monthyear[],

# At the Edge of a Cloud of Brownian Particles

Dom Brockington111Mathematics Institute, University of Warwick, Coventry CV4
7AL, UK. Email Address<EMAIL_ADDRESS>and Jon
Warren222Department of Statistics, University of Warwick, Coventry CV4 7AL,
UK. Email Address<EMAIL_ADDRESS>
###### Abstract

We study a simple model for the trajectory of a particle in a turbulent fluid,
where a Brownian motion travels through a random Gaussian velocity field. We
study the quenched law of the process and prove that in a weak environment
setting, the fluctuations at the transition from weak to strong disorder are
described by the KPZ equation. We conjecture the same is true without the
assumption of a weak environment.

##  1 Introduction

In this paper we consider a model of turbulent advection describing particle
trajectories in one dimensional turbulent fluids. The particles each have
their own, independent molecular diffusive motion, and are also transported
through a Gaussian random drift field which represents the effect of the
fluid. The field is Brownian in time and smooth, but rapidly decorrelating, in
space. Associated to this model is a stochastic flow of kernels, [LJR04],
which can be thought of as the density of a cloud of particles in the fluid,
or as the random transition density of a single particle running through a
realisation of the drift field. Using this interpretation, it is possible to
show the flow of kernels solve a stochastic partial differential equation
(SPDE) similar to a Fokker-Planck equation for a Brownian motion with drift.

The model is an example of the compressible Kraichnan model for turbulence,
see the review [FGmcV01], which models the trajectory of a particle in a
turbulent fluid, where the fluid is represented by a random evolving velocity
field that is white in time. In our case the spatial correlations are taken to
be of short length and smooth in space, similar to the case considered in
[GH04], where the authors showed that removing the molecular diffusivity, at
the same time as reducing the correlation length of the velocity field, with a
certain relation holding between the two, led to sticky interactions between
pairs of particles in the limiting process. This result was extended for the
model we consider, [War15], to a full description of the interactions between
any number of particles in the limiting process.

In [DG21] Dunlap and Gu show that the fluctuations of the density of the flow
of kernels are well approximated, at large times, by the product of the heat
kernel and the stationary solution to the SPDE solved by the flow of kernels.
We are instead interested in the fluctuations of the density in the tail.
Barraquand and Le Doussal, [BD20] conjecture that, at a cetain distance from
the origin, these fluctuations are governed by the Kardar-Parisi-Zhang
equation (KPZ equation). We will show that this is indeed the case in a regime
in which the strength of the environment is tending to zero. Further we
conjecture that the KPZ equation also appears when the diffusivity of the
molecular motion of the particles is taken to be small instead of the
environment strength.

We now briefly recall the stochastic heat equation, its connection to the KPZ
equation and describe some previous results on the universality of the KPZ
equation. The stochastic heat equation (SHE) is the stochastic partial
differential equation, driven by a space-time white noise $\dot{W}$, given
below

$\displaystyle\partial_{t}z=\frac{\nu}{2}\Delta z+\kappa z\dot{W}.$ (1.1)

The logarithm of the stochastic heat equation, $h=\frac{\nu}{2\delta}\log z$,
is the Cole-Hopf solution to the KPZ equation

$\displaystyle\partial_{t}h=\frac{\nu}{2}\Delta
h+\delta(\partial_{x}h)^{2}+\kappa\dot{W}.$ (1.2)

The KPZ equation describes the interface for surface growth model and is the
canonical model in the KPZ universality class [KPZ86]. In fact, the KPZ
equation is also a universal object itself; this is known as weak KPZ
universality and it has been shown that a large class of continuous surface
growth models lie in this class [HQ18]. In addition, it has been shown that
certain observables of some discrete models converge to the KPZ equation, via
convergence to the stochastic heat equation. The earliest such result was for
the height function of the weakly asymmetric exclusion process [BG97]. More
recently convergence to the KPZ equation has been shown for the free energy of
directed random polymers in the intermediate disorder limit [AKQ14]. Following
this result weak KPZ universality has been shown for a generalisation of ASEP
[CST18], for a class of weakly asymmetric non-simple exclusion processes
[DT16], and for Higher-Spin Exclusion process [CT17] and the related
Stochastic 6-vertex model [CGST20]. Most relevant for us, in [CG16] Corwin and
Gu studied a discrete version of our model, where the random drift field
appears instead as random transition probabilities for a nearest neighbour
random walk on $\mathbb{Z}$. They showed that the transition probabilities
evaluated in the large deviation regime, after rescaling, converge to the
solution to the stochastic heat equation.

Barraquand and Corwin, [BC17], showed that when the transition probabilities
in the random walk model are chosen to be Beta distributed the model becomes
exactly solvable. They then used their formulae to show that the tail
probabilities of the Beta Random walk in a random environment have Tracy-Widom
GUE fluctuations of size $N^{1/3}$, placing the model in the universality
class of the KPZ fixed point, called strong KPZ universality. The same result
is expected to hold for the density of the transition probabilities evaluated
at point in the tail, not just for the cumulative tail probabilities [TLD16].

Barraquand and Rychnovsky, [BR20], found similar results exist for sticky
Brownian motions and the associated Howitt-Warren flows; once again taking
advantage of exact solvability in a special case. Further they conjectured the
Howitt-Warren flows ( [HW09]) converge, under rescaling, to the stochastic
heat equation, based on the convergence of the moments. Note that by [War15]
this special case of the sticky flow is not the same one that arises in the
limit of our model, but it is reasonable to expect the results to hold more
generally.

In this paper we show that the flow of kernels associated with our model
converges to the solution to the stochastic heat equation, when the strength
of the random environment and its correlation length are both taken to zero,
with a certain relationship holding between the two. This result is similar to
the result of Corwin and Gu, [CG16], for the discrete random walk model. We
will also consider the behaviour of the second moment of the kernel in more
general regimes; Figure 1 shows the behavior we find to hold when the distance
from the origin and the ratio of the strength of environment to molecular
diffusivity vary. The stochastic heat equation occurs (or is conjectured to
occur) at a transition between weak and strong disorder regimes. In the former
the kernels converge to the deterministic heat equation. In the latter the
kernels converge to zero. This is analogous to the behaviour of the partition
function of a random polymer, [Com17].

###  1.1 The Model

Suppose that $W_{C}$ is a centred Gaussian field with covariance
$\mathbb{E}[W_{C}(s,x)W_{C}(t,y)]=(s\wedge t)C(x-y)$, where $C\in
C^{\infty}_{c}(\mathbb{R})$ is positive-definite. Let $\sigma>0$, we are
interested in the solutions to the SDE

$\displaystyle dX(t)=W_{C}(dt,X(t))+\sigma dB(t).$ (1.3)

In the above SDE, $B$ is a Brownian motion on $\mathbb{R}$ independent of
$W_{C}$ and both stochastic integrals are understood in the Itô sense. As
mentioned in the introduction, this is a simplified model for a particle
trajectory in a turbulent fluid. The fluid is represented by the noise,
$W_{C}$, and $\sigma$ is the molecular diffusivity of the particle itself.
Existence and uniqueness of solutions is proved in [Kun94b, Theorem 3.4.1]. In
fact, by [Kun94b, Theorem 4.5.1] the solution exists as a stochastic flow,
$X_{s,t}(x)$, such that for any $s\geq 0$, the process $(X_{s,t}(x))_{t>s}$
solves the SDE started from $x$ at time $s$. The solution is distributed as a
Brownian motion with diffusivity $\nu:=C(0)+\sigma^{2}$, which can be checked
directly by calculating the quadratic variation and recalling Lévy’s
characterisation of Brownian motion. Alternatively, the solution can be
thought of as a Brownian motion with diffusivity $\sigma^{2}$, running through
the time dependent Gaussian random field $W_{C}$, which we can think of as a
random velocity field. We then consider the transition probabilities
$(U_{s,t})_{s\leq t}$ of the process $(X_{s,t})_{s\leq t}$ conditional on
$W_{C}$:

$U_{s,t}(x,A):=\mathbb{P}(X_{s,t}(x)\in A|\ W),$ (1.4)

Note that the family of kernels $(U_{s,t})_{s<t}$ depend only on the field
$W$, and form a stochastic flow of kernels as introduced by Le Jan and Raimond
in [LJR04]. Further, because $X$ is itself a Brownian motion, if we average
$U_{s,t}$ over the law of $W$ we get the heat kernel,

$\displaystyle\mathbb{E}[U_{s,t}(x,A)]=P_{t-s}(x,A).$ (1.5)

Here, $P$ denotes the heat kernel with diffusivity $\nu$, the density of which
we will denote by
$p_{t-s}(x-y)=\frac{1}{\sqrt{2\pi\nu(t-s)}}e^{-\frac{(x-y)^{2}}{2\nu(t-s)}}$.

It is known from [DG21] that the family of probability kernels
$(U(s,t,x,dy))_{s<t}$ have continuous densities, $\mathcal{U}(s,t,x,\cdot)$,
with respect to the Lebesgue measure which solve the stochastic partial
differential equations

$\displaystyle\partial_{t}\mathcal{U}=\frac{\nu}{2}\partial^{2}_{y}\mathcal{U}-\partial_{y}\left(\mathcal{U}\dot{W}_{C}\right),$
(1.6)

together with the initial condition $\mathcal{U}(s,s,x,y)=\delta(x-y)$, where
$\delta$ is the Dirac delta. In the above equation, $\dot{W}_{C}$ is the
(formal) time derivative of $W_{C}$, so that it is white in time and smoothly
correlated in space. By solution, we mean it is a generalised solution in the
sense of [Kun94a], which we describe now by briefly recalling [DG21,
Proposition 2.1]. The process $\mathcal{U}(0,t,x,\cdot)$, considered as a
time-indexed family of tempered distributions on $\mathbb{R}$, is the unique
solution to (1.6). That is, for every $s,x\in\mathbb{R}$ and every Schwartz
function $f:\mathbb{R}\to\mathbb{R}$ the following equality holds almost
surely for every $t>s$

$\displaystyle\int_{\mathbb{R}}\mathcal{U}(s,t,x,y)f(y)dy$ $\displaystyle=$
$\displaystyle
f(x)+\frac{\nu}{2}\int_{s}^{t}\int_{\mathbb{R}}\mathcal{U}(s,r,x,y)f^{\prime\prime}(y)dydr+\int_{s}^{t}\int_{\mathbb{R}}\mathcal{U}(s,r,x,y)f^{\prime}(y)W_{C}(dr,y)dy.$
(1.7)

Here, the stochastic integral is interpreted in the Itô sense.

The same SPDE, in a slightly different formulation, was derived for the flow
of kernels (1.4) in [LJR04, Section 5]. The solution to the SPDE can be
constructed directly in terms of a Wiener chaos expansion, [LJR02, Theorem
3.2]. Note that (1.6) is similar to the Fokker-Planck equation for a Brownian
motion with diffusivity $\sigma^{2}$ moving through a velocity field
$\dot{W}_{C}$; however, the coefficient of the Laplacian part is $\nu$ instead
of $\sigma^{2}$ because of the Itô correction coming from the stochastic
integral.

###  1.2 The Stochastic Heat Equation as a Limit

We are interested in the fluctuations of $\mathcal{U}$ at large times far from
the origin. Hence, we look first at the effect of diffusive rescaling on the
kernel. Define for $\varepsilon>0$,

$\displaystyle\mathcal{U}^{\varepsilon}(s,t,x,y)=\varepsilon^{-1}\mathcal{U}(\varepsilon^{-2}s,\varepsilon^{-2}t,\varepsilon^{-1}x,\varepsilon^{-1}y).$

Then it is a straightforward calculation to show that
$\mathcal{U}^{\varepsilon}$ is a solution to the transport SPDE (1.1) in which
$W_{C}$ is replaced by its rescaled version $W^{\varepsilon}(t,y)=\varepsilon
W_{C}(\varepsilon^{-2}t,\varepsilon^{-1}y)$, which has spatial covariance
function $C^{\varepsilon}(x-y)=C(\varepsilon^{-1}(x-y))$. If we take
$\varepsilon\to 0$ then $\mathcal{U}^{\varepsilon}$ converges to the solution
to the Heat equation, as we will see in Proposition 1.3. This convergence is
only in a weak sense in the space variable; Dunlap and Gu, [DG21], show that
there is a non-trivial random fluctuation when the pointwise behaviour is
considered.

Rather than just considering in the diffusive regime, we are also interested
in the moderate deviation regime, where $y\approx t^{3/4}$ and Barraquand and
Le Doussal, [BD20], conjectured the SHE would appear. Thus, we introduce the
rescaled quantity, $\mathcal{V}=\mathcal{V}^{\varepsilon,\lambda}$, given by

$\displaystyle\mathcal{V}(s,t,x,y):=e^{\frac{\nu}{2}\lambda^{2}(t-s)+\lambda(y-x)}\mathcal{U}^{\varepsilon}(s,t,x+\lambda\nu
s,y+\lambda\nu t)$ (1.8)

where $\lambda\in\mathbb{R}$ is a parameter which we will vary with
$\varepsilon$ to control the distance from the origin. The prefactor is
motivated by fixing the expectation of $\mathcal{V}$:

$\displaystyle\mathbb{E}\left[\mathcal{V}(s,t,x,y)\right]=$ $\displaystyle
e^{\frac{\nu}{2}\lambda^{2}t+\lambda(y-x)}\mathbb{E}\left[\mathcal{U}^{\varepsilon}(s,t,x,y+\lambda\nu
t)\right]$ $\displaystyle=$ $\displaystyle
e^{\frac{\nu}{2}\lambda^{2}t-\lambda(y-x)}p_{t-s}(x-y)$ $\displaystyle=$
$\displaystyle p_{t-s}(x-y).$

In the most part, we will set $s,x=0$ and consider
$\mathcal{V}(t,y):=\mathcal{V}(0,t,0,y)$. According to Lemma 3.1 below,
$\mathcal{V}$ is the solution to the SPDE,

$\partial_{t}\mathcal{V}=\frac{\nu}{2}\partial^{2}_{y}\mathcal{V}+\lambda\mathcal{V}\dot{W}^{\varepsilon}-\partial_{y}\left(\mathcal{V}\dot{W}^{\varepsilon}\right),\quad\mathcal{V}(0,y)=\delta_{0}(y),$
(1.9)

where, here, $\dot{W}^{\varepsilon}$ is a shifted version of the noise,
$W^{\varepsilon}(t,y):=\varepsilon\int_{0}^{\varepsilon^{-2}t}W_{C}(ds,\varepsilon^{-1}y+\lambda\varepsilon^{-1}\nu
s)$. The covariance of the shifted noise is given by
$\mathbb{E}\left[W^{\varepsilon}(t,x)W^{\varepsilon}(s,y)\right]=(s\wedge
t)C^{\varepsilon}(x-y)$, so that $W^{\varepsilon}$ is equal in distribution to
the $W^{\varepsilon}$ discussed at the start of the section.

If $\lambda$ is chosen to depend on $\varepsilon$ in an appropriate manner,
namely $\lambda=\varepsilon^{-\frac{1}{2}}$, then
$\lambda^{2}C^{(\varepsilon)}$ approaches a multiple of the Dirac delta
distribution, and thus $\dot{W}^{\varepsilon}$ approaches a space-time white
noise as $\varepsilon\to 0$. Le Doussal and Barraquand, [BD20], argue that
taking $\varepsilon\to 0$ in (1.9) will lead to the final term vanishing,
suggesting the limit could be the stochastic heat equation.

One can also consider the moments of $\mathcal{V}$. First consider the moments
of $\mathcal{U}$, which have a representation in terms of solutions to (1.3).
More precisely, if $X^{1}$ and $X^{2}$ are two solutions to $(\ref*{system of
SDEs})$, each driven by independent Brownian motions $B^{1}$ and $B^{2}$ but
the same Gaussian velocity field, then for any $f\in C_{b}(\mathbb{R}^{2})$

$\displaystyle\mathbb{E}\left[\int_{\mathbb{R}^{2}}\mathcal{U}(0,t,0,y_{1})\mathcal{U}(0,t,0,y_{2})f(y_{1},y_{2})dy\right]=\mathbb{E}\left[f(X^{1}(t),X^{2}(t))\right].$

Notice that $(X^{1}(t),X^{2}(t))_{t\geq 0}$ is a diffusion with generator

$\displaystyle\mathcal{G}f=\frac{1}{2}\sum_{i,j=1}^{2}(\sigma^{2}\delta_{i,j}+C^{\varepsilon}(y_{i}-y_{j}))\frac{\partial^{2}f}{\partial
y_{i}\partial y_{j}}.$ (1.10)

To see that this representation is useful, we can note that it can be used to
show the convergence towards the heat equation, in the weak spatial sense,
simply by showing that $(\varepsilon X^{1}(\varepsilon^{-2}t),\varepsilon
X^{2}(\varepsilon^{-2}t))_{t\geq 0}$ converges towards a Brownian motion on
$\mathbb{R}^{2}$ with diffusivity $\nu$ as $\varepsilon\to 0$; we will do this
later in propositions 1.3 and 2.2. There is a similar representation for the
moments of $\mathcal{V}$,

$\displaystyle\mathbb{E}\left[\int_{\mathbb{R}^{2}}\mathcal{V}(t,y_{1})\mathcal{V}(t,y_{2})f(y_{1},y_{2})dy\right]=\mathbb{E}\left[e^{\lambda^{2}\int_{0}^{t}C^{\varepsilon}(Y^{1}(s)-Y^{2}(s))ds}f(Y^{1}(t),Y^{2}(t))\right],$
(1.11)

where $Y=(Y^{1},Y^{2})$ is a diffusion with generator

$\displaystyle\mathcal{G}f=\frac{1}{2}\sum_{i,j=1}^{2}(\sigma^{2}\delta_{i,j}+C^{\varepsilon}(y_{i}-y_{j}))\frac{\partial^{2}f}{\partial
y_{i}\partial y_{j}}+\sum_{i=1}^{2}\lambda
C^{\varepsilon}(y_{1}-y_{2})\frac{\partial f}{\partial y_{i}},\quad\text{for
}f\in C^{2}_{b}(\mathbb{R}^{2}),$

$\delta_{i,j}$ being the Kronecker delta.

By taking $\varepsilon\to 0$, $Y^{1}$ and $Y^{2}$ once again converge to
independent Brownian motions, and if we simultaneously take
$\lambda=\varepsilon^{-\frac{1}{2}}$, then the right hand side of (1.11)
approaches

$\displaystyle\mathbb{E}\left[\exp\left(\frac{\kappa^{2}}{2\nu}\mathcal{L}_{t}^{0}(B^{1}-B^{2})\right)f(B^{1}(t),B^{2}(t))\right],$
(1.12)

where $B^{1}$ and $B^{2}$ are independent Brownian motions with diffusivity
$\nu$, $\mathcal{L}^{0}_{t}(B^{1}-B^{2})$ is the local time at $0$ of the
process $B^{1}-B^{2}$ and
$\kappa^{2}=\nu\int_{\mathbb{R}}\frac{C(y)}{\sigma^{2}+C(0)-C(y)}dy$. This is
exactly the form of the second moment for the stochastic heat equation, (1.1).
Note that, importantly, this prediction for the coefficient, $\kappa^{2}$, of
the noise is different from that which one would obtain simply by neglecting
the transport noise in (1.9). The latter would be $\int_{\mathbb{R}}C(y)dy$,
which is strictly smaller. We can extend this argument by showing the
convergence of the higher moments as well. However, the distribution of the
stochastic heat equation is not determined by its moments, and unfortunately,
we do not have any proof of the convergence of $\mathcal{V}$ to a solution to
the stochastic heat equation.

We consider a more general setup where we change the balance between the
effects of the environment and the molecular diffusivity as we vary
$\varepsilon$. More precisely, we consider $\mathcal{V}$ solving (1.9) with
noise $\dot{W}^{\varepsilon}$ having covariance
$C^{\varepsilon}(\cdot)=\mu(\varepsilon)^{2}C(\varepsilon^{-1}\cdot)$, where
$C$ is a fixed positive definite function, as before. We can also let the
molecular diffusivity, $\sigma(\varepsilon)$, depend on $\varepsilon$ and then
$\nu=\sigma(\varepsilon)^{2}+\mu(\varepsilon)^{2}C(0)$. Two regimes are of
interest, in addition to case considered previously, where
$\mu(\varepsilon)\equiv 1$ and $\sigma(\varepsilon)^{2}=\sigma^{2}$. The weak
environment setting, where $\mu(\varepsilon)$ tends to $0$ and
$\sigma(\varepsilon)$ tends to some $\sigma>0$ as $\varepsilon\to 0$; and the
weak diffusivity, where $\mu(\varepsilon)$ tends to $1$ and
$\sigma(\varepsilon)$ tends to $0$ as $\varepsilon\to 0$.

In the weak environment setting, we will prove the convergence of
$\mathcal{V}^{\varepsilon}:=\mathcal{V}^{\varepsilon,\lambda(\varepsilon)}$
towards the solution to the stochastic heat equation when
$\lambda(\varepsilon)$ is chosen such that
$\lambda(\varepsilon)\mu(\varepsilon)\varepsilon^{\frac{1}{2}}$ converges to a
(non-zero) constant. In fact, we can couple the sequence of solutions,
$\mathcal{V}$, to (1.9) to the limiting solution of the stochastic heat
equation. This is achieved by constructing the driving noises from a common
space-time white noise, through appropriate mollifications.

Let $W$ be a cylindrical Brownian motion on $L^{2}(\mathbb{R})$, and suppose
that $C$ can be written as $C=\rho*\rho$ for a symmetric mollifier $\rho$.
Denote $C^{\varepsilon}(x-y):=\mu(\varepsilon)^{2}C(\varepsilon^{-1}(x-y))$
and
$\rho_{\varepsilon}(y):=\varepsilon^{-\frac{1}{2}}\mu(\varepsilon)\rho(\varepsilon^{-1}y)$,
so that $C^{\varepsilon}=\rho_{\varepsilon}*\rho_{\varepsilon}$. Define
$W^{\varepsilon}:=\rho_{\varepsilon}*W$, so that $W^{\varepsilon}$ is a
centred Gaussian field that is Brownian in time with spatial covariance
function $C^{\varepsilon}$. In the following theorem, we will consider
solutions to (1.9), $\mathcal{V}$, in the weak environment setting where
$\mu(\varepsilon)$ is taken to be converging to $0$ as $\varepsilon$ tends to
$0$. Note that whilst we have coupled the solutions, $\mathcal{V}$, they are
no longer connected by the diffusive scalings and shifts that we discussed
earlier.

######  Theorem 1.1.

Suppose that
$\mu(\varepsilon)=o\left(\log(\varepsilon^{-1})^{-\frac{1}{2}}\right)$ and
$\lambda(\varepsilon)\mu(\varepsilon)\varepsilon^{\frac{1}{2}}(\int_{\mathbb{R}}C(y)dy)^{1/2}\to\kappa$
as $\varepsilon\to 0$. Let $\mathcal{V}$ be the solution to the SPDE

$\displaystyle\partial_{t}\mathcal{V}=\frac{\nu}{2}\partial^{2}_{y}\mathcal{V}+\lambda(\varepsilon)\mathcal{V}\dot{W}^{\varepsilon}-\partial_{y}\left(\mathcal{V}\dot{W}^{\varepsilon}\right),\quad\mathcal{V}(0,\cdot)=\delta_{0},$
(1.13)

where $\nu>0$ is constant. Then, for any $t>0$, as $\varepsilon$ tends to
zero, the following convergence holds in $L^{2}(\mathbb{P}\times\mathbb{R})$

$\displaystyle\mathcal{V}(t)\rightarrow\mathcal{Z}(t),$

where $\mathcal{Z}\in C((0,T);C(\mathbb{R}))$ denotes the solution to the
stochastic heat equation,

$\displaystyle\partial_{t}\mathcal{Z}=\frac{\nu}{2}\partial^{2}_{y}\mathcal{Z}+\kappa\mathcal{Z}\dot{W},\quad\mathcal{Z}(0,\cdot)=\delta_{0}.$
(1.14)

The convergence in $L^{2}(\mathbb{P}\times\mathbb{R})$, established in Theorem
1.1, implies convergence in distribution of $\mathcal{V}(t)$ as random
elements of $L^{2}(\mathbb{R})$. If we adjust the definition of
${\mathcal{U}}$ to account for the weak environment ( noting this makes
$\mathcal{U}$ depend on $\varepsilon$) then (1.8) holds as an equality in
distribution, and we have the following corollary.

######  Corollary 1.2.

Suppose that ${\mathcal{U}}$ solves

$\partial_{t}\mathcal{U}=\frac{\nu}{2}\partial^{2}_{y}\mathcal{U}-\mu(\varepsilon)\partial_{y}\left(\mathcal{U}\dot{W}_{C}\right),\quad\mathcal{U}(0,\cdot)=\delta_{0},$

where $\mu(\varepsilon)=o\left(\log(\varepsilon^{-1})^{-\frac{1}{2}}\right)$.
Let $\lambda(\varepsilon)$ be chosen so that
$\lambda(\varepsilon)\mu(\varepsilon)\varepsilon^{\frac{1}{2}}\to\kappa\left(\int_{\mathbb{R}}C(y)dy\right)^{-1/2}$
as $\varepsilon\to 0$. Then,

$y\mapsto\varepsilon^{-1}\exp\bigl{(}{\lambda(\varepsilon)^{2}\nu
t/2+\lambda(\varepsilon)y}\bigr{)}\mathcal{U}\bigl{(}\varepsilon^{-2}t,\varepsilon^{-1}(y+\lambda(\varepsilon)\nu
t)\bigr{)}$

converges in distribution as a random element of $L^{2}({\mathbb{R}})$ to
$\mathcal{Z}(t)$, where ${\mathcal{Z}}$ solves the SHE with parameters $\nu$
and $\kappa$.

This extends simply to multiple times, and it is natural to expect that in
fact $\bigl{(}\mathcal{V}(t)\bigr{)}_{t>0}$ is converging as a process. We
have not established the tightness needed to prove this however.

Turning to the weak diffusivity regime, we can once again argue at the level
of the moments and show that if we set $\mu(\varepsilon)=1$ and suppose that
$\sigma(\varepsilon)$ tends to $0$ slowly enough that
$\sigma(\varepsilon)\varepsilon^{-1}$ tends to infinity as $\varepsilon$ tends
to $0$, and we choose $\lambda(\varepsilon)$ such that
$\lambda(\varepsilon)\sigma(\varepsilon)^{-\frac{1}{2}}\varepsilon^{\frac{1}{2}}$
is converging to a positive constant, $c$, then the second moment of
$\mathcal{V}$ converges to (1.12), this time with
$\kappa^{2}=\sqrt{2}c\nu\frac{C(0)}{|C^{\prime\prime}(0)|^{1/2}}$. We will
justify this prediction for $\kappa$ at the end of Section 2.1. The nature of
this convergence is expected to be quite different from in the weak
environment setting. Firstly, some smoothing in the spatial variable will be
needed to control oscillations in $\mathcal{V}$. Secondly, if the pair
$\bigl{(}\mathcal{V},W^{\varepsilon}\bigr{)}$ converges in distribution
(possibly down some subsequence) then the limit of the noises
$W^{\varepsilon}$ is not the noise driving the limiting SHE.

We summarize our predictions for general choices of
$\mu(\varepsilon),\lambda(\varepsilon)$ and $\sigma(\varepsilon)$ in Figure 1.
Let

$\displaystyle\beta=-\lim_{\varepsilon\to
0}\frac{\log(\lambda(\varepsilon))}{\log(\varepsilon)},$
$\displaystyle\alpha=-\lim_{\varepsilon\to
0}\frac{\log(\mu(\varepsilon)\sigma(\varepsilon)^{-1})}{\log(\varepsilon)},$

where we suppose that the above limits exist and that $\beta$ is positive.

Figure 1:  The limiting behaviour of the kernel $\mathcal{V}$ for varying
values of the exponents $\alpha$ and $\beta$. The convergence to the SHE
proven in Theorem 1.1 corresponds the line segment $\beta=\frac{1}{2}-\alpha$
in the left quadrant; convergence to the SHE is conjectured for the line
segment $\beta=\frac{1}{2}(1-\alpha)$ in the right quadrant, including at the
point $(0,\tfrac{1}{2})$ which corresponds to diffusive scaling of
${\mathcal{U}}$. Below these line segments is the weak disorder regime in
which the kernels converge to the deterministic heat equation (Propositions
1.3 and 1.4). The limit of the kernels is a sticky flow at $(1,0)$ ; for
$\alpha>1$ and $\beta=0$ the limit is the Arratia flow of coalescing Brownian
motions. Above the line segments lies the strong disorder regime in which the
limit of the kernels is conjectured to be zero; partially proven in
Proposition 1.5.

The next two results concern the behaviour of $\mathcal{V}$ in the weak
disorder regime, which in Figure 1 corresponds to the region below the line
segments.

######  Proposition 1.3.

Suppose we are in the setting of the preamble to Theorem 1.1. Suppose further
that $\mu(\varepsilon)$ is bounded and
$\lambda(\varepsilon)\mu(\varepsilon)\varepsilon^{\frac{1}{2}}\sigma^{-\tfrac{1}{2}}_{\varepsilon}$
tends to $0$ as $\varepsilon\to 0$. For any $t>0$ and $f\in
C^{2}_{b}(\mathbb{R})$ we have

$\displaystyle\int_{\mathbb{R}}\mathcal{V}(t,y)f(y)dy\to\int_{\mathbb{R}}p_{t}(y)f(y)dy\
\text{in }L^{2}(\mathbb{P}),\ \text{as }\varepsilon\to 0.$

As previously, $p_{t}(y)=\frac{1}{\sqrt{2\pi\nu t}}e^{-\frac{y^{2}}{2\nu t}}$
denotes the density of the heat kernel.

Notice that the above result involves spatial averaging; we expect that there
are random fluctuations of the type described by Dunlap and Gu, [DG21], which
mean that pointwise convergence fails. However, in the weak environment
setting these fluctuations vanish in the limit, and we obtain the following
result.

######  Proposition 1.4.

Suppose we are in the setting of the preamble to Theorem 1.1. Suppose further
that $\mu(\varepsilon)=o((\log(\varepsilon^{-1}))^{-\frac{1}{2}})$ and
$\lambda(\varepsilon)\mu(\varepsilon)\varepsilon^{\frac{1}{2}}\to 0$ as
$\varepsilon\to 0$. Then, for each $t>0$, the following convergence holds in
$L^{2}(\mathbb{P}\times\mathbb{R})$

$\displaystyle\mathcal{V}(t)\to p_{t},\text{ as }\varepsilon\to 0.$

In the strong disorder regime, corresponding to the region above the line
segments in Figure 1, the asymptotic behaviour of the kernels is expected to
be quite different. We give a first result in this direction, showing
convergence to zero; however the argument is only valid when
$\beta>\frac{1}{2}$, so that there is a region in Figure 1 on the right which
is not covered. The proof is an adaptation of an argument from Liggett
[Lig04], see also [CSY03].

######  Proposition 1.5.

Suppose we are in the setting of the preamble to Theorem 1.1. Suppose further
that $\mu(\varepsilon)$ is bounded and
$\lambda(\varepsilon)\mu(\varepsilon)\varepsilon^{\frac{1}{2}}$ tends to
infnity as $\varepsilon\to 0$. Then, for each $t>0$, the following convergence
holds in probability

$\displaystyle\int_{\mathbb{R}}\mathcal{V}(t,y)dy\to 0,\text{ as
}\varepsilon\to 0.$

##  2 The 2-Point Motion

###  2.1 The general case

Before we move on to the main part of the proof of Theorem 1.1, we will show
that the second moments of $\mathcal{V}$ can be written in terms of a two-
dimensional diffusion, which we will call the two-point motion associated to
$\mathcal{V}$.

######  Proposition 2.1.

Suppose that $\mathcal{V}^{\varepsilon,\lambda}$ is the solution to (1.9). Let
$Y=(Y^{1},Y^{2})$ be the diffusion in $\mathbb{R}^{2}$ with generator

$\displaystyle\mathcal{G}f=\frac{1}{2}\sum_{i,j=1}^{2}(\sigma^{2}\delta_{i,j}+C^{\varepsilon}(y_{i}-y_{j}))\frac{\partial^{2}f}{\partial
y_{i}\partial y_{j}}+\sum_{i=1}^{2}\lambda
C^{\varepsilon}(y_{1}-y_{2})\frac{\partial f}{\partial y_{i}},$

where $\delta_{i,j}$ is the Kronecker delta. Then we have the equality

$\displaystyle\mathbb{E}\left[\int_{\mathbb{R}^{2}}\mathcal{V}^{\varepsilon,\lambda}(t,y_{1})\mathcal{V}^{\varepsilon,\lambda}(t,y_{2})f(y_{1},y_{2})dy\right]=\mathbb{E}\left[e^{\lambda^{2}\int_{0}^{t}C^{\varepsilon}(Y^{1}(s)-Y^{2}(s))ds}f(Y^{1}(t),Y^{2}(t))\right].$
(2.1)

###### Proof.

As a consequence of (1.8) and the definition of the underlying flow of
kernels, (1.4), we have

$\displaystyle\mathbb{E}\left[\int_{\mathbb{R}^{2}}\mathcal{V}^{\varepsilon,\lambda}(t,y_{1})\mathcal{V}^{\varepsilon,\lambda}(t,y_{2})f(y_{1},y_{2})dy\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\left(\int_{\mathbb{R}}e^{-\frac{\nu}{2}\lambda^{2}(t-s)+\lambda
y}\mathcal{U}^{\varepsilon}(0,t,0,y)f(y-\lambda\nu t)dy\right)^{2}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[e^{-\frac{\nu}{2}\lambda^{2}(t-s)+\lambda(X^{1}(t)+X^{2}(t))}f(X^{1}(t)-\lambda\nu
t)f(X^{2}(t)-\lambda\nu t)\right],$

where the process $(X(t))_{t\geq 0}$ has the generator given in line (1.10).
The stated equality follows by applying Girsanov’s theorem with change of
measure given by the Doléans-Dade exponential of the process $(X^{1}+X^{2})$,
and then defining the process $(Y(t))_{t>0}$ as $Y^{i}(t)=X^{i}(t)-\lambda\nu
t$, for $i=1,2$. ∎

Note that both $Y^{1}$ and $Y^{2}$ are distributed as Brownian motions on
$\mathbb{R}$ with diffusivity $\nu=\sigma^{2}+C^{\varepsilon}(0)$ plus an
additional drift, which is the same for both processes. In fact, as
$\varepsilon\to 0$ the process $Y$ converges in distribution to two-
dimensional Brownian motion, which we will now prove. We will then use this
result to show that $\mathcal{V}$ converges in a weak sense towards the heat
kernel.

######  Proposition 2.2.

Suppose that $Y$ is the diffusion with the generator given in Line (2.1) for
parameters depending on $\varepsilon$, such that
$\lambda(\varepsilon)\mu(\varepsilon)\varepsilon^{\frac{1}{2}}\sigma(\varepsilon)^{-\frac{1}{2}}$
is bounded in $\varepsilon$ (i.e. we are on or below the line in Figure 1) and
$\sigma(\varepsilon)^{2}+C^{\varepsilon}(0)=\nu$ for all $\varepsilon>0$.
Then, $Y$ converges in distribution to a Brownian motion with diffusivity
$\nu$ on $\mathbb{R}^{2}$ as $\varepsilon\to 0$.

###### Proof.

We apply [EK09, Theorem 8.2]; this proves the convergence of finite
dimensional distributions by considering the behaviour of the generator. We
will use the following estimate, which we derive below. There is a constant
$C^{\prime}>0$ depending only on $t>0$ and $\nu$ such that

$\displaystyle\mathbb{E}\left[\int_{0}^{t}C^{\varepsilon}(Y^{1}(s)-Y^{2}(s))ds^{k}\right]^{1/k}\leq
C^{\prime\prime}\lambda(\varepsilon)^{-2}.$ (2.2)

For any $k\in\mathbb{N}$, the occupation times formula gives

$\displaystyle\mathbb{E}\left[\int_{0}^{t}C^{\varepsilon}(Y^{1}(s)-Y^{2}(s))ds^{k}\right]=\mathbb{E}\left[\left(\int_{\mathbb{R}}\frac{C^{\varepsilon}(y)}{\sigma(\varepsilon)^{2}+C^{\varepsilon}(0)-C^{\varepsilon}(y)}\mathcal{L}^{y}_{t}(Y^{1}-Y^{2})dy\right)^{k}\right],$

where $\mathcal{L}$ is the local time. If we apply Jensen’s inequality to the
right hand side, we get

$\displaystyle\mathbb{E}\left[\int_{0}^{t}C^{\varepsilon}(Y^{1}(s)-Y^{2}(s))ds^{k}\right]\leq
m_{\varepsilon}^{k-1}\int_{\mathbb{R}}\frac{C^{\varepsilon}(y)}{\sigma(\varepsilon)^{2}+C^{\varepsilon}(0)-C^{\varepsilon}(y)}\mathbb{E}\left[\mathcal{L}^{y}_{t}(Y^{1}-Y^{2})^{k}\right]dy,$

where
$m_{\varepsilon}=\int_{\mathbb{R}}\frac{C^{\varepsilon}(y)}{\sigma(\varepsilon)^{2}+C^{\varepsilon}(0)-C^{\varepsilon}(y)}dy$.
One can easily show that for each $t>0$, there is a constant $C>0$, depending
only on $\nu$, such that, for any $y\in\mathbb{R}$,
$\mathbb{E}[\mathcal{L}^{y}_{t}(Y^{1}-Y^{2})^{k}]^{1/k}\leq C$. Thus, we have
that for all $\varepsilon>0$

$\displaystyle\mathbb{E}\left[\int_{0}^{t}C^{\varepsilon}(Y^{1}(s)-Y^{2}(s))ds^{k}\right]^{\frac{1}{k}}$
$\displaystyle\leq Cm_{\varepsilon}$ $\displaystyle\leq
C^{\prime}\frac{\varepsilon\mu(\varepsilon)^{2}}{\sigma(\varepsilon)}.$

In the above inequality, $C^{\prime}>0$ is a new constant that depends only on
$t$ and $\nu$; the second inequality is a consequence of the assumption that
$C^{\varepsilon}=\rho_{\varepsilon}*\rho_{\varepsilon}$, where
$\rho_{\varepsilon}=\mu(\varepsilon)^{2}\varepsilon^{-\frac{1}{2}}\rho(\varepsilon^{-1}\cdot)$
for $\rho\in C^{\infty}_{c}(\mathbb{R})$. In particular, this means that the
main contribution to the integral comes from $y$ small enough that
$C^{\varepsilon}(0)-C^{\varepsilon}(y)$ looks quadratic. Our assumption that
$\lambda(\varepsilon)\mu(\varepsilon)\varepsilon^{\frac{1}{2}}\sigma(\varepsilon)^{-\frac{1}{2}}$
implies (2.2).

Estimate (2.2) allows us to apply [EK09, Theorem 8.2] by controlling the
difference between the action of the generator (2.1) and the Laplacian. The
Theorem gives us convergence of the finite dimensional distributions. The
proof is concluded by noting that since, for any $\varepsilon>0$, $Y^{1}$ and
$Y^{2}$ are both marginally Brownian motions with diffusivity $\nu$ the
sequence of processes $(Y)_{\varepsilon>0}$ is tight. ∎

Proposition 1.3, i.e. $\mathcal{V}(t)$ converges to the heat kernel in a weak
sense, is an immediate consequence of the above proposition.

###### Proof of Proposition 1.3.

First note that Proposition 2.1 implies

$\displaystyle\mathbb{E}\left[\int_{\mathbb{R}}\left(\mathcal{V}(t,y)-p_{t}(y)\right)f(y)dy^{2}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[e^{\lambda(\varepsilon)^{2}\int_{0}^{t}C^{\varepsilon}(Y^{1}(s)-Y^{2}(s))ds}f(Y^{1}(t))f(Y^{2}(t))\right]-\int_{\mathbb{R}}p_{t}(y)f(y)dy^{2}.$

Note that (2.2) implies that the exponential term is converging to $1$ in
$L^{p}(\mathbb{P})$, for any $p\in[1,\infty)$. Hence, Proposition 2.2 shows
the above expression vanishes as $\varepsilon\to 0$, and thus proves the
statement. ∎

Moving the discussion to the weak diffusivity regime, we will explain the
predictions we made in the introduction for the coefficients of the limiting
stochastic heat equation. Recall that in the weak diffusivity regime, we
assume that $\sigma(\varepsilon)$ tends to $0$ more slowly than $\varepsilon$,
as $\varepsilon$ tends to $0$. We also assume that
$\frac{\lambda(\varepsilon)^{2}\varepsilon}{\sigma(\varepsilon)}$ converges to
a positive constant $c$. The occupations times formula gives

$\lambda(\varepsilon)^{2}\int_{0}^{t}C^{\varepsilon}(Y^{1}(s)-Y^{2}(s))ds=\\\
\frac{\lambda(\varepsilon)^{2}\varepsilon}{2\sigma(\varepsilon)}\int_{\mathbb{R}}\frac{C(\sigma(\varepsilon)y)}{1+\underbrace{\sigma(\varepsilon)^{-2}(C(0)-C(\sigma(\varepsilon)y))}_{=-\frac{1}{2}C^{\prime\prime}(\xi_{\varepsilon}(y))y^{2}}}\mathcal{L}^{\sigma(\varepsilon)\varepsilon
y}_{t}(Y^{1}-Y^{2})dy,$ (2.3)

where the equality under the brace is due to Taylor’s expansion and
$|\xi_{\varepsilon}(y)|\leq\sigma(\varepsilon)|y|$. Proposition 2.2 shows that
the process $(Y^{1},Y^{2})$ converges in distribution to a Brownian motion
$(B^{1},B^{2})$; if we assume that the local times,
$\mathcal{L}^{y}_{t}(Y^{1}-Y^{2})$, are converging jointly with
$(Y^{1},Y^{2})$ to $\mathcal{L}^{y}_{t}(B^{1}-B^{2})$, then (2.3) suggests the
following convergence should hold as $\varepsilon\to 0$

$\displaystyle\mathbb{E}\left[e^{\lambda(\varepsilon)^{2}\int_{0}^{t}C^{\varepsilon}(Y^{1}(s)-Y^{2}(s))ds}f(Y^{1}(t))f(Y^{2}(t))\right]$
$\displaystyle\rightarrow\quad$
$\displaystyle\mathbb{E}[\exp\left(c\sqrt{2}|C^{\prime\prime}(0)|^{-\frac{1}{2}}C(0)\mathcal{L}^{0}_{t}(B^{1}-B^{2})\right)f(B^{1}(t))f(B^{2}(t))].$

This is the form of the second moment for the stochastic heat equation, with
the coefficients as predicted in the Introduction. This convergence can be
made rigorous, see the thesis [Bro22] for details.

###  2.2 In a Weak Environment

In the weak environment setting, we can make stronger statements about the
limiting behaviour of the $\mathcal{V}$ by looking at the behaviour of the
density of the law of the two-point motion. In fact, we will only need to look
at the difference between the two-point motions, which is a one-dimensional
diffusion. One explanation for the better behaviour in this setting is that
the diffusion coefficient in the generator of this difference is converging to
a constant function pointwise, rather than only pointwise almost everywhere,
as happens in the other two regimes. As a consequence of Proposition 2.1, we
can make the following statement about the process $D:=Y^{1}-Y^{2}$.

######  Proposition 2.3.

The process $D=Y^{1}-Y^{2}$ is a diffusion with generator

$\displaystyle(\sigma^{2}+(C^{\varepsilon}(0)-C^{\varepsilon}(\cdot)))\frac{d^{2}}{dx^{2}}.$
(2.4)

Let $q$ be the transition density of $D$, that is let $\mathbb{P}(D_{t}\in
dy|\ D_{s}=x)=q(t-s;x,y)$. The existence of this density is standard, see for
example [Bor17]. In the following, we collect some necessary estimates on $q$,
which hold uniformly in $\varepsilon>0$. In the rest of this section, we’ll
fix a constant $T>0$ and deal with a finite time window $(0,T)$.

######  Proposition 2.4.

Suppose that $\mu(\varepsilon)$ is a bounded sequence, there are constants
$c,C>0$ such that for any $\varepsilon>0$, $t\in(0,T)$, and $x,y\in\mathbb{R}$

$\displaystyle q(t;x,y)\leq Ct^{-\frac{1}{2}}e^{-c\frac{(x-y)^{2}}{2t}}.$

###### Proof.

We aim to apply the result [Aro67, Theorem 1], which provides a Gaussian
estimate for fundamental solutions to elliptic PDEs. To begin, we will show
that $q$ satisfies a PDE. Denote
$a_{\varepsilon}(y):=(\sigma^{2}+C^{\varepsilon}(0)-C^{\varepsilon}(y))$, and
define $F_{\varepsilon}(y):=\int_{0}^{y}\frac{1}{a_{\varepsilon}(x)}dx$. Note
that for each $\varepsilon>0$, $F_{\varepsilon}$ is a continuous and strictly
increasing function, and so has a well defined inverse. Let $\Delta$ be the
process defined by $\Delta(t):=F_{\varepsilon}(D(t))$, then it is a diffusion
with generator $\frac{d}{dx}\left((a_{\varepsilon}\circ
F_{\varepsilon}^{-1})^{-\frac{1}{2}}\frac{d}{dx}\right)$. If we denote the
transition density of $\Delta$ by $\tilde{q}$, then since $a$ is bounded above
and below uniformly over all $\varepsilon>0$, we can apply the result in
[Aro67, Theorem 1] to get that there are constants $c,C>0$ such that for all
$\varepsilon>0$, $t\in(0,T)$, and $x,y\in\mathbb{R}$

$\displaystyle\tilde{q}(t;x,y)\leq
Ct^{-\frac{1}{2}}e^{-c\frac{(x-y)^{2}}{2t}}.$

However, it is easy to see that
$q(t;x,y)=\tilde{q}(t;F_{\varepsilon}(x),F_{\varepsilon}(y))/a_{\varepsilon}(y)$,
so that we in fact have the estimate

$\displaystyle q(t;x,y)\leq
Ca(y)^{-1}t^{-\frac{1}{2}}e^{-c\frac{(F_{\varepsilon}(x)-F_{\varepsilon}(y))^{2}}{2t}}.$
(2.5)

The claimed estimate then follows from the fact that $0<\sigma^{2}\leq
a_{\varepsilon}(y)\leq\nu$ for all $\varepsilon>0$ and $y\in\mathbb{R}$. In
particular, the claim follows after noting

$\displaystyle|F_{\varepsilon}(y)-F_{\varepsilon}(x)|\geq\nu^{-1}|x-y|\quad\text{and
}|a_{\varepsilon}(y)|^{-1}\leq\sigma^{-2}.$

∎

The above estimate can be used to prove that
$\lambda(\varepsilon)\mathcal{V}*\rho_{\varepsilon}$ converges to a multiple
of the stochastic heat equation in the weak environment setting. In order to
study $\mathcal{V}$ itself, we must look at $q^{\lambda(\varepsilon)}$, which
we define through the following equality, which we impose for any $f\in
C_{b}(\mathbb{R})$

$\displaystyle\mathbb{E}\left[\int_{\mathbb{R}^{2}}\mathcal{V}(t,y_{1})\mathcal{V}(t,y_{2})f(y_{1}-y_{2})dy\right]=\int_{\mathbb{R}}q^{\lambda(\varepsilon)}(t,y)f(y)dy.$
(2.6)

We can also rewrite the expectation on the left-hand side in terms of the
difference of the two-point motion, $D=Y^{1}-Y^{2}$. This yields the equality

$\displaystyle\mathbb{E}\left[e^{\lambda(\varepsilon)^{2}\int_{0}^{t}C^{\varepsilon}(D(s))ds}f(D(t))\right]=\int_{\mathbb{R}}q^{\lambda(\varepsilon)}(t,y)f(y)dy.$

From which it is easily deduced that $q^{0}(t,y)=q(t;0,y)$. Suppose that $f\in
C^{2}_{b}(\mathbb{R})$, we can apply Itô’s formula, which shows us that
$q^{\lambda(\varepsilon)}$ satisfies the following PDE

$\displaystyle\partial_{t}q^{\lambda(\varepsilon)}=\partial_{y}^{2}\left(a_{\varepsilon}\cdot
q^{\lambda(\varepsilon)}\right)+\lambda(\varepsilon)^{2}C^{\varepsilon}q^{\lambda(\varepsilon)},\
q^{\lambda(\varepsilon)}(0,y)=\delta_{0}(y).$ (2.7)

As before, the function $a_{\varepsilon}:\mathbb{R}\to\mathbb{R}$ is defined
by

$\displaystyle
a_{\varepsilon}(y)=\sigma^{2}+C^{\varepsilon}(0)-C^{\varepsilon}(y).$

To simplify some notation we will introduce notation for the mass of
$\lambda(\varepsilon)\rho_{\varepsilon}$, which in most of the cases we
consider we will assume to be convergent as $\varepsilon$ tends to $0$.

$\displaystyle\kappa_{\varepsilon}:=\lambda(\varepsilon)\int_{\mathbb{R}}\rho_{\varepsilon}(y)dy.$

We can estimate the difference between $\mathcal{V}$ and its smoothing
$\kappa_{\varepsilon}^{-1}\lambda(\varepsilon)\mathcal{V}*\rho_{\varepsilon}$
in terms of $q^{\lambda(\varepsilon)}$ as follows.

$\displaystyle\mathbb{E}$
$\displaystyle\left[\|\mathcal{V}(t)-\kappa_{\varepsilon}^{-1}\lambda(\varepsilon)\mathcal{V}(t)*\rho_{\varepsilon}\|_{2}^{2}\right]$
$\displaystyle=$ $\displaystyle
q^{\lambda(\varepsilon)}(t,0)-2\kappa_{\varepsilon}^{-1}\lambda(\varepsilon)\int_{\mathbb{R}}q^{\lambda(\varepsilon)}(t,y)\rho_{\varepsilon}(y)dy+\kappa_{\varepsilon}^{-2}\lambda(\varepsilon)^{2}\int_{\mathbb{R}}q^{\lambda(\varepsilon)}(t,y)\rho_{\varepsilon}*\rho_{\varepsilon}(y)dy$
$\displaystyle=$ $\displaystyle
2\int_{\mathbb{R}}(q^{\lambda(\varepsilon)}(t,0)-q^{\lambda(\varepsilon)}(t,\varepsilon
y))\rho(y)dy+\int_{\mathbb{R}}(q^{\lambda(\varepsilon)}(t,\varepsilon
y)-q^{\lambda(\varepsilon)}(t,0))\rho*\rho(y)dy.$ (2.8)

Thus, we just need equicontinuity of $q^{\lambda(\varepsilon)}(t,\cdot)$, as a
sequence of functions depending on $\varepsilon$, to get the desired upgrade
for the convergence. We will begin proving this by considering the
${\lambda(\varepsilon)}=0$ case.

######  Proposition 2.5.

Suppose that $\mu(\varepsilon)^{2}$ converges to $0$ as $\varepsilon$ tends to
$0$, then for any $\delta>0$ the sequence of functions formed by $q$ is
equicontinuous on $[\delta^{2},\infty)\times\mathbb{R}\times\mathbb{R}$.

###### Proof.

From inequality (II.1.10) proved in [Str88, Theorem II.3.1] and the fact that
$\sigma^{2}\leq a_{\varepsilon}(y)\leq\nu$, we get that $\tilde{q}$ forms an
equicontinuous sequence of functions on
$[\delta^{2},\infty)\times\mathbb{R}\times\mathbb{R}$.
$(F_{\varepsilon})_{\varepsilon>0}$ is equicontinuous because of the upper and
lower bounds on $a_{\varepsilon}$, whereas $(a_{\varepsilon})_{\varepsilon>0}$
is equicontinous because of our assumption that $\mu(\varepsilon)$ is null.
The lower bounds on $a_{\varepsilon}$ then imply that
$(a_{\varepsilon}^{-1})_{\varepsilon>0}$ is equicontinuous. Thus, the equality
$q(t;x,y)=\tilde{q}(t;x,F_{\varepsilon}(y))/a_{\varepsilon}(y)$ shows that $q$
is also equicontinuous. ∎

We will use this result to prove the equicontinuity for
${\lambda(\varepsilon)}\neq 0$, but first we need a pointwise estimate on
$q^{\lambda(\varepsilon)}$.

######  Proposition 2.6.

Suppose that $\mu(\varepsilon)$ and $\kappa_{\varepsilon}$ are bounded for
$\varepsilon>0$, then there is a constant $C>0$ such that for all $t\in(0,T)$,
$y\in\mathbb{R}$ and $\varepsilon>0$

$\displaystyle\|q^{\lambda(\varepsilon)}(t)\|_{\infty}\leq
Ct^{-\frac{1}{2}}e^{Ct}.$

###### Proof.

In order to get the desired estimate, we use Duhamel’s principle to get an
integral equation for $q^{\kappa}$ in terms of $q$. In particular, Duhamel’s
principle tells us that

$\displaystyle
q^{\lambda(\varepsilon)}(t,y)=q(t;0,y)+\lambda(\varepsilon)^{2}\int_{0}^{t}\int_{\mathbb{R}}q(t-s;x,y)C^{\varepsilon}(x)q^{\lambda(\varepsilon)}(s,x)dxds.$
(2.9)

One can easily check, assuming all the involved functions are sufficiently
nice, that the solution to the above integral equation is also a solution to
(2.7). One can also verify the above equation by applying Itô’s formula in the
definition of $q^{\lambda(\varepsilon)}$ with a suitable test function, and
then taking a limit. We will use this equality to derive an inequality to
which we can apply Gronwall’s inequality.

Applying (2.4), we get that

$\displaystyle|q^{\lambda(\varepsilon)}(t,y)|\leq
Ct^{-\frac{1}{2}}e^{-c\frac{y^{2}}{2t}}+\lambda(\varepsilon)^{2}\int_{0}^{t}\int_{\mathbb{R}}C(t-s)^{-\frac{1}{2}}e^{-c\frac{(x-y)^{2}}{2(t-s)}}C^{\varepsilon}(x)q^{\lambda(\varepsilon)}(s,x)dxds.$
(2.10)

We can rewrite this in terms of the normalised version of $\rho_{\varepsilon}$
to get

$\displaystyle\|q^{\lambda(\varepsilon)}(t)\|_{\infty}$ $\displaystyle\leq
Ct^{-\frac{1}{2}}+C\int_{0}^{t}(t-s)^{-\frac{1}{2}}\|q^{\lambda(\varepsilon)}(s)\|_{\infty}ds.$

The inequality follows by estimating $q^{\lambda(\varepsilon)}$ by its
supremum and noting that
$\lambda(\varepsilon)^{2}\int_{\mathbb{R}}C^{\varepsilon}(x)dx=\kappa_{\varepsilon}^{2}$,
which is bounded in $\varepsilon$. Iterating this inequality, and allowing the
constants to change between lines, we get

$\displaystyle\|q^{\lambda(\varepsilon)}(t)\|_{\infty}$ $\displaystyle\leq
Ct^{-\frac{1}{2}}+C+C\int_{0}^{t}\int_{0}^{s}(t-s)^{-\frac{1}{2}}(s-u)^{-\frac{1}{2}}\|q^{\lambda(\varepsilon)}(u)\|_{\infty}dsdu$
$\displaystyle=Ct^{-\frac{1}{2}}+C+C\int_{0}^{t}\|q^{\lambda(\varepsilon)}(u)\|_{\infty}du.$

The proof is concluded by applying Gronwall’s inequality and making some
additional simplifications. To check we can apply the inequality we just need
to show that the integral on the right hand side is always finite. Note that
an $L^{1}$ bound on $q^{\lambda(\varepsilon)}$ provides an $L^{\infty}$
estimate through Equation (2.10), but an $L^{1}$ bound is easily provided by
setting $f(y)=1$ in Equation (2.6). ∎

######  Proposition 2.7.

Suppose that $\mu(\varepsilon)^{2}$ converges to $0$ as $\varepsilon$ tends to
$0$, and that $\kappa_{\varepsilon}$ is bounded, then, for each $t\in(0,T)$,
the sequence $(q^{\lambda(\varepsilon)}(t,\cdot))_{n\in\mathbb{N}}$ is
equicontinuous.

###### Proof.

We will use Equation (2.9) once more. Fix $t>0$ and let $\delta<t$ be
arbitrary, then for any $\varepsilon>0$, Proposition 2.5 tells us that there
exists a $\gamma_{\delta}>0$ such that for all $|x-y|<\gamma_{\delta}$ we have

$\displaystyle|q^{\kappa}(t,x)-q^{\kappa}(t,y)|$
$\displaystyle\leq\varepsilon+\lambda(\varepsilon)^{2}\int_{0}^{t-\delta^{2}}\int_{\mathbb{R}}\varepsilon
C^{\varepsilon}(z)q^{\lambda(\varepsilon)}(s,z)dzds$ (2.11) $\displaystyle\
+\lambda(\varepsilon)^{2}\int_{t-\delta^{2}}^{t}\int_{\mathbb{R}}(q(t-s;z,x)-q(t-s;z,y))C^{\varepsilon}(z)q^{\lambda(\varepsilon)}(s,z)dzds.$
(2.12)

If we apply Propositions 2.6 and 2.4, then we see that this is bounded above
by

$\displaystyle\varepsilon+\left(\varepsilon
Ct^{\frac{1}{2}}e^{Ct}+C^{2}e^{2Ct}\int_{t-\delta^{2}}^{t}(t-s)^{-\frac{1}{2}}s^{-\frac{1}{2}}ds\right)\underbrace{\lambda(\varepsilon)^{2}\int_{\mathbb{R}}C^{\varepsilon}(z)dz}_{=\kappa_{\varepsilon}^{2}}.$

The statement follows by noting that we are allowed to choose $\delta$ to be
sufficiently small. ∎

We conclude this section with a proof of Proposition 1.4, which follows
directly from the convergence of $q^{\lambda(\varepsilon)}$.

###### Proof of Proposition 1.4.

First note that

$\displaystyle\mathbb{E}\left[\|\mathcal{V}(t)-p_{t}\|_{2}^{2}\right]=q^{\lambda(\varepsilon)}(t,0)-p_{2t}(0).$

Hence, we only need to prove the convergence of
$q^{\lambda(\varepsilon)}(t)(0)$ towards $p_{2t}(0)$. We already have
equicontinuity and boundedness of $q^{\lambda(\varepsilon)}(t)$ from
propositions 2.7 and 2.10; thus, the Arzela-Ascoli theorem implies that the
sequence $q^{\lambda(\varepsilon)}$ is relatively compact in
$L^{\infty}([-K,K])$, where $K>0$ is arbitrarily chosen. From Equation (2.9),
we get the inequality

$\displaystyle\|q^{\lambda(\varepsilon)}(t)-q(t;0,\cdot)\|_{\infty}\leq\kappa_{\varepsilon}^{2}\int_{0}^{t}\sup_{(x,y)\in\mathbb{R}^{2}}|q(t-s;x,y)|\|q^{\lambda(\varepsilon)}(s)\|_{\infty}ds.$

Applying Propositions 2.4 and 2.10, and using that $\kappa_{\varepsilon}$ is a
null sequence, we see that the right hand side vanishes. Finally, we note that
propositions 2.4 and 2.5 prove that, for each $T>\delta>0$, $q$ is relatively
compact as a sequence in $\varepsilon$ on
$L^{\infty}([\delta^{2},T]\times[-K,K]\times[-K,K])$. Due to Proposition 2.2,
the process $D$ converges in distribution to a Brownian motion with
diffusivity $2\nu$, combining this with the above relative compactness proves
uniform convergence on compact sets of $q(t)$ towards $p_{2t}$. ∎

##  3 The Mild Equation

In the following, we will use the notation for $f,g\in L^{2}(\mathbb{R})$

$\displaystyle(f,g)_{2}=\int_{\mathbb{R}}f(y)g(y)dy,\quad\|f\|_{2}=(f,f)_{2}^{\frac{1}{2}},$
$\displaystyle(f,g)_{2,\rho_{\varepsilon}}=\int_{\mathbb{R}}(f*\rho_{\varepsilon})(y)(g*\rho_{\varepsilon})(y)dy,\quad\|f\|_{2,\rho_{\varepsilon}}=(f,f)_{2,\rho_{\varepsilon}}^{\frac{1}{2}}.$
(3.1)

We will also use the notation below for stochastic integrals with respect to
$W^{\varepsilon}$.

$\displaystyle\int_{0}^{t}(f(s),W^{\varepsilon}(ds))_{2}$
$\displaystyle:=\int_{0}^{t}\int_{\mathbb{R}}f(s,y)W^{\varepsilon}(ds,y)dy$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}}f(s,\cdot)*\rho_{\varepsilon}(y)W(ds,dy).$
(3.2)

The second inequality is a consequence of the coupling between the fields
$W^{\varepsilon}$ that we chose in the setup of Theorem 1.1. Note that we have
the isometry

$\displaystyle\mathbb{E}\left[\left(\int_{0}^{t}(f(s),W^{\varepsilon}(ds))_{2}\right)^{2}\right]=\int_{0}^{t}\|f(s)\|_{2,\rho_{\varepsilon}}^{2}ds.$
(3.3)

It is important to note here that $\rho_{\varepsilon}$ is not normalised to
have unit mass, so that for a fixed function $f\in L^{2}(\mathbb{R})$,
$\|f\|_{2,\rho_{\varepsilon}}$ converges to $0$ as $\varepsilon$ tends to $0$.
However, under the choice of $\lambda(\varepsilon)$ made in Theorem 1.1, the
function $\lambda(\varepsilon)\rho_{\varepsilon}$ has mass
$\kappa_{\varepsilon}$, which is converging to a positive constant. We can
derive the SPDE for $\mathcal{V}$, (1.9), directly from the SPDE for
$\mathcal{U}$.

######  Lemma 3.1.

With $\mathcal{V}$ and $\mathcal{U}^{\varepsilon}$ defined as in Section 1.1,
we have that for each $f\in C^{\infty}_{c}(\mathbb{R})$

$\displaystyle\int_{\mathbb{R}}\mathcal{V}(t,y)f(y)dy-f(x)$ (3.4)
$\displaystyle=$
$\displaystyle\frac{\nu}{2}\int_{0}^{t}\int_{\mathbb{R}}\mathcal{V}(s,y)f^{\prime\prime}(y)dyds+\int_{0}^{t}\int_{\mathbb{R}}\mathcal{V}(s,y)\left(\lambda
f(y)+f^{\prime}(y)\right)W^{\varepsilon}(ds,y)dy.$ (3.5)

###### Proof.

Let $f\in C^{\infty}_{c}(\mathbb{R})$ and define
$g(t,y):=e^{-\frac{\nu}{2}\lambda^{2}t+\lambda(y-x)}f(y-\lambda\nu t)$. By
definition, we have
$\int_{\mathbb{R}}\mathcal{V}(t,x,y)f(y)dy=\int_{\mathbb{R}}\mathcal{U}(0,t,x,y)g(t,y)dy$.
Thus, a straightforward calculation using Equation (1.1) and the stochastic
and deterministic Fubini theorems leads to the equality

$\displaystyle\int_{0}^{t}\int_{\mathbb{R}}\partial_{s}g(s,y)\mathcal{U}^{\varepsilon}(0,s,x,y)dyds+f(x)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}}\mathcal{V}(t,y)f(y)dy-\frac{\nu}{2}\int_{0}^{t}\int_{\mathbb{R}}\partial_{y}^{2}g(s,y)\mathcal{U}^{\varepsilon}(0,s,x,y)dyds$
$\displaystyle-\int_{0}^{t}\int_{\mathbb{R}}e^{-\frac{\nu}{2}\lambda^{2}s+\lambda(y-x)}(\lambda
f(y-\lambda\nu s)+f^{\prime}(y-\lambda\nu
s))\mathcal{U}^{\varepsilon}(0,s,x,y)W^{\varepsilon}(ds,y)dy.$

Noticing that
$\partial_{t}g(t,y)=-\frac{\nu}{2}\partial_{y}^{2}g(t,y)-\frac{\nu}{2}e^{-\frac{\nu}{2}\lambda^{2}t+\lambda(y-x)}f^{\prime\prime}(y-\lambda\nu
t)$, we get the desired equation with
$\mathcal{W}^{\varepsilon}(ds,y+\lambda\nu t)$ in place of
$\mathcal{W}^{\varepsilon}(ds,y)$. As discussed in Section 1.1, we replace
$W^{\varepsilon}$ by its shifted version, which has the same distribution, to
get the result as stated.

∎

From Equation (1.9), one can derive the following mild equation for
$\mathcal{V}$. For any $f\in C^{\infty}_{c}(\mathbb{R})$

$\displaystyle\left(\mathcal{V}(t),f\right)=P^{\nu}_{t}f(x)+\int_{0}^{t}\left(\mathcal{V}(s)\cdot
P_{t-s}^{\nu}(\lambda(\varepsilon)f+f^{\prime}),W^{\varepsilon}(ds)\right)_{2}.$
(3.6)

The solution to the stochastic heat equation,

$\displaystyle\partial_{t}\mathcal{Z}_{\varepsilon}=\frac{\nu}{2}\partial^{2}_{y}\mathcal{Z}_{\varepsilon}+\lambda(\varepsilon)\mathcal{Z}_{\varepsilon}\dot{W}^{\varepsilon},$

satisfies its own mild equation. For any $f\in C^{\infty}_{c}(\mathbb{R})$

$\displaystyle\left(\mathcal{Z}_{\varepsilon}(t),f\right)=P^{\nu}_{t}f(x)+\lambda(\varepsilon)\int_{0}^{t}\left(\mathcal{Z}_{\varepsilon}(s)\cdot
P_{t-s}^{\nu}f,W^{\varepsilon}(ds)\right)_{2}.$

We will use these two mild equations to prove Theorem 1.1.

###  3.1 Proof of Theorem 1.1

The proof of Theorem 1.1 relies on a Gronwall argument that will yield the
following inequality.

######  Proposition 3.2.

There are constants $c,C>0$ such that for every $\varepsilon\in(0,1)$ and
$t>0$

$\displaystyle\mathbb{E}\left[\|\mathcal{V}(t)-\mathcal{Z}_{\varepsilon}(t)\|_{2,\rho_{\varepsilon}}^{2}\right]$
$\displaystyle\leq$ $\displaystyle
C\lambda(\varepsilon)^{-2}t^{-\frac{1}{2}}e^{ct}\left(\varepsilon\log(\varepsilon^{-1})+\mu(\varepsilon)^{2}\log(\varepsilon^{-1})\right).$
(3.7)

Before the proof of this statement, we will first derive an estimate to which
we can apply Gronwall’s inequality, and then collect some lemmas to get an
estimate of the form in the proposition.

Also note that the estimate in Proposition 3.2 shows that the
$\|\cdot\|_{\rho_{\varepsilon}}$ norm tends to $0$ faster than
$\lambda(\varepsilon)^{2}$. as $\varepsilon\to 0$. This is necessary for the
estimate to be meaningful because of the fact that the mass of
$\rho_{\varepsilon}$ is proportional to $\lambda(\varepsilon)^{-1}$.

######  Lemma 3.3.

There is a constant $C>0$ such that for all $\varepsilon>0$ and $\eta\in(0,1)$

$\displaystyle\mathbb{E}\left[\left\|\mathcal{V}(t)-\mathcal{Z}_{\varepsilon}(t)\right\|^{2}_{2,\rho_{\varepsilon}}\right]\leq$
$\displaystyle
C\mathbb{E}\left[\int_{0}^{t}(t-s)^{-\frac{1}{2}}\|\mathcal{V}(s)-\mathcal{Z}_{\varepsilon}(s)\|_{2,\rho_{\varepsilon}}^{2}ds\right]$
$\displaystyle+C\lambda(\varepsilon)^{-2}\varepsilon^{1-\eta}\mathbb{E}\left[\int_{0}^{t}(t-s)^{\frac{1}{2}\eta-1}\|\mathcal{V}(s)-\mathcal{Z}_{\varepsilon}(s)\|_{2}^{2}ds\right]$
$\displaystyle+\mathbb{E}\left[\int_{0}^{t}\int_{\mathbb{R}}\left\|\mathcal{V}(s)P^{\nu}_{t-s}(\rho_{\varepsilon}^{\prime}(\cdot-x))\right\|_{2,\rho_{\varepsilon}}^{2}dxds\right].$
(3.8)

###### Proof.

From the previous section, for the difference between
$\mathcal{Z}_{\varepsilon}$ and $\mathcal{V}$, we have

$\displaystyle\left(\mathcal{V}(t)-\mathcal{Z}_{\varepsilon}(t),f\right)_{2}=$
$\displaystyle\lambda(\varepsilon)\int_{0}^{t}\left(\left(\mathcal{V}(s)-\mathcal{Z}_{\varepsilon}(s)\right)\cdot
P_{t-s}^{\nu}f,W^{\varepsilon}(ds)\right)_{2}$
$\displaystyle+\int_{0}^{t}\left(\mathcal{V}(s)\cdot
P_{t-s}^{\nu}f^{\prime},W^{\varepsilon}(ds)\right)_{2}.$

Thus, we have

$\displaystyle\mathbb{E}\left[\left(\mathcal{V}(t)-\mathcal{Z}_{\varepsilon}(t),f\right)^{2}\right]$
(3.9) $\displaystyle\leq$
$\displaystyle\lambda(\varepsilon)^{2}\mathbb{E}\left[\int_{0}^{t}\left\|(\mathcal{V}(s)-\mathcal{Z}_{\varepsilon}(s))P^{\nu}_{t-s}f\right\|_{2,\rho_{\varepsilon}}^{2}ds\right]+\mathbb{E}\left[\int_{0}^{t}\left\|\mathcal{V}(s)P^{\nu}_{t-s}f^{\prime}\right\|_{2,\rho_{\varepsilon}}^{2}ds\right].$
(3.10)

This is not yet in a suitable form for Gronwall’s inequality. However, if we
choose $f=\rho_{\varepsilon}(\cdot-x)=:\rho_{\varepsilon,x}$, and then
integrate out the $x$ variable, inequality (3.9) becomes

$\displaystyle\mathbb{E}\left[\left\|\mathcal{V}(t)-\mathcal{Z}_{\varepsilon}(t)\right\|^{2}_{2,\rho_{\varepsilon}}dx\right]$
(3.11) $\displaystyle\leq$
$\displaystyle\lambda(\varepsilon)^{2}\mathbb{E}\left[\int_{0}^{t}\int_{\mathbb{R}}\left\|(\mathcal{V}(s)-\mathcal{Z}_{\varepsilon}(s))P^{\nu}_{t-s}(\rho_{\varepsilon,x})\right\|_{2,\rho_{\varepsilon}}^{2}dxds\right]$
$\displaystyle+\mathbb{E}\left[\int_{0}^{t}\int_{\mathbb{R}}\left\|\mathcal{V}(s)P^{\nu}_{t-s}(\rho_{\varepsilon,x}^{\prime})\right\|_{2,\rho_{\varepsilon}}^{2}dxds\right].$

We will show that the first term after the inequality can be bounded above by

$\displaystyle
C\mathbb{E}\left[\int_{0}^{t}(t-s)^{-\frac{1}{2}}\|\mathcal{V}(s)-\mathcal{Z}_{\varepsilon}(s)\|_{2,\rho_{\varepsilon}}^{2}ds\right]$
$\displaystyle+C\lambda(\varepsilon)^{-2}\varepsilon^{1-\eta}\mathbb{E}\left[\int_{0}^{t}(t-s)^{\frac{1}{2}\eta-1}\|\mathcal{V}(s)-\mathcal{Z}_{\varepsilon}(s)\|_{2}^{2}ds\right].$

We will also show that the second term after the inequality in (3.11) tends to
$0$ as $\varepsilon\to 0$. This, together with bounds on the $L^{2}$ norms of
$\mathcal{V}$ and $\mathcal{Z}$, will allow us to apply Gronwall’s inequality
to (3.11) and prove Proposition 3.2. Recall that
$C^{\varepsilon}:=\rho_{\varepsilon}*\rho_{\varepsilon}$, so that we may
rewrite as follows

$\displaystyle\int_{\mathbb{R}}\left\|(\mathcal{V}(s)-\mathcal{Z}_{\varepsilon}(s))P^{\nu}_{t-s}(\rho_{\varepsilon}(\cdot-x))\right\|_{2,\rho_{\varepsilon}}^{2}dx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}p^{\nu}_{t-s}*\rho_{\varepsilon}(x-y_{1})p^{\nu}_{t-s}*\rho_{\varepsilon}(x-y_{2})dx\left(\mathcal{V}(s)-\mathcal{Z}_{\varepsilon}(s)\right)^{\otimes
2}(y_{1},y_{2})C^{\varepsilon}(y_{1}-y_{2})dy$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{2}}(p^{\nu}_{t-s}*\rho_{\varepsilon})*(p^{\nu}_{t-s}*\rho_{\varepsilon})(y_{1}-y_{2})\left(\mathcal{V}(s)-\mathcal{Z}_{\varepsilon}(s)\right)^{\otimes
2}(y_{1},y_{2})C^{\varepsilon}(y_{1}-y_{2})dy.$

By rearranging the convolutions and using the convolution property of the heat
kernel, we get that the above expression is equal to

$\displaystyle\int_{\mathbb{R}}\left(\mathcal{V}(s)-\mathcal{Z}_{\varepsilon}(s)\right)*\left((p^{\nu}_{2(t-s)}*C^{\varepsilon})\cdot
C^{\varepsilon}\right)(y)\left(\mathcal{V}(s,y)-\mathcal{Z}_{\varepsilon}(s,y)\right)dy$
$\displaystyle=$
$\displaystyle\left(\left(\mathcal{V}(s)-\mathcal{Z}_{\varepsilon}(s)\right)*\left((p^{\nu}_{2(t-s)}*C^{\varepsilon})\cdot
C^{\varepsilon}\right),\mathcal{V}(s)-\mathcal{Z}_{\varepsilon}(s)\right)_{2}.$
(3.12)

Young’s inequality gives us the estimate

$\displaystyle\left(\left(\mathcal{V}(s)-\mathcal{Z}_{\varepsilon}(s)\right)*\left((p^{\nu}_{2(t-s)}*C^{\varepsilon}-p^{\nu}_{2(t-s)}*C^{\varepsilon}(0))\cdot
C^{\varepsilon}\right),\mathcal{V}(s)-\mathcal{Z}_{\varepsilon}(s)\right)_{2}$
$\displaystyle\leq$
$\displaystyle\|\mathcal{V}(s)-\mathcal{Z}_{\varepsilon}(s)\|_{2}^{2}\|(p^{\nu}_{2(t-s)}*C^{\varepsilon}-p^{\nu}_{2(t-s)}*C^{\varepsilon}(0))\cdot
C^{\varepsilon}\|_{1}.$

For each $\eta\in[0,1]$ there is a $C>0$ such that we have the following
inequality

$\displaystyle\|(p^{\nu}_{2(t-s)}*C^{\varepsilon}-p^{\nu}_{2(t-s)}*C^{\varepsilon}(0))\cdot
C^{\varepsilon}\|_{1}$ $\displaystyle\leq$
$\displaystyle(2\pi\nu)^{-\frac{1}{2}}\lambda(\varepsilon)^{-2}(t-s)^{\frac{1}{2}\eta-1}\int_{\mathbb{R}}|y|^{1-\eta}C^{\varepsilon}(y)dy$
$\displaystyle\leq$ $\displaystyle
C\varepsilon^{1-\eta}\lambda(\varepsilon)^{-4}(t-s)^{\frac{1}{2}\eta-1}.$

The last inequality is a consequence of the assumption that
$C^{\varepsilon}=\rho_{\varepsilon}*\rho_{\varepsilon}$ for
$\rho_{\varepsilon}=\mu(\varepsilon)^{2}\varepsilon^{-\frac{1}{2}}\rho(\varepsilon^{-1}\cdot)$,
where $\rho\in C^{\infty}_{c}(\mathbb{R})$, and the assumption that
$\lambda(\varepsilon)^{2}\mu(\varepsilon)^{2}\varepsilon$ is bounded in
$\varepsilon$. Another consequence is that
$(p^{\nu}_{2(t-s)}*C^{\varepsilon})(0)\leq
C^{\prime}\lambda(\varepsilon)^{-2}(t-s)^{-\frac{1}{2}}$ for some constant
$C^{\prime}>0$. Hence, line (3.12) is bounded above by

$\displaystyle
C^{\prime}\lambda(\varepsilon)^{-2}(t-s)^{-\frac{1}{2}}\|\mathcal{V}(s)-\mathcal{Z}_{\varepsilon}(s)\|_{2,\rho_{\varepsilon}}^{2}+C\varepsilon^{1-\eta}\lambda(\varepsilon)^{-4}(t-s)^{\frac{1}{2}\eta-1}\|\mathcal{V}(s)-\mathcal{Z}_{\varepsilon}(s)\|_{2}^{2}.$

Applying this to (3.11) yields the desired estimate. ∎

So that we just need a good estimate on the second and third lines of (3.8).
Then, we can apply Gronwall’s inequality and get a meaningful estimate. Note
the that the second line of (3.8) is bounded above by

$\displaystyle
2C\lambda(\varepsilon)^{2}\varepsilon^{1-\eta}\int_{0}^{t}(t-s)^{\frac{1}{2}\eta-1}\mathbb{E}\left[\int_{\mathbb{R}}\mathcal{V}(s,y)^{2}+\mathcal{Z}_{\varepsilon}(s,y)^{2}dy\right]ds.$
(3.13)

In the following lemma, we provide estimates on the $L^{2}$ norms of
$\mathcal{V}$ and $\mathcal{Z}_{\varepsilon}$.

######  Lemma 3.4.

There are constants $c,C>0$ such that for all $t>0$

$\displaystyle\mathbb{E}\left[\int_{\mathbb{R}}\mathcal{V}(t,y)^{2}dy\right]$
$\displaystyle\leq Ct^{-\frac{1}{2}}e^{ct};$
$\displaystyle\mathbb{E}\left[\int_{\mathbb{R}}\mathcal{Z}_{\varepsilon}(t,y)^{2}dy\right]$
$\displaystyle\leq Ct^{-\frac{1}{2}}e^{ct}.$

###### Proof.

For the second inequality we can square and integrate the mild equation for
$\mathcal{Z}_{\varepsilon}$ to get

$\displaystyle\mathbb{E}\left[\int_{\mathbb{R}}\mathcal{Z}_{\varepsilon}(t,y)^{2}dy\right]$
$\displaystyle\leq$ $\displaystyle
C(t-s)^{-\frac{1}{2}}+\lambda(\varepsilon)^{2}\int_{0}^{t}\int_{\mathbb{R}}\int_{\mathbb{R}^{2}}p_{t-s}(x-y_{1})p_{t-s}(x-y_{1})\mathcal{Z}_{\varepsilon}(s,y_{1})\mathcal{Z}_{\varepsilon}(s,y_{2})C^{\varepsilon}(y_{1}-y_{2})dydxds$
$\displaystyle\leq$ $\displaystyle
C(t-s)^{-\frac{1}{2}}+\kappa_{\varepsilon}^{2}\int_{0}^{t}\int_{\mathbb{R}}\int_{\mathbb{R}}p_{t-s}(x-y)^{2}\mathcal{Z}_{\varepsilon}(s,y)^{2}dydxds$
$\displaystyle\leq$ $\displaystyle
C(t-s)^{-\frac{1}{2}}+C\int_{0}^{t}\int_{\mathbb{R}}(t-s)^{-\frac{1}{2}}\mathcal{Z}_{\varepsilon}(s,y)^{2}dyds.$

The desired inequality follows by iterating the inequality derived above, and
then applying Gronwall’s inequality.

For the first inequality, we instead estimate in terms of the two point
motions through the following equality.

$\displaystyle\mathbb{E}\left[\int_{\mathbb{R}}\mathcal{V}(t,y)^{2}dy\right]=q^{\lambda(\varepsilon)}(t,0),$

where $q^{\varepsilon,\lambda(\varepsilon)}$ is as defined in (2.6). The
required estimate follows from Proposition 2.6. ∎

This result allows us to estimate the second line in (3.8), we now move on to
estimating the third line.

######  Lemma 3.5.

There is a constant $C>0$ such that for every $n\in\mathbb{N}$ with $n>1$ and
$t>0$

$\displaystyle\mathbb{E}\left[\int_{0}^{t}\int_{\mathbb{R}}\left\|\mathcal{V}(s)P_{t-s}(\rho_{\varepsilon}^{\prime}(\cdot-x))\right\|_{2,\rho_{\varepsilon}}^{2}dxds\right]\leq
C\lambda(\varepsilon)^{-2}t^{\frac{1}{2\log(\varepsilon^{-1})}-\frac{1}{2}}e^{ct}\mu(\varepsilon)^{2}\log(\varepsilon^{-1}).$

###### Proof.

Some simple rearrangements using that
$C^{\varepsilon}=\rho_{\varepsilon}*\rho_{\varepsilon}$ give the equality

$\displaystyle\int_{\mathbb{R}}\left\|\mathcal{V}(s)P_{t-s}(\rho_{\varepsilon}^{\prime}(\cdot-x))\right\|_{2,\rho_{\varepsilon}}^{2}dx$
$\displaystyle=$ $\displaystyle\int_{\mathbb{R}^{2}}(\mathcal{V}(s))^{\otimes
2}(y)\int_{\mathbb{R}^{2}}p^{\otimes
2}_{t-s}(y-z)(C^{\varepsilon})^{\prime\prime}(z_{1}-z_{2})dzC^{\varepsilon}(y_{1}-y_{2})dy.$

We will apply a combination of the following two bounds.

$\displaystyle\int_{\mathbb{R}^{2}}p^{\otimes
2}_{t-s}(y-z)(C^{\varepsilon})^{\prime\prime}(z_{1}-z_{2})dz$
$\displaystyle\leq$
$\displaystyle\begin{cases}C\mu(\varepsilon)^{2}\varepsilon^{-2}\\\
C\mu(\varepsilon)^{2}(t-s)^{-1}.\end{cases}$

Thus, we see that the right hand side in the lemma is bounded, for any
$\gamma\in[0,1)$, by

$\displaystyle
C\mathbb{E}\left[\int_{0}^{t}\int_{\mathbb{R}^{2}}\mu(\varepsilon)^{2}\varepsilon^{-2\gamma}(t-s)^{\gamma-1}\mathcal{V}(s,y_{1})\mathcal{V}(s,y_{2})C^{\varepsilon}(y_{1}-y_{2})dyds\right].$

Which, by the Cauchy-Schwartz inequality, is bounded above by

$\displaystyle
C\mu(\varepsilon)^{2}\varepsilon^{-2\gamma}\underbrace{\int_{\mathbb{R}}C^{\varepsilon}(y)dy}_{=\kappa_{\varepsilon}^{2}\lambda(\varepsilon)^{-2}}\int_{0}^{t}(t-s)^{\gamma-1}\mathbb{E}\left[\|\mathcal{V}(s)\|_{2}^{2}\right]ds.$
(3.14)

We recall that we assumed $\kappa_{\varepsilon}$ is bounded, so that we can
apply Lemma 3.4 to get the following upper bound on (3.14), for a new constant
$C>0$.

$\displaystyle
C\mu(\varepsilon)^{2}\varepsilon^{-2\gamma}\lambda(\varepsilon)^{-2}\int_{0}^{t}(t-s)^{\gamma-1}s^{-\frac{1}{2}}e^{cs}ds$
$\displaystyle\leq
Ct^{\gamma-\frac{1}{2}}e^{ct}\mu(\varepsilon)^{2}\varepsilon^{-2\gamma}\lambda(\varepsilon)^{-2}B(\gamma,\tfrac{1}{2}),$
(3.15)

where $B$ is the Beta function. Note that
$B(\gamma,\frac{1}{2})=\frac{\Gamma(\gamma)\Gamma(\frac{1}{2})}{\Gamma(\gamma+\frac{1}{2})}$,
which is bounded above for all $\gamma\in(0,1]$ by $C\gamma^{-1}$, for some
constant $C>0$. Thus, Line (3.15) is bounded above by

$\displaystyle
Ct^{\gamma-\frac{1}{2}}e^{ct}\gamma^{-1}\mu(\varepsilon)^{2}\varepsilon^{-2\gamma}\lambda(\varepsilon)^{-2},$

where, once again, the constant $C>0$ has changed between lines. If we set
$\gamma=\frac{1}{2\log(\varepsilon^{-1})}$, which we can do for
$\varepsilon\in(0,1)$, then we get the desired upper bound. ∎

###### Proof of Proposition 3.2.

Thus, applying Lemmas 3.4 and 3.5 to (3.8), we get the inequality (for a new
constant $C>0$)

$\displaystyle\mathbb{E}\left[\|\mathcal{V}(t)-\mathcal{Z}_{\varepsilon}(t)\|_{2,\rho_{\varepsilon}}^{2}\right]$
(3.16) $\displaystyle\leq$ $\displaystyle
C\lambda(\varepsilon)^{-2}\mathbb{E}\left[\int_{0}^{t}(t-s)^{-\frac{1}{2}}\|\mathcal{V}(s)-\mathcal{Z}_{n}(s)\|_{2,\rho_{\varepsilon}}^{2}ds\right]$
$\displaystyle+C\lambda(\varepsilon)^{-2}\varepsilon^{1-\eta}\int_{0}^{t}(t-s)^{\frac{1}{2}\eta-1}s^{-\frac{1}{2}}e^{cs}ds+C\lambda(\varepsilon)^{-2}t^{\frac{1}{2\log(\varepsilon^{-1})}-\frac{1}{2}}e^{ct}\mu(\varepsilon)^{2}\log(\varepsilon^{-1})$
$\displaystyle\leq$ $\displaystyle
C\lambda(\varepsilon)^{-2}\mathbb{E}\left[\int_{0}^{t}(t-s)^{-\frac{1}{2}}\|\mathcal{V}(s)-\mathcal{Z}_{\varepsilon}(s)\|_{2,\rho_{\varepsilon}}^{2}ds\right]$
$\displaystyle+C\lambda(\varepsilon)^{-2}t^{\frac{1}{2\log(\varepsilon^{-1})}-\frac{1}{2}}e^{ct}\left(\varepsilon\log(\varepsilon^{-1})+\mu(\varepsilon)^{2}\log(\varepsilon^{-1})\right).$
(3.17)

Where for the last inequality, we have repeated our estimate for the Beta
function and set $\eta=\frac{1}{\log(\varepsilon^{-1})}$. If we apply this
inequality, we can see

$\displaystyle\mathbb{E}\left[\int_{0}^{t}(t-s)^{-\frac{1}{2}}\|\mathcal{V}(s)-\mathcal{Z}_{\varepsilon}(s)\|_{2,\rho_{\varepsilon}}^{2}ds\right]$
$\displaystyle\leq$ $\displaystyle
C\lambda(\varepsilon)^{-2}\int_{0}^{t}(t-s)^{-\frac{1}{2}}\mathbb{E}\bigg{[}s^{\frac{1}{2\log(\varepsilon^{-1})}-\frac{1}{2}}e^{cs}\left(\varepsilon\log(\varepsilon^{-1})+\mu(\varepsilon)^{2}\log(\varepsilon^{-1})\right)$
$\displaystyle+\int_{0}^{s}(s-u)^{-\frac{1}{2}}\|\mathcal{V}(u)-\mathcal{Z}_{n}(u)\|_{2,\rho_{\varepsilon}}^{2}du\bigg{]}ds$
$\displaystyle\leq$ $\displaystyle
C\lambda(\varepsilon)^{-2}t^{\frac{1}{2\log(\varepsilon^{-1})}}e^{ct}\left(\varepsilon\log(\varepsilon^{-1})+\mu(\varepsilon)^{2}\log(\varepsilon^{-1})\right)+C\lambda(\varepsilon)^{-2}\int_{0}^{t}\mathbb{E}\left[\|\mathcal{V}(u)-\mathcal{Z}_{\varepsilon}(u)\|_{2,\rho_{\varepsilon}}^{2}\right]ds.$

Applying this bound to (3.17), we can apply Grönwall’s inequality. Which,
after some elementary simplifications, yields the desired inequality. ∎

###### Proof.

Proof of Theorem 1.1 As a consequence of the previous proposition, we have
proved that if
$\mu(\varepsilon)=o\left(\log(\varepsilon^{-1})^{-\frac{1}{2}}\right)$ then

$\displaystyle\lambda(\varepsilon)^{2}\mathbb{E}\left[\|(\mathcal{V}(t)-\mathcal{Z}_{\varepsilon}(t))*\rho_{\varepsilon}\|_{2}^{2}\right]\to
0,\ \text{as }\varepsilon\to 0.$

We also have that for each $t>0$,
$\mathcal{Z}_{\varepsilon}(t)\to\mathcal{Z}(t)$ in
$L^{2}(\mathbb{P}\times\mathbb{R})$, this follows from estimates in [BC95].
Therefore,

$\displaystyle\mathbb{E}\left[\|\mathcal{V}(t)*\hat{\rho}_{\varepsilon}-\mathcal{Z}_{\varepsilon}(t)\|_{2}^{2}\right]\to
0,\ \text{as }\varepsilon\to 0,$

where
$\hat{\rho}_{\varepsilon}:=\kappa_{\varepsilon}^{-1}\lambda(\varepsilon)\rho_{\varepsilon}$
is the normalised version of $\rho_{\varepsilon}$. We now just need to show
that the mollified version of $\mathcal{V}$ is close to $\mathcal{V}$ itself.
From the triangle inequality, we have

$\displaystyle\mathbb{E}\left[\|\mathcal{V}(t)-\mathcal{Z}(t)\|_{2}^{2}\right]\leq
2\mathbb{E}\left[\|\mathcal{V}(t)-\mathcal{V}(t)*\hat{\rho}_{\varepsilon}\|_{2}^{2}\right]+2\mathbb{E}\left[\|\mathcal{V}(t)*\hat{\rho}_{\varepsilon}-\mathcal{Z}(t)\|_{2}^{2}\right].$
(3.18)

Theorem 1.1 tells us that the latter term on the right hand side tends to $0$
as $\varepsilon\to 0$, and we discussed the former term in the previous
section. Recall from (2.8) that we only need to control the the following
increment of $q^{\lambda(\varepsilon)}$ to get the desired convergence.

$\displaystyle
2\int_{\mathbb{R}}(q^{\lambda(\varepsilon)}(t,0)-q^{\lambda(\varepsilon)}(t,\varepsilon
y))\rho(y)dy+\int_{\mathbb{R}}(q^{\lambda(\varepsilon)}(t,\varepsilon
y)-q^{\lambda(\varepsilon)}(t,0))\rho*\rho(y)dy.$

Since both $\rho$ and $\rho*\rho$ have compact support, we only need to apply
Proposition 2.7 to finish the proof. ∎

##  4 Proof of Proposition 1.5

Let $\mathcal{V}$ be the solution to the SPDE (1.9). Then
$v_{t}:=\int_{\mathbb{R}}\mathcal{V}(t,y)dy$ is a martingale with quadratic
variation

$\lambda^{2}\int v(t,x)v(t,y)C^{\varepsilon}(x-y)\;dxdy.$

Applying Itô’s formula and taking expectations gives

$\frac{d}{dt}{\mathbb{E}}[v_{t}^{1/2}]=-\frac{\lambda^{2}}{4}{\mathbb{E}}\left[v_{t}^{-3/2}\int\mathcal{V}(t,x)\mathcal{V}(t,y)C^{\varepsilon}(x-y)\;dxdy\right]$

Now, similarly to previous sections, assume the covariance can be expressed as

$C^{\varepsilon}(x)=\int\rho_{\varepsilon}(z)\rho_{\varepsilon}(x-z)dz$

with
$\rho_{\varepsilon}:=\mu(\varepsilon)^{2}\varepsilon^{-\frac{1}{2}}\rho(\varepsilon^{-1}\cdot)$
for a positive and symmetric function $\rho:\mathbb{R}\to\mathbb{R}$. Then,
for any $a>0$, by the Cauchy-Schwartz inequality,

$\int\mathcal{V}(t,x)\mathcal{V}(t,y)C^{\varepsilon}(x-y)\;dxdy=\int\bigl{(}(\mathcal{V}(t)*\rho_{\varepsilon})(z)\bigr{)}^{2}dz\geq\frac{1}{2a}\left(\int_{-a}^{a}(\mathcal{V}(t)*\rho_{\varepsilon})(z)dz\right)^{2}.$

Let $\kappa_{\varepsilon}=\int_{\mathbb{R}}\rho_{\varepsilon}(y)dy$, so that
$\int_{-\infty}^{\infty}(\mathcal{V}(t)*\rho_{\varepsilon})(z)dz=\kappa_{\varepsilon}v_{t}$,
and the right-hand side above can be written as

$\frac{1}{2a}\left(\kappa_{\varepsilon}v_{t}-\int_{{\mathbb{R}}\setminus[-a,a]}(v_{t}*\rho_{\varepsilon})(z)dz\right)^{2}$

Since $(x-y)^{2}=x^{2}(1-y/x)^{2}\geq x^{2}(1-2y/x)\geq
x^{2}(1-2(y/x)^{1/2})=x^{2}-2y^{1/2}x^{3/2}$ whenever $0\leq y\leq x$, we
deduce that

$\displaystyle
v_{t}^{-3/2}\int\mathcal{V}(t,x)\mathcal{V}(t,y)C^{\varepsilon}(x-y)\;dxdy$
$\displaystyle\geq$
$\displaystyle\frac{\kappa_{\varepsilon}^{2}}{2a}v_{t}^{1/2}-\frac{\kappa_{\varepsilon}^{3/2}}{a}\left(\int_{{\mathbb{R}}\setminus[-a,a]}(v_{t}*\rho_{\varepsilon})(z)dz\right)^{1/2}.$

Taking expectations, and using Jensen’s inequality finally gives

$\frac{d}{dt}{\mathbb{E}}[v_{t}^{1/2}]\leq-\frac{\lambda^{2}\kappa_{\varepsilon}^{2}}{8a}{\mathbb{E}}\bigl{[}v_{t}^{1/2}\bigr{]}+\frac{\lambda^{2}\kappa_{\varepsilon}^{3/2}}{4a}{\mathbb{E}}\left[\int_{{\mathbb{R}}\setminus[-a,a]}(v_{t}*\rho_{\varepsilon})(z)dz\right]^{1/2}$

Express $\rho_{\varepsilon}$ as $\kappa_{\varepsilon}\hat{\rho}_{\varepsilon}$
where $\hat{\rho}_{\varepsilon}$ has unit mass. Then integrating the
inequality gives

${\mathbb{E}}[v_{t}^{1/2}]\leq\exp\left\\{-\frac{\lambda^{2}\kappa_{\varepsilon}^{2}}{8a}t\right\\}{\mathbb{E}}[v_{0}^{1/2}]\\\
+\frac{\lambda^{2}\kappa_{\varepsilon}^{2}}{4a}\int_{0}^{t}ds\;\exp\left\\{-\frac{\lambda^{2}\kappa_{\varepsilon}^{2}}{8a}(t-s)\right\\}{\mathbb{E}}\left[\int_{{\mathbb{R}}\setminus[-a,a]}(v_{s}*\hat{\rho}_{\varepsilon})(z)dz\right]^{1/2}$
(4.1)

Now we can move on to the proof of Proposition 1.5.

###### Proof of Proposition 1.5.

The idea is to use the estimate (4.1) to deduce that
${\mathbb{E}}[v_{t}^{1/2}]$ tends to zero with $\varepsilon$.

First notice that Jensen’s inequality and the initial condition implies that

${\mathbb{E}}[v_{t}^{1/2}]\leq{\mathbb{E}}[v_{t}]^{1/2}=1.$

We can write

${\mathbb{E}}\left[\int_{{\mathbb{R}}\setminus[-a,a]}(v_{s}(\cdot)*\hat{\rho}_{\varepsilon})(z)dz\right]$

as

${\mathbb{P}}\bigl{(}|\sqrt{s}Z+\varepsilon W|>a\bigr{)}$

where $W$ and $Z$ are independent random variables, the distribution of $W$
having density $\hat{\rho}$ and $Z$ being a Gaussian with variance $\nu$.
Consequently the second term on the RHS of (4.1) can be made arbitraily small,
for all $\varepsilon$ by choosing $a$ large enough.

The hypothesis that
$\varepsilon^{1/2}\lambda(\varepsilon)\mu(\varepsilon)\rightarrow\infty$
ensures that
$\lambda(\varepsilon)^{2}\kappa_{\varepsilon}^{2}/a\rightarrow\infty$ for any
fixed $a$, which in turn implies that, for a fixed $a$, the first term on the
RHS of (4.1) is tending to zero.

∎

Acknowledgement. D.B. was supported by EPSRC grant No. EP/W522594/1.

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