# KPZ equation limit of sticky Brownian motion

Sayan Das  S. Das, Department of Mathematics, Columbia University,
S. Das 2990 Broadway, New York, NY 10027 USA<EMAIL_ADDRESS>,  Hindy
Drillick  H. Drillick, Department of Mathematics, Columbia University,
H. Drillick 2990 Broadway, New York, NY 10027 USA<EMAIL_ADDRESS>and  Shalin Parekh  S. Parekh, Department of Mathematics, University of
Maryland,
S. Parekh 4176 Campus Dr, College Park, MD 20742 USA<EMAIL_ADDRESS>
###### Abstract.

We consider the motion of a particle under a continuum random environment
whose distribution is given by the Howitt-Warren flow. In the moderate
deviation regime, we establish that the quenched density of the motion of the
particle (after appropriate centering and scaling) converges weakly to the
$(1+1)$ dimensional stochastic heat equation driven by multiplicative space-
time white noise. Our result confirms physics predictions and computations in
[LDT17, BLD20, HCGCC23] and is the first rigorous instance of such weak
convergence in the moderate deviation regime. Our proof relies on a certain
Girsanov transform and works for all Howitt-Warren flows with finite and
nonzero characteristic measures. Our results capture universality in the sense
that the limiting distribution depends on the flow only via the total mass of
the characteristic measure. As a corollary of our results, we prove that the
fluctuations of the maximum of an $N$-point sticky Brownian motion are given
by the KPZ equation plus an independent Gumbel on timescales of order $(\log
N)^{2}.$

###### Key words and phrases:

Kardar–Parisi–Zhang equation, stochastic heat equation, Howitt-Warren flow,
sticky Brownian motion, local time, extreme value theory.

###### 2020 Mathematics Subject Classification:

Primary 60K37, 82B21, 82C22, Secondary 60G70.

###### Contents

  1. 1 Introduction
  2. 2 Girsanov transform and sticky Brownian motion estimates
  3. 3 Proof of Proposition 1.3
  4. 4 Martingales associated to the sticky kernels
  5. 5 Analysis of the quadratic martingale field
  6. 6 Solving the martingale problem for the SHE
  7. 7 Quenched tail field
  8. A Technical lemmas on Brownian bridges

##  1\. Introduction

###  1.1. Preface

Diffusion in time-dependent random environments has been a subject of intense
investigation recently due to its connection with the KPZ universality class
[BC17]. It is well known that the quenched density of the position of a
particle in the diffusive regime (when its location $=O(\sqrt{\mbox{time}})$)
converges to the Gaussian distribution with a second-order correction given by
the fixed point of the Edwards-Wilkinson (EW) universality class [FK99, KO01,
BMP04, RAS05, KLO12, Yu16, DG22]. Meanwhile in the large deviation (LD) regime
(when location $=O({\mbox{time}})$), the quenched density is expected to admit
a large deviation principle with linear speed, and the second-order correction
is given by the Tracy-Widom (TW) distribution, the one-point marginal of the
fixed point in the Kardar-Parisi-Zhang (KPZ) universality class (proven
rigorously only for a few special models in [BC17, BR20]). The goal of this
paper is to show that for a large class of such diffusions in the moderate
deviation (MD) regime (when location $=O({\mbox{time}}^{3/4})$), the crossover
distribution between these two classes, namely the KPZ equation, arises as a
second-order correction.

The (1+1) dimensional KPZ equation is a stochastic partial differential
equation (SPDE) given by

$\displaystyle\partial_{t}\mathcal{H}=\tfrac{1}{2}\partial_{xx}\mathcal{H}+\tfrac{1}{2}(\partial_{x}\mathcal{H})^{2}+\sigma^{1/2}\cdot\xi,\qquad\mathcal{H}=\mathcal{H}_{t}(x),$
(KPZ)

where $\sigma>0$ and $\xi=\xi(t,x)$ is a space-time white noise. The KPZ
equation was first introduced in [KPZ86] as a prototypical model for
interfaces of random growth. Since then, the model has been studied
intensively in both the mathematics and physics literature. We refer to [FS10,
Qua11, Cor12, QS15, CW17, CS20] for some surveys of the mathematical studies
of the KPZ equation.

As an SPDE, (KPZ) is ill-posed due to the presence of the non-linear term
$(\partial_{x}\mathcal{H})^{2}$. One way to make sense of the equation is to
consider $\mathcal{Z}:=e^{\mathcal{H}}$ which formally solves the stochastic
heat equation (SHE) with multiplicative noise:

$\partial_{t}\mathcal{Z}=\tfrac{1}{2}\partial_{xx}\mathcal{Z}+\sigma^{1/2}\cdot\mathcal{Z}\xi,\qquad\mathcal{Z}=\mathcal{Z}_{t}(x).$
(SHE)

The SHE is known to be well-posed and has a well-developed solution theory
based on the Itô integral and chaos expansions [Wal86, BC95, Qua11, Cor12]. In
this paper, we will consider the solution of the (SHE) started with Dirac
delta initial data $\mathcal{Z}_{0}(x)=\delta_{0}(x)$. For this initial data,
[Flo14] established that $\mathcal{Z}_{t}(x)>0$ for all $t>0$ and
$x\in\mathbb{R}$ almost surely (see also [Mue91]). Thus
$\mathcal{H}=\log\mathcal{Z}$ is well-defined and is called the Cole-Hopf
solution of the KPZ equation. This is the notion of solution that we will work
with in this paper, and it coincides with other existing notions of solutions
[Hai13, Hai14, GIP15, GP17, GJ14, GP18], under suitable assumptions.

The work of [ACQ11, CLDR10, Dot10, SS10] demonstrates that the one-point
distribution of the rescaled KPZ equation as $t\to\infty$ goes to the TW
distribution, whereas as $t\to 0$ then under a different rescaling, the KPZ
equation converges to the Gaussian distribution (the one-point distribution of
the EW fixed point). Thus, the KPZ equation serves as a mechanism for crossing
over between the EW and the KPZ universality classes. Going back to the
diffusion models in time-dependent random environments, in [LDT17] it was
argued that these diffusion models are rich enough to admit the KPZ equation
as limiting statistics. By physical arguments (explained briefly in Section
1.5.2), [LDT17] derived that in the moderate deviation (MD) regime (when
location $=O({\mbox{time}}^{3/4})$), (KPZ) arises as a second-order correction
for the quenched density (see Figure 1). Their heuristic arguments were later
supported by [BLD20] via rigorous moment-level computations for certain
integrable discrete and continuous diffusion models. More recently, using
high-precision numerical simulations, [HCGCC23] provided strong numerical
evidence for this KPZ equation limiting behavior.

Diffusive regimeMD regimeLD
regime$e^{-I(c)t+\operatorname{TW}}$$e^{-\frac{c^{2}\sqrt{t}}{2}+\operatorname{KPZ}}$$e^{-\frac{x^{2}}{2t}}\cdot\operatorname{EW}$$x\propto\sqrt{t}$$x=ct^{3/4}$$x=ct$
Figure 1. Schematic diagram of the quenched density $p(x)$ of the position of
a particle at time $t$. The results in the diffusive regime need to be
interpreted appropriately and may be found in [Yu16]. The fluctuations in the
Tracy-Widom regime are known only in some exactly solvable cases [BR20], but
are conjectured to hold generally.

In this paper, we work with diffusion in continuum random environments. We
consider the stochastic flow of kernels whose $k$-point motions solve the
Howitt-Warren martingale problem [HW09a]. We call such stochastic flows
Howitt-Warren flows. We show that the logarithm of the quenched density of the
motion of a particle under the Howitt-Warren flow, upon appropriate centering
and scaling, converges weakly to the KPZ equation.

Our work is the first rigorous instance of weak convergence to the KPZ
equation for diffusion in time-dependent random media under the moderate
deviation regime. We mention that such weak convergence to the KPZ equation
has been shown in the large deviation regime under weak random environment
settings [CG17, BW22]. Our proof techniques rely on a certain Girsanov
transform related to sticky Brownian motions (see Section 1.5 for details). In
particular, we do not rely on tools from integrable probability, and our
results hold for all Howitt-Warren flows with finite and positive
characteristic measures.

###  1.2. The model: Sticky Brownian motion

In order to define random motions in a continuum random environment, we need
to introduce the notion of stochastic flows of kernels. For $s\leq t$, a
random probability kernel, denoted $K_{s,t}(x,A)$, is a measurable function
defined on some underlying probability space $\Omega$, such that it defines a
probability measure on $\mathbb{R}$ for each
$(x,\omega)\in\mathbb{R}\times\Omega$. $K_{s,t}(x,A)$ is interpreted as the
random probability to arrive in $A$ at time $t$ starting at $x$ at time $s$.

######  Definition 1.1.

A family of random probability kernels $(K_{s,t})_{s\leq t}$ on $\mathbb{R}$
is called a stochastic flow of kernels if

  1. (a)

For any $s\leq t\leq u$ and $x\in\mathbb{R}$, almost surely
$K_{s,s}(x,A)=\delta_{x}(A)$, and

$\displaystyle\int_{\mathbb{R}}K_{t,u}(y,A)K_{s,t}(x,dy)=K_{s,u}(x,A)$

for all $A$ in the Borel $\sigma$-algebra of $\mathbb{R}$.

  2. (b)

For any $t_{1}\leq t_{2}\leq\cdots\leq t_{k}$, the
$(K_{t_{i},t_{i+1}})_{i=1}^{k-1}$ are independent.

  3. (c)

For any $s\leq u$ and $t\in\mathbb{R}$, $K_{s,u}$ and $K_{s+t,u+t}$ have the
same finite dimensional distributions.

The general theory of stochastic flows was developed by Le Jan and Raimond in
[LJR04a]; see also Tsirelson [Tsi04b]. For the stochastic flow that we
consider in this text, one can ensure that the random set of probability $1$
on which (a) holds is independent of $x\in\mathbb{R}$ and $s\leq t\leq u$.
This allows us to interpret $(K_{s,t})_{s\leq t}$ as bona fide transition
kernels of a random motion in a continuum random environment. The annealed law
of such a motion is called the $1$-point motion associated to
$(K_{s,t})_{s\leq t}$. More generally, the $k$-point motion of a stochastic
flow of kernels is defined as the $\mathbb{R}^{k}$ valued stochastic process
$\mathbf{X}=(X^{1},\ldots,X^{k})$ with transition probabilities given by

$\displaystyle
P_{t}(\vec{x},d\vec{y})=\mathbb{E}\left[\prod_{i=1}^{k}K_{0,t}(x_{i},dy_{i})\right].$

We will be interested in a particular random motion in a continuum random
environment originating from the Howitt-Warren flow of kernels. Its
corresponding $k$-point motion solves a well-posed martingale problem that was
first studied by Howitt and Warren in [HW09a]. Below, we introduce the
$k$-point motion by stating the martingale problem formulated in [SSS14].

######  Definition 1.2.

We say an $\mathbb{R}^{k}$-valued process
$\mathbf{X}_{t}=(X_{t}^{1},\ldots,X_{t}^{k})$ solves the Howitt-Warren
martingale problem with characteristic measure $\nu$ if $\mathbf{X}$ is a
continuous, square-integrable martingale with the covariance process between
$X^{i}$ and $X^{j}$ given by

$\displaystyle\langle
X^{i},X^{j}\rangle_{t}=\int_{0}^{t}\mathbf{1}_{\\{X_{s}^{i}=X_{s}^{j}\\}}ds,$
(1.1)

and furthermore it satisfies the following condition:

Consider any nonempty $\Delta\subset\\{1,2,\ldots,k\\}$. For
$\mathbf{x}\in\mathbb{R}^{k}$, let

$\displaystyle
f_{\Delta}(\mathbf{x}):=\max\\{x_{i}:i\in\Delta\\},\qquad\mbox{and}\qquad
g_{\Delta}(\mathbf{x}):=\big{|}\\{i\in\Delta\mid
x_{i}=f_{\Delta}(\mathbf{x})\\}\big{|}.$

Then the process
$f_{\Delta}(\mathbf{X}_{t})-\int_{0}^{t}\beta_{+}\big{(}g_{\Delta}(\mathbf{X}_{s})\big{)}ds$
is a martingale with respect to the filtration generated by $\mathbf{X}$,
where $\beta_{+}(1):=0$ and

$\displaystyle\beta_{+}(m):={\frac{1}{2}}\int\sum_{k=0}^{m-2}(1-y)^{k}\nu(dy),\quad
m\geq 2.$

We remark that the factor of $1/2$ before the integral deviates slightly from
the literature, which usually has a factor of 2 instead. Note that from the
covariance process formula above, we see that each $X^{i}$ is marginally a
Brownian motion. Focusing on the $k=2$ case, one can check that the last
condition in Definition 1.2 is equivalent to

$|X_{t}^{1}-X_{t}^{2}|-{\nu([0,1])}\int_{0}^{t}\mathbf{1}_{\\{X_{s}^{1}=X_{s}^{2}\\}}ds$
(1.2)

being a martingale. Using Tanaka’s formula, we see that this forces

$\displaystyle
L_{0}^{X^{1}-X^{2}}(t)={\nu([0,1])}\int_{0}^{t}\mathbf{1}_{\\{X_{s}^{1}=X_{s}^{2}\\}}ds$

where $L_{0}^{X^{1}-X^{2}}(t)$ denotes the local time accrued by $X^{1}-X^{2}$
at zero by time $t$. Thus, from the above formula, we see that $X^{1}$ and
$X^{2}$ can be interpreted as Brownian motions evolving independently of each
other when apart, but when they meet there is some stickiness. Due to this
stickiness, the two motions momentarily move together in the sense that they
are equal on a nowhere-dense set of positive measure. The $k$-point motion
defined above is a generalization of this stickiness phenomenon and is thus
referred to as sticky Brownian motion ($\mathsf{SBM}$) in the literature
(Figure 2).

Figure 2. 500-point $\mathsf{SBM}$ with weak and strong stickiness simulated
from discrete random walks in random environments.

The study of Brownian motions with stickiness goes back to the work of Feller
[Fel52], where he studied general boundary conditions for diffusions on the
half line. Since then, sticky Brownian motion has been observed to arise as a
diffusive scaling limit of various models: storage processes [HL81], discrete
random walks in random environments [Ami91, HW09b], and certain families of
exclusion processes with a tunable interaction [RS15]. An $\mathsf{SBM}$ with
a uniform characteristic measure inherits integrability from the beta random
walk in random environment model studied in [BC17]. This was exploited in
[BR20, BLD20, BW21] to extract various exact formulas and asymptotics.
$\mathsf{SBM}$ models also bear connections to the Kraichnan model of
turbulent flow [Kra68]. Indeed, in the works [GH04, War15], the sticky
behavior of particles was observed under certain fine-tuning in the Kraichnan
model. We refer to the series of the physics works [CFKL95, GK95, BGK98, GK06,
GV00] and expository notes [SS00, Kup10] on the Kraichnan model for more
details in this direction.

Howitt and Warren [HW09a, Proposition 8.1] proved that the martingale problem
in Definition 1.2 is well-posed, and its solutions form a consistent family of
Feller processes. By a remarkable result of Le Jan and Raimond [LJR04a,
Theorem 2.1], any consistent family of Feller processes can be viewed as a
$k$-point motion of some stochastic flow of kernels, unique in finite-
dimensional distributions. Thus, in particular, the solution of the Howitt-
Warren martingale problem can be viewed as the $k$-point motion of some
stochastic flow of kernels. We call this stochastic flow of kernels the
Howitt-Warren flow. When referring to the $k$-point motions, we will continue
to use $\mathsf{SBM}$. We refer to [SSS09, SSS14, SSS17] for more background
on how $\mathsf{SBM}$ can be viewed as random motions in continuum random
environments, and how to give a concrete construction of such a flow in a
space of measure-valued flows using a coupling with the Brownian web and net.

###  1.3. Main results

Let $K_{0,t}(dx)$ denote the Howitt-Warren flow started from a Dirac mass at
$0$ whose characteristic measure $\nu$ is non-degenerate in the sense that
$\nu([0,1])>0$. As mentioned before, $K_{0,t}(x)$ can be interpreted as the
random probability of a particle hitting $x$ at time $t$. The goal of this
paper is to study the density of this quenched probability in the moderate
deviation regime where we take $(t,x)\mapsto(Nt,tN^{3/4}+xN^{1/2})$. Formally
speaking, we show that the log of the quenched density after appropriate
centering:

$\displaystyle{\log K_{0,Nt}\big{(}tN^{3/4}+xN^{1/2}\big{)}}+{\tfrac{1}{2}\log
N+\tfrac{t}{2}N^{1/2}+xN^{1/4}}$

converges in law to the solution $\mathcal{H}_{t}(x)$ of the KPZ equation
defined in (KPZ).

It is actually the case that $K_{0,t}$ is singular with respect to the
Lebesgue measure. In fact, by [SSS14, Theorem 2.8] it is almost surely purely
atomic at deterministic times, and may thus be viewed (formally) as a large
system of interacting Brownian particles of different masses that dynamically
split and recombine according to a time-homogeneous rule determined by the
characteristic measure $\nu$. Since $K$ only exists as a measure and not as a
function in general, some care must be taken in order to make sense of the
convergence statement written above. To do this, we introduce the fields
$\\{\mathscr{X}^{N}\\}_{N\geq 1}$ below.

For $t\geq 0$ and a bounded test function $\phi:\mathbb{R}\to\mathbb{R}$ we
first define

$\mathscr{X}^{N}_{t}(\phi):=\int_{\mathbb{R}}e^{-\frac{t}{2}N^{1/2}+uN^{-1/4}}\phi(N^{-1/2}(u-N^{3/4}t))K_{0,Nt}(du)$
(1.3)

so that on a purely formal level, one has (via the substitution
$x=N^{-1/2}(u-N^{3/4}t)$)

$\mathscr{X}^{N}_{t}(\phi)=\int_{\mathbb{R}}\phi(x)N^{1/2}e^{\frac{t}{2}N^{1/2}+xN^{1/4}}K_{0,Nt}(tN^{3/4}+xN^{1/2})dx.$

The above formally defines a spatial pairing of $\mathscr{X}_{t}^{N}$ with
$\phi$ in $L^{2}(\mathbb{R})$, and we can also define space-time pairings of
smooth compactly supported test functions
$\varphi:\mathbb{R}^{2}\to\mathbb{R}$ by the formula

$(\mathscr{X}^{N},\varphi)_{L^{2}(\mathbb{R}^{2})}:=\int_{0}^{\infty}\mathscr{X}_{t}^{N}(\varphi(t,\cdot))dt,$
(1.4)

though again we emphasize that $\mathscr{X}^{N}$ is not an element of
$L^{2}(\mathbb{R}^{2})$ and the subscript here is merely suggestive. Our first
result shows that for a fixed $t>0$ and a spatial test function $\phi$, the
moments of $\mathscr{X}_{t}^{N}(\phi)$ converge to the moments of the
stochastic heat equation paired with $\phi$.

######  Proposition 1.3 (Convergence of moments).

Fix $t>0$, and $k\in\mathbb{N}$. For all $\phi\in\mathcal{S}(\mathbb{R})$ (the
Schwartz space on $\mathbb{R}$), one has the following limit:

$\displaystyle\lim_{N\to\infty}\mathbf{\mathbb{E}}[\mathscr{X}_{t}^{N}(\phi)^{k}]=\mathbf{E}_{B^{\otimes
k}}\bigg{[}\prod_{j=1}^{k}\phi(B^{j}_{t})e^{\sigma\sum_{i<j}L^{B^{i}-B^{j}}_{0}(t)}\bigg{]}=\mathbf{E}[\mathcal{Z}_{t}(\phi)^{k}].$
(1.5)

Here $\mathbb{E}$ denotes the annealed expectation over the environment
$\omega$ and
$\mathcal{Z}_{t}(\phi):=\int_{\mathbb{R}}\mathcal{Z}_{t}(x)\phi(x)dx$, where
$(t,x)\mapsto\mathcal{Z}_{t}(x)$ solves (SHE) with
$\sigma=\frac{1}{\nu([0,1])}$ under Dirac delta initial condition. The
expectation of the middle term is with respect to $k$ independent Brownian
motions, and $L_{0}^{Y}(t)$ denotes the local time accrued by $Y$ at zero by
time $t$.

Using different methods, a similar multipoint moment convergence result is
established in [BLD20] for the case where the characteristic measure $\nu$ is
a uniform measure. However, in contrast to the field (1.3) that we use in this
paper, [BLD20] used a slight variant which we refer to as the “quenched tail
field.” We refer the reader to Section 1.3.1 where we define the quenched tail
field and discuss our results related to it.

We will now describe our weak convergence result for the above field under the
appropriate topology. Note that $\mathcal{Z}_{t}(x)$ is not uniquely
characterized by its moments, since they grow too fast. Therefore, the
convergence of moments alone will not be enough to establish weak convergence
of $\mathscr{X}^{N}$. However, Proposition 1.3 will still be relevant and will
help us to identify the limit points of $\mathscr{X}^{N}$ once we show
tightness in an appropriate Banach space.

We next introduce these suitable topologies for $\mathscr{X}^{N}$. Fix any
$T>0$ and set $\Lambda_{T}:=[0,T]\times\mathbb{R}$. We denote by
$C_{c}^{\infty}(\Lambda_{T})$ the set of functions $\Lambda_{T}\to\mathbb{R}$
that are restrictions to $\Lambda_{T}$ of some function in
$C_{c}^{\infty}(\mathbb{R}^{2})$. For $\alpha<0$ we let
$r=-\lfloor\alpha\rfloor$ and define the weighted parabolic Hölder space
$C^{\alpha,\tau}_{\mathfrak{s}}(\Lambda_{T})$ to be the closure of
$C_{c}^{\infty}(\Lambda_{T})$ with respect to the norm given by

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

where the scaling operators are defined by
$S^{\lambda}_{(t,x)}\varphi(s,y)=\lambda^{-3}\varphi(\lambda^{-2}(t-s),\lambda^{-1}(x-y))$,
and where $B_{r}$ is the set of all smooth functions of $C^{r}$ norm (see
(6.1)) less than $1$ with support contained in the unit ball of
$\mathbb{R}^{2}$. These spaces are separable and embed naturally into
$\mathcal{S}^{\prime}(\mathbb{R}^{2})$ (the space of tempered distributions).

Similarly, for $\alpha<0,r=-\lfloor\alpha\rfloor$, we define the weighted
elliptic Hölder space $C^{\alpha,\tau}(\mathbb{R})$ to be the closure of
$C_{c}^{\infty}(\mathbb{R})$ with respect to the norm given by

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

where the scaling operators are defined by
$S^{\lambda}_{x}\phi(s,y)=\lambda^{-1}\phi(\lambda^{-1}(x-y))$, and where
$B_{r}$ is now the set of all smooth functions of $C^{r}$-norm (see (6.1))
less than $1$ with support contained in the unit ball of $\mathbb{R}$. As
before, these spaces are separable and embed naturally into
$\mathcal{S}^{\prime}(\mathbb{R})$. Finally, for a Banach space $\Xi$ we
define the function space $C([0,T],\Xi)$ containing all continuous paths
$[0,T]\to\Xi$, equipped with a norm given by
$\|v\|:=\sup_{t\in[0,T]}\|v(t)\|_{\Xi}.$ In particular, we will consider the
spaces $C([0,T],C^{\gamma,\tau}(\mathbb{R}))$.

Our main result, stated below, shows that the collection
$\\{\mathscr{X}^{N}\\}_{N\geq 1}$ converges weakly to the stochastic heat
equation when viewed as elements of certain weighted parabolic Hölder spaces
or certain function spaces.

######  Theorem 1.4 (Weak Convergence).

Fix any $T>0$, and $\tau>1$.

  1. (a)

Fix any $\alpha<-1$. For each $N\geq 1$, $\mathscr{X}^{N}$ as defined by (1.3)
and (1.4) can be viewed as an element of
$C_{\mathfrak{s}}^{\alpha,\tau}(\Lambda_{T})$. Furthermore, the collection
$\\{\mathscr{X}^{N}\\}_{N\geq 1}$ is tight with respect to the above topology
of $C_{\mathfrak{s}}^{\alpha,\tau}(\Lambda_{T})$. Additionally, any limit
point as $N\to\infty$ is concentrated on $C((0,T],C(\mathbb{R}))$ and
coincides with the law of (SHE) with $\sigma=\frac{1}{\nu([0,1])}$, started
from Dirac delta initial condition.

  2. (b)

Fix any $\gamma<-2$. For each $N\geq 1$, $(\mathscr{X}_{t}^{N})_{t\in[0,T]}$
as defined by (1.3) can be viewed as an element of the space
$C([0,T],C^{\gamma,\tau}(\mathbb{R}))$. Furthermore, the collection
$\\{\mathscr{X}^{N}\\}_{N\geq 1}$ is tight with respect to that topology. Any
limit point as $N\to\infty$ is concentrated on $C((0,T],C(\mathbb{R}))$ and
coincides with the law of (SHE) with $\sigma=\frac{1}{\nu([0,1])}$, started
from Dirac delta initial condition.

######  Remark 1.5.

A few remarks related to the above theorem are now in order.

  1. (a)

We explain how to interpret the above results in terms of test functions. For
any $\varphi_{1},\ldots,\varphi_{m}\in C_{c}^{\infty}(\mathbb{R}^{2})$, either
part of Theorem 1.4 implies the joint convergence of
$\big{(}(\mathscr{X}^{N},\varphi_{j})_{L^{2}(\mathbb{R}^{2})}\big{)}_{1\leq
j\leq m}$ as defined in (1.4) to
$\big{(}\int_{\mathbb{(}0,\infty)\times\mathbb{R}}\mathcal{Z}_{t}(x)\varphi_{j}(t,x)dtdx\big{)}_{1\leq
j\leq m}$.

  2. (b)

Theorem 1.4(b) implies convergence of
$(\mathscr{X}_{t}^{N}(\phi))_{t\in[0,T]}$ to
$\big{(}\int_{\mathbb{R}}\mathcal{Z}_{t}(x)\phi(x)dx\big{)}_{t\in[0,T]}$,
viewed as random variables in $C[0,T]$, for any $\phi\in
C_{c}^{\infty}(\mathbb{R})$. In particular, one may evaluate the field at some
finite collection of fixed times to obtain multi-time convergence.

  3. (c)

There is a tradeoff between the two parts of the theorem. Theorem 1.4(a) is a
statement about the convergence of the field $\mathscr{X}^{N}$ when tested
against smooth functions in both space and time, and it does not imply
convergence in law of $\mathscr{X}_{t}^{N}(\phi)$ for fixed $t>0$. However,
$\alpha<-1$ is indeed the optimal Hölder exponent that one could hope to
obtain for convergence of the fields $\mathscr{X}^{N}$ in the parabolic spaces
(the heat kernel itself does not have better regularity). On the other hand,
Theorem 1.4(b) implies convergence of the spatial field for fixed $t>0$, but
we believe $\gamma<-2$ is no longer the optimal Hölder exponent for the
function space.

  4. (d)

The weights $\tau>1$ are not optimal. It should be possible to get rid of the
weights altogether, since (SHE) started from Dirac initial condition is known
to have nice decay properties in both space and time, but some technical
aspects of the paper are simplified by using weights.

Theorem 1.4 is part of a series of efforts that have sought to show the weak
KPZ universality conjecture, which postulates that a large class of weakly
asymmetric models rescale to the KPZ equation (see the introduction of [HQ18]
for a brief background). For instance, convergence to the KPZ equation has
been established in a variety of models: directed polymers in the intermediate
disorder regime [AKQ14], exclusion processes [BG97, ACQ11, DT16, CT17, GJ17,
CST18, CGST20, Yan22, Yan23b], and a large class of stochastic growth models
[HQ18, AC22, Cha22, Yan23a]. In the context of diffusion in time-dependent
random environments, [CG17] studied nearest neighbor random walk on
$\mathbb{Z}$ in random environments. They showed that under the weak scaling
of the environment, the rescaled quenched transition probability evaluated in
the large deviation regime converges to the solution of the stochastic heat
equation. In a similar spirit, [BW22] considered a continuous SPDE model that
models the trajectory of a particle in a turbulent fluid. They showed that
under a weak environment setting, the limiting fluctuations of the quenched
law of the underlying process are given by the KPZ equation.

We emphasize that we do not deliberately introduce any weak asymmetry into our
model, i.e., the environment is independent of $N$ and there are no parameters
of the model that are being tuned. Rather, the KPZ fluctuations suggest that
the weak asymmetry is somehow introduced naturally as a consequence of the
moderate deviation scaling. In fact, by the scaling property of $\mathsf{SBM}$
[SSS14, Proposition 2.4], our result can be converted to a large deviation
regime result under weak stickiness.

####  1.3.1. Quenched tail field and connection to extreme value theory

In this subsection, we describe how KPZ equation convergence can be
established for the quenched tail probability from our results on the density
field stated in the previous section.

######  Definition 1.6.

We define the quenched tail field by

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

We remark that although the $F_{N}$ are function-valued, they are
discontinuous functions due to the atomic nature of $K_{0,t}$ [SSS14, Theorem
2.8]. Our next theorem states that the family $\\{\log F_{N}\\}_{N\geq 1}$ of
space-time processes converges to the KPZ equation, in the sense of finite-
dimensional distributions of pointwise values $(t,x)$.

######  Theorem 1.7.

For any finite collection of space-time points
$\\{(t_{i},x_{i})\\}_{i=1}^{m}\in((0,\infty)\times\mathbb{R})^{m}$ one has the
joint convergence

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

where $\mathcal{H}_{t}(x)$ solves (KPZ) with $\sigma=\frac{1}{\nu([0,1])}$.

The above theorem was conjectured in [BLD20] where the authors established a
multipoint moment convergence of the field $F_{N}(t,x)$ to that of the
stochastic heat equation for the case where the characteristic measure $\nu$
is a uniform measure. Again, since the moments do not determine the
distribution of the stochastic heat equation, the results in [BLD20] do not
yield Theorem 1.7 even for the uniform case.

To prove Theorem 1.7, we rely on Theorem 1.4(b) and an integration by parts
argument to first obtain the joint convergence

$\displaystyle\bigg{(}\int_{\mathbb{R}}\phi_{i}(x)F_{N}(t_{i},x)dx\bigg{)}_{i=1}^{m}\stackrel{{\scriptstyle
d}}{{\to}}\bigg{(}\int_{\mathbb{R}}\phi_{i}(x)\mathcal{Z}_{t_{i}}(x)dx\bigg{)}_{i=1}^{m}.$
(1.6)

for $\phi_{1},\ldots,\phi_{m}\in C_{c}^{\infty}(\mathbb{R})$ and
$t_{1},\ldots,t_{m}>0$. We then establish regularity bounds for the two-point
spatial difference of the quenched tail field. This essentially follows from
the work of [SSS14], [Yu16], and [BR20]. Given this regularity bound, the
finite-dimensional convergence in (1.6) can be upgraded to finite-dimensional
convergence of pointwise space-time values by an application of Fatou’s lemma.
The full details of the proof of Theorem 1.7 are presented in Section 7.

One could go even further and ask about the convergence of $F_{N}$ in a
stronger topology such as the Skorohod topology (recall the $F_{N}$ are
discontinuous), which implies the multipoint result. We do not pursue this in
the present paper and leave this as a future work.

As a consequence of Theorem 1.7, we obtain the limiting distribution for the
maximum particle of a $k$-point sticky Brownian motion
$(X_{t}^{1},\ldots,X_{t}^{k})$ in the regime $t=O((\log k)^{2})$.

######  Theorem 1.8.

Fix $c,t>0$ and $d\in\mathbb{R}$. Let $(X_{t}^{1},\ldots,X_{t}^{k})$ be a
$k$-point sticky Brownian motion with characteristic measure $\nu$. Set the
number of particles
$k=k(N):=\lfloor\exp(\frac{1}{2}cN^{1/2}+dN^{1/4}+r_{N})\rfloor$ where $r_{N}$
can be any sequence satisfying $r_{N}=o(N^{1/4})$. Then

$\displaystyle\max_{1\leq i\leq
k(N)}\big{\\{}N^{-\frac{1}{4}}X^{i}(Nt)\big{\\}}-a_{N}(c,d,t)\stackrel{{\scriptstyle
d}}{{\to}}\sqrt{\tfrac{t}{c}}\cdot G+\mathcal{H}_{c}(d),$

where

$\displaystyle
a_{N}(c,d,t):=\sqrt{ctN}-dN^{\frac{1}{4}}\sqrt{\tfrac{t}{c}}-\sqrt{\tfrac{c}{t}}\big{(}r_{N}-\tfrac{1}{4}\log
N\big{)}.$

Here $G$ is a Gumbel random variable (i.e., $P(G\leq a)=e^{-e^{-a}}$) which is
independent of $\mathcal{H}$, the solution to (KPZ) with
$\sigma=\frac{\sqrt{c}}{\nu([0,1])\sqrt{t}}$.

We remark that $k$-point sticky Brownian motion refers to the annealed law of
the $X^{i}$: we are not making a pathwise statement about the maximum for each
realization of the kernels $K_{s,t}$, which is consistent with the fact that
Theorem 1.4 is a weak convergence statement, not an almost sure convergence
statement. We also remark that rather than allowing $\sigma$ to depend on $t$
and $c$, one may instead take $\sigma=1/\nu([0,1])$ but then
$\mathcal{H}_{c}(d)$ must be changed accordingly to
$\mathcal{H}_{\frac{t^{2}}{c}}\big{(}\frac{td}{c}\big{)}+\log\big{(}\frac{t}{c}\big{)}.$

Taking $d=r_{N}=0$ and $t=1$, we see that the above statement is a result of
the maximum of $e^{\frac{1}{2}c\sqrt{N}}$ many sticky Brownian motion
particles at time $N$. This is the same as understanding the maximum of $N$
particles when time is of the order $(\log N)^{2}.$ This leads to the question
of what happens when (for a fixed characteristic measure $\nu$) one looks at
the maximum of $N$ particles at timescales different from $(\log N)^{2}$. At
timescales of order $1$, we do not expect universality, as the answer may
depend on the characteristic measure. If the characteristic measure satisfies
$\int_{[0,1]}q^{-1}\nu(dq)<\infty$, then the support result of [SSS14, Theorem
2.5a] implies that the maximum of $N$ particles at time $t=1$ converges in law
(without any centering or scaling) to a Gaussian of mean
$\int_{[0,1]}q^{-1}\nu(dq)$. If $\int_{[0,1]}q^{-1}\nu(dq)=\infty$, we have no
conjecture what happens.

At timescales of order $\log N$, the maximum of $N$ particles fluctuates like
$(\log N)^{1/3}$ times a Tracy-Widom distribution. This is conjectured to be
universal, but it is currently only provable in certain exactly solvable cases
(see [BC17, Corollary 5.8] or [BR20]). Finally, on timescales greater than
$(\log N)^{2}$ we believe that the Gumbel term will dominate, since in this
regime the motions closely resemble i.i.d, Brownian motions. We conjecture
that $(\log N)^{2}$ is the unique timescale at which one sees a mix of Gumbel
with the KPZ equation. It remains to explore what happens on timescales
between 1 and $\log N$, or between $\log N$ and $(\log N)^{2}$.

The physics paper [HCGCC23] contains numerical simulations which explore these
regimes and seem to support some of our conjectures, although they consider
the random walk in random environment which is a discrete analogue of our
model. The physics paper [KLD23] also contains interesting conjectures related
to timescales slightly larger than $(\log N)^{2}$.

###  1.4. Issues with the chaos expansion technique

Before explaining the core ideas and novel techniques of the proof, it is
important to highlight the constraints of traditional methods used in showing
convergence to the (SHE). Among the existing methods, the polynomial chaos
method is a widely used approach in establishing weak convergence to the
(SHE). In this method, the prelimiting object is first identified as a
sequence of multi-linear polynomials of independent random variables (called
polynomial chaos expansions). Then each term of the chaos series is shown to
converge in $L^{2}$ to that of the Wiener chaos series of (SHE). This idea was
first implemented by Alberts, Khanin, and Quastel [AKQ14] for directed
polymers. Later, [CSZ16] set up a general framework, formulating general
conditions under which a polynomial chaos series converges in law to a Wiener
chaos expansion. This framework has since been utilized extensively to show
that (SHE) arises as a limit from several models of interest. In particular,
Corwin and Gu [CG17] used the framework of [CSZ16] to obtain KPZ equation
convergence for the random walk in a random environment (RWRE) model in the
large deviation regime under a weak environment scaling. Although sticky
Brownian motion bears a strong resemblance to the RWRE model and can be
realized as its diffusive limit [SSS17], there are two serious obstacles in
carrying out the polynomial chaos approach in our setting.

  1. (1)

Firstly, it is not clear how to set up the polynomial chaos for the quenched
density in the context of a continuum random environment. Indeed, as shown in
[LJR04b] the noise generated by the Howitt-Warren flow is a black noise in the
sense of Tsirelson [Tsi04b] (see also [LJR04b]). Black noises arise as a
scaling limit in various discrete models, such as systems of coalescing random
walks [Tsi04a, EF16] and 2D critical percolation [SSG11]. These non-classical
noises are a much more subtle and less understood subject than white noise. A
stochastic calculus for black noise is not known, and in particular, there is
no notion of iterated stochastic integrals with respect to black noise.

  2. (2)

Secondly, even for the discrete RWRE model, it is not straightforward to
replicate the ideas of Corwin and Gu [CG17] to prove KPZ equation convergence
under the moderate deviation regime. Although a polynomial chaos expansion for
the quenched density is available in this regime, taking a naive limit of this
discrete chaos expansion interestingly gives a noise coefficient in the
limiting stochastic heat equation which is strictly smaller than the physics
prediction from [BLD20], see the introduction of our upcoming work [DDP23].
This suggests that this polynomial chaos does not satisfy the conditions in
[CSZ16, Theorem 2.3] needed to apply their framework. In this particular
scenario, a nonzero proportion of the $L^{2}$-mass of the polynomial chaos
series escapes into the tails of the series in the $N\to\infty$ limit,
suggesting that additional independent noise is generated in the limit.

As far as we know the latter phenomenon has not been observed previously. We
study this phenomenon in our upcoming work [DDP23] where we prove a similar
KPZ equation universality result for the quenched transition kernel of the
RWRE using a similar strategy. Like this paper and in contrast with [CG17],
the environment law will be fixed under the scalings, and we will focus on the
moderate deviation setting.

###  1.5. Proof Idea

In this section, we describe the broad ideas of the proof of our main theorem:
Theorem 1.4. We focus on the proof of Theorem 1.4(a). The proof of Theorem
1.4(b) will follow readily from Theorem 1.4(a), together with a short
embedding lemma about these Hölder spaces under the action of the heat kernel.

####  1.5.1. Girsanov’s formula

The main technique in our analysis will be a Girsanov-type formula for sticky
Brownian motion. For simplicity, we illustrate here the $1$-point case. Using
the definition of $\mathscr{X}_{t}^{N}(\phi)$ from (1.3) we may write

$\displaystyle\mathscr{X}_{t}^{N}(\phi)=\mathbb{E}_{\omega}^{(1)}\left[e^{-\frac{1}{2}t\sqrt{N}+N^{-1/4}X_{Nt}}\cdot\phi\left(N^{-1/2}(X_{Nt}-N^{3/4}t)\right)\right],$
(1.7)

where $\mathbb{E}_{\omega}^{(1)}$ denotes the quenched expectation with
respect to a single motion $X$ sampled from the environment
$\omega=\\{K_{s,t}:-\infty\leq s\leq t<\infty\\}.$ Note that in the annealed
sense $K_{0,Nt}$ is simply the law of $B_{Nt}$ for a standard Brownian motion
$B$. Thus, taking the annealed expectation on both sides of the above
equation, and by the tower property for conditional expectation, we obtain

$\displaystyle\mathbb{E}[\mathscr{X}_{t}^{N}(\phi)]$
$\displaystyle=\mathbf{E}_{BM}\left[e^{-\frac{1}{2}t\sqrt{N}+N^{-1/4}X_{Nt}}\cdot\phi\left(N^{-1/2}(X_{Nt}-N^{3/4}t)\right)\right]$
$\displaystyle=\mathbf{E}_{BM}\left[e^{-\frac{1}{2}t\sqrt{N}+N^{1/4}X_{t}}\cdot\phi\left(X_{t}-N^{1/4}t\right)\right].$

Here the expectations are taken with respect to a standard Brownian motion
$B$. In the last line, we used the scale invariance of Brownian motion to say
that $X\stackrel{{\scriptstyle d}}{{=}}(N^{-1/2}X_{Nt})_{t\geq 0}$. Note that
$Z_{t}:=e^{-\frac{t}{2}\sqrt{N}+N^{1/4}X_{t}}$ is the stochastic exponential
of $N^{1/4}X_{t}$. By Girsanov’s theorem for Brownian motion, under the
changed measure $Q(A):=\mathbb{E}_{BM}(Z_{T}\mathbf{1}_{A})$, the process
$(X_{t}-N^{1/4}t)_{t\in[0,T]}$ is again a Brownian motion. Thus, the last
expression in the above equation is precisely equal to
$\mathbf{E}_{BM}[\phi(B_{t})]$ which no longer depends on $N$. This matches
the first moment of $\int_{\mathbb{R}}\mathcal{Z}_{t}(x)\phi(x)dx$ where
$(t,x)\mapsto\mathcal{Z}_{t}(x)$ is defined in Proposition 1.3.

In the case of higher moments, $\mathscr{X}_{t}^{N}(\phi)^{k}$ can be viewed
as the quenched expectation with respect to $k$-point motion sample from the
environment $\omega$. Then taking the annealed expectation over the quenched
expectation will lead to expressions in terms of a $k$-point sticky Brownian
motion. Then the key idea is to use a Girsanov-type formula for sticky
Brownian motion (see Lemma 2.2) to get rid of divergent terms appearing in the
annealed expectation expression. The resulting higher moment formulas appear
in Lemma 3.1. Unlike the first-moment computation, the resulting expressions
for higher moments still depend on $N$. However, the expressions are amenable
to taking the large $N$ limit. The expressions obtained in Lemma 3.1 are
essentially annealed expectations with respect to a $k$-point sticky Brownian
motion with characteristic measure $N^{1/2}\nu$. As $N\to\infty$, the
stickiness disappears, and we are left with expectations with respect to a
standard Brownian motion on $\mathbb{R}^{k}$. In Theorem 3.2 we compute these
limits and show that they indeed match with the moments of (SHE) defined in
Proposition 1.3.

Through our method, the $N^{3/4}$ term appearing in the scaling is seen to be
the unique and natural choice of exponent that universally gives KPZ
fluctuations. Indeed, following (1.7), one could potentially consider a model
with more general exponents:

$\displaystyle\mathscr{X}_{t}^{N,c_{N}}(\phi):=\mathbb{E}_{\omega}^{(1)}\left[e^{-\frac{1}{2}c_{N}^{2}Nt+c_{N}X_{Nt}}\cdot\phi\left(N^{-1/2}(X_{Nt}-c_{N}Nt)\right)\right],$

and certain aspects of the paper would still go through. Indeed, following the
arguments in the proof of Lemma 3.1, one can check that the second moment is
given by

$\displaystyle\mathbb{E}\big{[}\mathscr{X}_{t}^{N,c_{N}}(\phi)^{2}\big{]}=\mathbf{E}\left[\phi(X_{t})\phi(Y_{t})\exp\bigg{(}c_{N}^{2}N\int_{0}^{t}\mathbf{1}_{\\{X_{s}=Y_{s}\\}}ds\bigg{)}\right],$

where $(X,Y)$ is a certain tilted version of $2$-point $\mathsf{SBM}$ with
characteristic measure $N^{1/2}\nu$. The key observation here is that
$c_{N}=N^{-1/4}$ is the unique choice for which local times appear in the
limiting expressions of the intersection times (see Theorem 3.2). When
$c_{N}\ll N^{-1/4}$, the contribution of the intersection times would vanish
in the limit, whereas for $c_{N}\gg N^{-1/4}$ the expressions blow up.

####  1.5.2. Tightness

We now explain the main idea used in proving the tightness of the field
$\mathscr{X}^{N}$. Roughly speaking, the original conjecture made in [BLD20]
interpreted the Howitt-Warren flows $K_{0,t}$ as a Kolmogorov forward equation
associated to an SDE with drift coefficient formally given by space-time white
noise. They then apply a shear transform of space-time given by
$(t,x)\mapsto(Nt,N^{3/4}t+N^{1/2}x)$ and note that (at least formally) this
transforms the Kolmogorov forward equation into an SPDE which is essentially
(SHE) plus some term that should vanish in the limit. Their derivation is non-
rigorous because such a Kolmogorov SPDE is ill-posed due to the roughness of
the noise. The main idea in our proof is to use a rigorous variant of this
idea.

More precisely, in Lemma 4.3 we will show that the fields $\mathscr{X}^{N}$
satisfy a forced heat equation of the form

$\displaystyle(\partial_{t}-\tfrac{1}{2}\partial_{x}^{2})\mathscr{X}^{N}=dM^{N}$
(1.8)

in the sense of space-time Schwartz distributions, where $M^{N}$ is a
martingale forcing that is constructed in Section 4 below. We do not aim to
explicitly describe $M^{N}$ but simply work with it as though (1.8) is the
definition. Despite this non-explicit description of $M^{N}$, we can
nonetheless show that the quadratic variations of $M^{N}$ admit the following
nice decomposition:

$\langle
M^{N}(\phi)\rangle_{t}=Q_{t}^{N}(\phi^{2})+\mathcal{E}_{t}^{N}(\phi),$ (1.9)

where $Q^{N}$ is the quadratic variation field introduced in (5.1) and
$\mathcal{E}_{t}^{N}(\phi)$ is an error term defined in (5.2) that goes to
zero in $L^{2}$ norm. Our tightness proof then proceeds in two steps:

  * •

In the first step, we obtain various moment estimates for $Q^{N}$. This is
done by the same method outlined in Section 1.5.1. Indeed, the Girsanov
approach allows us to get precise expressions for other relevant observables
related to the field $\mathscr{X}^{N}$, not just the moments. In particular,
it gives us access to moment estimates for $Q^{N}$ as well (Proposition 5.6).

  * •

The next step is to use the moment estimates for $Q^{N}$ to obtain tightness
estimates on the fields $\mathscr{X}^{N}$. Indeed, since the fields $M^{N}$
have a martingale structure, the Burkholder-Davis-Gundy inequality yields
moment estimates for $M^{N}$ from moment estimates for $Q^{N}$. By Schauder
estimates for the heat equation, we may, in turn, translate moment estimates
for $M^{N}$ into tightness estimates for the fields $\mathscr{X}^{N}$ using
(1.8).

From the above steps, we obtain tightness for the fields $\mathscr{X}^{N}$,
$Q^{N}$, and $M^{N}$ in an appropriate topology (see Propositions 6.12 and
6.13). This roundabout method turns out to be much more tractable than trying
to obtain tightness for the fields $\mathscr{X}^{N}$ directly, see Proposition
5.6 below. This type of method is somewhat similar to that used in [BG97]
where the authors proved KPZ fluctuations for WASEP.

####  1.5.3. Identification of the limit points

After tightness is obtained, it remains to identify the limit points. To do
this, we will use the martingale characterization of the solution of the
multiplicative noise stochastic heat equation. Specifically, consider a
measure $\mu$ on $C([0,T],C(\mathbb{R}))$, and let $(X_{t})_{t\in[0,T]}$
denote the canonical process on that space. The canonical filtration
$\mathcal{F}_{t}$ on $C([0,T],C(\mathbb{R}))$ is the one generated by
$\\{X_{s}:s\leq t\\}$. A result of [BG97, Proposition 4.11] inspired by the
work of [KS88] says that if for all $\phi\in C_{c}^{\infty}(\mathbb{R})$ the
processes

$\displaystyle
M_{t}(\phi):=(X_{t},\phi)_{L^{2}(\mathbb{R})}-\frac{1}{2}\int_{0}^{t}(X_{s},\phi^{\prime\prime})_{L^{2}(\mathbb{R})}ds$
(1.10)

are $(\mathcal{F}_{t},\mu)$-martingales with quadratic variation given by

$\displaystyle\langle
M(\phi)\rangle_{t}=\sigma\int_{0}^{t}(X_{t}^{2},\phi^{2})_{L^{2}(\mathbb{R})}ds,$
(1.11)

then (under reasonable assumptions on the spatial growth of $X_{t}$ at
infinity) the measure $\mu$ necessarily coincides with the law of (SHE)
started from an initial condition that is distributed as $X_{0}$ under $\mu$.

Let $(X^{\infty},Q^{\infty},M^{\infty})$ be a limit point of
$(\mathscr{X}^{N},Q^{N},M^{N})$. Since in the prelimit the observables satisfy
(1.8), from that equation it is not hard to deduce that
$(X^{\infty},M^{\infty})$ satisfies (1.10) with
$(X,M)=(X^{\infty},M^{\infty})$. To show (1.11), we again rely on the Girsanov
approach to extract moment formulas for certain observables in the prelimit.
Using these formulas, loosely speaking, we shall show in Proposition 5.3 that
as $N\to\infty$

$\displaystyle
Q_{t}^{N}(a)-\sigma\int_{0}^{t}\left(\mathscr{X}_{s}^{N}(a)\right)^{2}ds\stackrel{{\scriptstyle
L^{2}}}{{\to}}0$

for each $a\in\mathbb{R}\setminus\\{0\\}$. The precise formulation of the
above equation requires more care, as $Q_{t}^{N}$ and $\mathscr{X}_{t}^{N}$
exist only as distributions. Assuming this, thanks to the decomposition in
(1.9) and the fact that $\mathcal{E}_{t}^{N}(\phi)$ vanishes in the limit, we
get (1.11) with $(X,M)=(X^{\infty},M^{\infty})$.

The proof strategy outlined above has the potential to generalize to the
random walk in random environment (RWRE) setting. In an upcoming work, we plan
to prove a similar KPZ equation universality result for the quenched
transition kernel of the RWRE in the moderate deviation regime using the same
strategy.

######  Remark 1.9 (Universality).

From the proof outlined above, we see that only the 1-point and 2-point
motions associated with the kernels $K_{s,t}$ are consequential in the limit.
The 1-point motion is simply Brownian motion which is why $M_{t}(\phi)$ as
defined above is a martingale, while the 2-point motions appear in the
expressions for the quadratic variations of those martingales. Indeed the
kernels $K_{s,t}$ and their “squares” $K_{s,t}^{Sq}$ (see (4.1)) which appear
in the expressions for the martingale $M^{N}$ and the quadratic martingale
field $Q^{N}$ are purely in terms of the quenched expectations of at most two-
point motions of $\mathsf{SBM}$ (see for instance (5.3)). Since the 1-point
and 2-point motions of $\mathsf{SBM}$ are completely determined in law by just
the total mass $\nu([0,1])$ of the characteristic measure (this can be seen
for the 2-point motion by looking at (1.2)), the limiting law in Theorem 1.4
only depends on $\mathsf{SBM}$ via $\nu([0,1])$. In other words, our result is
universal in the sense that it yields the same limit for all characteristic
measures $\nu$ with the same total mass.

### Organization

The rest of the article is organized as follows. In Section 2 we describe the
Girsanov transform and collect estimates related to sticky Brownian motion. In
Section 3 we prove the moment convergence (Proposition 1.3). In Section 4 we
identify the martingale $M^{N}$ and show that the field $\mathscr{X}^{N}$
satisfies a forced heat equation with forcing $M^{N}$. Section 5 is devoted to
analyzing the quadratic variation of $M^{N}$. Finally, in Section 6 we prove
Theorem 1.4 by utilizing the estimates from the previous sections to obtain
tightness estimates and to identify the limit points for the fields
$\mathscr{X}^{N}$. In Section 7, we prove results related to the quenched tail
field discussed in Section 1.3.1. In Appendix A we prove a few technical
estimates related to Brownian bridges which are used in proving our main
theorems.

### Notation and Conventions

Throughout this paper we use $\mathrm{C}=\mathrm{C}(a,b,c,\ldots)>0$ to denote
a generic deterministic positive finite constant depending on $a,b,c,\ldots$
that may change from line to line. We write $\mathcal{S}(\mathbb{R}^{d})$ to
denote the space of all Schwartz functions on $\mathbb{R}^{d}$ and use
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$ to denote it’s dual: the space of all
tempered distributions. We write $\mathbb{E}$ and $\mathbb{E}_{\omega}$ for
annealed and quenched expectations in the context of random motions in random
environments. We use $\mathbf{E}$ to denote expectation under path measures
such as Brownian motion, sticky Brownian motion, etc.

### Acknowledgements

We thank Ivan Corwin, Yu Gu, and Jon Warren for useful discussions related to
this project and for comments on an earlier draft of the paper. We thank
Rongfeng Sun for helpful discussions related to chaos expansion issues for
sticky Brownian motion. We also thank Guillaume Barraquand for several
illuminating discussions that eventually led us to derive results related to
the quenched tail field. The project was initiated during the authors’
participation in the “Universality and Integrability in Random Matrix Theory
and Interacting Particle Systems” semester program at MSRI in the fall of
2021. The authors thank the program organizers for their hospitality and
acknowledge the support from NSF DMS-1928930. SD and HD’s research was
partially supported by Ivan Corwin’s NSF grant DMS-1811143, the Fernholz
Foundation’s “Summer Minerva Fellows” program, and also the W.M. Keck
Foundation Science and Engineering Grant on “Extreme diffusion”. HD was also
supported by the NSF Graduate Research Fellowship under Grant No. DGE-2036197.

##  2\. Girsanov transform and sticky Brownian motion estimates

In this section, we develop the basic framework of our proof. As mentioned in
the introduction, our proof relies on a certain Girsanov-type transform for
sticky Brownian motion. In Section 2.1 we describe this transform that we will
use repeatedly in our later analysis. In Section 2.2, we collect several
estimates related to sticky Brownian motion.

###  2.1. Girsanov transform

We begin with some necessary notation and definitions. Throughout this paper
we assume $\nu$ is a fixed finite measure on $[0,1]$ with $\nu([0,1])>0$. Fix
$T>0$. For each $k\in\mathbb{N}$, we denote

  * •

$\mathbf{P}_{B^{\otimes k}}$ : the law on the canonical space
$C([0,T],\mathbb{R}^{k})$ of $k$ independent Brownian motions.

  * •

$\mathbf{P}_{SB_{\nu}^{(k)}}$ : the law on the canonical space
$C([0,T],\mathbb{R}^{k})$ of the $k$-point motion of a sticky Brownian motion
with characteristic measure $\nu$.

Given $(X^{1},\ldots,X^{k})$ distributed as $\mathbf{P}_{SB_{\nu}^{(k)}}$,
using (1.1) we have

$\displaystyle\bigg{\langle}\sum_{j=1}^{k}X^{j}\bigg{\rangle}(t)=kt+\sum_{i\neq
j}\int_{0}^{t}\mathbf{1}_{\\{X_{s}^{i}=X_{s}^{j}\\}}ds\leq k^{2}t.$ (2.1)

Thus by Novikov’s criterion for each $\lambda\in\mathbb{R}$

$\displaystyle
Z(t):=\exp\bigg{[}\lambda\sum_{j=1}^{k}X^{j}(s)-\frac{k\lambda^{2}t}{2}-\lambda^{2}\sum_{i<j}\int_{0}^{t}\mathbf{1}_{\\{X^{i}_{s}=X^{j}_{s}\\}}ds\bigg{]}.$
(2.2)

is a martingale. With this information in hand, we now introduce two more
measures in the following definition.

######  Definition 2.1.

We denote by $\mathbf{P}_{T_{\lambda}SB_{\nu}^{(k)}}$ the measure which is
absolutely continuous with respect to $\mathbf{P}_{SB_{\nu}^{(k)}}$ with
density
$\mathbf{P}_{T_{\lambda}SB_{\nu}^{(k)}}=Z(t)\cdot\mathbf{P}_{SB_{\nu}^{(k)}}$
on $\mathcal{F}_{t}$ where $(\mathcal{F}_{t})_{t\in[0,T]}$ is the filtration
generated by $\mathbf{X}$. We define the measure
$\mathbf{P}_{D_{\lambda}SB^{(k)}_{\nu}}$ to be the law on the canonical space
$C([0,T],\mathbb{R}^{k})$ of the process $t\mapsto(X^{i}_{t}-\lambda
t)_{i=1}^{k}$ where $(X^{i})_{i=1}^{k}$ is distributed as
$\mathbf{P}_{T_{\lambda}SB^{(k)}_{\nu}}.$ We shall often refer to them as the
$\mathsf{T}_{\lambda}\mathsf{SB}_{\nu}^{(k)}$ and
$\mathsf{D}_{\lambda}\mathsf{SB}_{\nu}^{(k)}$ measures.

The following lemma shows that $\mathbf{P}_{D_{\lambda}SB^{(k)}_{\nu}}$ is
also absolutely continuous with respect to $\mathbf{P}_{SB^{(k)}_{\nu}}$ on
the interval $[0,T]$ with an explicit Radon-Nikodym derivative. To state the
lemma, we introduce a few more pieces of notation. For
$\mathbf{x}\in\mathbb{R}^{k}$, define $m_{i}(\mathbf{x})$ to be the
cardinality of the set $\\{j\leq k:x_{i}=x_{j}\\}.$ Define

$\displaystyle
G(t):=\sum_{j=1}^{k}\int_{0}^{t}\bigg{[}1-\frac{1}{m_{j}(\mathbf{X}(s))}\bigg{]}dX^{j}(s),\quad\mathcal{M}(t):=\exp\bigg{(}\lambda
G(t)-\frac{\lambda^{2}}{2}\langle G\rangle(t)\bigg{)},$ (2.3)

where $\mathbf{X}=(X^{1},\ldots,X^{k})$ is distributed as
$\mathbf{P}_{SB_{\nu}^{(k)}}$.

######  Lemma 2.2.

The measure $\mathbf{P}_{D_{\lambda}SB_{\nu}^{(k)}}$ defined in Definition 2.1
is absolutely continuous with respect to $\mathbf{P}_{SB_{\nu}^{(k)}}$ with
density
$\mathbf{P}_{D_{\lambda}SB_{\nu}^{(k)}}=\mathcal{M}(t)\cdot\mathbf{P}_{SB_{\nu}^{(k)}}$
on $\mathcal{F}_{t}$ where $(\mathcal{F}_{t})_{t\in[0,T]}$ is the filtration
generated by $\mathbf{X}$.

###### Proof.

Define the measure $\mathbf{P}_{Q_{\lambda}SB^{(k)}_{\nu}}$ to be the law on
the canonical space $C([0,T],\mathbb{R}^{k})$ of the process
$t\mapsto(X^{i}_{t}-\lambda t)_{i=1}^{k}$ where $(X^{i})_{i=1}^{k}$ is
distributed as $\mathbf{P}_{SB^{(k)}_{\nu}}.$ By [HW09a, Lemma 5.4], we know
that $\mathbf{P}_{Q_{\lambda}SB^{(k)}_{\nu}}$ is absolutely continuous with
respect to $\mathbf{P}_{SB^{(k)}_{\nu}}$ on the interval $[0,T]$ with Radon-
Nikodym derivative given by

$\displaystyle\left.\frac{d\mathbf{P}_{Q_{\lambda}SB^{(k)}_{\nu}}}{d\mathbf{P}_{SB^{(k)}_{\nu}}}\right|_{\mathcal{F}_{t}}(\mathbf{X})=\exp\bigg{(}-\lambda
G_{1}(t)-\frac{\lambda^{2}}{2}\langle G_{1}\rangle(t)\bigg{)},$ (2.4)

where

$G_{1}(t):=\sum_{j=1}^{k}\int_{0}^{t}\frac{1}{m_{i}(\mathbf{X}(s))}dX^{i}(s).$

Take any bounded $\mathcal{F}_{t}$-measurable functional
$F:C([0,T],\mathbb{R}^{k})\to\mathbb{R}$. We have

$\displaystyle\mathbf{E}_{D_{\lambda}SB^{(k)}_{\nu}}[F(\mathbf{X})]$
$\displaystyle=\mathbf{E}_{T_{\lambda}SB^{(k)}_{\nu}}\big{[}F\big{(}(\mathbf{X}_{u}-\boldsymbol{\lambda}u)_{0\leq
u\leq t}\big{)}\big{]}$
$\displaystyle=\mathbf{E}_{SB^{(k)}_{\nu}}\bigg{[}\exp\bigg{(}\lambda\sum_{j=1}^{k}X_{t}^{j}-\frac{\lambda^{2}}{2}\bigg{\langle}\sum_{j=1}^{k}X^{j}\bigg{\rangle}(t)\bigg{)}F\big{(}(\mathbf{X}_{u}-\boldsymbol{\lambda}u)_{0\leq
u\leq t}\big{)}\bigg{]}$
$\displaystyle=\mathbf{E}_{Q_{\lambda}SB^{(k)}_{\nu}}\bigg{[}\exp\bigg{(}\lambda\sum_{j=1}^{k}\big{(}X_{t}^{j}+\lambda
t\big{)}-\frac{\lambda^{2}}{2}\bigg{\langle}\sum_{j=1}^{k}X^{j}\bigg{\rangle}(t)\bigg{)}F(\mathbf{X})\bigg{]}$
$\displaystyle=\mathbf{E}_{SB^{(k)}_{\nu}}\bigg{[}\exp\bigg{(}\lambda\sum_{j=1}^{k}X_{t}^{j}-\lambda
G_{1}(t)+\lambda^{2}kt-\frac{\lambda^{2}}{2}\bigg{\langle}\sum_{j=1}^{k}X^{j}\bigg{\rangle}(t)-\frac{\lambda^{2}}{2}\langle
G_{1}\rangle(t)\bigg{)}F(\mathbf{X})\bigg{]}.$

Here $\boldsymbol{\lambda}u$ denotes the vector $(\lambda u,\ldots,\lambda
u)\in\mathbb{R}^{k}$ so that
$(\mathbf{X}_{u}-\boldsymbol{\lambda}u)=(X_{u}^{i}-\lambda
u)_{i=1}^{k}\in\mathbb{R}^{k}$. The first three equalities in the above
equation follow directly from Definition 2.1 and the definition of
$\mathbf{P}_{Q_{\lambda}SB_{\nu}^{(k)}}$. The last equality uses (2.4). Note
that $G(t)$, defined in (2.3), equals to $\sum_{j=1}^{k}X^{j}(t)-G_{1}(t)$.
Using (2.1) and the fact that $\langle G_{1},X^{j}\rangle(t)=t$ for each $j$,
we observe that the last expression in the above equation is precisely equal
to $\mathbf{E}_{SB^{(k)}_{\nu}}[\mathcal{M}(t)\cdot F(\mathbf{X})]$ where
$\mathcal{M}(t)$ is defined in (2.3). This completes the proof. ∎

######  Remark 2.3.

In view of Lemma 2.2, a bit of definition chasing shows that for all bounded
$\mathcal{F}_{t}$-measurable $F$

$\displaystyle\mathbf{E}_{SB_{\nu}^{(k)}}\left[Z(t)F\big{(}\left(\mathbf{X}_{u}-\boldsymbol{\lambda}u\right)_{0\leq
u\leq
t}\big{)}\right]=\mathbf{E}_{SB_{\nu}^{(k)}}\left[\mathcal{M}(t)F\big{(}(\mathbf{X}_{u})_{0\leq
u\leq t}\big{)}\right],$

where $Z$ and $\mathcal{M}$ are defined in (2.2) and (2.3) respectively. Using
the same argument, one also has

$\displaystyle\mathbf{E}_{SB_{\nu}^{(k)}}\left[\frac{Z(t)}{Z(s)}F\big{(}\left(\mathbf{X}_{u}-\boldsymbol{\lambda}(u-s)\right)_{s\leq
u\leq
t}\big{)}\mid\mathcal{F}_{s}\right]=\mathbf{E}_{SB_{\nu}^{(k)}}\left[\frac{\mathcal{M}(t)}{\mathcal{M}(s)}F\big{(}\left(\mathbf{X}_{u}\right)_{s\leq
u\leq t}\big{)}\mid\mathcal{F}_{s}\right]$

almost surely. This implies that for all bounded $\mathcal{F}_{t}$-measurable
$H$

$\displaystyle\mathbf{E}_{SB_{\nu}^{(k)}}\left[\exp\bigg{(}\lambda\sum_{j=1}^{k}(X_{t}^{j}-X_{s}^{j})-\tfrac{k}{2}\lambda^{2}(t-s)\bigg{)}H\big{(}\left(\mathbf{X}_{u}-\boldsymbol{\lambda}(u-s)\right)_{s\leq
u\leq t}\big{)}\mid\mathcal{F}_{s}\right]$ (2.5)
$\displaystyle=\mathbf{E}_{SB_{\nu}^{(k)}}\left[e^{\lambda(G(t)-G(s))-\tfrac{\lambda^{2}}{2}[\langle
G\rangle(t)-\langle G\rangle(s)]}H\big{(}\left(\mathbf{X}_{u}\right)_{s\leq
u\leq
t}\big{)}\exp\bigg{(}\lambda^{2}\sum_{i<j}\int_{s}^{t}\mathbf{1}_{\\{X_{u}^{i}=X_{u}^{j}\\}}du\bigg{)}\mid\mathcal{F}_{s}\right]\\!.\\!$

###  2.2. Sticky Brownian motion estimates

In this section, we gather various estimates related to sticky Brownian motion
that are necessary for our later proofs. In our analysis, we will often
encounter intersection times of a $k$-point sticky Brownian motion. For ease
of notation, for each $k\in\mathbb{N}$ and $1\leq i\neq j\leq k$ we define the
functional $V^{ij}:C([0,T],\mathbb{R}^{k})\to C([0,T],[0,\infty))$ by

$\displaystyle
V^{ij}_{t}(\mathbf{X}):=\int_{0}^{t}\mathbf{1}_{\\{X^{i}_{s}=X^{j}_{s}\\}}ds.$
(2.6)

We remark that the processes $V^{ij}$ are Borel-measurable and adapted to the
canonical filtration on $C([0,T],\mathbb{R}^{k})$. The following two lemmas
record moment estimates related to these functionals.

######  Lemma 2.4.

There exists an absolute constant $\mathrm{C}>0$ such that for all $k>1$,
$\theta,t>0$ and for all characteristic measures $\nu$ we have

$\mathbf{E}_{SB_{\nu}^{(k)}}\bigg{[}\exp{\bigg{(}\theta\sum_{i=1}^{k}\big{[}\big{(}\sup_{s\leq
t}|X^{i}_{s}|\big{)}+\nu([0,1])\sum_{j\neq
i}V^{ij}_{t}(\mathbf{X})}\big{]}\bigg{)}\bigg{]}\leq\mathrm{C}\cdot\exp(\mathrm{C}\cdot
k^{4}\theta^{2}t).$

###### Proof.

Applying Hölder’s inequality we get

$\displaystyle\mathbf{E}_{SB_{\nu}^{(k)}}\bigg{[}\exp{\bigg{(}\theta\sum_{i=1}^{k}\big{(}\sup_{s\leq
t}|X^{i}_{s}|\big{)}+\nu([0,1])\sum_{j\neq
i}V^{ij}_{t}(\mathbf{X})]}\bigg{)}\bigg{]}$ (2.7) $\displaystyle\hskip
14.22636pt\leq\prod_{i=1}^{k}\mathbf{E}_{SB_{\nu}^{(k)}}\bigg{[}\exp\bigg{(}2k\theta\sup_{s\leq
t}|X^{i}_{s}|\bigg{)}\bigg{]}^{\frac{1}{2k}}\cdot\prod_{i=1}^{k}\mathbf{E}_{SB_{\nu}^{(k)}}\big{[}e^{2k\theta\nu([0,1])\sum_{j\neq
i}V^{ij}_{t}(\mathbf{X})}\big{]}^{\frac{1}{2k}}.$

We now proceed to bound the above two product terms individually. Note that
marginally each $X^{i}$ is a Brownian motion under
$\mathbf{P}_{SB_{\nu}^{(k)}}$. For a Brownian motion, one has exponential tail
estimates of the form

$\displaystyle\mathbf{P}_{B}\bigg{[}\sup_{s\leq t}|X_{s}|>a\bigg{]}\leq
2e^{-a^{2}/2t}.$ (2.8)

We thus have

$\displaystyle\prod_{i=1}^{k}\mathbf{E}_{SB_{\nu}^{(k)}}\bigg{[}\exp\bigg{(}2k\theta\sup_{s\leq
t}|X^{i}_{s}|\bigg{)}\bigg{]}^{\frac{1}{2k}}=\mathbf{E}_{B}\bigg{[}\exp\bigg{(}2k\theta\sup_{s\leq
t}|X_{s}|\bigg{)}\bigg{]}^{\frac{1}{2}}\leq\mathrm{C}e^{\mathrm{C}k^{2}\theta^{2}t},$
(2.9)

for some absolute constant $\mathrm{C}>0$.

To deal with the second product in (2.7), another application of the Hölder
inequality on each of the expectation terms in the product shows that for each
$i=1,\ldots,k$ we have

$\mathbf{E}_{SB_{\nu}^{(k)}}\bigg{[}\exp\bigg{(}2k\theta\nu([0,1])\sum_{j\neq
i}V^{ij}_{t}(\mathbf{X})\bigg{)}\bigg{]}\leq\prod_{j\neq
i}\mathbf{E}_{SB_{\nu}^{(k)}}\bigg{[}\exp\bigg{(}2k(k-1)\theta\nu([0,1])\int_{0}^{t}\mathbf{1}_{\\{X_{s}^{i}=X_{s}^{j}\\}}ds\bigg{)}\bigg{]}^{\frac{1}{k-1}}.$

Notice that for all $i\neq j$ the pair $(X^{i},X^{j})$ is a 2-point sticky
Brownian motion under $\mathbf{P}_{SB_{\nu}^{(k)}}$ and therefore the latter
product is the same as

$\mathbf{E}_{SB_{\nu}^{(2)}}\bigg{[}\exp\bigg{(}2k(k-1)\theta\nu([0,1])\int_{0}^{t}\mathbf{1}_{\\{X_{s}^{1}=X_{s}^{2}\\}}ds\bigg{)}\bigg{]}.$

Summarizing, we find that

$\displaystyle\prod_{i=1}^{k}\mathbf{E}_{SB_{\nu}^{(k)}}\big{[}e^{2k\theta\nu([0,1])\sum_{j\neq
i}V^{ij}_{t}(\mathbf{X})}\big{]}^{\frac{1}{2k}}\leq\mathbf{E}_{SB_{\nu}^{(2)}}\bigg{[}\exp\bigg{(}2k(k-1)\theta\nu([0,1])\int_{0}^{t}\mathbf{1}_{\\{X_{s}^{1}=X_{s}^{2}\\}}ds\bigg{)}\bigg{]}^{\frac{1}{2}}.$

Note that for a 2-point sticky Brownian motion $(X,Y)$, recall from (1.2) that
the process $\nu([0,1])\int_{0}^{t}\mathbf{1}_{\\{X_{s}^{1}=Y_{s}\\}}ds$
agrees with the local time $L_{0}^{X-Y}(t).$ By [IM63], $X-Y$ can be viewed as
a time changed Brownian motion. Indeed, given a Brownian motion $B$, if we
define

$\displaystyle T(t)=t+\frac{1}{\nu([0,1])}L_{0}^{B}(t),$ (2.10)

the process $\sqrt{2}B_{T^{-1}(\cdot)}$ has the same distribution as $X-Y$.
Note that for $0\leq s\leq t$ we have

$\displaystyle T(T^{-1}(s)+t-s)$
$\displaystyle=T^{-1}(s)+t-s+\tfrac{1}{\nu([0,1])}L_{0}^{X-Y}(T^{-1}(s)+t-s))$
$\displaystyle\geq
T^{-1}(s)+t-s+\tfrac{1}{\nu([0,1])}L_{0}^{X-Y}(T^{-1}(s)))\geq s+t-s=t.$

Writing $t=T(T^{-1}(t))$, we see that

$\displaystyle T^{-1}(t)-T^{-1}(s)\leq t-s.$ (2.11)

This means that the time increments of the time-changed process are slower
than that of the standard time. In particular, $T^{-1}(t)\leq t$, hence
$L_{0}^{X-Y}(t)$ is stochastically dominated by
$L_{0}^{\sqrt{2}B}(t)={\sqrt{2}}L_{0}^{B}(t)$. By Levy’s identity for Brownian
local time, we have $L_{0}^{B}(t)\stackrel{{\scriptstyle d}}{{=}}\max_{s\leq
t}B(t)$ for each fixed $t>0$. Using (2.8) we thus have

$\displaystyle\prod_{i=1}^{k}\mathbf{E}_{SB_{\nu}^{(k)}}\big{[}e^{2k\theta\nu([0,1])\sum_{j\neq
i}V^{ij}_{t}(\mathbf{X})}\big{]}^{\frac{1}{2k}}$
$\displaystyle\leq\mathbf{E}_{SB_{\nu}^{(2)}}\bigg{[}\exp\bigg{(}2k(k-1)\theta\nu([0,1])\int_{0}^{t}\mathbf{1}_{\\{X_{s}^{1}=X_{s}^{2}\\}}ds\bigg{)}\bigg{]}^{\frac{1}{2}}$
$\displaystyle\leq\mathrm{C}e^{\mathrm{C}k^{2}(k-1)^{2}\theta^{2}t}$

for some absolute constant $\mathrm{C}>0$. Inserting the above bound and the
bound in (2.9) back in (2.7) we get the desired bound. This completes the
proof. ∎

######  Lemma 2.5.

Fix $k\in\mathbb{N}$ and $p\geq 1$. There exists a constant
$\mathrm{C}=\mathrm{C}(p,k)>0$ such that for all $t>s\geq 0$

$\sup_{\nu\in\mathcal{M}([0,1])}\mathbf{E}_{SB_{\nu}^{(k)}}\bigg{[}\bigg{(}\nu([0,1])\sum_{i<j}(V^{ij}_{t}(\mathbf{X})-V^{ij}_{s}(\mathbf{X}))\bigg{)}^{p}\bigg{]}\leq\mathrm{C}|t-s|^{p/2},$

where $\mathcal{M}([0,1])$ is the space of finite and non-negative Borel
measures on $[0,1].$

###### Proof.

Use Minkowski’s inequality and the fact that the 2-point motions of
$\mathbf{P}_{SB_{\nu}^{(k)}}$ are distributed as $\mathbf{P}_{SB_{\nu}^{(2)}}$
to write

$\displaystyle\mathbf{E}_{SB_{\nu}^{(k)}}\bigg{[}\bigg{(}\nu([0,1])\sum_{i<j}(V^{ij}_{t}(\mathbf{X})-V^{ij}_{s}(\mathbf{X}))\bigg{)}^{p}\bigg{]}^{1/p}$
$\displaystyle\leq\sum_{i<j}\mathbf{E}_{SB_{\nu}^{(k)}}\bigg{[}\bigg{(}\nu([0,1])(V^{ij}_{t}(\mathbf{X})-V^{ij}_{s}(\mathbf{X}))\bigg{)}^{p}\bigg{]}^{1/p}$
$\displaystyle=\sum_{i<j}\mathbf{E}_{SB_{\nu}^{(2)}}\bigg{[}\bigg{(}\nu([0,1])\int_{s}^{t}\mathbf{1}_{\\{X_{u}=Y_{u}\\}}du\bigg{)}^{p}\bigg{]}^{1/p}$
$\displaystyle=\frac{k(k-1)}{2}\mathbf{E}_{SB_{\nu}^{(2)}}\bigg{[}\big{(}L_{0}^{X-Y}(t)-L_{0}^{X-Y}(s)\big{)}^{p}\bigg{]}^{1/p}.$
(2.12)

The last identity follows from the fact that for a 2-point sticky Brownian
motion $(X,Y)$ the process
$\nu([0,1])\int_{0}^{t}\mathbf{1}_{\\{X_{s}^{1}=Y_{s}\\}}ds$ agrees with the
local time $L_{0}^{X-Y}(t)$. Recall that $X-Y\stackrel{{\scriptstyle
d}}{{=}}\sqrt{2}B_{T^{-1}(\cdot)}$ where $B$ is a standard Brownian motion and
$T$ is defined in (2.10). It follows from (2.11) that
$L_{0}^{X-Y}(t)-L_{0}^{X-Y}(s)$ is stochastically dominated by
${\sqrt{2}}(L_{0}^{B}(t)-L_{0}^{B}(s))$. By Levy’s identity for Brownian local
time, we have $L_{0}^{B}(t)-L_{0}^{B}(s)\stackrel{{\scriptstyle
d}}{{=}}M(t)-M(s)$ where $M(u):=\max_{v\leq u}B(v)$. Since
$\mathbb{E}[|M(t)-M(s)|^{p}]\leq\mathbb{E}[\sup_{s\leq u\leq
t}|B(u)-B(s)|^{p}]=\mathrm{C}|t-s|^{p/2}$ for some constant $\mathrm{C}>0$
depending on $p$, we have $\mbox{\eqref{tr}}\leq\mathrm{C}|t-s|^{1/2}$ for
some constant $\mathrm{C}>0$ depending on $p,k$. This completes the proof. ∎

We end this section by recording a uniform exponential moment estimate for a
certain class of $\mathbf{P}_{SB^{(k)}_{N^{1/2}\nu}}$-martingales that will be
very important.

######  Proposition 2.6.

Fix $k\in\mathbb{N}$. Let $\mathcal{G}_{N}=\mathcal{G}_{N}(\mathbf{X})$ be any
family of continuous $\mathbf{P}_{SB^{(k)}_{N^{1/2}\nu}}$-martingales
satisfying

$\displaystyle\langle\mathcal{G}_{N}\rangle(t)\leq\mathrm{C}N^{1/2}\sum_{i<j}V_{t}^{ij}(\mathbf{X})$
(2.13)

for some deterministic constant $\mathrm{C}>0$ independent of $N$. For all
$p,t>0$ we have

$\displaystyle\sup_{N\geq
1}\mathbf{E}_{SB^{(k)}_{N^{1/2}\nu}}\big{[}e^{p\mathcal{G}_{N}(t)-\frac{p}{2}\langle\mathcal{G}_{N}\rangle(t)}\big{]}<\infty.$
(2.14)

######  Remark 2.7.

Suppose $\mathbf{X}=(X^{1},\ldots,X^{k})$ is distributed as
$\mathbf{P}_{SB_{N^{1/2}\nu}^{(k)}}$. Consider $G$ defined in (2.3). Clearly
$\mathcal{G}_{N}:=N^{1/4}G$ satisfies (2.13). Let us write

$\displaystyle\mathcal{M}_{N}(t):=\exp\left(N^{1/4}G(t)-\tfrac{1}{2}N^{1/2}\langle
G\rangle(t)\right).$ (2.15)

Note that $\mathcal{M}_{N}$ is the same as $\mathcal{M}$ defined in (2.3) with
$\lambda\mapsto N^{1/4}$ and $\nu\mapsto N^{1/2}\nu$. The above proposition
shows that the $L^{p}$ norms of $\mathcal{M}_{N}(t)$ are uniformly bounded.

###### Proof of Proposition 2.6.

For convenience, we write $\mathbf{E}$ for
$\mathbf{E}_{SB^{(k)}_{N^{1/2}\nu}}$. Due to the hypothesis (2.13), for every
$\lambda>0$ we have

$\displaystyle\sup_{N\geq
1}\mathbf{E}\left[e^{\lambda\langle\mathcal{G}_{N}\rangle(t)}\right]\leq\sup_{N\geq
1}\mathbf{E}\bigg{[}\exp\bigg{(}\lambda\mathrm{C}N^{1/2}\sum_{i<j}V_{t}^{ij}(\mathbf{X})\bigg{)}\bigg{]}<\infty.$
(2.16)

where the last inequality is due to Lemma 2.4 (applied with $\nu\mapsto
N^{1/2}\nu$). Thus,

$\displaystyle\mathbf{E}\big{[}e^{p\mathcal{G}_{N}(t)-\frac{p}{2}\langle\mathcal{G}_{N}\rangle(t)}\big{]}$
$\displaystyle\leq\mathbf{E}\big{[}e^{p\mathcal{G}_{N}(t)}\big{]}$ (2.17)
$\displaystyle=\mathbf{E}\big{[}e^{p\mathcal{G}_{N}(t)}\mathbf{1}_{\\{p\mathcal{G}_{N}(t)\leq
2p^{2}\langle\mathcal{G}_{N}\rangle(t)\\}}\big{]}+\mathbf{E}\big{[}e^{p\mathcal{G}_{N}(t)}\mathbf{1}_{\\{p\mathcal{G}_{N}(t)>2p^{2}\langle\mathcal{G}_{N}\rangle(t)\\}}\big{]}$
$\displaystyle\leq\mathbf{E}\big{[}e^{2p^{2}\langle\mathcal{G}_{N}\rangle(t)}\big{]}+\mathbf{E}\big{[}e^{2p\mathcal{G}_{N}(t)-2p^{2}\langle\mathcal{G}_{N}\rangle(t)}\big{]}.$

Due to (2.16), Novikov’s condition holds and thus
$\mathbf{E}\big{[}e^{2p\mathcal{G}_{N}(t)-2p^{2}\langle\mathcal{G}_{N}\rangle(t)}\big{]}=1$.
Thus, taking supremum over $N\geq 1$ on both sides of (2.17), in view of
(2.16), we get (2.14). This establishes the proposition. ∎

##  3\. Proof of Proposition 1.3

The goal of this section is to prove Proposition 1.3, the moment convergence
of the field $\mathscr{X}_{t}^{N}$ (defined in (1.3)) to that of the
stochastic heat equation. The key idea is to observe that the moments of
$\mathscr{X}_{t}^{N}$ can be written as expectations of certain functionals of
a tilted version of a sticky Brownian motion (see Lemma 3.1). We then
carefully analyze sticky Brownian motions and the associated tilted measures
to obtain a weak convergence result for the tilted measure. This eventually
leads to the moment convergence for the field $\mathscr{X}_{t}^{N}$.

###  3.1. Moment computations and moment convergence

We first identify moments of $\mathscr{X}_{t}^{N}(\phi)$ as expectations of
certain functions under the
$\mathsf{D}_{N^{1/4}}\mathsf{SB}_{N^{1/2}\nu}^{(k)}$ measure introduced in
Definition 2.1.

######  Lemma 3.1.

For all bounded functions $\phi$ on $\mathbb{R}$, $t>s\geq 0$, and
$k\in\mathbb{N}$ one has that

$\displaystyle\mathbb{E}[\mathscr{X}_{t}^{N}(\phi)^{k}]$
$\displaystyle=\mathbf{E}_{D_{N^{1/4}}SB_{N^{1/2}\nu}^{(k)}}\bigg{[}\prod_{j=1}^{k}\phi(X^{j}_{t})\exp\bigg{(}N^{1/2}\sum_{1\leq
i<j\leq k}V^{ij}_{t}(\mathbf{X})\bigg{)}\bigg{]},$ (3.1)

where the functionals $V^{ij}$ are defined in (2.6).

###### Proof.

The proof essentially follows by keeping track of all the definitions. Indeed,
using the definition of $\mathscr{X}_{t}^{N}(\phi)$ from (1.3) we may write

$\displaystyle\mathscr{X}_{t}^{N}(\phi)=\mathbb{E}_{\omega}^{(1)}\left[e^{-\frac{1}{2}t\sqrt{N}+N^{-1/4}X_{Nt}}\cdot\phi\left(N^{-1/2}(X_{Nt}-N^{3/4}t)\right)\right],$
(3.2)

where $\mathbb{E}_{\omega}^{(1)}$ denotes the quenched expectation with
respect to a single motion $X$ sampled from the environment
$\omega=\\{K_{s,t}:-\infty\leq s\leq t<\infty\\}.$ Taking the $k^{th}$ power
we find that

$\mathscr{X}_{t}^{N}(\phi)^{k}=\mathbb{E}_{\omega}^{(k)}\left[e^{-\frac{k}{2}t\sqrt{N}+N^{-1/4}(X^{1}_{Nt}+\cdots+X^{k}_{Nt})}\prod_{j=1}^{k}\phi\big{(}N^{-1/2}(X^{j}_{Nt}-N^{3/4}t)\big{)}\right],$

where $\mathbb{E}_{\omega}^{(k)}$ denotes the quenched expectation with
respect to a $k$-point motion $(X^{1},\ldots,X^{k})$ sampled from the
environment $\omega=\\{K_{s,t}:-\infty\leq s\leq t<\infty\\}.$ Taking an
annealed expectation over the above expression and then using the fact that
the annealed law of $(X^{1},\ldots,X^{k})$ is a $k$-point sticky Brownian
motion with characteristic measure $\nu$ we obtain that

$\mathbb{E}\big{[}\mathscr{X}_{t}^{N}(\phi)^{k}\big{]}=\mathbf{E}_{SB^{(k)}_{\nu}}\left[e^{-\frac{k}{2}t\sqrt{N}+N^{-1/4}(X^{1}_{Nt}+\cdots+X^{k}_{Nt})}\prod_{j=1}^{k}\phi\big{(}N^{-1/2}(X^{j}_{Nt}-N^{3/4}t)\big{)}\right].$

A standard fact about sticky Brownian motion is that
$N^{-1/2}(X^{1}_{Ns},\ldots,X^{k}_{Ns})$ is distributed as
$SB^{(k)}_{N^{1/2}\nu}$ whenever $(X_{s}^{1},\ldots,X_{s}^{k})$ is distributed
as $SB^{(k)}_{\nu}$ (see [SSS14, Proposition 2.4]). Therefore the expectation
in the last math display can be rewritten as

$\mathbf{E}_{SB^{(k)}_{N^{1/2}\nu}}\left[e^{-\frac{k}{2}t\sqrt{N}+N^{1/4}(X^{1}_{t}+\cdots+X^{k}_{t})}\prod_{j=1}^{k}\phi(X^{j}_{t}-N^{1/4}t)\right].$

Using the definition of $\mathsf{D}_{N^{1/4}}\mathsf{SB}_{N^{1/2}\nu}^{(k)}$
measure from Definition 2.1 and the definition of $V^{ij}$ from (2.6), it is
not hard to check that the above expression is precisely equal to the right-
hand side of (3.1). This establishes the lemma. ∎

Thus, in view of the above lemma, it suffices to study weak convergence of
$\mathsf{D}_{N^{1/4}}\mathsf{SB}_{N^{1/2}\nu}^{(k)}$ measures to extract
moment convergence for $\mathscr{X}_{t}^{N}(\phi)$. Our next theorem is the
main technical result of this section. It shows that if we consider the path
$\mathbf{X}$ to be distributed according to the
$\mathsf{D}_{N^{1/4}}\mathsf{SB}_{N^{1/2}\nu}^{(k)}$ measure, as $N\to\infty$,

  * •

$\mathbf{X}$ converges to $k$ independent Brownian motions.

  * •

On the $N^{1/2}$ scale, pairwise intersection times converge to pairwise local
times of the corresponding Brownian motions.

Throughout the theorem and its proof, we will diverge from our usual
notational conventions by describing the random variables instead of referring
to the path measures on the canonical space explicitly. In particular, we will
write the associated expectations generically as $\mathbf{E}$, rather than
$\mathbf{E}_{SB_{\nu}^{(k)}}$ and $\mathbf{E}_{B^{\otimes k}}$. This will
avoid heavy notation, and hopefully does not cause confusion.

######  Theorem 3.2.

Fix any $k\in\mathbb{N}$ and constants $A,b,C,T>0$. Fix any continuous
function $f:C([0,T],\mathbb{R}^{k}\times\mathbb{R}^{k(k-1)/2})\to\mathbb{R}$
such that $|f(p)|\leq Ae^{b\|p\|_{\infty}}$ for all paths
$p:[0,T]\to\mathbb{R}^{k}\times\mathbb{R}^{k(k-1)/2}$. Suppose
$\mathbf{X}^{N}$ is distributed according to the
$\mathsf{SB}_{N^{1/2}\nu}^{(k)}$ measure from Definition 2.1. Set
$V^{ij}=V^{ij}(\mathbf{X}^{N})$ as defined in (2.6). The following holds.

  1. We have the following convergence

$\displaystyle\lim_{N\to\infty}\mathbf{E}\big{[}f\big{(}\mathbf{X}^{N}\;,\;N^{1/2}\nu([0,1])\big{(}V^{ij}\big{)}_{1\leq
i<j\leq
k}\big{)}\big{]}=\mathbf{E}\big{[}f\big{(}\mathbf{U}\;,\;\big{(}L^{ij}\big{)}_{1\leq
i<j\leq k}\big{)}\big{]},$

where $\mathbf{U}$ is a standard $k$-dimensional Brownian motion on $[0,T]$,
and $L^{ij}(t):=L^{U^{i}-U^{j}}_{0}(t)$ is the local time accrued by
$U_{i}-U_{j}$ at zero by time $t$.

  2. (a)

Let $\mathcal{G}_{N}=\mathcal{G}_{N}(\mathbf{X}^{N})$ be any family of
continuous $\mathbf{P}_{SB^{(k)}_{N^{1/2}\nu}}$-martingales with
$\mathcal{G}_{N}(0)=0$ satisfying

$\int_{s}^{t}d\langle\mathcal{G}_{N}\rangle(u)\leq
CN^{1/2}\sum_{i<j}(V_{t}^{ij}(\mathbf{X}^{N})-V_{s}^{ij}(\mathbf{X}^{N})\big{)},$
(3.3)

for all $0\leq s<t\leq T$. Then $\\{\mathcal{G}_{N}\\}_{N}$ is a tight family
of random variables in $C[0,T]$. Furthermore, any limit point of the triple
$\big{(}\mathcal{G}_{N},\;\mathbf{X}^{N},\;N^{1/2}\nu([0,1])\big{(}V^{ij}\big{)}_{1\leq
i<j\leq k}\big{)}$ is of the form
$(\mathcal{G},\mathbf{U},\big{(}L^{ij}\big{)}_{1\leq i<j\leq k}),$ where
$\mathbf{U},L^{ij}$ are as in part (a), and $\mathcal{G}$ is a $C[0,T]$-valued
random variable satisfying
$\mathbf{E}[e^{p{\mathcal{G}}(T)-\frac{p}{2}\langle{\mathcal{G}}\rangle(T)}]<\infty$
for all $p>0$ as well as

$\displaystyle\mathbf{E}[\exp(\mathcal{G}(T)-\tfrac{1}{2}\langle\mathcal{G}\rangle(T))\mid\mathcal{F}_{T}(\mathbf{U})]=1,$
(3.4)

where $\mathcal{F}_{T}(\mathbf{U})$ denotes the $\sigma$-algebra generated by
$\mathbf{U}$. In particular, we have

$\displaystyle\lim_{N\to\infty}\mathbf{E}\bigg{[}e^{\mathcal{G}_{N}(T)-\frac{1}{2}\langle\mathcal{G}_{N}\rangle(T)}f\bigg{(}\mathbf{X}^{N}\;,\;N^{1/2}\nu([0,1])\big{(}V^{ij}\big{)}_{1\leq
i<j\leq
k}\bigg{)}\bigg{]}=\mathbf{E}\big{[}f\big{(}\mathbf{U},\big{(}L^{ij}\big{)}_{1\leq
i<j\leq k}\big{)}\big{]}.$ (3.5)

  3. (b)

Suppose $k=2m$ is even. Let us write

$\displaystyle\Delta_{m}(s,t):=\\{(s_{1},s_{2},\ldots,s_{m})\in[s,t]^{m}\mid
s\leq s_{1}\leq s_{2}\leq\cdots\leq s_{m}\leq t\\}$ (3.6)

for the set of all $m$ ordered points in $[s,t]$. We define
$\mathcal{M}(\Delta_{m}(0,T))$ to be the space of finite and non-negative
Borel measures on that simplex, equipped with the topology of weak
convergence. Consider the following sequence of
$\mathcal{M}(\Delta_{m}(0,T))$-valued random variables

$\displaystyle\gamma_{N}(du_{1},\ldots,du_{m}):=N^{m/2}\prod_{j=1}^{m}\mathbf{1}_{\\{X^{2j-1}_{u_{j}}=X^{2j}_{u_{j}}\\}}\,du_{1}\ldots\,du_{m}.$
(3.7)

The random variables $\\{\gamma_{N}\\}_{N\geq 1}$ are tight in
$\mathcal{M}(\Delta_{m}(0,T))$. Moreover, any limit point of the triple
$(\mathbf{X}^{N},\;N^{1/2}\nu([0,1])\big{(}V^{ij}\big{)}_{1\leq i<j\leq
k},\;\gamma_{N})$ is of the form

$\displaystyle\left(\mathbf{U},\;\big{(}L^{ij}\big{)}_{1\leq i<j\leq
k},\;\sigma^{m}\prod_{j=1}^{m}dL_{0}^{U^{2j-1}-U^{2j}}(u_{j})\right)$

where $\mathbf{U},L^{ij}$ are as in part (a), and $dL(t)$ denotes the
Lebesgue-Stiltjes measure induced by the increasing function $t\mapsto L(t)$.

By Remark 2.7, taking $\mathcal{G}_{N}:=N^{1/4}G$ with $G$ defined as in
(2.3), we see that equation (3.5) in part (a) of the above theorem deals with
the weak convergence of $\mathsf{D}_{N^{1/4}}\mathsf{SB}_{N^{1/2}\nu}^{(k)}$
measures. Let us first show how it implies Proposition 1.3.

###### Proof of Proposition 1.3.

Fix any $\phi\in\mathcal{S}(\mathbb{R})$. Given $\mathbf{X}$ distributed as
$\mathbf{P}_{SB_{N^{1/2}\nu}^{(k)}}$, the martingales

$\mathcal{G}_{N}(t)=N^{1/4}\sum_{j=1}^{k}\int_{0}^{t}\bigg{[}1-\frac{1}{m_{j}(\mathbf{X}(s))}\bigg{]}dX^{j}(s)$

clearly satisfy (3.3). Thus, Theorem 3.2 part (a) holds for this choice of
$\mathcal{G}_{N}$. In view of Lemma 2.2, equation (3.5) yields that

$\lim_{N\to\infty}\mathbf{E}_{D_{N^{1/4}}SB_{N^{1/2}\nu}^{(k)}}\bigg{[}f\bigg{(}\mathbf{X}\;,\;N^{1/2}\nu([0,1])\big{(}V^{ij}\big{)}_{1\leq
i<j\leq k}\bigg{)}\bigg{]}=\mathbf{E}_{B^{\otimes
k}}\bigg{[}f\bigg{(}\mathbf{U}\;,\;\big{(}L^{ij}\big{)}_{1\leq i<j\leq
k}\bigg{)}\bigg{]},$

for all continuous functions
$f:C([0,T],\mathbb{R}^{k}\times\mathbb{R}^{k(k-1)/2})\to\mathbb{R}$ with
$|f(p)|\leq Ae^{b\|p\|_{\infty}}$ for some $A,b>0$. Specializing $f$ to a
particular choice given by product of $\phi$ multiplied by the exponential of
the intersection times, we get

$\displaystyle\lim_{N\to\infty}\mathbf{E}_{D_{N^{1/4}}SB_{N^{1/2}\nu}^{(k)}}\bigg{[}\prod_{j=1}^{k}\phi(X^{j}_{t})e^{N^{1/2}\sum_{i<j}V_{t}^{ij}(\mathbf{X})}\bigg{]}=\mathbf{E}_{B^{\otimes
k}}\bigg{[}\prod_{j=1}^{k}\phi(B^{j}_{t})e^{\sigma\sum_{i<j}L^{B^{i}-B^{j}}_{0}(t)}\bigg{]}.$

where $\sigma=\frac{1}{\nu([0,1])}$. Thanks to the moment formula from Lemma
3.1, we thus have the first equality in (1.5) from the above equation.

By the Feynman-Kac formula, the stochastic heat equation (SHE) admits well-
known moment formulas in terms of local times of Brownian bridges (see [BC95]
for example). In particular, from [BC95] we have

$\displaystyle\mathbf{E}\left[\prod_{i=1}^{k}\mathcal{Z}_{t}(x_{i})\right]=p_{t}(x_{1})\cdots
p_{t}(x_{n})\mathbf{E}_{\vec{x}}\bigg{[}\prod_{j=1}^{k}e^{\sigma\sum_{i<j}L^{B^{i}-B^{j}}_{0}(t)}\bigg{]},$

where $p$ is the standard heat kernel, and the $B^{i}$ are independent
Brownian bridges on $[0,t]$ from $0$ to $x_{i}$ respectively. From here one
arrives at the same moment formula for
$\mathcal{Z}_{t}(\phi):=\int_{\mathbb{R}}\mathcal{Z}_{t}(x)\phi(x)dx$,
yielding the second equality in (1.5). This completes the proof. ∎

###  3.2. Proof of Theorem 3.2

Throughout the proof, we will write $X^{N,i}$ and $U^{i}$ for the $i^{th}$
coordinate of the $\mathbb{R}^{k}$-valued processes $\mathbf{X}^{N}$ and
$\mathbf{U}$ respectively.

Proof of 3.2. We prove part 3.2 for any bounded continuous function $f$. In
view of Lemma 2.4 and Proposition 2.6, a uniform integrability argument then
extends the result to all continuous functions $f$ with exponential growth.
Recall that the stickiness of the sticky Brownian motion has an inverse
relationship to the characteristic measure, so intuitively, as we take
$N\to\infty$, the Brownian motions will no longer stick together, resulting in
$k$ independent Brownian motions in the limit.

We now flesh out the technical details of the above-claimed Brownian motion
convergence. We assume for convenience that all $(\mathbf{X}^{N})_{N\geq 1}$
are coupled onto the same probability space $(\Omega,\mathcal{F},\mathbf{P})$.
For each $N$, we partition $[0,T]$ into the two random sets $A_{N},B_{N}$ as

$\displaystyle A_{N}$
$\displaystyle:=\left\\{s\in[0,T]\mid\\#\\{X_{s}^{N,1},X_{s}^{N,2},\ldots,X_{s}^{N,k}\\}=k\right\\},$
(3.8) $\displaystyle B_{N}$
$\displaystyle:=\left\\{s\in[0,T]\mid\\#\\{X_{s}^{N,1},X_{s}^{N,2},\ldots,X_{s}^{N,k}\\}\leq
k-1\right\\}.$

Let us use $|S|$ to denote the Lebesgue measure of a set $S$. Clearly by the
definition of $V_{T}^{ij}$ (see (2.6)), we have
$|B_{N}|\leq\sum_{i<j}V_{T}^{ij}(\mathbf{X}^{N})$. Using $a\leq e^{a}$ for
$a>0$, we see that

$\displaystyle\mathbf{E}\left[|B_{N}|\right]\leq
N^{-1/2}\cdot\mathbf{E}\bigg{[}\exp\bigg{(}N^{1/2}\sum_{i<j}V_{T}^{ij}(\mathbf{X}^{N})\bigg{)}\bigg{]}\leq\mathrm{C}\cdot
N^{-1/2},$ (3.9)

where $\mathrm{C}=\mathrm{C}(k,T)>0$ may be chosen free of $N$. The last
inequality above is a consequence of Lemma 2.4 with $\nu\mapsto N^{1/2}\nu$.
Let $\mathbf{W}$ be a Brownian motion independent of $\mathbf{X}^{N}$ defined
on the same probability space. Let us set

$\mathbf{Y}^{N}_{t}:=\int_{0}^{t}\mathbf{1}_{\\{s\in
A_{N}\\}}d\mathbf{X}^{N}_{s}+\int_{0}^{t}\mathbf{1}_{\\{s\in
B_{N}\\}}d\mathbf{W}_{s}.$

As $X_{s}^{N,i}\neq X_{s}^{N,j}$ for all $s\in A_{N}$, we get that

$\displaystyle\langle Y^{N,i},Y^{N,j}\rangle(t)$
$\displaystyle=\int_{0}^{t}\mathbf{1}_{\\{s\in A_{N}\mid
X_{s}^{N,i}=X_{s}^{N,j}\\}}ds+\mathbf{1}_{\\{i=j\\}}\int_{0}^{t}\mathbf{1}_{\\{s\in
B_{N}\\}}ds=\mathbf{1}_{\\{i=j\\}}\cdot t.$

for all $t\in[0,T]$. Thus, by Levy’s criterion, we find that the
$\mathbf{Y}^{N}$ is a $k$-dimensional standard Brownian motion in the combined
filtration of $(\mathbf{X}^{N},\mathbf{W})$. We claim that

$\displaystyle\mathbf{P}\bigg{[}\sup_{t\leq
T}\big{\|}\mathbf{Y}_{t}^{N}-\mathbf{X}_{t}^{N}\big{\|}>N^{-1/6}\bigg{]}\leq
2k\cdot\mathrm{C}N^{-1/6}.$ (3.10)

where the $\mathrm{C}$ is the same as in (3.9). An application of Doob’s
submartingale inequality followed by Itô’s isometry yields

$\displaystyle\mathbf{P}\bigg{[}\sup_{t\leq
T}\big{\|}\mathbf{Y}_{t}^{N}-\mathbf{X}_{t}^{N}\big{\|}>N^{-1/6}\bigg{]}$
$\displaystyle\leq
N^{1/3}\cdot\mathbf{E}\left[\big{\|}\mathbf{Y}_{T}^{N}-\mathbf{X}_{T}^{N}\big{\|}^{2}\right]$
$\displaystyle=N^{1/3}\cdot\mathbf{E}\bigg{[}\bigg{\|}\int_{0}^{T}\mathbf{1}_{\\{s\in
B_{N}\\}}d\big{(}\mathbf{W}_{s}-\mathbf{X}^{N}_{s}\big{)}\bigg{\|}^{2}\bigg{]}$
$\displaystyle=2kN^{1/3}\cdot\mathbf{E}\bigg{[}\int_{0}^{T}\mathbf{1}_{\\{s\in
B_{N}\\}}ds\bigg{]}=2kN^{1/3}\cdot\mathbf{E}[|B_{N}|].$

Inserting the bound from (3.9) leads to (3.10). As $\mathbf{Y}^{N}$ is a
standard $k$-dimensional Brownian motion, (3.10) implies $\mathbf{X}^{N}$
converges weakly to a standard $k$-dimensional Brownian motion.

Let us now settle the weak convergence of pairwise intersection times. Observe
that by Lemma 2.5, $N^{1/2}\nu([0,1])V^{ij}(\mathbf{X}^{N})$ is a tight
sequence in $C[0,T]$ for each $i<j$. Let us consider any limit point
$(\mathbf{U},(K^{ij})_{1\leq i<j\leq k})$ of

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

By the property (1.2) of sticky Brownian motion, for each $i<j$ we have that

$|X^{N,i}(t)-X^{N,j}(t)|-N^{1/2}\nu([0,1])V^{ij}(\mathbf{X}^{N})$

is a martingale for each $N$. Thanks to the estimates in Lemma 2.4, the above
expression is uniformly integrable. Thus in the limit point, the process
$|U^{i}-U^{j}|-K^{ij}$ is a martingale as well. However, we have already
established that $\mathbf{U}$ is a standard $k$-dimensional Brownian motion.
Thus, by Tanaka’s formula, $K^{ij}(\cdot)$ has to be
$L_{0}^{U^{i}-U^{j}}(\cdot)$.

Thus, summarizing, we have identified the limit point as
$(\mathbf{U},(L^{ij})_{1\leq i<j\leq k})$ where $\mathbf{U}$ is a standard
$k$-dimensional Brownian motion and $L^{ij}(t):=L_{0}^{U^{i}-U^{j}}(t)$. This
verifies part 3.2.

Proof of (a). Let us take any $\mathcal{G}_{N}$ satisfying the assumption of
the theorem. We first prove that $\mathcal{G}_{N}$ is tight in $C[0,T]$.
Indeed, by Burkholder-Davis-Gundy inequality, we have that

$\mathbf{E}\big{[}|\mathcal{G}_{N}(t)-\mathcal{G}_{N}(s)|^{p}\big{]}\leq\mathbf{E}\bigg{[}\bigg{(}\int_{s}^{t}d\langle\mathcal{G}_{N}\rangle(u)\bigg{)}^{\frac{p}{2}}\bigg{]}\leq
C\cdot\mathbf{E}\bigg{[}\bigg{(}N^{1/2}\sum_{i<j}\big{(}V_{t}^{ij}(\mathbf{X}^{N})-V_{s}^{ij}(\mathbf{X}^{N})\big{)}\bigg{)}^{\frac{p}{2}}\bigg{]}.$

The second inequality above is due to the assumption (3.3). Lemma 2.5 implies
that the term on the right-hand side of the above equation is bounded by
$\mathrm{C}|t-s|^{p/4}$ for some constant $\mathrm{C}>0$ free of $N$. This
verifies the tightness of $\mathcal{G}_{N}$.

Note that (3.5) would be immediate from (3.4), because the latter would imply
that any subsequence of the left side of (3.5) has a further subsequence which
converges to the right side of (3.5). Let us therefore prove (3.4). Consider
any joint limit point $(\mathcal{G},\mathbf{U},(K^{ij})_{1\leq i<j\leq k})$ of

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

By part 3.2, $\mathbf{U}$ is a standard $k$-dimensional Brownian motion and
$K^{ij}=L_{0}^{U^{i}-U^{j}}$. By our moment estimates, we have uniform
integrability of both prelimiting martingales $\mathcal{G}_{N}$ and
$\mathbf{X}^{N}$. Thus in the limit, $\mathcal{G}$ and $\mathbf{U}$ are
martingales in their joint filtrations. By Proposition 2.6 and the assumption
(3.3), we have boundedness of $p^{th}$ moments (hence uniform integrability)
of
$\\{e^{\mathcal{G}_{N}(T)-\frac{1}{2}\langle\mathcal{G}_{N}\rangle_{T}}\\}_{N\geq
1}.$ Thus, in the limit,
$\exp(\mathcal{G}(t)-\frac{1}{2}\langle\mathcal{G}\rangle(t))$ is a martingale
as well.

To complete the proof of (3.4), the main idea is to use (3.3) to prove that
the martingales $\mathcal{G}_{N}$ are asymptotically decoupled from the sticky
Brownian motions. To be precise, we shall prove that for each $i$ and for all
$0\leq t\leq T$

$\displaystyle\lim_{N\to\infty}\mathbf{E}\bigg{[}\int_{0}^{t}d|\langle
X^{N,i},\mathcal{G}_{N}\rangle(s)|\bigg{]}=0.$ (3.11)

Let us assume (3.11) for the moment and complete the proof of (3.4). Note that
in the prelimit $X^{N,i}\cdot\mathcal{G}_{N}-\langle
X^{N,i},\mathcal{G}_{N}\rangle$ are martingales and (3.11) shows that the
covariations vanish in probability. Therefore, in the limit, we can conclude
that $\mathcal{G}\cdot U^{i}$ is a martingale so that $\langle
U^{i},\mathcal{G}\rangle=0.$

Take any bounded $\mathcal{F}_{T}(\mathbf{U})$-measurable functional
$H:C([0,T],\mathbb{R}^{k})\to\mathbb{R}$. Note that
$t\mapsto\mathbf{E}[H(\mathbf{U})|\mathcal{F}_{t}(\mathbf{U})]$ is a
martingale in the filtration of $\mathbf{U}$. As $\mathbf{U}$ is a standard
$k$-dimensional Brownian motion, by the martingale representation theorem we
have

$H(\mathbf{U})=\mathbf{E}[H(\mathbf{U})]+\sum_{i=1}^{k}\int_{0}^{T}h_{s}^{i}dU_{s}^{i}$

for some adapted $\mathbb{R}$-valued processes $h^{1},\ldots,h^{k}$. For
stochastic exponential we have

$\exp\left(\mathcal{G}(T)-\frac{1}{2}\langle\mathcal{G}\rangle(T)\right)=1+\int_{0}^{T}\exp\left(\mathcal{G}(s)-\frac{1}{2}\langle\mathcal{G}\rangle(s)\right)d\mathcal{G}(s).$

Using this we find that

$\displaystyle\mathbf{E}\left[\big{(}e^{\mathcal{G}(T)-\frac{1}{2}\langle\mathcal{G}\rangle(T)}-1\big{)}\big{(}H(\mathbf{U})-\mathbf{E}[H(\mathbf{U})]\big{)}\right]$
$\displaystyle=\sum_{i=1}^{k}\mathbf{E}\bigg{[}\bigg{(}\int_{0}^{T}e^{\mathcal{G}(s)-\frac{1}{2}\langle\mathcal{G}\rangle(s)}d\mathcal{G}(s)\bigg{)}\bigg{(}\int_{0}^{T}h_{s}^{i}dU^{i}_{s}\bigg{)}\bigg{]}$
$\displaystyle=\sum_{i=1}^{k}\mathbf{E}\bigg{[}\int_{0}^{T}h_{s}^{i}e^{\mathcal{G}(s)-\frac{1}{2}\langle\mathcal{G}\rangle(s)}d\langle\mathcal{G},U^{i}\rangle(s)\bigg{]}=0,$

where the last equality is due to the fact $\langle
U^{i},\mathcal{G}\rangle\equiv 0$ almost surely. This proves that for all
bounded measurable $H$ we have
$\mathbf{E}[e^{\mathcal{G}(T)-\frac{1}{2}\langle\mathcal{G}\rangle(T)}H(\mathbf{U})]=\mathbf{E}[H(\mathbf{U})]$,
establishing (3.4) modulo (3.11).

Let us now explain why (3.11) holds. Note that by assumption (3.3)

$\mathcal{G}_{N}(t)=\int_{0}^{t}\mathbf{1}_{\\{s\in
B_{N}\\}}d\mathcal{G}_{N}(s),$

where $B_{N}$ is defined in (3.8). By the Kunita-Watanabe inequality, we have
that

$\displaystyle\mathbf{E}\bigg{[}\int_{0}^{t}d|\langle
X^{N,i},\mathcal{G}_{N}\rangle(s)|\bigg{]}$
$\displaystyle=\mathbf{E}\bigg{[}\int_{0}^{t}\mathbf{1}_{\\{s\in
B_{N}\\}}d|\langle X^{N,i},\mathcal{G}_{N}\rangle(s)|\bigg{]}$
$\displaystyle\leq\mathbf{E}\bigg{[}\bigg{(}\int_{0}^{t}\mathbf{1}_{\\{s\in
B_{N}\\}}ds\bigg{)}^{\frac{1}{2}}\sqrt{\langle\mathcal{G}_{N}\rangle(t)}\bigg{]}.$
(3.12)

By definition, $\int_{0}^{t}\mathbf{1}_{\\{s\in
B_{N}\\}}ds\leq\sum_{i<j}V_{t}^{ij}$. Using assumption (3.3) we get that

$\displaystyle\mbox{r.h.s.~{}of \eqref{w1}}\leq
N^{1/4}\cdot\mathbf{E}\bigg{[}\sum_{i<j}V^{ij}_{t}\bigg{]}\leq
N^{-1/4}\cdot\mathbf{E}\bigg{[}\exp\bigg{(}N^{1/2}\sum_{i<j}V^{ij}_{t}\bigg{)}\bigg{]}\leq\mathrm{C}\cdot
N^{-1/4},$

where we used $a\leq e^{a}$ and then applied Lemma 2.4 in the last two bounds.
This verifies (3.11), completing the proof of the theorem.

Proof of (b). Note that

$\displaystyle\gamma_{N}(\Delta_{m}(0,T))\leq\prod_{j=1}^{m}\left[N^{\frac{1}{2}}\int_{0}^{T}\mathbf{1}_{\\{X^{2j-1}_{u}=X^{2j}_{u}\\}}du\right].$

From Lemma 2.4 we know the exponential moments for each of the terms in the
product are uniformly bounded under $\mathsf{SB}_{N^{1/2}\nu}^{(k)}$ measure.
Thus for all $p\geq 1$,

$\displaystyle\sup_{N\geq
1}\mathbf{E}_{SB^{(k)}_{N^{1/2}\nu}}[\gamma_{N}(\Delta_{m}(0,T))^{p}]<\infty.$
(3.13)

Hence the laws of $\\{\gamma_{N}\\}_{N\geq 1}$ are tight, because the total
mass of $\gamma_{N}$ is a tight family of random variables and because
$\Delta_{m}(0,T)$ as defined in (3.6) is a compact space. Let
$\left(\mathbf{U},\big{(}K^{ij}\big{)}_{1\leq i<j\leq k},\gamma\right)$ be any
limit point of the sequence
$(\mathbf{X}^{N},N^{1/2}\nu([0,1])\big{(}V^{ij}\big{)}_{1\leq i<j\leq
k},\gamma_{N})$. By part 3.2, we know $\mathbf{U}$ is a standard
$k$-dimensional Brownian motion and $K^{ij}=L^{ij}$. Note that the joint
cumulative distribution function of the measure $\gamma$ is necessarily given
by $\sigma^{m}$ times a product of the $K^{ij}$. Since we know that
$K^{ij}=L^{ij}$, this immediately implies that
$\gamma=\sigma^{m}\prod_{j=1}^{m}dL_{0}^{U^{2j-1}-U^{2j}}(u_{j})$. This
establishes part (b).

##  4\. Martingales associated to the sticky kernels

In this section, we begin our analysis of the field $\mathscr{X}_{t}^{N}$ by
identifying some useful martingale observables which will later be used to
identify the limit in Section 6.

######  Definition 4.1.

For a finite measure $\nu=\lambda+\sum_{k\in\mathbb{N}}p_{k}\delta_{x_{k}}$ on
the real number line with $\lambda$ atomless, we define its “square”

$\displaystyle\nu^{Sq}:=\sum_{k}p_{k}^{2}\delta_{x_{k}}.$ (4.1)

For a continuous function $\phi$ we also denote by
$(\nu,\phi):=\int_{\mathbb{R}}\phi\;d\nu$ the natural (spatial) pairing. When
$\phi$ is smooth, we will use $\phi^{\prime},\phi^{\prime\prime}$ to denote
the first and second (spatial) derivatives of $\phi$ respectively. We then
have the following lemma.

######  Lemma 4.2.

Let $K_{0,t}$ denote the Howitt-Warren flow. Then for all $\phi\in
C_{c}^{\infty}(\mathbb{R})$, the process

$S_{t}(\phi):=(K_{0,t},\phi)-\frac{1}{2}\int_{0}^{t}(K_{0,s},\phi^{\prime\prime})ds$

is a continuous martingale in the filtration
$\mathcal{F}_{t}:=\sigma(\\{K_{a,b}(x,A):0\leq a<b\leq
t,x\in\mathbb{R},A\mbox{ Borel}\\}).$ Furthermore its quadratic variation is
given by

$\langle S(\phi)\rangle_{t}=\int_{0}^{t}(K_{0,s}^{Sq},(\phi^{\prime})^{2})ds.$

From [SSS14, Theorem 2.8], we know for each fixed $t>0$, $K_{0,t}$ is atomic
almost surely. Thus $K_{0,s}^{Sq}$ is nonzero for almost every $s>0$.

###### Proof.

For the moment being, we consider a fixed realization of the kernels
$\\{K_{s,t}:0\leq s\leq t\\}.$ By [LJR04a, Section 2.6], we can sample paths
$B^{1},B^{2},B^{3},\ldots$ starting from 0 in such a way that:

  * •

given $\\{K_{s,t}:0\leq s\leq t\\},$ the paths $\\{B^{i}\\}_{i\geq 1}$ are all
independent.

  * •

given $\\{K_{s,t}:0\leq s\leq t\\},$ the law of $B^{i}_{t}$ given
$(B_{u}^{i})_{0\leq u\leq s}$ is $K_{s,t}(B_{s},\cdot)$ for all $0\leq s\leq
t.$

It is clear that the joint quenched law of these paths $\\{B^{i}\\}_{i\geq 1}$
is a deterministic function of the collection $\\{K_{s,t}:0\leq s\leq t\\}$,
because the two bullet points automatically determine the multi-time marginal
laws of any finite subcollection $\\{B^{i}\\}_{i=1}^{r}$ in terms of the
kernels $K_{s,t}$. It is also clear that the annealed law of
$(B^{1},\ldots,B^{r})$ is just $r$-point sticky Brownian motion (see the
discussion after Definition 1.1). We define
$\mu_{t}^{r}:=\frac{1}{r}\sum_{i=1}^{r}\delta_{B_{t}^{i}}.$ By the law of
large numbers, as $r\to\infty$ we have

$\mu_{t}^{r}\to K_{0,t}\;\;\;\;\;\;\text{ and
}\;\;\;\;\;\;\int_{0}^{t}\mu_{s}^{r}ds\to\int_{0}^{t}K_{0,s}ds$

almost surely in the topology of weak convergence of Borel measures, for each
$t>0$. Notice that

$|(\mu^{r}_{t},\phi)-(\mu^{r}_{s},\phi)|\leq\frac{1}{r}\sum_{i=1}^{r}|\phi(B_{t}^{i})-\phi(B^{i}_{s})|\leq\frac{\|\phi^{\prime}\|_{L^{\infty}(\mathbb{R})}}{r}\sum_{i=1}^{r}|B_{t}^{i}-B_{s}^{i}|$

for all $s,t\geq 0$, all $r\in\mathbb{N}$, and all $\phi\in
C_{c}^{\infty}(\mathbb{R})$. By Minkowski’s inequality and the fact that the
$B^{i}$ are individually Brownian motions in the annealed sense, we obtain

$\mathbb{E}[|(\mu^{r}_{t},\phi)-(\mu^{r}_{s},\phi)|^{p}]^{1/p}\leq\frac{\|\phi^{\prime}\|_{L^{\infty}(\mathbb{R})}}{r}\sum_{i=1}^{r}\mathbb{E}[|B_{t}^{i}-B_{s}^{i}|^{p}]^{1/p}=C_{p}\|\phi^{\prime}\|_{L^{\infty}}|t-s|^{1/2},$
(4.2)

for any $p\geq 1$. Define

$\displaystyle\widetilde{S}_{t}^{r}(\phi):=(\mu_{t}^{r},\phi)-\frac{1}{2}\int_{0}^{t}(\mu_{s}^{r},\phi^{\prime\prime})ds=\frac{1}{r}\sum_{i=1}^{r}\int_{0}^{t}\phi^{\prime}(B_{s}^{i})dB_{s}^{i}.$
(4.3)

where the second equality above is due to Itô’s formula. Hence,
$\widetilde{S}_{t}^{r}(\phi)$ is a martingale. Clearly
$|(\mu^{r}_{t},\phi)|\leq\|\phi\|_{L^{\infty}(\mathbb{R})}$. Combining this
with (4.2) and the first equality in (4.3), one has

$\mathbb{E}[|\widetilde{S}_{t}^{r}(\phi)-\widetilde{S}_{s}^{r}(\phi)|^{p}]^{1/p}\leq
C_{p}\|\phi^{\prime}\|_{L^{\infty}}|t-s|^{1/2}+\frac{1}{2}\|\phi^{\prime\prime}\|_{L^{\infty}}|t-s|,$

uniformly over $s,t\geq 0$. On any compact interval $[0,T]$ the right side can
be bounded above by $\mathrm{C}|t-s|^{1/2}$ for some constant
$\mathrm{C}=\mathrm{C}(p,T,\phi)$ independent of $s,t\in[0,T]$. Since
$S_{t}(\phi)=\lim_{r\to\infty}\widetilde{S}_{t}^{r}(\phi)$ almost surely, this
$L^{p}$ bound implies that the limit $S_{t}(\phi)$ is a continuous
$L^{p}$-martingale for each $\phi\in C_{c}^{\infty}(\mathbb{R})$. The
quadratic variations of $\widetilde{S}^{r}(\phi)$ can be computed explicitly
as

$\displaystyle\langle\widetilde{S}^{r}(\phi)\rangle_{t}=\frac{1}{r^{2}}\sum_{i,j}\int_{0}^{t}\phi^{\prime}(B_{s}^{i})\phi^{\prime}(B_{s}^{j})d\langle
B^{i},B^{j}\rangle_{s}=\frac{1}{r^{2}}\sum_{i,j}\int_{0}^{t}\phi^{\prime}(B_{s}^{i})^{2}\mathbf{1}_{\\{B_{s}^{i}=B_{s}^{j}\\}}ds=\int_{0}^{t}((\mu_{s}^{r})^{Sq},(\phi^{\prime})^{2})ds.$

We consider what happens to the quadratic variation as we take $r\to\infty$.
By [SSS14], we have that for each $t>0$, $K_{0,t}$ is a.s. atomic. Using this,
it is easy to show that for each $s>0$, the quantity
$((\mu_{s}^{r})^{Sq},(\phi^{\prime})^{2})$ will converge almost surely as
$r\to\infty$ to $(K_{0,s}^{Sq},(\phi^{\prime})^{2}).$ Since $\mu^{r}_{t}$ is a
probability measure, we have
$((\mu_{s}^{r})^{Sq},(\phi^{\prime})^{2})\leq(\mu_{s}^{r},(\phi^{\prime})^{2})\leq\|\phi^{\prime}\|_{L^{\infty}}^{2}$
deterministically, and likewise
$(K_{0,s}^{Sq},(\phi^{\prime})^{2})\leq(K_{0,s},(\phi^{\prime})^{2})\leq\|\phi^{\prime}\|_{L^{\infty}}^{2}$.
Therefore the dominated convergence theorem applied on the product space
$\mathbb{P}\otimes ds$ gives

$\lim_{r\to\infty}\mathbb{E}\bigg{[}\int_{0}^{t}\big{|}((\mu_{s}^{r})^{Sq},(\phi^{\prime})^{2})-(K_{0,s}^{Sq},(\phi^{\prime})^{2})\big{|}ds\bigg{]}=0.$

Here as always, $\mathbb{E}$ is annealed expectation. In other words, the
expression for $\langle\widetilde{S}^{r}(\phi)\rangle_{t}$ will converge in
$L^{1}(\mathbb{P})$ as $r\to\infty$ to the quantity
$\int_{0}^{t}(K_{0,s}^{Sq},(\phi^{\prime})^{2})ds.$ Putting all of this
together, we obtain that
$S_{t}(\phi)=\lim_{r\to\infty}\widetilde{S}_{t}^{r}(\phi)$ is a martingale
with quadratic variation

$\langle
S(\phi)\rangle_{t}=\lim_{r\to\infty}\langle\widetilde{S}^{r}(\phi)\rangle_{t}=\int_{0}^{t}(K_{0,s}^{Sq},(\phi^{\prime})^{2})ds,$

where the limit is interpreted in $L^{1}(\mathbb{P}).$ This completes the
proof. ∎

Given the above lemma, our next goal is to derive a (local) martingale that is
related to the rescaled field $\mathscr{X}^{N}$ from (1.3) instead of
$K_{0,t}$. To motivate the choice of our next martingale, we first perform
some martingale calculations on a formal level. The process $S$ from Lemma 4.2
resembles an orthomartingale in the sense that the cross-variation vanishes
whenever $\phi$ and $\psi$ have disjoint support, and the above lemma states
that $K_{0,t}$ solves the SPDE

$dK=\partial_{x}^{2}Kdt+dS.$

Note that if $u$ solves the forced heat equation
$\partial_{t}u=\frac{1}{2}\partial_{x}^{2}u+F$ and if $a,b,c\in\mathbb{R}$
then $e^{ax+bt}u(t,x+ct)$ solves the equation $\partial_{t}v=Lv+\overline{F}$
where

$L=\frac{1}{2}\big{(}\partial_{x}-(a-c)I\big{)}^{2}+\big{(}b-\frac{1}{2}c^{2}\big{)}I,$
$\overline{F}(t,x)=e^{ax+bt}F(t,x+ct).$

This makes sense even when $F$ is a Schwartz distribution. Note that when
$a=c$ and $b=\frac{1}{2}c^{2}$ then $L=\frac{1}{2}\partial_{x}^{2}$ is
unchanged.

We now formally go from $K_{0,t}$ to $\mathscr{X}^{N}_{t}$ defined in (1.3) in
two steps. We first apply the above argument for $u=K$ and $F=dS$, with
$a=c=N^{-1/4}$ and $b=\frac{1}{2}{N^{-1/2}}$. This gives us that the measure-
valued process

$\displaystyle\widetilde{K}^{N}(t,x)$
$\displaystyle:=e^{ax+bt}K_{0,t}(x+ct)=e^{\frac{t}{2}N^{-1/2}+xN^{-1/4}}K_{0,t}(x+N^{-1/4}t)$

solves the SPDE

$\partial_{t}\widetilde{K}^{N}=\partial_{x}^{2}\tilde{K}^{N}dt+d{S}^{N}$

where ${S}^{N}$ is the orthomartingale measure defined by

$d{S}^{N}(t,x)=e^{\frac{t}{2}N^{-1/2}+xN^{-1/4}}\cdot dS(t,x+N^{-1/4}t),$

to be interpreted by integration against test functions. Next, we apply a
diffusive scaling to $\widetilde{K}^{N}$. Let us set

$\displaystyle\mathscr{X}^{N}_{t}(x)$
$\displaystyle:=N^{1/2}\widetilde{K}_{N}(Nt,N^{1/2}x)=N^{1/2}e^{\frac{t}{2}N^{1/2}+xN^{1/4}}K_{0,Nt}(tN^{3/4}+xN^{1/2}).$

This solves the SPDE

$\partial_{t}\mathscr{X}^{N}=\partial_{x}^{2}\mathscr{X}^{N}dt+dM^{N}$

where $M^{N}$ is the orthomartingale measure defined by

$\displaystyle dM^{N}(t,x)$
$\displaystyle=N^{3/2}d{S}^{N}(Nt,N^{1/2}x)=N^{3/2}e^{\frac{t}{2}N^{1/2}+xN^{1/4}}\cdot
dS(Nt,tN^{3/4}+xN^{1/2}).$

The above formal calculations suggest the following lemma.

######  Lemma 4.3.

With $\mathscr{X}^{N}$ as defined in (1.3), for all $\phi\in
C_{c}^{\infty}(\mathbb{R})$ the processes

$M_{t}^{N}(\phi):=\mathscr{X}_{t}^{N}(\phi)-\phi(0)-\frac{1}{2}\int_{0}^{t}\mathscr{X}_{s}^{N}(\phi^{\prime\prime})ds$
(4.4)

are continuous martingales with

$\displaystyle\langle
M^{N}(\phi)\rangle_{t}=\int_{0}^{t}\big{(}(\mathscr{X}_{s}^{N})^{Sq},(\phi^{\prime}+N^{1/4}\phi)^{2}\big{)}ds$

where $(\mathscr{X}_{s}^{N})^{Sq}$ is defined via (4.1).

We remark that the $M^{N}(\phi)$ are actually tight in $C[0,T]$ as
$N\to\infty$. This will follow from the Burkholder-Davis-Gundy inequality as
well as moment bounds on the quadratic variation that are proved in
Proposition 5.6. Tightness will be proved in Proposition 6.13, using moment
formulas and bounds that we will derive in Section 5. This will be crucial in
solving the martingale problem for (SHE).

###### Proof.

The proof follows in a similar manner to the proof of Lemma 4.2. We omit the
details. ∎

Lemma 4.3 can be extended to include time-dependent test functions as well.

######  Lemma 4.4.

Let $\varphi\in\mathcal{S}(\mathbb{R}^{2})$. Consider $M_{t}^{N}(\phi)$ as
defined in (4.4). The process

$\displaystyle
R_{t}^{N}(\varphi):=M^{N}_{t}\big{(}\varphi(t,\cdot)\big{)}-\int_{0}^{t}M^{N}_{s}\big{(}\partial_{s}\varphi(s,\cdot)\big{)}ds$
(4.5)

is a continuous martingale, and its quadratic variation up to time $t$ is
given by

$\displaystyle\langle
R^{N}(\varphi)\rangle_{t}=N^{1/2}\int_{0}^{t}\big{(}(\mathscr{X}_{s}^{N})^{Sq},(\varphi(s,\cdot)+N^{-1/4}\partial_{x}\varphi(s,\cdot))^{2}\big{)}ds.$
(4.6)

Intuitively, one should think of $R_{t}^{N}(\varphi)$ as being equal to
$\int_{0}^{t}(\varphi(s,\cdot),dM^{N}_{s})_{L^{2}(\mathbb{R})},$ which yields
(4.5) by a formal integration-by-parts.

###### Proof.

Let $\mathcal{F}_{t}:=\sigma(\\{K_{Na,Nb}(x,A):0\leq a<b\leq
t,x\in\mathbb{R},A\mbox{ Borel}\\})$. Fix $0\leq u\leq t$. By Lemma 4.3 we
have

$\displaystyle\mathbb{E}[R_{t}^{N}(\varphi)\mid\mathcal{F}_{u}]=M_{u}^{N}\big{(}\varphi(t,\cdot)\big{)}-\int_{0}^{u}M^{N}_{s}\big{(}\partial_{s}\varphi(s,\cdot)\big{)}ds-\int_{u}^{t}M^{N}_{u}\big{(}\partial_{s}\varphi(s,\cdot)\big{)}ds.$
(4.7)

Note that

$\displaystyle\int_{u}^{t}M^{N}_{u}\big{(}\partial_{s}\varphi(s,\cdot)\big{)}ds=M_{u}^{N}\big{(}\varphi(t,\cdot)\big{)}-M_{u}^{N}\big{(}\varphi(u,\cdot)\big{)}.$

Inserting this identity back in (4.7), we get
$\mathbb{E}[R_{t}^{N}(\varphi)\mid\mathcal{F}_{u}]=R_{u}^{N}(\varphi)$. This
verifies that $R_{t}^{N}(\varphi)$ is a martingale with respect to
$\mathcal{F}_{t}$. The claimed quadratic variation for $R^{N}_{t}(\varphi)$ in
(4.6) can be verified similarly. ∎

##  5\. Analysis of the quadratic martingale field

Recall the martingale $M_{t}^{N}$ from (4.3). As explained in the
introduction, in order to identify limit points of the field
$M_{t}^{N}(\phi)$, one must study the quadratic variation of this martingale.
In this section, we identify the leading order contribution of $\langle
M^{N}(\phi)\rangle$ as a functional which we call the quadratic martingale
field—introduced below.

######  Definition 5.1.

Recall the field $\mathscr{X}_{t}^{N}$ from (1.3), as well as the squaring
operation from Definition 4.1. For a bounded test function $\phi$ on
$\mathbb{R}$, we define the quadratic martingale field (QMF) as

$\displaystyle Q_{t}^{N}(\phi)$
$\displaystyle:=N^{1/2}\int_{0}^{t}\big{(}(\mathscr{X}_{s}^{N})^{Sq},\phi\big{)}ds$
(5.1)
$\displaystyle=N^{1/2}\int_{0}^{t}\int_{\mathbb{R}}e^{-s\sqrt{N}+2uN^{-1/4}}\phi(N^{-1/2}(u-N^{3/4}s))K_{0,Ns}^{Sq}(du)ds,$

From Lemma 4.3, in view of the above definition, one has that

$\langle
M^{N}(\phi)\rangle_{t}=Q_{t}^{N}(\phi^{2})+\mathcal{E}_{t}^{N}(\phi),$

where $\mathcal{E}_{t}^{N}(\phi)$ is the “error term” given by

$\displaystyle\mathcal{E}_{t}^{N}(\phi)$
$\displaystyle:=\int_{0}^{t}\big{(}(\mathscr{X}_{s}^{N})^{Sq},2N^{1/4}\phi\phi^{\prime}+(\phi^{\prime})^{2}\big{)}ds=Q_{t}^{N}\big{(}N^{-1/2}(\phi^{\prime})^{2}+2N^{-1/4}\phi\phi^{\prime}\big{)}.$
(5.2)

The main goal of this section is to show that only $Q_{t}^{N}$ contributes to
the limit (in a very specific way, see Proposition 5.3) and that
$\mathcal{E}_{t}^{N}$ vanishes in the limit (Proposition 5.5). The starting
point of our analysis is a probabilistic interpretation of $Q_{t}^{N}(\phi)$.
Indeed, similar to the representation (3.2) for $\mathscr{X}_{t}^{N}(\phi)$,
the field $Q_{t}^{N}(\phi)$ also admits a formula in terms of a quenched
expectation of $2$-point motion sampled from the environment
$\omega:=\\{K_{s,t}:-\infty<s<t<\infty\\}$. Given a $2$-point motion sampled
from the environment $\omega:=\\{K_{s,t}:-\infty<s<t<\infty\\}$, we may view
$Q_{t}^{N}(\phi)$ as

$\displaystyle Q_{t}^{N}(\phi)$
$\displaystyle=N^{1/2}\int_{0}^{t}\mathbb{E}^{(2)}_{\omega}\bigg{[}e^{-s\sqrt{N}+2N^{-1/4}X_{Ns}}\phi(N^{-1/2}(X_{Ns}-N^{3/4}s))\mathbf{1}_{\\{X_{Ns}=Y_{Ns}\\}}\bigg{]}ds$
$\displaystyle=N^{1/2}\int_{0}^{t}\mathbb{E}^{(2)}_{\omega}\bigg{[}e^{-s\sqrt{N}+N^{-1/4}(X_{Ns}+Y_{Ns})}\phi(N^{-1/2}(X_{Ns}-N^{3/4}s))\mathbf{1}_{\\{X_{Ns}=Y_{Ns}\\}}\bigg{]}ds$
(5.3)

where the second equality follows by replacing $2N^{-1/4}X_{Ns}$ in the
exponential with $N^{-1/4}(X_{Ns}+Y_{Ns})$ due to the presence of the
indicator $\mathbf{1}_{\\{X_{Ns}=Y_{Ns}\\}}.$ The representation (5.3) will be
used as a starting point in the next section, Section 5.1, in extracting
formulas and estimates for the QMF.

###  5.1. Formulas and limits for the Quadratic Martingale Field

Recall the field $\mathscr{X}_{t}^{N}$ from (1.3). The goal of this section is
to show that the limit of $Q_{t}^{N}$ can be related to that of
$\mathscr{X}_{t}^{N}$ in a precise way (see Proposition 5.3). To do this, we
first extract moment formulas of certain observables of
$(\mathscr{X}_{t}^{N},Q_{t}^{N})$ in terms of sticky Brownian motion in the
same spirit as Lemma 3.1.

######  Lemma 5.2 (Moment formulas).

Fix any bounded functions $\psi,\phi$ on $\mathbb{R}$ and $N\geq 1$. Suppose
that $(X^{1},\ldots,X^{2k})$ denotes a $2k$-point sticky Brownian motion with
characteristic measure $N^{1/2}\nu$. Recall $\Delta_{m}(s,t)$ from (3.6) and
$\sigma=\frac{1}{\nu([0,1])}$. We have the following moment formulas.

  1. (a)

For all $t>0$, we have

$\displaystyle\mathbb{E}\left[\bigg{(}Q_{t}^{N}(\psi)-\sigma\int_{0}^{t}\mathscr{X}_{s}^{N}(\phi)^{2}ds\bigg{)}^{k}\right]$
(5.4) $\displaystyle\hskip
28.45274pt=k!\int_{\Delta_{k}(0,t)}\mathbf{E}_{SB_{N^{1/2}\nu}^{(2k)}}\bigg{[}\prod_{i=1}^{k}e^{-s_{i}\sqrt{N}+N^{1/4}(X_{s_{i}}^{2i-1}+X_{s_{i}}^{2i})}\Upsilon_{s_{i}}^{N}(X^{2i-1},X^{2i})\bigg{]}\prod_{i=1}^{k}ds_{i},$

where

$\displaystyle\Upsilon_{u}^{N}(X,Y):=N^{1/2}\psi\big{(}X_{u}-N^{1/4}u\big{)}\mathbf{1}_{\\{X_{u}=Y_{u}\\}}-\sigma\phi\big{(}X_{u}-N^{1/4}u\big{)}\phi\big{(}Y_{u}-N^{1/4}u\big{)}.$

  2. (b)

For all $0\leq s<t$ we have the following moment formula for the increment of
the QMF

$\displaystyle\mathbb{E}\left[\bigg{(}Q_{t}^{N}(\psi)-Q_{s}^{N}(\psi)\bigg{)}^{k}\right]$
(5.5)
$\displaystyle=N^{\frac{k}{2}}k!\int_{\Delta_{k}(s,t)}\mathbf{E}_{SB_{N^{1/2}\nu}^{(2k)}}\bigg{[}\prod_{i=1}^{k}e^{-s_{i}\sqrt{N}+N^{\frac{1}{4}}(X_{s_{i}}^{2i-1}+X_{s_{i}}^{2i})}\phi(X_{s}^{i}-N^{1/4}s_{i})\mathbf{1}_{\\{X_{s_{i}}^{2i-1}=X_{s_{i}}^{2i}\\}}\bigg{]}\prod_{i=1}^{k}ds_{i}.$

###### Proof.

Recall the probabilistic interpretation of $\mathscr{X}_{t}^{N}(\phi)$ and
$Q_{t}^{N}(\psi)$ from (3.2) and (5.3) respectively. Combining them we get
that

$\displaystyle
Q_{t}^{N}(\psi)-\sigma\int_{0}^{t}\mathscr{X}_{s}^{N}(\phi)^{2}ds=\int_{0}^{t}\mathbb{E}^{(2)}_{\omega}\left[e^{-s\sqrt{N}+N^{-1/4}(X_{Ns}^{1}+X_{Ns}^{2})}\widetilde{\Upsilon}_{s}^{N}(X^{1},X^{2})\right]ds$
(5.6)

where $\mathbb{E}_{\omega}^{(2)}$ denotes the quenched expectation of a
$2$-point motion sampled from the environment
$\omega:=\\{K_{s,t}:-\infty<s<t<\infty\\}$ and

$\displaystyle\widetilde{\Upsilon}_{s}^{N}(X^{1},X^{2}):=N^{1/2}\psi\left(N^{-1/2}(X_{Ns}^{1}-N^{3/4}s)\right)\mathbf{1}_{\\{X_{Ns}^{1}=X_{Ns}^{2}\\}}-\sigma\prod_{i=1}^{2}\phi\left(N^{-1/2}(X_{Ns}^{i}-N^{3/4}s)\right).$

Taking the $k$-th power of (5.6) we see that

$\displaystyle\left(Q_{t}^{N}(\psi)-\sigma\int_{0}^{t}\mathscr{X}_{s}^{N}(\phi)^{2}ds\right)^{k}$
(5.7)
$\displaystyle\qquad\qquad=k!\int_{\Delta_{k}(0,t)}\mathbb{E}^{(2k)}_{\omega}\left[\prod_{i=1}^{k}e^{-s_{i}\sqrt{N}+N^{-1/4}(X_{Ns_{i}}^{2i-1}+X_{Ns_{i}}^{2i})}\widetilde{\Upsilon}_{s_{i}}^{N}(X^{2i-1},X^{2i})\right]\prod_{i=1}^{k}ds_{i}.$

In the above line, we have also used the fact that
$\int_{[0,t]^{k}}\prod_{i=1}^{k}F(x_{i})dx_{i}=k!\int_{\Delta_{k}(0,t)}\prod_{i=1}^{k}F(x_{i})dx_{i}$
(recall $\Delta_{k}$ from (3.6)). Here $\mathbb{E}_{\omega}^{(2k)}$ denotes
the quenched expectation of a $2k$-point motion sampled from the environment
$\omega:=\\{K_{s,t}:-\infty<s<t<\infty\\}$. Taking annealed expectation on
both sides of the above equation, interchanging the order of integration and
expectation, and then using the fact that the annealed law of
$(X^{1},\ldots,X^{2k})$ is a $2k$-point sticky Brownian motion with
characteristic measure $\nu$, we get

$\displaystyle\mathbb{E}\left[\left(Q_{t}^{N}(\psi)-\sigma\int_{0}^{t}\mathscr{X}_{s}^{N}(\phi)^{2}ds\right)^{k}\right]$
$\displaystyle\qquad\qquad=k!\int_{\Delta_{k}(0,t)}\mathbf{E}_{SB_{\nu}^{(2k)}}\left[\prod_{i=1}^{k}e^{-s_{i}\sqrt{N}+N^{-1/4}(X_{Ns_{i}}^{2i-1}+X_{Ns_{i}}^{2i})}\widetilde{\Upsilon}_{s_{i}}^{N}(X^{2i-1},X^{2i})\right]\prod_{i=1}^{k}ds_{i}.$

The interchanging of the order of integration and expectation is permissible
by Fubini’s theorem. Indeed, Fubini’s theorem is applicable as applying Lemma
2.4, one can check that for each fixed $N$, the expectation of the absolute
value of the integrand on the right-hand side of (5.7) is uniformly bounded as
$\vec{s}$ varies in $\Delta_{k}(0,t)$. Using the fact that
$N^{-1/2}(X_{Ns}^{1},\ldots,X_{Ns}^{2k})$ is distributed as
$\mathsf{SB}^{(2k)}_{N^{1/2}\nu}$ measure, (5.4) follows from the above
formula. Relying on (5.3) alone, one can derive the formula in (5.5) by the
exact same argument. This completes the proof. ∎

We now come to the main result of this section, which shows that the quadratic
martingale field $Q_{t}^{N}$ is well approximated by the integrated version of
the square of $\mathscr{X}_{t}^{N}$. Loosely speaking, for each
$a\in\mathbb{R}\setminus\\{0\\}$ we shall show as $N\to\infty$

$\displaystyle
Q_{t}^{N}(a)-\sigma\int_{0}^{t}\left(\mathscr{X}_{s}^{N}(a)\right)^{2}ds\stackrel{{\scriptstyle
L^{2}}}{{\to}}0.$

Since $Q_{t}^{N}$ and $\mathscr{X}_{s}^{N}$ exist as distributions, to make
sense of the above display, we work with a sequence of Gaussian test functions
converging to $\delta_{a}$.

######  Proposition 5.3 (Limiting behavior of the QMF).

Let $a\in\mathbb{R}$ and let $\xi(x):=\frac{1}{\sqrt{\pi}}e^{-x^{2}}$ be the
Gaussian test function and let
$\xi_{\varepsilon}^{a}(x):=\varepsilon^{-1}\xi(\varepsilon^{-1}(x-a))$. Then
for all $t>0$ and $a\in\mathbb{R}\setminus\\{0\\}$,

$\displaystyle\limsup_{\varepsilon\to
0}\limsup_{N\to\infty}\mathbb{E}\bigg{[}\bigg{(}Q_{t}^{N}(\xi_{\varepsilon}^{a})-\sigma\int_{0}^{t}\mathscr{X}_{s}^{N}\big{(}\xi_{\varepsilon\sqrt{2}}^{a}\big{)}^{2}ds\bigg{)}^{2}\bigg{]}=0,$
(5.8)

where as usual, $\sigma=\frac{1}{\nu([0,1])}$. Furthermore, we have the bound

$\sup_{\begin{subarray}{c}\varepsilon>0\\\
a\in\mathbb{R}\setminus\\{0\\}\end{subarray}}\limsup_{N\to\infty}\left[1\wedge(|\log
a|^{-2})\right]\cdot\mathbb{E}\bigg{[}\bigg{(}Q_{t}^{N}(\xi_{\varepsilon}^{a})-\sigma\int_{0}^{t}\mathscr{X}_{s}^{N}\big{(}\xi_{\varepsilon\sqrt{2}}^{a}\big{)}^{2}ds\bigg{)}^{2}\bigg{]}<\infty.$
(5.9)

We remark that the relevance of the second bound (5.9) is that even though we
may not have convergence to 0 in (5.8) when $a=0$, we can still control the
behavior near $a=0$ by a blow-up that is polylogarithmic at worst. In the
proof of (6.15) in Theorem 6.16 below, we will see that such blow-ups are
irrelevant in the limit.

The starting point of the proof of Proposition 5.3 is the moment formula in
Lemma 5.2 (a). As noted in (5.4), the moments can be expressed as integral of
the expectation of certain observables under sticky Brownian motion. Recall
that in Theorem 3.2 we saw that the expectation of a certain class of
observables under sticky Brownian motion converges to the expectation of those
observables under standard $k$ dimensional Brownian motion. That theorem will
be used to prove a similar convergence result for the expectations of the type
of observables that appear in (5.4).

######  Proposition 5.4.

Fix any $k\in\mathbb{N}$ and $0=t_{0}<t_{1}<t_{2}<\cdots<t_{k}<\infty$. Set
$s_{0}=t_{0}$ and $s_{2i-1}=s_{2i}=t_{i}$ for $1\leq i\leq k$. Suppose
$(X^{1},\ldots,X^{2k})$ is distributed according
$\mathsf{SB}_{N^{1/2}\nu}^{(2k)}$. Let $(\phi_{i})_{i=1}^{2k}$ be bounded
continuous functions on $\mathbb{R}$. We have

$\displaystyle\lim_{N\to\infty}\mathbf{E}_{SB^{(2k)}_{N^{1/2}\nu}}\bigg{[}\prod_{i=1}^{2k}e^{-\frac{1}{2}s_{i}\sqrt{N}+N^{\frac{1}{4}}X_{s_{i}}^{i}}\phi_{i}(X_{s_{i}}^{i}-N^{\frac{1}{4}}s_{i})\bigg{]}=\mathbf{E}_{B^{\otimes
2k}}\bigg{[}e^{\sigma\mathcal{L}_{k}(\vec{t})}\prod_{i=1}^{2k}\phi_{i}(U_{s_{i}}^{i})\bigg{]}$
(5.10)

where

$\displaystyle\mathcal{L}_{k}(\vec{t}):=\sum_{r=1}^{k}\sum_{2r-1\leq p<q\leq
2k}\left[L_{0}^{U^{p}-U^{q}}(t_{r})-L_{0}^{U^{p}-U^{q}}(t_{r-1})\right].$
(5.11)

Furthermore, recalling $\Delta_{k}(s,t)$ from (3.6), the integrated version of
(5.10) also holds:

$\displaystyle\lim_{N\to\infty}\int_{\Delta_{k}(0,t)}\mathbf{E}_{SB^{(2k)}_{N^{1/2}\nu}}\bigg{[}\prod_{i=1}^{2k}e^{-\frac{1}{2}s_{i}\sqrt{N}+N^{\frac{1}{4}}X_{s_{i}}^{i}}\phi_{i}(X_{s_{i}}^{i}-N^{\frac{1}{4}}s_{i})\bigg{]}\prod_{i=1}^{k}dt_{i}$
(5.12) $\displaystyle\qquad\qquad=\int_{\Delta_{k}(0,t)}\mathbf{E}_{B^{\otimes
2k}}\bigg{[}e^{\sigma\mathcal{L}_{k}(\vec{t})}\prod_{i=1}^{2k}\phi_{i}(U_{s_{i}}^{i})\bigg{]}\prod_{i=1}^{k}dt_{i}.$

Additionally, for each $0\leq s<t\leq T<\infty$ we have

$\displaystyle\lim_{N\to\infty}N^{\frac{k}{2}}\cdot\mathbf{E}_{SB^{(2k)}_{N^{1/2}\nu}}\bigg{[}\int_{\Delta_{k}(s,t)}\prod_{i=1}^{k}e^{-t_{i}\sqrt{N}+N^{\frac{1}{4}}(X^{2i-1}_{t_{i}}+X^{2i}_{t_{i}})}\phi_{i}(X_{t_{i}}^{2i}-N^{\frac{1}{4}}t_{i})\mathbf{1}_{\\{X_{t_{i}}^{2i-1}=X_{t_{i}}^{2i}\\}}dt_{i}\bigg{]}$
(5.13) $\displaystyle\hskip 113.81102pt=\sigma^{k}\mathbf{E}_{B^{\otimes
2k}}\bigg{[}\int_{\Delta_{k}(s,t)}e^{\sigma\mathcal{L}_{k}(\vec{t})}\prod_{i=1}^{k}\phi_{i}(U_{t_{i}}^{2i})dL_{0}^{U^{2i-1}-U^{2i}}(t_{i})\bigg{]}.$

Here $\int_{0}^{t}f(s)dL_{0}^{U^{i}-U^{j}}(s)$ denotes the integration of the
continuous function $f:[0,t]\to\mathbb{R}$ against the random Lebesgue-
Stiltjes measure $dL_{0}^{U^{i}-U^{j}}$ induced from the increasing function
$t\mapsto L_{0}^{U^{i}-U^{j}}(t).$

The proof of Proposition 5.4 builds up on the estimates proved in Section 2
and uses Theorem 3.2. We postpone the proof of Proposition 5.4 to Section 5.2
and complete the proof of Proposition 5.3 assuming it.

###### Proof of Proposition 5.3.

For clarity, we split the proof into three steps. We first provide a brief
outline of the steps below. Note that, given the moment formulas from Lemma
5.2 (a) and the limit of those formulas from Proposition 5.4, one can compute
the $N\to\infty$ limit of the expectation in (5.9) in terms of expectation of
certain observables under a $4$D Brownian motion measure
$(X^{1},X^{2},Y^{1},Y^{2})$. We then proceed to carefully massage this
expectation formula to obtain the desired result.

  * In Step 1, we use a simple transformation to express the limit in terms of an expectation under a different $4$D Brownian motion measure:

$(X^{1}-Y^{1},X^{2}-Y^{2},X^{1}+Y^{1},X^{2}+Y^{2}).$

  * •

In Step 2, we show how local time heuristics (which are rigorously shown in
Appendix A) can be used to rewrite the expectation formula in terms of a
certain concatenation of Brownian bridge and Brownian motion measures.

  * •

In Step 3, we use local time estimates for concatenated processes (Lemma A.6)
and heat kernel calculations to bound the formula obtained in the previous
step. The bound obtained in this step is sharp enough to conclude the proof of
Proposition 5.3.

Step 1. Applying Lemma 5.2 (a) and Proposition 5.4 with $k=2$ we get

$\displaystyle\lim_{N\to\infty}\mathbb{E}\left[\bigg{(}Q_{t}^{N}(\psi)-\sigma\int_{0}^{t}\mathscr{X}_{s}^{N}(\phi)^{2}ds\bigg{)}^{2}\right]$
$\displaystyle=2\sigma^{2}\cdot\mathbf{E}_{B^{\otimes
4}}\left[\int_{\Delta_{2}(0,t)}e^{\sigma\mathcal{L}_{2}(s_{1},s_{2})}\prod_{i=1}^{2}\left(\psi(X_{s_{i}}^{i})dL_{0}^{X^{i}-Y^{i}}(s_{i})-\phi(X^{i})\phi(Y^{i})ds_{i}\right)\right]$
$\displaystyle=2\sigma^{2}\cdot\mathbf{E}_{B^{\otimes
4}}\left[\int_{\Delta_{2}(0,t)}e^{\sigma\mathcal{L}_{2}(s_{1},s_{2})}\prod_{i=1}^{2}\left(\psi\big{(}\tfrac{1}{2}(X_{s_{i}}^{i}+Y_{s_{i}}^{i})\big{)}dL_{0}^{X^{i}-Y^{i}}(s_{i})-\phi(X^{i})\phi(Y^{i})ds_{i}\right)\right]$
(5.14)

for all $\psi,\phi\in\mathcal{S}(\mathbb{R})$, where $\mathcal{L}_{2}$ is
defined in (5.11). The second equality in the above equation follows by
observing that $X_{u}^{i}=Y_{u}^{i}$ for $u$ in the support of
$L_{0}^{X^{i}-Y^{i}}(du)$.

We shall now write $\mathbf{E}$ instead of $\mathbf{E}_{B^{\otimes 4}}$ for
convenience. Let us now take

$\psi(x):=\xi^{a}_{\varepsilon}(x)=\frac{1}{\sqrt{\pi\varepsilon^{2}}}e^{-(x-a)^{2}/\varepsilon^{2}},\quad\phi(x):=\xi^{a}_{\varepsilon\sqrt{2}}(x)=\frac{1}{\sqrt{2\pi\varepsilon^{2}}}e^{-(x-a)^{2}/2\varepsilon^{2}},$

in (5.14). Using the identity
$\xi^{a}_{\varepsilon\sqrt{2}}(x)\xi^{a}_{\varepsilon\sqrt{2}}(y)=\xi^{a}_{\varepsilon}((x+y)/2)\xi^{0}_{2\varepsilon}(x-y)$,
we may now write (5.14) as

$\displaystyle
2\sigma^{2}\cdot\mathbf{E}\left[\int_{\Delta_{2}(0,t)}e^{\sigma\mathcal{L}_{2}(s_{1},s_{2})}\prod_{i=1}^{2}\left(\xi_{\varepsilon}^{a}\big{(}\tfrac{1}{2}(X_{s_{i}}^{i}+Y_{s_{i}}^{i})\big{)}\cdot(dL_{0}^{X^{i}-Y^{i}}(s_{i})-\xi_{2\varepsilon}^{0}(X_{s_{i}}^{i}-Y_{s_{i}}^{i})ds_{i})\right)\right].$
(5.15)

Let us write $U^{i,-}:=X^{i}-Y^{i}$ and $U^{i,+}:=X^{i}+Y^{i}$. Note that
under $\mathbf{P}_{B^{\otimes 4}}$ the four processes
$U^{1,-},U^{1,+},U^{2,-},U^{2,+}$ are independent Brownian motions with
diffusion coefficient $2$. This enables us to view (5.15) as

$\displaystyle
2\sigma^{2}\cdot\mathbf{E}\left[\int_{\Delta_{2}(0,t)}e^{\sigma\mathcal{L}_{2}(s_{1},s_{2})}\prod_{i=1}^{2}\left(\xi_{\varepsilon}^{a}\big{(}\tfrac{1}{2}U_{s_{i}}^{i,+})\big{)}\cdot(dL_{0}^{U^{i,-}}(s_{i})-\xi_{2\varepsilon}^{0}(U_{s_{i}}^{i,-})ds_{i})\right)\right]$
$\displaystyle=:2\sigma^{2}[A_{1}(\varepsilon)-A_{2}(\varepsilon)-A_{3}(\varepsilon)+A_{4}(\varepsilon)],$
(5.16)

where

$\displaystyle A_{1}(\varepsilon)$
$\displaystyle:=\mathbf{E}\left[\int_{\Delta_{2}(0,t)}e^{\sigma\mathcal{L}_{2}(s_{1},s_{2})}\xi_{\varepsilon}^{a}\big{(}\tfrac{1}{2}U_{s_{1}}^{1,+}\big{)}\xi_{\varepsilon}^{a}\big{(}\tfrac{1}{2}U_{s_{2}}^{2,+}\big{)}\,dL_{0}^{U^{1,-}}(s_{1})\,dL_{0}^{U^{2,-}}(s_{2})\right]$
(5.17) $\displaystyle A_{2}(\varepsilon)$
$\displaystyle:=\mathbf{E}\left[\int_{\Delta_{2}(0,t)}e^{\sigma\mathcal{L}_{2}(s_{1},s_{2})}\xi_{\varepsilon}^{a}\big{(}\tfrac{1}{2}U_{s_{1}}^{1,+}\big{)}\xi_{\varepsilon}^{a}\big{(}\tfrac{1}{2}U_{s_{2}}^{2,+}\big{)}\xi_{2\varepsilon}^{0}(U_{s_{2}}^{2,-})\,dL_{0}^{U^{1,-}}(s_{1})\,ds_{2}\right]$
(5.18) $\displaystyle A_{3}(\varepsilon)$
$\displaystyle:=\mathbf{E}\left[\int_{\Delta_{2}(0,t)}e^{\sigma\mathcal{L}_{2}(s_{1},s_{2})}\xi_{\varepsilon}^{a}\big{(}\tfrac{1}{2}U_{s_{1}}^{1,+}\big{)}\xi_{\varepsilon}^{a}\big{(}\tfrac{1}{2}U_{s_{2}}^{2,+}\big{)}\xi_{2\varepsilon}^{0}(U_{s_{1}}^{1,-})\,dL_{0}^{U^{2,-}}(s_{2})\,ds_{1}\right]$
(5.19) $\displaystyle A_{4}(\varepsilon)$
$\displaystyle:=\mathbf{E}\left[\int_{\Delta_{2}(0,t)}e^{\sigma\mathcal{L}_{2}(s_{1},s_{2})}\xi_{\varepsilon}^{a}\big{(}\tfrac{1}{2}U_{s_{1}}^{1,+}\big{)}\xi_{\varepsilon}^{a}\big{(}\tfrac{1}{2}U_{s_{2}}^{2,+}\big{)}\xi_{2\varepsilon}^{0}(U_{s_{1}}^{1,-})\xi_{2\varepsilon}^{0}(U_{s_{2}}^{2,-})\,ds_{1}\,ds_{2}\right].$
(5.20)

Step 2. In this step we focus on each of the $A_{i}(\varepsilon)$ terms
separately. We drop the $\varepsilon$ and write $A_{i}$ for simplicity. Note
that informally $dL_{0}^{U^{i,-}}(s_{i})$ may be written as
$\delta_{0}(U^{i,-})ds_{i}$ which suggests that each of the $A_{i}$ may be
written in terms of Brownian bridge expectations. To this end, we introduce
the function

$\displaystyle
H_{s_{1},s_{2}}(\vec{x},\vec{y}):=\mathbf{E}\left[e^{\sigma\mathcal{L}_{2}(s_{1},s_{2})}\mid
U_{s_{i}}^{i,-}=x_{i},U_{s_{i}}^{i,+}=y_{i},i=1,2\right]$ (5.21)

where the above conditional expectation is interpreted as taking expectation
under the measure where $U^{i,-}$ ($U^{i,+}$ resp.) is a concatenation of a
Brownian bridge from $0$ to $x_{i}$ ($0$ to $y_{i}$ resp.) over the interval
$[0,s_{i}]$ and an independent Brownian motion started from $x_{i}$ ($y_{i}$
resp.) on $[s_{i},t]$. All Brownian objects considered are independent with
diffusion coefficient $2$.

$A_{1}$ term: Let us consider the $A_{1}$ term from (5.17). Let
$\mathcal{F}_{t}^{-}$ be the $\sigma$-field generated by
$\\{U_{s}^{1,-},U_{s}^{2,-}\mid 0\leq s\leq t\\}$. By the tower property of
conditional expectation, we have

$\displaystyle A_{1}$
$\displaystyle:=\mathbf{E}\left[\int_{\Delta_{2}(0,t)}\mathbf{E}\left[e^{\sigma\mathcal{L}_{2}(s_{1},s_{2})}\prod_{i=1}^{2}\xi_{\varepsilon}^{a}\big{(}\tfrac{1}{2}U_{s_{i}}^{i,+}\big{)}\mid\mathcal{F}_{t}^{-}\right]dL_{0}^{U^{1,-}}(s_{1})\,dL_{0}^{U^{2,-}}(s_{2})\right].$

By Lemma A.2, the right-hand side of the above equation simplifies to

$\displaystyle
A_{1}=\int_{\Delta_{2}(0,t)}\mathbf{E}\left[e^{\sigma\mathcal{L}_{2}(s_{1},s_{2})}\cdot\xi_{\varepsilon}^{a}(\tfrac{1}{2}U_{s_{i}}^{1,+})\xi_{\varepsilon}^{a}(\tfrac{1}{2}U_{s_{i}}^{2,+})\mid
U_{s_{i}}^{i,-}=0,i=1,2\right]p_{2s_{i}}(0)\,\,ds_{i}$ (5.22)

where again the conditional expectation is interpreted via concatenation of
Brownian bridges and Brownian motions. Note that via Brownian motion
decomposition, we have

$\displaystyle\mathbf{E}\left[e^{\sigma\mathcal{L}_{2}(s_{1},s_{2})}\cdot\xi_{\varepsilon}^{a}(\tfrac{1}{2}U_{s_{i}}^{1,+})\xi_{\varepsilon}^{a}(\tfrac{1}{2}U_{s_{i}}^{2,+})\mid
U_{s_{i}}^{i,-}=x_{i}.i=1,2\right]$ (5.23) $\displaystyle\hskip
56.9055pt=\int_{\mathbb{R}^{2}}H_{s_{1},s_{2}}(\vec{x},\vec{y})\prod_{i=1}^{2}\xi_{\varepsilon}^{a}(\tfrac{1}{2}y_{i})p_{2s_{i}}(y_{i})dy_{i}$

where $H$ is defined in (5.21). Inserting the above formula back in (5.22) we
get

$\displaystyle
A_{1}=\int_{\Delta_{2}(0,t)}\int_{\mathbb{R}^{2}}H_{s_{1},s_{2}}((0,0),(y_{1},y_{2})))\prod_{i=1}^{2}\xi_{\varepsilon}^{a}(\tfrac{1}{2}y_{i})p_{2s_{i}}(y_{i})p_{2s_{i}}(0)\,dy_{i}\,ds_{i}.$
(5.24)

$A_{2}$ and $A_{3}$ terms: Recall $A_{2}$ from (5.18). We condition only on
$\mathcal{F}_{t}^{1,-}:=\sigma\\{U_{s}^{1,-}\mid 0\leq s\leq t\\}$ to get

$\displaystyle A_{2}$
$\displaystyle:=\mathbf{E}\left[\int_{0}^{t}F(U^{1,-})dL_{0}^{U^{1,-}}(s_{1})\right]$

where

$\displaystyle
F(U^{1,-}):=\int_{0}^{s_{1}}\mathbf{E}\left[e^{\sigma\mathcal{L}_{2}(s_{1},s_{2})}\xi_{\varepsilon}^{a}\big{(}\tfrac{1}{2}U_{s_{1}}^{1,+}\big{)}\xi_{\varepsilon}^{a}\big{(}\tfrac{1}{2}U_{s_{2}}^{2,+}\big{)}\xi_{2\varepsilon}^{0}(U_{s_{2}}^{2,-})\mid\mathcal{F}_{t}^{1,-}\right]\,ds_{2}$

Applying Lemma A.2 again, we see that

$\displaystyle A_{2}=\int_{0}^{t}\mathbb{E}[F(U^{1,-})\mid
U_{s_{1}}^{1,-}=0]p_{2s_{1}}(0)ds_{2}.$

where again the conditional expectation is interpreted in a similar manner.
Performing a similar trick as in (5.23), we see that

$\displaystyle
A_{2}=\int_{\Delta_{2}(0,t)}\int_{\mathbb{R}^{3}}H_{s_{1},s_{2}}((0,x_{2});\vec{y})\left[\prod_{i=1}^{2}\xi_{\varepsilon}^{a}\big{(}\tfrac{1}{2}y_{i}\big{)}p_{2s_{i}}(y_{i})\right]\xi_{2\varepsilon}^{0}(x_{2})p_{2s_{2}}(x_{2})p_{2s_{1}}(0)d\vec{y}\,dx_{2}\,d\vec{s}.$
(5.25)

Similarly for $A_{3}$ defined in (5.19) one has

$\displaystyle
A_{3}=\int_{\Delta_{2}(0,t)}\int_{\mathbb{R}^{3}}H_{s_{1},s_{2}}((x_{1},0);\vec{y})\left[\prod_{i=1}^{2}\xi_{\varepsilon}^{a}\big{(}\tfrac{1}{2}y_{i}\big{)}p_{2s_{i}}(y_{i})\right]\xi_{2\varepsilon}^{0}(x_{1})p_{2s_{1}}(x_{1})p_{2s_{2}}(0)d\vec{y}\,dx_{1}\,d\vec{s}.$
(5.26)

$A_{4}$ term: Recall $A_{4}$ from (5.20). Note that $A_{4}$ does not involve
any integration with respect to local times. Thus we may use $H$ from (5.21)
directly and tricks similar to (5.23) to get

$\displaystyle
A_{4}:=\int_{\Delta_{2}(0,t)}\int_{\mathbb{R}^{4}}H_{s_{1},s_{2}}(\vec{x},\vec{y})\prod_{i=1}^{2}\xi_{\varepsilon}^{a}\big{(}\tfrac{1}{2}y_{i}\big{)}\xi_{2\varepsilon}^{0}(x_{i})p_{2s_{i}}(x_{i})p_{2s_{i}}(y_{i})dx_{i}\,dy_{i}\,ds_{i}.$
(5.27)

Step 3. In this step, we determine convergence and provide bounds for each of
the $A_{i}$ terms defined at the end of Step 1. We claim that for each $i$,

$\displaystyle\lim_{\varepsilon\to
0}A_{i}(\varepsilon)=\int_{\Delta_{2}(0,t)}H_{s_{1},s_{2}}((0,0),(2a,2a))\prod_{i=1}^{2}p_{2s_{i}}(0)p_{2s_{i}}(2a)ds_{i},$
(5.28)

where $H$ is defined in (5.21). Inserting this limit back in (5.16) verifies
(5.8). Let us consider $A_{1}(\varepsilon)$ from (5.17). Let us consider the
integrand on the right-hand side of (5.24):

$\displaystyle
p_{2s_{i}}(0)\int_{\mathbb{R}^{2}}H_{s_{1},s_{2}}((0,0),(y_{1},y_{2})))\prod_{i=1}^{2}\xi_{\varepsilon}^{a}(\tfrac{1}{2}y_{i})p_{2s_{i}}(y_{i})\,dy_{i}.$

Clearly as $\varepsilon\to 0$, it converges to
$p_{2s_{i}}(0)\int_{\mathbb{R}^{2}}H_{s_{1},s_{2}}((0,0),(a_{1},a_{2})))p_{2s_{1}}(a_{1})p_{2s_{2}}(a_{2})$.
Thus, to show (5.28) holds for $A_{1}(\varepsilon)$, it suffices by dominated
convergence to show that for all $a\neq 0$,

$\displaystyle\sup_{\varepsilon\in(0,1]}p_{2s_{i}}(0)\int_{\mathbb{R}^{2}}H_{s_{1},s_{2}}((0,0),(y_{1},y_{2})))\prod_{i=1}^{2}\xi_{\varepsilon}^{a}(\tfrac{1}{2}y_{i})p_{2s_{i}}(y_{i})\,dy_{i}$
(5.29)

is dominated by some integrable function of $s_{1},s_{2}$. Towards this end,
note that $H$ is defined as the expectation of exponentials of local times of
certain linear functions of concatenated processes introduced at the beginning
of Step 2. In Lemma A.6, we study these expectations and in particular show
that for all $s_{1},s_{2}\in[0,t]$ and for all
$\vec{x},\vec{y}\in\mathbb{R}^{2}$,
$H_{s_{1},s_{2}}(\vec{x},\vec{y})\leq\mathrm{C}$ for some constant
$\mathrm{C}>0$ that depends only on $t$. As
$\xi_{\varepsilon}^{a}(\frac{1}{2}y)=2p_{2\varepsilon^{2}}(y-2a)$, using
$H_{s_{1},s_{2}}(\vec{x},\vec{y})\leq\mathrm{C}$ and then using the semigroup
property of the heat kernel (i.e.,
$\int_{\mathbb{R}}p_{u}(x-b)p_{t}(x)dx=p_{u+t}(b)$) we have from (5.24) that

$\displaystyle\mbox{\eqref{supbd}}\leq\mathrm{C}\sup_{\varepsilon\in(0,1]}\prod_{i=1}^{2}p_{2s_{i}+2\varepsilon^{2}}(2a)p_{2s_{i}}(0).$
(5.30)

Note that the heat kernel globally satisfies the bound
$p_{t}(x)\leq\mathrm{C}(t+x^{2})^{-1/2}\mathbf{1}_{\\{t>0\\}}$ for some large
enough constant $\mathrm{C}>0.$ Consequently we have

$\displaystyle\int_{\Delta_{2}(0,t)}\sup_{\varepsilon\in(0,1]}\prod_{i=1}^{2}p_{2s_{i}+2\varepsilon^{2}}(2a)p_{2s_{i}}(0)ds_{i}$
$\displaystyle\leq\mathrm{C}\int_{\Delta_{2}(0,t)}\prod_{i=1}^{2}s_{i}^{-1/2}(s_{i}+a^{2})^{-1/2}\,ds_{i}$
$\displaystyle=\mathrm{C}\int_{\Delta_{2}(0,a^{-2}t)}\prod_{i=1}^{2}v_{i}^{-1/2}(v_{i}+1)^{-1/2}\,dv_{i}.$

where the equality above follows by making the substitution
$s_{i}=a^{2}v_{i}$. Extending the range of integration we have

$\displaystyle\int_{\Delta_{2}(0,a^{-2}t)}\prod_{i=1}^{2}v_{i}^{-1/2}(v_{i}+1)^{-1/2}\,dv_{i}$
$\displaystyle\leq\int_{[0,a^{-2}t]^{2}}\prod_{i=1}^{2}v_{i}^{-1/2}(v_{i}+1)^{-1/2}\,dv_{i}\leq\mathrm{C}\max\\{1,|\log
a|^{2}\\}.$ (5.31)

This verifies the integrability of the bound in (5.30) and consequently proves
(5.28) for $A_{1}$. Starting from (5.25), (5.26), and (5.27), an analogous
computation verifies (5.28) for $A_{2},A_{3}$ and $A_{4}$ respectively. This
establishes (5.8). From the polylogarithmic bound in (5.31) (and its analogous
counterparts for $A_{2}$, $A_{3}$, $A_{4}$) one arrives at (5.9). This
completes the proof. ∎

######  Proposition 5.5 (Error term limit).

For any $t\geq 0$ and $\phi\in\mathcal{S}(\mathbb{R})$, we have that

$\displaystyle\limsup_{N\to\infty}\mathbb{E}[\mathcal{E}_{t}^{N}(\phi)^{2}]=0.$
(5.32)

###### Proof.

Note that $\phi\phi^{\prime}\leq\frac{1}{2}(\phi^{2}+(\phi^{\prime})^{2})$
thus

$|\mathcal{E}_{t}^{N}(\phi)|\leq
2N^{1/4}\left[\int_{0}^{t}((\mathscr{X}_{s}^{N})^{Sq},\phi^{2})ds+\int_{0}^{t}((\mathscr{X}_{s}^{N})^{Sq},(\phi^{\prime})^{2})ds\right]=2N^{-1/4}\left[Q_{t}^{N}(\phi^{2})+Q_{t}^{N}(\phi^{\prime
2})\right].$

Now by Lemma 5.2 (b) and Proposition 5.4 ((5.10) specifically), we see that
for every $\psi\in\mathcal{S}(\mathbb{R})$, $L^{2}$ moments of
$Q_{t}^{N}(\psi)$ are uniformly bounded in $N$. Since $\phi^{2}$ and
$\phi^{\prime 2}$ are also Schwartz functions, in view of the above
inequality, we readily have (5.32). ∎

###  5.2. Supporting estimates

In this section we prove Proposition 5.4 and establish an estimate for the
moments of the increments of the quadratic martingale field which will be
useful in Section 6.2.

###### Proof of Proposition 5.4.

Proof of (5.10). We continue with the same notation as in the statement of the
Proposition 5.4. Let us set

$\displaystyle\mathcal{G}_{N}^{r}(t)=\sum_{j=2r-1}^{2k}\int_{0}^{t}\bigg{[}1-\frac{1}{m_{j}^{r}(\mathbf{X}(u))}\bigg{]}dX^{j}(u)$

where $m_{j}^{r}(\mathbf{x}):=\\#\\{i\in\\{2r-1,\ldots,2k\\}:x_{i}=x_{j}\\}$.
Note that $\mathcal{G}_{N}^{r}$ are martingales of the form (2.3) where only
the last $2k-2r$ of the $2k$ particles are taken into account. Define the
martingale

$\displaystyle
d\overline{\mathcal{G}}_{N}(t):=N^{1/4}\sum_{r=1}^{k}\mathbf{1}_{\\{t\in[t_{r-1},t_{r})\\}}d\mathcal{G}_{N}^{r}(t).$
(5.33)

It turns out that the stochastic exponential of $\overline{\mathcal{G}}_{N}$