_{\mathbb{R}}\mathcal{Z}_{t}(y)\phi^{2^{-k}}_{x}(y)dy\big{)}_{k\geq
1}$. We now claim that

$\displaystyle\limsup_{\lambda\to
0}\limsup_{N\to\infty}\mathbb{E}\bigg{|}F_{N}(t,x)-\int_{\mathbb{R}}F_{N}(t,y)\phi^{\lambda}_{x}(y)dy\bigg{|}=0.$
(7.2)

Assuming this fact, one finds that such a measure $\mu$ is necessarily
supported on those sequences $(a_{k})_{k\geq 0}$ which satisfy
$a_{0}=\lim_{k\to\infty}a_{k}$ in $L^{1}(\mu)$, which means that $a_{0}$ must
have the law of $\mathcal{Z}_{t}(x)$ under $\mu$. We are thus left to check
(7.2). By Lemma 7.3, we have that

$\limsup_{N\to\infty}\mathbb{E}[(F_{N}(t,x)-F_{N}(t,y))^{2}]=\mathbf{E}[(\mathcal{Z}_{t}(x)-\mathcal{Z}_{t}(y))^{2}]\leq\mathrm{C}|x-y|$

uniformly over compacts sets of $(t,x,y)\in(0,\infty)\times\mathbb{R}^{2}$
where the above inequality is a known estimate for (SHE) (see [Das22,
Proposition 2.4-nw] for example). Thus, by Fatou’s lemma, we note that

$\displaystyle\limsup_{N\to\infty}\mathbb{E}\bigg{|}F_{N}(t,x)-\int_{\mathbb{R}}F_{N}(t,y)\phi^{\lambda}_{x}(y)dy\bigg{|}$
$\displaystyle\leq\limsup_{N\to\infty}\int_{\mathbb{R}}\mathbb{E}|F_{N}(t,x)-F_{N}(t,y)|\phi^{\lambda}_{x}(y)dy$
$\displaystyle\leq\int_{\mathbb{R}}\limsup_{N\to\infty}\mathbb{E}|F_{N}(t,x)-F_{N}(t,y)|\phi^{\lambda}_{x}(y)dy$
$\displaystyle\leq\int_{\mathbb{R}}\limsup_{N\to\infty}\mathbb{E}[(F_{N}(t,x)-F_{N}(t,y))^{2}]^{1/2}\phi^{\lambda}_{x}(y)dy$
$\displaystyle\leq\mathrm{C}\int_{\mathbb{R}}|x-y|^{1/2}\phi^{\lambda}_{x}(y)dy=\mathrm{C}\lambda^{1/2}\int_{\mathbb{R}}|u|^{1/2}\phi(u)du.$

Taking $\lambda\to 0$ leads to (7.2). To justify the above application of
Fatou’s lemma with $\limsup$, one needs a bound on the integrand which is
independent of $N$ and is in $L^{1}(\phi^{\lambda}_{x}(y)dy)$, for which we
may use Lemma 7.1 to get

$\mathbb{E}|F_{N}(t,x)-F_{N}(t,y)|\leq\mathbb{E}[F_{N}(t,x)+F_{N}(t,y)]\leq
2(2\pi t)^{-1/2}.$

Finally, the case when $x_{i}^{N}$ varies with $N$ follows from the fact that
as $N\to\infty$ we have

$\mathbb{E}\big{[}\big{(}F_{N}(t_{i},x_{i}^{N})-F_{N}(t_{i},x_{i})\big{)}^{2}\big{]}\to
0$

due to the uniform convergence in Lemma 7.3. This completes the proof. ∎

######  Remark 7.5.

We remark that one can allow $t_{1},\ldots,t_{m}$ to depend on $N$ as well,
and the multi-point convergence result analogous to Proposition 7.4 would
still be true. However, the proof is more involved than just studying the
correlation functions: first one modifies the proof of Proposition 7.2 to
allow the $t_{i}$ to depend on $N$. This is fairly immediate since Theorem
1.4(b) gives convergence in a topology that is uniform in time. Then notice
that in Lemma 7.3 we have already allowed the $t_{i}$ to depend on $N$. Thus
we can replace $t_{i}$ by $t_{i}^{N}$ in Proposition 7.4 as well (the estimate
using Fatou’s Lemma will still work), which is enough to establish the claim.
After the claim is established, some interesting effects can be observed by
setting $t^{N}_{i}:=t_{i}+\alpha N^{-1/4}$ for $\alpha\in\mathbb{R}$. Using
Definition 1.6 one sees in this case that $\alpha$ gets reinterpreted as a
shift of the $x$ variable, but with different scaling values.

###### Proof of Theorem 1.8.

We continue with the same notations as in the statement of the Theorem 1.8.
Fix any $a\in\mathbb{R}$. Set $v:=\sqrt{ct}$ and $x:=d\sqrt{\frac{t}{c}}$ so
that $c=\frac{v^{2}}{t}$ and $d=vx/t$. It suffices to show that

$\displaystyle\lim_{N\to\infty}\mathbf{P}_{SB^{(k)}_{\nu}}\bigg{(}\max_{1\leq
i\leq k(N)}\big{\\{}X^{i}(Nt)\big{\\}}-vN^{3/4}-x\sqrt{N}$
$\displaystyle-\tfrac{tN^{1/4}}{v}\big{(}r_{N}-\tfrac{1}{4}\log N\big{)}\leq
aN^{1/4}\bigg{)}$ (7.3)
$\displaystyle=\mathbb{P}(tG/v+\log\mathcal{Z}_{v^{2}/t}(vx/t)\leq a),$

where $\mathcal{Z}$ solves (SHE) with $\sigma=\frac{v}{t\nu([0,1])}$ and $G$
is an independent Gumbel random variable.

For now, fix any $k\in\mathbb{N}$ which may be arbitrary. Fix a realization of
the kernels $\omega=\\{K_{s,t}:-\infty<s<t<\infty\\}.$ As before, we let
$\mathbb{P}_{\omega}^{(k)}$ be the quenched probability of $k$ random motions
$(X^{i}(t))_{i=1}^{k}$ sampled from these kernels. The random variable

$\mathbb{P}_{\omega}^{(k)}\bigg{(}\max_{1\leq i\leq
k}\big{\\{}X^{i}(Nt)\big{\\}}\leq
vN^{3/4}+x\sqrt{N}+\frac{t}{v}N^{1/4}(r_{N}-\frac{1}{4}\log
N)+aN^{1/4}\bigg{)}$

has the same law for any $y>0$ as the random variable

$\mathbb{P}_{\omega}^{(k)}\bigg{(}\max_{1\leq i\leq
k}\big{\\{}X^{i}(Ny^{2}t)\big{\\}}\leq
y\big{[}vN^{3/4}+x\sqrt{N}+\frac{t}{v}N^{1/4}(r_{N}-\frac{1}{4}\log
N)+aN^{1/4}\big{]}\bigg{)}$

except that the characteristic measure in the latter expression has changed to
$y\cdot\nu$. Now we make a specific choice of $y:=v/t$ and we get by
Definition 1.6 that it is equal (pathwise, not just in distribution) to the
quantity

$\displaystyle\bigg{(}1-\exp\bigg{[}-\frac{v^{2}}{2t}\sqrt{N}-N^{\frac{1}{4}}\frac{vx}{t}-\frac{av}{t}-r_{N}\bigg{]}\cdot
F_{N}\big{(}v^{2}/t,\;\tfrac{vx}{t}+N^{-\frac{1}{4}}(\tfrac{av}{t}+r_{N}-\tfrac{1}{4}\log
N)\big{)}\bigg{)}^{k}.$ (7.4)

where it is implicit that the characteristic measure associated to $F_{N}$ is
$(v/t)\cdot\nu$. Then we may choose

$k=k(N)=\bigg{\lfloor}\exp\bigg{[}\frac{v^{2}}{2t}\sqrt{N}+\frac{vx}{t}N^{1/4}+r_{N}\bigg{]}\bigg{\rfloor}$

as stated in the theorem to obtain that (7.4) is a quantity of the form
$(1-\frac{u(N)}{k(N)+O_{N}(1)})^{k(N)}$, where the $O_{N}(1)$ term is
deterministic and bounded between $0$ and $1$, and $u(N)$ is a sequence of
random variables which by Proposition 7.4 converges in law to the strictly
positive random variable $e^{-\frac{av}{t}}\mathcal{Z}_{v^{2}/t}(vx/t)$. Since
the functions $u\mapsto(1-\frac{u}{k(N)+O_{N}(1)})^{k(N)}$ converge uniformly
to $u\mapsto e^{-u}$ on compact subsets of $[0,\infty)$, we can conclude that
the expression in (7.4), therefore converges in law to the random variable

$\exp\big{[}-e^{-\frac{av}{t}}\mathcal{Z}_{v^{2}/t}(vx/t)\big{]},$

where $\mathcal{Z}$ solves (SHE) with $\sigma$ equal to $v/(t\nu([0,1]))$. Now
this convergence in law also implies convergence of the associated annealed
expectations thanks to the boundedness of all quantities involved, which
implies that the limit of the left side of (7.3) equals

$\mathbb{E}\bigg{[}\exp\big{[}-e^{-\frac{av}{t}}\mathcal{Z}_{v^{2}/t}(vx/t)\big{]}\bigg{]},$

which agrees with the right-hand side of (7.3) by the explicit form of the
Gumbel distribution. ∎

Using Remark 7.5, one could also consider replacing time $Nt$ by time $Nt$
plus a correction of order $N^{\alpha}$ for some $\alpha<1$, and this would
change the constants appearing in the proof in an explicit manner and
introduce even more terms in the recentering if $\alpha\in[1/2,1)$. For
brevity, we will not explore this.

##  Appendix A Technical lemmas on Brownian bridges

######  Definition A.1 (Brownian concatenated process).

Fix any $x\in\mathbb{R}$, and $s\in[0,1]$. Let $B_{\operatorname{br}}$ be a
Brownian bridge on $[0,s]$ from $0$ to $x$. Let $W$ be a standard Brownian
motion independent of $B_{\operatorname{br}}$. We define the Brownian
concatenated process $B_{s,x}$ with anchor at $(s,x)$ as

$\displaystyle B_{s,x}(y):=\begin{cases}B_{\operatorname{br}}(y)&y\in[0,s]\\\
W(y-s)+x&y\in[s,1].\end{cases}$

######  Lemma A.2.

Let $W=(W_{s})_{s\in[0,1]}$ be a Brownian motion on $[0,1]$. Let
$F=(F_{s})_{s\in[0,1]}:C[0,1]\to C[0,1]$ be a Borel-measurable functional.
Suppose that $\mathbf{E}[\sup_{s\leq 1}F_{s}(W)^{2}]<\infty$. Suppose further
that the function $g(s,x):=\mathbf{E}\left[F_{s}(B_{s,x})\right]$ is bounded
on $[0,1]\times\mathbb{R}$ and continuous in $x$ for each $s>0$. Then for all
compactly supported smooth functions $\phi$ we have the identity

$\mathbf{E}\bigg{[}\int_{0}^{1}F_{s}(W)\phi(W_{s})ds\bigg{]}=\int_{0}^{1}\int_{\mathbb{R}}g(s,x)p_{s}(x)\phi(x)\,dx\,ds,$

and moreover for all $x\in\mathbb{R}$ we have

$\mathbf{E}\bigg{[}\int_{0}^{1}F_{s}(W)dL_{x}^{W}(s)\bigg{]}=\int_{0}^{1}g(s,x)p_{s}(x)ds,$

where $dL^{W}_{x}(s)$ is the Lebesgue-Stiltjes measure associated to the
nondecreasing function $s\mapsto L_{x}^{W}(s).$

We remark that all expectations appearing above are finite. Let
$Z:=\sup_{s\leq 1}|F_{s}(W)|$. Note that
$\big{|}\int_{0}^{1}F_{s}(W)\phi(W_{s})ds\big{|}\leq
Z\cdot\int_{0}^{1}|\phi(W_{s})|ds$ and
$\big{|}\int_{0}^{1}F_{s}(W)dL_{x}^{W}(s)\big{|}\leq Z\cdot L^{W}_{x}(1).$
Therefore a simple application of Cauchy-Schwarz (noting that
$\mathbf{E}[Z^{2}]<\infty$ by assumption) shows that all expectations above
are finite.

###### Proof.

Note that for all $\phi$ continuous and compactly supported, by the tower
property of conditional expectation we have

$\mathbf{E}\bigg{[}\int_{0}^{1}F_{s}(W)\phi(W_{s})ds\bigg{]}=\int_{0}^{1}\mathbf{E}[F_{s}(W)\phi(W_{s})]ds=\int_{0}^{1}\mathbf{E}\big{[}\mathbf{E}[F_{s}(W)|W_{s}]\phi(W_{s})\big{]}ds.$

Now the law of $W$ conditioned on $W_{s}$ is precisely given by Brownian
concatenated process with the anchor at $(s,W_{s}).$ Thus,
$\mathbf{E}[F_{s}(W)|W_{s}]=g(s,W_{s})$. Therefore, the above expectation
equals

$\int_{0}^{1}\mathbf{E}[g(s,W_{s})\phi(W_{s})]ds=\int_{0}^{1}\int_{\mathbb{R}}g(s,x)p_{s}(x)\phi(x)dxds,$

proving the first identity. To prove the second identity, we will let $\phi$
converge weakly to a Dirac mass and use the continuity in $x$ of $g$. Let
$\phi$ be smooth, nonnegative, even, supported in $[-1,1]$, and suppose that
it integrates to 1. Let
$\phi^{\varepsilon}(u):=\varepsilon^{-1}\phi(\varepsilon^{-1}u)$. It is
standard that the random measure on $[0,1]$ given by
$\phi^{\varepsilon}(x-W_{s})ds$ converges almost surely as $\varepsilon\to 0$
to $L^{W}_{x}(ds),$ with respect to the topology of weak convergence on the
space of finite measures on $[0,1].$ Consequently, since we assume continuity
of $s\mapsto F_{s}(W)$ we have the almost sure convergence

$\int_{0}^{1}F_{s}(W)\phi^{\varepsilon}(x-W_{s})ds\stackrel{{\scriptstyle\varepsilon\to
0}}{{\longrightarrow}}\int_{0}^{1}F_{s}(W)L^{W}_{x}(ds).$ (A.1)

Now by the first identity, the expectation of the left-hand side equals

$\int_{0}^{1}H_{x}^{\varepsilon}(s)ds,\ \ \ \mbox{ where
}H_{x}^{\varepsilon}(s):=\int_{\mathbb{R}}g(s,u)p_{s}(u)\phi^{\varepsilon}(x-u)du.$

It is clear from the continuity of $g$ in $x$ that $H_{x}^{\varepsilon}(s)\to
g(s,x)p_{s}(x)$ as $\varepsilon\to 0$, and moreover since $g$ is assumed to be
globally bounded and $\phi$ is supported in $[-1,1]$, we have that

$|H_{x}^{\varepsilon}(s)|\leq\max_{u\in[x-1,x+1]}|g(s,u)|p_{s}(u)\leq\|g\|_{L^{\infty}}s^{-1/2}.$

Hence by dominated convergence we find that

$\mathbf{E}\bigg{[}\int_{0}^{1}F_{s}(W)\phi^{\varepsilon}(x-W_{s})ds\bigg{]}=\int_{0}^{1}H^{\varepsilon}(s)ds\to\int_{0}^{1}g(s,x)p_{s}(x)ds.$

Thus it suffices to prove that (A.1) also holds in $L^{1}$. To prove that, we
note that

$\bigg{|}\int_{0}^{1}F_{s}(W)\phi^{\varepsilon}(x-W_{s})ds\bigg{|}\leq\int_{0}^{1}|F_{s}(W)|\phi^{\varepsilon}(x-W_{s})ds\leq
Z\int_{0}^{1}\phi^{\varepsilon}(x-W_{s})ds,$

where $Z:=\sup_{s\in[0,1]}|F_{s}(W)|$. By the occupation time formula for the
local time, we have

$\sup_{\varepsilon\in(0,1]}\int_{0}^{1}\phi^{\varepsilon}(x-W_{s})ds=\sup_{\varepsilon\in(0,1]}\int_{\mathbb{R}}\phi^{\varepsilon}(x-u)L^{W}_{u}(1)du\leq\sup_{u\in\mathbb{R}}L^{W}_{u}(1).$

Summarizing the last two expressions and applying the Cauchy-Schwarz
inequality, we have that

$\mathbf{E}_{BM}\bigg{[}\sup_{\varepsilon\in(0,1]}\bigg{|}\int_{0}^{1}F_{s}(W)\phi^{\varepsilon}(x-W_{s})ds\bigg{|}\bigg{]}\leq\mathbf{E}[Z^{2}]^{1/2}\mathbf{E}\big{[}\sup_{u\in\mathbb{R}}L^{W}_{u}(1)^{2}\big{]}^{1/2}.$

The expected square of the sup over local times is finite, e.g., by [RY99,
Chapter XI, Theorem (2.4)]. Therefore, we have uniform integrability and thus
$L^{1}$ convergence in (A.1) as desired. ∎

######  Lemma A.3.

Let $B^{s,x}(t)$ be a Brownian bridge on $[0,s]$ from $0$ to $x$. Denote by
$L^{B^{s,x}}_{0}(t)$ the local time at 0 of $B^{s,x}$ at time $t$. Then the
map $(s,x)\mapsto L^{B^{s,x}}_{0}(s)$ has a modification which is almost
surely continuous in both variables on $(0,1]\times\mathbb{R}$.

###### Proof.

Let $G(s,x):=L^{B^{s,x}}_{0}(s)$. Note that the zero set of $B^{s,xs^{1/2}}$
is just a reparametrization of that of $B^{x,1}$, in fact $B^{x,xs^{1/2}}$ is
just $s^{1/2}$ multiplied by a reparametrization of $B^{x,1}$, and so we have
that $G(s,xs^{1/2})=G(1,x)$. Thus $G(s,x)=G(1,xs^{-1/2})$ and therefore it
suffices to show a.s. Hölder continuity of $x\mapsto G(1,x)$.

To prove continuity of $x\mapsto G(1,x)$, let us simplify notation and
henceforth write $B^{1,x}=B^{x}$. Now write
$L_{0}^{B^{x}}(1)=L_{0}^{B^{x}}(1/2)+L_{0}^{W^{x}}(1/2)$, where
$W^{x}(t)=B^{x}(1-t).$ Note that on the time interval $[0,1/2]$, $B^{x}$ is
absolutely continuous with respect to a Brownian motion of drift $x$ started
from 0, and moreover the Radon-Nikodym derivative is independent of $x$.
Likewise, $W^{x}$ is absolutely continuous with respect to a Brownian motion
of drift $-x$ started from $x$, and similarly, the Radon-Nikodym derivative is
independent of $x$. Consequently, for a standard Brownian motion $Z$ if we
define $Z^{a,\lambda}(t):=a+\lambda t+Z(t),$ then it suffices to show that
$(a,\lambda)\mapsto L_{0}^{Z^{a,\lambda}}(1/2)$ is a.s. jointly Hölder
continuous as this would imply the desired result for both $B^{x}$ and
$W^{x}$. Note that $L_{0}^{Z^{a,\lambda}}(1/2)=L_{-a}^{Z^{0,\lambda}}(1/2).$
For Brownian motion, joint continuity of local times in the drift and the
spatial variable is now a standard result. ∎

######  Lemma A.4.

Fix $a,b,c,d,x\in\mathbb{R}$ with $c^{2}+d^{2}=1$. Let $B$ be a Brownian
bridge from $0$ to $x$ on $[0,1]$ and let $W$ be an independent standard
Brownian motion. Consider the process $X:=a+cB+dW$. For all $\theta>0$, we
have

$\mathbf{E}[e^{\theta L_{0}^{X}(1)}]\leq\theta\sqrt{2\pi}e^{\theta^{2}/2}.$
(A.2)

######  Remark A.5.

Since the above bound is uniform in $a,b,c,d,x$, the same bound continues to
hold when $a,b,c,d,x$ are random variables (with $c^{2}+d^{2}=1$) independent
of $B$ and $W$.

###### Proof.

Note that conditioned on $W(1)$, $X$ is a standard Brownian bridge from $a$ to
$a+cx+dW(1)$. In [Pit99, (3)] it is shown that for a standard Brownian bridge
$Y$ from $a$ to $a+b$ one has for all $v>0$ that

$\mathbb{P}(L_{0}^{Y}(1)>v)=e^{b^{2}/2}e^{-(|a|+|a+b|+v)^{2}/2}$

and therefore if $\theta>0$ then

$\mathbf{E}[e^{\theta L_{0}^{Y}(1)}]=\int_{0}^{\infty}\theta e^{\theta
y}\mathbb{P}(L_{0}^{Y}(1)>y)dy\leq\theta e^{b^{2}/2}\int_{\mathbb{R}}e^{\theta
y}e^{-(|a|+|a+b|+y)^{2}/2}dy=\theta\sqrt{2\pi}e^{\theta^{2}/2}e^{-(|a|+|a+b|)\theta}.$

Thus returning to the process $X$ we simply take $b=cx+dW(1)$, and we find
that

$\mathbf{E}[e^{\theta
L_{0}^{X}(1)}|W(1)]\leq\theta\sqrt{2\pi}e^{\theta^{2}/2}e^{-(|a|+|a+cx+dW(1)|)\theta}.$

Taking the expectation again and noting that
$e^{-(|a|+|a+cx+dW(1)|)\theta}\leq 1$ we arrive at (A.2), proving the claim. ∎

######  Lemma A.6.

Fix $r,m\in\mathbb{Z}_{>0}$. Fix
$\mathbf{x}=(x_{1},x_{2},\ldots,x_{r})\in\mathbb{R}^{r}$,
$\mathbf{t}=(t_{1},t_{2},\ldots,t_{m})\in[0,1]^{m}$, and
$\mathbf{s}=(s_{1},s_{2},\ldots,s_{r})\in[0,1]^{r}$. Let
$\mathbf{B}^{\mathbf{s},\mathbf{x}}=(B^{i})_{i=1}^{r}$ be $r$-many independent
processes with each $B^{i}$ being distributed as a Brownian concatenated
process with the anchor at $(s_{i},x_{i})$ (recall Definition A.1). Let
$\ell_{k}:\mathbb{R}^{n}\to\mathbb{R}$ be linear functions for $1\leq k\leq
m$. Define

$f(\mathbf{x},\mathbf{s},\mathbf{t}):=\mathbf{E}\bigg{[}\exp\bigg{(}\sum_{k=1}^{m}L_{0}^{\ell_{k}(\mathbf{B}^{\mathbf{s},\mathbf{x}})}(t_{k})\bigg{)}\bigg{]}.$

Then $f$ is globally bounded and continuous on
$\mathbb{R}^{n}\times(0,1]^{r}\times(0,1]^{m}.$

###### Proof.

We use a coupling argument based on Lemma A.3. Take $W^{1},\ldots,W^{n}$ to be
i.i.d. Brownian motions on $[0,1]$ and then define

$B^{j,s,x}(t):=W^{j}(t)-(\tfrac{t}{s}\wedge 1)(W^{j}(s)-x).$

Note that for all $r$-tuples of pairs $(s_{j},x_{j})$ the processes
$(B^{j,s_{j},x_{j}})_{j=1}^{r}$ are distributed as $(B^{i})_{i=1}^{r}.$
Applying Lemma A.3 we see that if
$(\mathbf{x}_{n},\mathbf{s}_{n},\mathbf{t}_{n})_{n\in\mathbb{N}}$ converges to
$(\mathbf{x},\mathbf{s},\mathbf{t})\in\mathbb{R}^{n}\times(0,1]^{r}\times(0,1]^{m},$
we have

$L_{0}^{\ell_{k}(\mathbf{B}^{\mathbf{s}_{n},\mathbf{x}_{n}})}(t_{n,k})\to
L_{0}^{\ell_{k}(\mathbf{B}^{\mathbf{s},\mathbf{x}})}(t_{k})$

for each $1\leq k\leq n$ almost surely. Consequently, the exponential of the
sum over $k$ also converges almost surely. To show convergence of the
expectations of those exponentials we need uniform integrability of the
exponentials. We will prove this in a way that also gives global boundedness
of the function $f.$ Since local time is an increasing process we have the
automatic bound $f(\mathbf{x},\mathbf{s},\mathbf{t})\leq
f(\mathbf{x},\mathbf{s},\mathbf{1}).$ To prove uniform integrability we will
upper-bound $f(\mathbf{x},\mathbf{s},\mathbf{1})$ in $L^{2}$ independently of
$s_{j},x_{j}$. First, apply Hölder’s inequality to obtain

$\mathbf{E}\bigg{[}\exp\bigg{(}\sum_{k=1}^{m}L_{0}^{\ell_{k}(\mathbf{B}^{\mathbf{s},\mathbf{x}})}(1)\bigg{)}\bigg{]}\leq\prod_{k=1}^{m}\mathbf{E}\bigg{[}\exp\bigg{(}mL_{0}^{\ell_{k}(\mathbf{B}^{\mathbf{s},\mathbf{x}})}(1)\bigg{)}\bigg{]}^{1/m}.$

Thus, it suffices to show each term in the product above is finite. Let us
take any $\ell(\mathbf{v}):=a_{1}v_{1}+\cdots+a_{r}v_{r}$. Assume without loss
of generality that $s_{1}\leq\cdots\leq s_{r}.$ Then on each subinterval,
$[s_{j},s_{j+1}]$, we have

$\ell(\mathbf{B}^{\mathbf{s},\mathbf{x}})(t)=\sum_{i=1}^{j}a_{i}[W^{i}(t)-(W^{i}(s_{i})-x_{i})]+\sum_{i=j+1}^{r}a_{i}[W^{i}(t)-\tfrac{t}{s_{i}}(W^{i}(s_{i})-x_{i})].$

Let us set

$\displaystyle D_{1,j}:=\sum_{i=1}^{j}a_{i}^{2},\qquad
D_{2,j}:=\sum_{i=j+1}^{r}a_{i}^{2},\qquad\gamma:=D_{2,j}\cdot\big{[}\sum_{i=j+1}^{r}a_{i}^{2}/s_{i}\big{]}^{-1}.$
(A.3)

Note that
$\ell(\mathbf{B}^{\mathbf{s},\mathbf{x}})(t+s_{j})-\ell(\mathbf{B}^{\mathbf{s},\mathbf{x}})(s_{j})$
has mean $t\sum_{i=j+1}^{r}\frac{a_{i}x_{i}}{s_{i}}$, and for
$s,t\in[0,s_{j+1}-s_{j}]$ its covariance function is given by

$\displaystyle C(s,t)$ $\displaystyle:=(s\wedge t)\cdot
D_{1,j}+\sum_{i=j+1}^{r}a_{i}^{2}\big{[}(s\wedge
t)-\tfrac{ts}{s_{i}}\big{]}=(s\wedge t)\cdot D_{1,j}+\big{[}(s\wedge
t)-\tfrac{ts}{\gamma}\big{]}\cdot D_{2,j}.$

As $s_{1}\leq\cdots\leq s_{r}$, one can check that
$\gamma\geq(s_{j+1}-s_{j})$. Thus,

$\ell(\mathbf{B}^{\mathbf{s},\mathbf{x}})(t+s_{j})-\ell(\mathbf{B}^{\mathbf{s},\mathbf{x}})(s_{j})\stackrel{{\scriptstyle
d}}{{=}}U_{1}(t)+U_{2}(t)$

as processes on $[0,s_{j+1}-s_{j}]$ where $U_{1}$ is a Brownian bridge with
diffusion rate $D_{2,j}$ from $0$ to $\gamma\sum_{i=j+1}^{r}a_{i}x_{i}/s_{i}$
on $[0,\gamma]$ and $U_{2}$ is an independent Brownian motion with diffusion
rate $D_{1,j}$.

On $[0,s_{j+1}-s_{j}],$ we have shown that
$\ell(\mathbf{B}^{\mathbf{s},\mathbf{x}})-\ell(\mathbf{B}^{\mathbf{s},\mathbf{x}})(s_{j})$
is distributed as the independent sum of a Brownian bridge (not necessarily
ending at zero) and a Brownian motion whose diffusivities sum to
$\|\ell\|_{2}^{2}=a_{1}^{2}+\cdots+a_{r}^{2}.$ Consequently, by breaking up
the expectation on each of these subintervals (by again applying Hölder’s
inequality) and using the fact that each of these subintervals has size at
most 1, it suffices to bound $\mathbf{E}[e^{\theta L_{0}^{X}(1)}]$ for all
$\theta>0$ where $X:=a+cB+dW$, where $B$ is a standard Brownian bridge from
$0$ to $x$ on $[0,1]$ and $W$ is an independent standard Brownian motion on
$[0,1]$. More precisely, we need a bound which is uniform over all
$a\in\mathbb{R}$ and $c^{2}+d^{2}=1$. This is precisely done in Lemma A.4.
This completes the proof. ∎

We remark that Definition A.1 can be naturally extended to Brownian
concatenated processes defined on $[0,T]$ for any fixed $T>0$. Both Lemma A.2
and Lemma A.6 continue to hold under this setting.

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