# An SPDE with Robin-type boundary for a system of elastically killed
diffusions on the positive half-line

Ben Hambly111Mathematical Institute, University of Oxford, Email:
<EMAIL_ADDRESS>Julian Meier222Mathematical Institute, University of
Oxford, Email<EMAIL_ADDRESS>(corresponding author) and Andreas
Søjmark333Department of Statistics, London School of Economics, Email:
<EMAIL_ADDRESS>
###### Abstract

We consider a system of particles undergoing correlated diffusion with elastic
boundary conditions on the half-line. By taking the large particle limit we
establish existence and uniqueness for the limiting empirical measure valued
process for the surviving particles. This process can be viewed as the weak
form for an SPDE with a noisy Robin boundary condition satisfied by the
particle density. We establish results on the $L^{2}$-regularity properties of
this density process, showing that it is well behaved in the interior of the
domain but may exhibit singularities on the boundary at a dense set of times.
We also show existence of limit points for the empirical measure in the non-
linear case where the particles have a measure dependent drift. We make
connections for our linear problem to the corresponding absorbing and
reflecting SPDEs, as the elastic parameter takes its extreme values.

Keywords— Particle System, Common Noise, Mean-field type SPDE, Elastic
killing, Robin boundary

##  1 Introduction

Consider a conditionally i.i.d. system of $N$ reflected Itô diffusions
$\\{X^{i}\\}_{1\leqslant i\leqslant N}$ living on the positive half-line
$[0,\infty)$ with a given finite time horizon $T$. The initial values
$\\{X_{0}^{i}\\}_{1\leqslant i\leqslant N}$ are chosen independently from a
common distribution $\nu_{0}$ on $(0,\infty)$, and each $X^{i}$ has diffusive
dynamics

$\displaystyle\begin{split}X_{t}^{i}&=X_{0}^{i}+\int_{0}^{t}\mu(s,X_{s}^{i})ds+\int_{0}^{t}\sigma(s,X_{s}^{i})\rho(s)dW_{s}^{0}\\\
&+\int_{0}^{t}\sigma(s,X_{s}^{i})(1-\rho(s)^{2})^{\frac{1}{2}}dW_{s}^{i}+L_{t}^{i},\quad
i=1,\ldots,n,\end{split}$ (1.1)

where $L^{i}$ is the local time of $X^{i}$ at the origin and
$W^{0},W^{1},\ldots,W^{N}$ are independent Brownian motions. The precise
assumptions on the coefficients are left for Section 2.

We shall also refer to each $X^{i}$ as the $i$’th particle. On top of the
dynamics (1.1), we wish to consider the ‘killing’ of each particle after a
suitable (random) amount of time has been spent at the boundary. To this end,
we fix a parameter $\kappa>0$ and define

$\tau^{i}\coloneqq\inf\\{t>0:L_{t}^{i}>\chi^{i}\\},\quad i=1,\ldots,n,$ (1.2)

for a family $(\chi^{i})_{1\leqslant i\leqslant N}$ of i.i.d. exponential
random variables with rate $\kappa>0$. This captures the idea of each particle
being killed elastically at the origin with parameter $\kappa>0$, and we say
$\tau^{i}$ is the elastic killing time of $X^{i}$. Note that the measure
$dL^{i}_{t}$, induced by the local time, is supported on the set
$\\{t:X_{t}^{i}=0\\}$, so we know that $X_{\tau^{i}}^{i}=0$, meaning that each
particle can only be killed when it is at the boundary.

The main object of study in this paper is the family of empirical measures for
the surviving particles, for every time $t$ and any number of particles $N$,
given by

$\nu_{t}^{N}(dx):=\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{t}^{i}}(dx)\mathbbm{1}_{t<\tau^{i}},\quad
t\in[0,T],\quad N\geqslant 1.$ (1.3)

A similar particle system with elastic killing but without a common noise has
been studied in [14] to model an epidemic advancing through a population.

In financial applications, $\tau^{i}$ could represent the _default time_ of an
entity whose _financial health_ is modelled by $X^{i}$. We then say that an
entity is in _distress_ when $X^{i}_{t}$ is zero. In practice, an entity could
for example be an asset such as a credit instrument forming part of a larger
credit default obligation or a defaultable loan in a mortgage backed security.
It could also represent an entire company or a financial institution. Either
way, if enough time is spent in this distressed state, the inevitable will
happen: the clock $\tau^{i}$ rings and the $i$’th entity is declared to be in
default. Due to the structure of (1.2), where the underlying exponential
random variables are not observed, we obtain the desirable properties that the
default times are not predictable and their laws are memoryless.

In order to mange the risk of investments and regulate or monitor a financial
system, it becomes important to understand, respectively, the pricing of
credit derivatives on portfolios of a large number of defaultable entities or
the computation of risk measures related to the overall health of a large
number of financial institutions. Both cases will typically involve the
computation of expected values $\mathbb{E}[F(\nu^{N})]$ for some functional
$F$ of the paths $t\mapsto\nu_{t}^{N}$ up until a terminal time $T$. Viewing
$\nu^{N}:=(\nu_{t}^{N})_{t\geqslant 0}$ as a measure-valued càdlàg stochastic
process, it is then natural to look for a functional limit theorem as
$N\rightarrow\infty$ and thereby seek to identify a good approximation
$\mathbb{E}[F(\nu)]$, where the limit $\nu=(\nu_{t})_{t\geqslant 0}$ satisfies
an evolution equation driven only by the common factor $W^{0}$.

Closely related to our setting, [9] provides such a result motivated by the
pricing of credit default swap indices, which involves computing the expected
value of suitable functionals of the loss process
$t\mapsto\mathcal{L}^{N}_{t}:=1-\nu^{N}_{t}(0,\infty)$, giving the proportion
of defaults. However, [9] does not work with elastic default times for
reflected dynamics as we do here. Instead, they consider a constant
coefficient version of the system (1.1) without reflection, where defaults are
declared to happen immediately at the first time the financial health $X^{i}$
hits zero. The paper [24] studies precise regularity properties of solutions
to the resulting SPDE with absorbing boundary satisfied by $\nu$ (in the limit
as $N\rightarrow\infty$). Moreover, [15, 29] have proposed and analysed
efficient numerical schemes for simulating this SPDE, based on finite
difference schemes in combination with multi-level or multi-index Monte Carlo
methods, which can yield substantial gains over the simulation of the finite
system for reasonable sizes of $N$. Finally, [16, 17] have considered the
well-posedness of broader classes of such parabolic SPDEs with absorbing
boundary and [1] considers an SPDE with an absorbing boundary on a compact
interval which is applied to the pricing of Mortgage Backed Securities.

In a class of sufficiently regular solutions, uniqueness of the SPDE with
absorbing boundary in [9] can be deduced from the general Sobolev theory on
Dirichlet problems for SPDEs developed in [21, 22]. As far as we are aware, no
such theory exists for the type of SPDEs with reflecting or elastic boundaries
that we consider in the present paper. For details on the specific type, see
(1.4)-(1.5) below with $\kappa=0$ or $\kappa>0$, respectively.

In [9], adapting the approach of [20, 23] for the whole space to a half-line
with absorbing boundary, uniqueness of a priori measure-valued solutions is
proved by means of uniform $L^{2}$ energy estimates for suitable
mollifications of the solutions. This method only works because the authors
can quantify the second moment of the mass near the boundary as having decay
of at least order
$\mathbb{E}[\int_{0}^{T}|\nu_{t}(0,\varepsilon)|^{2}dt]=O(\varepsilon^{3+\gamma})$,
as $\varepsilon\downarrow 0$, for some $\gamma>0$. Even with constant
coefficients except for a time dependent correlation $t\mapsto\rho(t)$, it
becomes unclear how to obtain such a strong order of decay. In [16], it was
instead realised that, by working in the dual, $H^{-1}$, of the first Sobolev
space, $H^{1}$, uniqueness in the case of an absorbing boundary can be
obtained from energy estimates in $H^{-1}$ as soon as we know that the first
moment of the mass near the boundary satisfies
$\mathbb{E}[\int_{0}^{T}|\nu_{t}(0,\varepsilon)|dt]=O(\varepsilon^{1+\gamma})$,
as $\varepsilon\downarrow 0$, for some $\gamma>0$, which is very easy to
verify for the limit points of the empirical measures in the absorbing case.

The aforementioned first moment control, however, is much too strong a
requirement for elastic and reflecting boundaries. Indeed, the absolute best
one could hope for is that
$\mathbb{E}[\int_{0}^{T}|\nu_{t}(0,\varepsilon)|dt]$ is of order
$O(\varepsilon)$ as $\varepsilon\downarrow 0$. Fortunately, returning instead
to work with the second moment, we are able to implement the $H^{-1}$ approach
from [16] by showing that limit points of (1.3) must satisfy
$\mathbb{E}[\int_{0}^{T}|\nu_{t}(0,\varepsilon)|^{2}dt]=O(\varepsilon^{1+\gamma})$,
as $\varepsilon\downarrow 0$, for some $\gamma>0$. This is enough for us to
succeed in establishing uniqueness within a broad class of measure-valued
solutions that includes the limit points of the empirical measures.

###  1.1 Summary of main results

By analogy with the connection between an elastically killed Brownian motion
and the heat equation with a convective Robin boundary, one heuristically
expects limit points of (1.3) to take the form of a stochastic process
$(V_{t})_{t\geqslant 0}$, where each $V_{t}$ is a random sub-probability
density on $[0,\infty)$ governed, in a suitable sense, by the evolution
equation

$dV_{t}=\bigl{(}\partial_{xx}\bigl{(}\frac{\sigma^{2}_{t}}{2}V_{t}\bigr{)}-\partial_{x}(\mu_{t}V_{t})\bigr{)}dt-\rho_{t}\partial_{x}\left(\sigma_{t}V_{t}\right)dW^{0}_{t},$
(1.4)

in the interior, together with the noisy Robin-type boundary condition

$\partial_{x}\bigl{(}\frac{\sigma^{2}(\cdot,t)}{2}V_{t}\bigr{)}(0)=\kappa\frac{\sigma^{2}(t,0)}{2}V_{t}(0)+\mu(t,0)V_{t}(0)+\rho(t)\sigma(t,0)V_{t}(0)\dot{W}^{0}_{t}.$
(1.5)

Note that, to simplify notation, we will often suppress the spatial dependence
and simply write $\mu_{t}=\mu(t,x)$ and $\sigma_{t}=\sigma(t,x)$ as well as
$\rho_{t}=\rho(t)$.

Our main contributions in this paper are as follows. Firstly, we identify a
suitable weak formulation of the SPDE (1.4)-(1.5), which is shown to uniquely
characterize a functional limit theorem for the empirical measures (1.3) as
$N\rightarrow\infty$ (Theorems 2.4 and 2.5). Secondly, we show that the unique
solution $\nu$ of our weak formulation has a density $V_{t}$ at all times,
that this density is square-integrable for almost all times, and that it is
square-integrable away from the boundary at all times (Theorem 2.6 and
Proposition 2.7). Thirdly, we connect the study of absorbing, reflecting, and
elastic boundaries within the class of linear parabolic SPDEs on a half-line
analysed here. We do this by showing that well-posedness with a reflecting
boundary is also captured by our arguments (Theorem 2.8) and that the cases of
a reflecting or an absorbing boundary emerge as limiting cases when sending
the parameter of elastic killing $\kappa$ to $0$ or $+\infty$, respectively
(Theorem 2.9). Finally, towards the end of the paper, we show that our
arguments for identifying a limiting SPDE extend to drifts depending
continuously on the empirical measures (Theorem 7.2), and we show that the
resulting limiting solutions have McKean–Vlasov type probabilistic
representations (Theorem 7.3). However, it does not seem straightforward for
our $H^{-1}$ approach to yield uniqueness in that setting.

###  1.2 Making sense of the elastic boundary at the origin

In this section, we present some heuristic arguments that motivate our notion
of solution in Section 2.1 and highlight the reasons for working with a
relaxed interpretation of the boundary condition (1.5).

If we had a smooth noise $W^{0}$ and (1.4)-(1.5) hold in the classical sense,
then they imply a loss of mass at the rate

$\frac{d}{dt}\nu_{t}([0,\infty))=-\kappa\frac{\sigma^{2}(t,0)}{2}V_{t}(0),\quad\text{for
all}\quad t>0,$ (1.6)

almost surely, after integration by parts. This reveals a natural and slightly
weaker way of imposing the elastic boundary condition without asking for
spatial differentiability.

Nonetheless, we will have to drop the differentiability in time implied by
(1.6), and, moreover, we will have to avoid evaluating the solution at the
boundary, for reasons we discuss below. Thus, rather than taking (1.6) as a
definition we derive a relaxed version for the case where the noise $W^{0}$ is
a Brownian motion. Consider the density $G^{E,\kappa}_{\varepsilon}(x,y)$, at
time $\varepsilon>0$, of a Brownian motion on $[0,\infty)$ started at
$y\geqslant 0$ and killed elastically at $0$ with rate $\kappa>0$ (see (A.3)).
Then $y\mapsto G_{\varepsilon}^{E,\kappa}(0,y)$ approximates a Dirac mass at
$0$ from the right, tending to infinity at $y=0$, while the function

$\phi^{E,\kappa}_{\varepsilon}(x):=\int_{0}^{\infty}G^{E,\kappa}_{\varepsilon}(x,y)\mathbbm{1}_{[0,\infty)}(y)dy=\int_{0}^{\infty}G^{E,\kappa}_{\varepsilon}(x,y)dy.$
(1.7)

defines a particular smooth approximation of $\mathbbm{1}_{[0,\infty)}$ which
is close to $1$ at $x=0$. In Theorem 2.1, we introduce a succinct weak
formulation of (1.4)-(1.5) that will hold for the limit points of our particle
system. As we will see in Section 5.1, a key step in our uniqueness proof is
that these weak solutions can be shown to satisfy

$d\langle\nu_{t},\phi^{E,\kappa}_{\varepsilon}\rangle=-\kappa\langle\nu_{t},\frac{\sigma_{t}^{2}}{2}G_{\varepsilon}^{E,\kappa}(0,\cdot)\rangle
dt+\langle\nu_{t},\mu_{t}\bar{g}_{\varepsilon}^{E,\kappa}\rangle
dt+\langle\nu_{t},\rho_{t}\sigma_{t}\bar{g}_{\varepsilon}^{E,\kappa}\rangle
dW_{t}^{0}$ (1.8)

for any $\varepsilon>0$, for a suitable correction function
$\bar{g}_{\varepsilon}^{E,\kappa}:=g_{\varepsilon}^{E,\kappa}(0,\cdot)$
defined in (A.3).

######  Remark.

Through out we use the notation $\langle\xi,\phi\rangle$ for the application
of a distribution $\xi$ to the test function $\phi$. In the case where $\xi$
is a measure this represents the integration of the test function against the
measure.

Equation (1.8) may be viewed as a relaxed version of (1.6), where we have
shifted attention from the boundary to arbitrarily small neighbourhoods of the
boundary, at the cost of introducing two correction terms: an absolutely
continuous term due to the drift $\mu$ and a local martingale term due to the
driving noise $W^{0}$. As per the arguments in Section 5.1, we can take an
expectation and send $\varepsilon$ to zero, to obtain

$\mathbb{E}[\nu_{t}([0,\infty))]=1-\lim_{\varepsilon\downarrow
0}\int_{0}^{t}\kappa\mathbb{E}\bigl{[}\langle\nu_{s},\frac{\sigma_{s}^{2}}{2}G_{\varepsilon}^{E,\kappa}(0,\cdot)\rangle\bigr{]}ds.$
(1.9)

Therefore, we may instructively think of (1.8) as enforcing (1.6) in a
generalized sense in time and space, subject to taking expectations, when we
assume nothing about the existence or regularity of a density or the noise
being smooth.

While this is a much weaker formulation of the boundary condition (1.5) than
(1.6), it will be enough for us to carry through our uniqueness proof, as we
see in Section 5.2. Besides allowing for uniqueness, the relaxed formulation
(1.8) is crucial for two reasons. Firstly, we need a sufficiently relaxed
formulation so that it can be guaranteed to be satisfied by weak limit points
of the particle system, for which it would seem intractable to ask for much
more than the above. Secondly, even if we had more precise knowledge of the
density for the limiting solutions, it would not be possible to have (1.6)
holding for all $t\in[0,T]$ with probability 1. As it turns out, for each
realisation, the density can blow up as we approach the boundary for certain
times (albeit a set of times of measure zero). We discuss this further in
Section 2.1 together with our main results.

Intuitively, suppose $dW^{0}_{t}$ is badly behaved at some time $t$, pushing a
non-negligible amount of mass towards the boundary in an infinitesimal amount
of time while not pushing any mass the other way (due to a Brownian path for
which $B_{s}-B_{t}<0$ for all $s$ in some small right-neighbourhood of $t$).
Since only a fraction of this mass leaves the domain, the rest must be
instantaneously accommodated for near the boundary, which may then force
$\limsup_{\varepsilon\downarrow
0}\langle\nu_{t},G^{E,\kappa}_{\varepsilon}(0,\cdot)\rangle=+\infty$. In
particular, this will also imply non-differentiability of the loss of mass at
that time. As the paths $t\mapsto\nu_{t}([0,\infty))$ are decreasing, they are
automatically differentiable at almost every time, but, for any given
realisation, there can still be a dense set of times of measure zero where
this fails for the aforementioned reasons.

###  1.3 Literature on SPDEs with noisy boundary conditions

Our focus in this paper is on identifying the limit of the empirical measures
(1.3) as the unique solution of a suitable weak formulation of the SPDE
(1.4)-(1.5). Working with the relaxed condition (1.8), we will make no attempt
at understanding (1.5) in a stronger sense. Nevertheless, it is worth
emphasising that there are several results in the literature on SPDEs with
irregular noise terms in the boundary condition.

Alòs and Bonaccorsi [2] study SPDEs with a white noise Dirichlet boundary
condition on the half-line, using the theory of fundamental solutions for
linear stochastic evolution equations from [28] and techniques from Malliavin
calculus. They prove existence of solutions in a weighted $L^{p}$ space and
determine the blow-up rate at the boundary. Da Prato and Zabczyk [12] study
nonlinear stochastic evolution equations with white noise boundary conditions.
They employ a stochastic version of the semigroup approach developed in [3] to
show global existence and uniqueness of mild solutions. Moreover, DaPrato and
Zabczyk establish that the solution in the Neumann case exhibits a higher
regularity near the boundary. Maslowski [27] uses similar techniques based on
the semigroup approach to analyse SPDEs on a bounded domain driven by space-
dependent Gaussian noise and Robin-type boundary condition with bounded
operators and independent noise at the boundary and shows existence and
uniqueness of mild solutions. Sowers [33] studies stochastic reaction
diffusion equations on an $n$-dimensional manifold with additive noise in the
boundary conditions showing existence and uniqueness of solutions. In work on
stochastic dynamical boundary conditions, Chueshow and Schmalfus [10] [11]
show well-posedness of a system of quasi-linear parabolic SPDEs on bounded
domains with noisy boundary dynamics, and verify that solutions give rise to a
random dynamical system. In more recent work, Shirikyan [32] studies 2d
Navier–Stokes systems on a bounded domain driven by boundary noise with a
piecewise independent structure.

The SPDE (1.4)-(1.5) differs from these problems in a number of ways. The
equation involves a stochastic transport term where the derivative of the
solution appears in front of the Brownian motion. Moreover, the boundary noise
is the same as the noise driving the equation in the interior while most of
the literature is focused on the case of an independent noise at the boundary.

##  2 Convergence to an SPDE with elastic boundary

To have a unique strong solution of the finite particle system and establish
the desired results on convergence and well-posedness of the limit, we impose
the following assumptions on the initial condition and the coefficients of the
system.

######  Assumption 2.1 (Initial Condition).

The initial condition $\nu_{0}$ is supported on $(0,\infty)$, and satisfies

$\nu_{0}(\lambda,\infty)=o(\exp(-\alpha\lambda))\quad\text{ as
}\lambda\to+\infty,$

for every $\alpha>0$. Furthermore, we assume that $\nu_{0}$ has an
$L^{2}$-density $V_{0}$.

######  Assumption 2.2 (Regularity in Space and Time).

The coefficients $\mu$, $\sigma$ and $\rho$ have the following regularity in
the space and time variables

  1. (i)

The maps $x\mapsto\mu(t,x)$ and $x\mapsto\sigma(t,x)$ are in
$C^{2}([0,\infty))$ with

$\lvert\partial_{x}^{n}\mu(t,x)\rvert,\lvert\partial_{x}^{n}\sigma(t,x)\rvert\leqslant
C_{\sigma,\mu}$

for all $t\in[0,T],x\in\mathbb{R}$ and $n=0,1,2$, for a constant
$C_{\sigma,\mu}>0$,

  2. (ii)

For all $t\in[0,T],x\in\mathbb{R}$, $\sigma(t,x)\geqslant
C^{-1}_{\sigma,\mu}>0$ and $0\leqslant\rho(t)<1$,

  3. (iii)

The map $t\mapsto\sigma(t,x)$ is in $C^{1}([0,T])$ and
$\sup_{s\in[0,T]}\int_{0}^{\infty}\lvert\partial_{t}\sigma(s,y)\rvert
dy<\infty$.

Given these assumptions, strong existence and uniqueness for the system (1.1)
follows by classical results. Next, we define a class of processes capturing
the fundamental regularity conditions satisfied by limit points of the
empirical measure processes. Our uniqueness result will be for solutions to
the limiting SPDE belonging to this class.

######  Definition 2.3 (The class $\Lambda$).

We say that a distribution-valued càdlàg process $(\nu_{t})_{t\in[0,T]}$ is of
class $\Lambda$ if $\nu$ takes values in the space of sub-probability measures
$\mathbf{M}_{\leqslant 1}(\mathbb{R})$ and the following conditions are
satisfied:

  1. (i)

(Support on positive half-line) For every $t\in[0,T]$, the sub-probability
measure $\nu_{t}$ is supported on the positive half-line $[0,\infty)$,

  2. (ii)

(Exponential tails) For every $\alpha>0$, we have

$\mathbb{E}\left[\int_{0}^{T}\nu_{t}(\lambda,+\infty)dt\right]=o(e^{-\alpha\lambda}),\quad\text{as
}\lambda\to\infty,$

  3. (iii)

(Boundary decay) There exists $\gamma>0$ such that

$\mathbb{E}\left[\int_{0}^{T}(\nu_{t}(0,\varepsilon))^{2}dt\right]=O(\varepsilon^{1+\gamma}),\quad\text{as
}\varepsilon\to 0,$

  4. (iv)

(Spatial concentration) There exist constants $C>0$ and $\delta\in(0,1)$ such
that

$\mathbb{E}\left[\int_{0}^{T}\nu_{t}(a,b)dt\right]\leqslant C\lvert
b-a\rvert^{\delta},\quad\text{for all pairs }0<a<b.$

###  2.1 Functional convergence and well-posedness of the SPDE

As discussed in the introduction, we are interested in a functional limit
theorem for the empirical measures $\nu^{N}=(\nu^{N}_{t})_{t\in[0,T]}$ seen as
càdlàg stochastic processes. To this end, we proceed as in [16, 17, 25] and
establish weak convergence on the Skorokhod space of
$\mathcal{S}^{\prime}$-valued càdlàg paths, denoted
$D_{\mathcal{S}^{\prime}}=D_{\mathcal{S}^{\prime}}[0,T]$. As usual,
$\mathcal{S}^{\prime}$ is the space of tempered distributions, forming the
dual of $\mathcal{S}$, the space of Schwarz functions on $\mathbb{R}$. As per
[25], we endow $D_{\mathcal{S}^{\prime}}$ with Skorokhod’s M1 topology and the
corresponding Borel $\sigma$-field.

######  Theorem 2.4 (Functional limit theorem).

Let $\nu^{N}=(\nu^{N}_{t})_{t\in[0,T]}$ be given by (1.3) with Assumptions
2.1-2.2 in place. Every subsequence of $(\nu^{N},W^{0})$ has a further
subsequence converging in law on
$(D_{\mathcal{S}^{\prime}},\text{M1})\times(C_{\mathbb{R}},\left\|\right\|_{\infty})$.
Moreover, for any limit point $(\nu,W^{0})$, the marginal $\nu$ is in the
class $\Lambda$ and there is a filtration $\mathcal{F}^{\nu,W^{0}}$, for which
the marginal $W^{0}$ is a Brownian motion and $\nu$ is adapted, so that the
pair $(\nu,W^{0})$ satisfies the SPDE

$\displaystyle\begin{split}\langle\nu_{t},\phi\rangle=\langle\nu_{0},\phi\rangle&+\int_{0}^{t}\langle\nu_{s},\mu(s,\cdot)\phi^{\prime}\rangle
ds+\frac{1}{2}\int_{0}^{t}\langle\nu_{s},\sigma^{2}(s,\cdot)\phi^{\prime\prime}\rangle
ds\\\ &+\int_{0}^{t}\langle\nu_{s},\rho(s)\sigma(s,\cdot)\phi^{\prime}\rangle
dW_{s}^{0}\end{split}$ (2.1)

for all times $t\in[0,T]$ and all test functions
$\phi\in\mathcal{C}^{E,\kappa}_{0}(\mathbb{R})$, with probability 1, where

$\mathcal{C}^{E,\kappa}_{0}(\mathbb{R})=\\{\phi\in\mathcal{S}:\partial_{x}\phi(0)=\kappa\phi(0)\\}.$

Our next result establishes uniqueness for measure-valued solutions to the
SPDE (2.1) in the regularity class $\Lambda$. In particular, we thus obtain
the full convergence in law of the empirical measure processes to the unique
solution of (2.1) in this class.

######  Theorem 2.5 (Uniqueness).

Let $(\nu,W^{0})$ and $(\tilde{\nu},\tilde{W}^{0})$ be solutions to the SPDE
(2.1) such that $\nu$ and $\tilde{\nu}$ are in the class $\Lambda$, for
Brownian motions $W^{0}$ and $\tilde{W}^{0}$. Then, we have

  1. (i)

pathwise uniqueness if $W^{0}=\tilde{W}^{0}$, that is,

$\nu_{t}(S)=\tilde{\nu}_{t}(S)\quad\text{ for every }t\in[0,T]\text{ and every
Borel sets }S\subseteq\mathbb{R}.$ (2.2)

  2. (ii)

uniqueness in law, that is,
$\mathrm{Law}((\nu,W^{0}))=\mathrm{Law}((\tilde{\nu},\tilde{W}^{0}))$.

So far, our existence and uniqueness statements have been phrased solely in
terms of properties of measure-valued solutions. Our third result concerns the
extent to which we can guarantee $L^{2}$-regularity of the unique solution to
the SPDE in the class $\Lambda$.

######  Theorem 2.6 ($L^{2}$-regularity).

Let $(\nu,W^{0})$ be the unique solution to the SPDE (2.1) in the class
$\Lambda$. With probability 1, it holds for all $t\in[0,T]$ that the measure
$\nu_{t}$ restricted to $(0,\infty)$ has a density $V_{t}$, which is in
$L^{2}$ away from the origin. Furthermore, it holds for almost every
$t\in[0,T]$ that $V_{t}$ is the density of $\nu_{t}$ on all of $[0,\infty)$,
and we have

$\mathbb{E}\Bigl{[}\int_{0}^{T}\|V_{t}\|_{L^{2}(0,\infty)}dt\Bigr{]}<\infty.$

The above theorem highlights two interesting issues. Firstly, for a set of
times of measure zero, there could fail to be a density of $\nu_{t}(\omega)$
on $[0,\infty)$. Secondly, even if we have a density for all times, it could
fail to be in $L^{2}$ up to the boundary. The former turns out to not be an
issue, as results of [6] on Brownian motion reflected off of a path of another
Brownian motion will guarantee that there cannot be a point mass at zero.
Moreover, based on the results of [8], we should expect to have
$\limsup_{x\downarrow 0}V_{t}(x)=+\infty$ for a dense set of times. In line
with this, we believe it is unlikely that $L^{2}$ integrability of the density
can be guaranteed all the way up to the boundary for those times.

######  Proposition 2.7 (Density on all of $[0,\infty)$).

With probability 1, there is no atom of $\nu_{t}$ at the origin, for any
$t\in[0,T]$. Consequently, each $V_{t}$ from Theorem 2.6 is in
$L^{1}(0,\infty)$ and gives the density of $\nu_{t}$ on $[0,\infty)$ for all
$t\in[0,T]$, with probability 1.

While all our other results are proved in a self-contained manner, using
mollification and energy estimates, we were not able to exploit this technique
to rule out atoms at the origin, and hence a result of [6] is needed for the
above proposition. We do not use this proposition for any of our other
results, but we believe it is important to emphasise that there is indeed a
density on all of $[0,\infty)$. In particular, we note that our proof of
Theorem 2.4 deals explicitly with the a priori possibility of having atoms at
zero.

###  2.2 Reflecting and absorbing boundaries as limiting cases

The elastic boundary condition in the SPDE (2.1) is encoded in the space of
test functions. Changing the space of test functions, we in turn obtain an
evolution equation with a different boundary behaviour. Specifically, the
space of test functions

$\mathcal{C}^{A}_{0}(\mathbb{R})\coloneqq\\{\phi\in\mathcal{S}(\mathbb{R}):\phi(0)=0\\}.$
(2.3)

corresponds to an absorbing boundary, while the space of test functions

$\mathcal{C}^{R}_{0}(\mathbb{R})\coloneqq\\{\phi\in\mathcal{S}(\mathbb{R}):\partial_{x}\phi(0)=0\\}$
(2.4)

captures a reflecting boundary. The well-posedness of (2.1) with a reflecting
boundary can be inferred from our proofs in this paper, and so we also get the
following result.

######  Theorem 2.8.

The SPDE (2.1) with a reflecting boundary at zero, captured by the test
function space $\mathcal{C}^{R}_{0}(\mathbb{R})$, has a unique solution in the
class $\Lambda$, as in Theorem 2.5.

Existence and uniqueness in the case of an absorbing boundary has been studied
in [9, 24, 16, 17] with various additional features. Our final result shows
that the elastic boundary gives rise to the absorbing and reflecting
boundaries as limiting cases.

######  Theorem 2.9.

Let $(\nu^{\kappa},W^{0})$ denote the unique weak solution with $\nu$ in the
class $\Lambda$ to the SPDE (2.1) with elastic boundary for a given
$\kappa>0$. Then we have that

  1. (i)

as $\kappa\to\infty$, $(\nu^{\kappa},W^{0})$ converges in law to
$(\nu^{\infty},W^{0})$ which is a solution to the SPDE (2.1) with test
function space $\mathcal{C}^{A}_{0}(\mathbb{R})$,

  2. (ii)

as $\kappa\to 0$, $(\nu^{\kappa},W^{0})$ converges in law to $(\nu^{0},W^{0})$
which is a solution to the SPDE (2.1) with test function space
$\mathcal{C}^{R}_{0}(\mathbb{R})$.

##  3 Probabilistic estimates for the particle system

In this section we show a series of probabilistic estimates that will be used
in Section 4.3 to prove the regularity of limit points. As a first step, we
note that all particles in the system
$\\{(X^{i}_{t}\mathbbm{1}_{t<\tau^{i}})_{t\in[0,T]}\\}_{1\leqslant i\leqslant
N}$ are identically distributed. As a result, we obtain for an arbitrary
measurable set $S\subseteq\mathbb{R}$, $N\geqslant 1$ and $t\in[0,T]$

$\mathbb{E}\left[\nu^{N}_{t}(S)\right]=\frac{1}{N}\sum_{i=1}^{N}\mathbb{E}\left[\mathbbm{1}_{\\{X_{t}^{i}\in
S\\}}\mathbbm{1}_{t<\tau^{1}}\right]=\mathbb{P}(X_{t}^{i}\in S,t<\tau^{1}).$
(3.1)

We show how we can use a scale transformation and a change of measure to
estimate (3.1) in terms of the distribution of a reflected Brownian motion
$\lvert W\rvert$.

######  Lemma 3.1 (Scale transformation).

Let $X_{t}^{1}$ be a particle from the particle system. Define the
transformation function $\zeta:[0,T]\times\mathbb{R}\to\mathbb{R}$ as

$\zeta(t,x)\coloneqq\int_{0}^{x}\frac{dy}{\sigma(y,t)}$

then the process $Z_{t}\coloneqq\zeta(t,X_{t}^{1})$ is a semimartingale and
its dynamics are given by $dZ_{t}=\tilde{\mu}_{t}dt+dB_{t}+dL_{t}(Z)$. The
stochastic process B is a Brownian motion

$B_{t}=\int_{0}^{t}\rho_{s}dW_{s}^{0}+\int_{0}^{t}(1-\rho_{s}^{2})^{\frac{1}{2}}dW_{s}^{1},$

L is the local time of Z at zero and $\tilde{\mu}$ a drift coefficient given
by

$\tilde{\mu}_{t}=(\frac{\mu}{\sigma}-\partial_{x}\sigma)(t,X_{t}^{1})-\int_{0}^{X_{t}^{1}}\frac{\partial_{t}\sigma}{\sigma^{2}}(t,y)dy.$

The coefficient $\tilde{\mu}_{t}$ is uniformly bounded in N and t.

###### Proof.

We apply Itô’s formula to obtain the dynamics of $Z$

$\displaystyle dZ_{t}$
$\displaystyle=\partial_{t}\zeta(t,X_{t}^{1})dt+\partial_{x}\zeta(t,X_{t}^{1})dX_{t}^{1}+\partial_{xx}\zeta(t,X_{t}^{1})d[X^{1}]_{t}$
$\displaystyle=\tilde{\mu}(t,X_{t}^{1})dt+dB_{t}+\frac{1}{\sigma(t,X_{t}^{1})}dL_{t}$

with $\tilde{\mu}$ as defined above. By [14, Proposition 4.2 and Proposition
4.4] we can identify the last term as

$\int_{0}^{t}\frac{1}{\sigma(s,X_{s}^{1})}dL_{s}=L(Z_{t})$

establishing the equation for Z. The uniform boundedness for the drift
parameter $\tilde{\mu}$ follows from the conditions on the coefficients in
Assumption 2.2. ∎

######  Remark.

The boundedness assumptions on $\sigma$ give that, for all $t\in[0,T]$,

$\left\lvert\zeta(t,y)-\zeta(t,x)\right\rvert\leqslant C_{\mu,\sigma}\lvert
y-x\rvert,\qquad x,y\geqslant 0$ (3.2)

with the constant $C_{\mu,\sigma}>0$ from Assumption 2.2. Moreover, note that
the transformation $\zeta(t,\cdot)$ is invertible for all $t\in[0,T]$.

######  Remark.

Note that the process $Z_{t}$ obtained from the scale transformation is a
reflected Brownian motion with drift. Thus, the process Z solves the Skorokhod
problem for the process $dY_{t}=\tilde{\mu}dt+dB_{t}$ and, in particular,
stays non-negative (see [30]).

After removing the volatility factor $\sigma$ through the scale
transformation, the next step is to remove the drift and obtain a reflected
Brownian motion. This can be done using Girsanov’s theorem.

######  Lemma 3.2 (Removing the drift).

For all $\delta\in(0,1)$ there is a constant $c_{\delta}>0$ so that

$\mathbbm{P}(X_{t}^{1}\in S)\leqslant
c_{\delta}F_{t}(\zeta(t,S))^{\delta},\qquad{\text{ for every measurable
}S\subseteq\mathbb{R}},$

where $F_{t}$ is the marginal law of a reflected Brownian motion at time t.

###### Proof.

We know that $Z$ is the solution to the Skorokhod problem for the process $Y$
with $dZ_{t}=\tilde{\mu}_{t}dt+dB_{t}+dL_{t}(Z)$. As $\tilde{\mu}$ is bounded,
we can apply Girsanov’s Theorem with the change of measure

$\left.\frac{d\mathbb{Q}}{d\mathbbm{P}}\right|_{\mathcal{F}_{t}}=\exp\left(-\int_{0}^{t}\tilde{\mu}_{s}dB_{s}-\frac{1}{2}\int_{0}^{t}(\tilde{\mu_{s}})^{2}ds\right)$

to see that there is a measure $\mathbb{Q}$ with a Brownian motion
$B^{\mathbb{Q}}$ under which the dynamics of $Z$ are those of a reflected
Brownian motion

$dZ_{t}=dB_{t}^{\mathbb{Q}}+dL_{t}(B^{\mathbb{Q}}).$

The result then follows by proceeding as in the proof of [16, Lemma 4.2]. ∎

As an easy corollary we obtain good spatial concentration of the empirical
measures.

######  Lemma 3.3.

There exist constants $c>0$ and $\delta\in(0,1)$ such that property (iv) of
Definition 2.3 is satisfied by $(\nu_{t}^{N})_{t\in[0,T]}$ uniformly in
$N\geqslant 1$.

###### Proof.

By Lemma 3.2 we have

$\displaystyle\mathbb{E}\left[\nu_{t}^{N}(a,b)\right]\leqslant
c_{\delta}F_{t}(\zeta(t,(a,b)))^{\delta}.$

We note that $\zeta(t,(a,b))\subseteq[\zeta(t,a),\zeta(t,b)]$ where
$\zeta(t,(a,b))=\\{\zeta(t,x):x\in(a,b)\\}$. Using the reflecting heat kernel
$G_{\varepsilon}^{R}$ given in (A.2) we estimate

$\displaystyle F_{t}(\zeta(t,(a,b)))$
$\displaystyle\leqslant\int_{0}^{\infty}\int_{\zeta(t,a)}^{\zeta(t,b)}\frac{1}{\sqrt{2t\pi}}\Bigg{[}\exp\left(-\frac{(x-\zeta(0,x_{0}))^{2}}{2t}\right)$
$\displaystyle\qquad\qquad+\exp\left(-\frac{(x+\zeta(0,x_{0}))^{2}}{2t}\right)\Bigg{]}dx\nu_{0}(dx_{0})$
$\displaystyle\leqslant 2(2\pi t)^{-1/2}(\zeta(t,b)-\zeta(t,a))\leqslant
2C_{\mu,\sigma}(2\pi t)^{-1/2}(b-a)$

where the last inequality follows from (3.2). Since the map $t\mapsto
t^{-\delta/2}$ is integrable, the result then follows. ∎

Next we prove the boundary estimate that is a crucial part of the proof of
uniqueness for the SPDE. First, we establish an estimate in the simple case
where the particles follow a reflected Brownian motion with zero drift and
$\sigma\equiv 1$.

######  Lemma 3.4.

Assume that we have two particles X and Y following a reflected Brownian
motion with correlation $\rho$ with $\lvert\rho\rvert<1$ and initial values
$X_{0}$ and $Y_{0}$. i.e. $X=\lvert X_{0}+W^{1}\rvert$ and $Y=\lvert
Y_{0}+W^{2}\rvert$ Then we have the following estimate

$\mathbb{P}(0<\lvert X_{0}+W_{t}^{1}\rvert<\varepsilon,0<\lvert
Y_{0}+W_{t}^{2}\rvert<\varepsilon)\leqslant\frac{2}{\pi\sqrt{1-\rho^{2}}t}\varepsilon^{2}.$
(3.3)

###### Proof.

We have, using the explicit form of the folded bivariate Gaussian density,

$\displaystyle\mathbb{P}(0<\lvert X_{0}+W_{t}^{1}\rvert<\varepsilon,0<\lvert
Y_{0}+W_{t}^{2}\rvert<\varepsilon)$
$\displaystyle=\int_{0}^{\varepsilon}\int_{0}^{\varepsilon}\int_{0}^{\infty}\int_{0}^{\infty}\frac{1}{2\pi\sqrt{1-\rho^{2}}t}\times$
$\displaystyle\Bigg{[}\exp\left(-\frac{1}{2(1-\rho^{2})}\left(\frac{(x-x_{0})^{2}}{t}-2\rho\frac{(x-x_{0})(y-y_{0})}{t}+\frac{(y-y_{0})^{2}}{t}\right)\right)$
$\displaystyle+\exp\left(-\frac{1}{2(1-\rho^{2})}\left(\frac{(x+x_{0})^{2}}{t}-2\rho\frac{(x+x_{0})(y+y_{0})}{t}+\frac{(y+y_{0})^{2}}{t}\right)\right)$
$\displaystyle+\exp\left(-\frac{1}{2(1-\rho^{2})}\left(\frac{(x+x_{0})^{2}}{t}+2\rho\frac{(x+x_{0})(y-y_{0})}{t}+\frac{(y-y_{0})^{2}}{t}\right)\right)$
$\displaystyle+\exp\left(-\frac{1}{2(1-\rho^{2})}\left(\frac{(x-x_{0})^{2}}{t}+2\rho\frac{(x-x_{0})(y+y_{0})}{t}+\frac{(y+y_{0})^{2}}{t}\right)\right)\Bigg{]}\nu_{0}(dx_{0})\nu_{0}(dy_{0})dxdy$

Simple bounds on the Gaussian density give the result ∎

######  Proposition 3.5 (Boundary estimate).

For any $q>1$ and $t\in[0,T]$ we have as $\varepsilon\to 0$ that for
$i,j=1,\ldots,N$ with $i\neq j$

$\mathbbm{P}(0<X_{t}^{i}<\varepsilon,t<\tau^{i},0<X_{t}^{j}<\varepsilon,t<\tau^{j})=t^{-\frac{1}{q}}O(\varepsilon^{\frac{2}{q}}).$

###### Proof.

It is clear that we can estimate the probability from above by dropping the
stopping time conditions

$\mathbbm{P}(0<X_{t}^{i}<\varepsilon,t<\tau^{i},0<X_{t}^{j}<\varepsilon,t<\tau^{j})\leqslant\mathbbm{P}(0<X_{t}^{i}<\varepsilon,0<X_{t}^{j}<\varepsilon).$

We apply the scale transformation $\zeta$ and then use the multidimensional
version of Girsanov’s theorem (see [18, §1.7.4]) that preserves the
correlation structure between the Brownian motions. Note furthermore that
$\zeta(t,0)=0$ and by (3.2) we have $\zeta(t,\varepsilon)\leqslant
C_{\mu,\sigma}\varepsilon$. As a result, we have

$\mathbbm{P}(0<X_{t}^{i}<\varepsilon,0<X_{t}^{j}<\varepsilon)\leqslant\tilde{C}\mathbb{Q}(0<\lvert
X_{0}^{i}+W_{t}^{i}\rvert<C_{\mu,\sigma}\varepsilon,0<\lvert
X_{0}^{j}+W_{t}^{j}\rvert<C_{\mu,\sigma}\varepsilon)^{1/q}$

for $q>1$ and a constant $\tilde{C}>0$. Then, by Lemma 3.4 we get the desired
estimate. ∎

Note that $t\mapsto t^{-\frac{1}{q}}$ is integrable because $q>1$. A decay
result can also be found for the mass escaping to infinity.

######  Proposition 3.6 (Tail estimate).

For every $\alpha>0$, as $\lambda\to+\infty$

$\mathbb{E}\left[\nu^{N}_{t}(\lambda,+\infty)\right]=o(\exp(-\alpha\lambda)),\quad\text{uniformly
in $N\geqslant 1$ and $t\in[0,T]$}.$

###### Proof.

We will again rely on the results and notation of Lemma 3.2. Setting
$\pi_{0}=\nu_{0}\circ\zeta(0,\cdot)^{-1}$ we can estimate for $\lambda>0$

$\displaystyle F_{t}((\lambda,\infty))$
$\displaystyle=\int_{0}^{\infty}\mathbb{P}(\lvert W_{t}\rvert>\lambda|\lvert
W_{0}\rvert=x_{0})\pi_{0}(dx_{0})$ $\displaystyle=\frac{1}{\sqrt{2\pi
t}}\int_{0}^{\infty}\int_{\lambda}^{\infty}\left[e^{-\frac{(x-x_{0})^{2}}{2t}}+e^{-\frac{(x+x_{0})^{2}}{2t}}\right]dx\pi_{0}(dx_{0})$
$\displaystyle\leqslant\frac{2}{\sqrt{2\pi
t}}\int_{0}^{\infty}\int_{\lambda}^{\infty}e^{-\frac{(x-x_{0})^{2}}{2t}}dxd\pi_{0}(dx_{0})=2\int_{0}^{\infty}\mathbb{P}(W_{t}>\lambda|W_{0}=x_{0})\pi_{0}(dx_{0}).$

Now we can argue as in [16, Proposition 4.5] using Assumption 2.1 for the
initial condition and applying Lemma 3.2. ∎

##  4 Convergence of the particle system

In this section we show that limit points of the particle system satisfy the
SPDE (2.1). Furthermore, we verify that all limit points satisfy the
regularity conditions of the class $\Lambda$ and thus complete the proof of
Theorem 2.4.

###  4.1 Evolution equation for the empirical measures

We begin by finding an evolution equation for the empirical-measure process
$\nu^{N}$ of the particle system.

######  Proposition 4.1 (Evolution equation for $\nu^{N}$).

For all $N\geqslant 1$ the empirical measure process $\nu^{N}$ satisfies the
evolution equation

$\displaystyle\begin{split}\langle\nu_{t}^{N},\phi\rangle=&\langle\nu_{0}^{N},\phi\rangle+\int_{0}^{t}\langle\nu_{s}^{N},\mu(s,\cdot)\phi^{\prime}\rangle
ds+\frac{1}{2}\int_{0}^{t}\langle\nu_{s}^{N},\sigma^{2}(s,\cdot)\phi^{\prime\prime}\rangle
ds\\\
&+\int_{0}^{t}\langle\nu_{s}^{N},\rho_{s}\sigma(s,\cdot)\phi^{\prime}\rangle
dW_{s}^{0}+I_{t}^{N}(\phi)+J_{t}^{N}(\phi),\end{split}$ (4.1)

with

$I_{t}^{N}(\phi)=\frac{1}{N}\sum_{i=1}^{N}\int_{0}^{t\wedge\tau^{i}}(1-\rho_{s}^{2})^{\frac{1}{2}}\sigma(s,X_{s}^{i})\phi^{\prime}(X_{s}^{i})dW_{s}^{i}$

and

$J_{t}^{N}(\phi)=\phi(0)\left(\frac{\kappa}{N}\sum_{i=1}^{N}L_{t\wedge\tau^{i}}^{i}-\frac{1}{N}\sum_{i=1}^{N}\mathbbm{1}_{\tau^{i}\leqslant
t}\right),$

for all $\phi\in\mathcal{C}^{E,\kappa}_{0}(\mathbb{R})$, where
$I_{t}^{N}(\phi)\to 0$ and $J_{t}^{N}(\phi)\to 0$ as $N\to\infty$.

###### Proof.

Apply Itô’s formula to $\phi(X_{t\wedge\tau^{i}}^{i})$ to get

$\displaystyle\phi(X_{t\wedge\tau^{i}}^{i})=\phi(X_{0}^{i})+\int_{0}^{t\wedge\tau^{i}}\\!\\!\phi^{\prime}(X_{s}^{i})\mu(s,X_{s}^{i})ds+\int_{0}^{t\wedge\tau^{i}}\\!\\!\phi^{\prime}(X_{s}^{i})(1-\rho_{s}^{2})^{\frac{1}{2}}\sigma(s,X_{s}^{i})dW_{s}^{i}$
$\displaystyle\quad+\int_{0}^{t\wedge\tau^{i}}\\!\\!\phi^{\prime}(X_{s}^{i})\rho_{s}\sigma(s,X_{s}^{i})dW_{s}^{0}+\frac{1}{2}\int_{0}^{t\wedge\tau^{i}}\\!\\!\phi^{\prime\prime}(X_{s}^{i})\sigma^{2}(s,X_{s}^{i})ds+\int_{0}^{t\wedge\tau^{i}}\\!\\!\phi^{\prime}(X_{s}^{i})dL_{s}^{i}.$

Recall that

$\langle\nu_{t}^{N},\phi\rangle=\frac{1}{N}\sum_{i=1}^{N}\phi(X_{t}^{i})\mathbbm{1}_{t<\tau^{i}}.$

Using
$\phi(X_{t\wedge\tau^{i}})=\phi(X_{t})\mathbbm{1}_{t<\tau^{i}}+\phi(X_{\tau^{i}})\mathbbm{1}_{t\geqslant\tau^{i}}$,
the elastic boundary condition of the test function $\phi$ and integrating
against $\nu^{N}$ we obtain (4.1).

It remains to prove that the two terms $I^{N}(\phi)$ and $J^{N}(\phi)$ vanish
in a suitable way as $N\to\infty$. We start by computing the quadratic
variation of $I^{N}(\phi)$. By independence of the idiosyncratic Brownian
motions $W^{i}$ we obtain

$[I_{\cdot}^{N}(\phi)]_{t}=\frac{1}{N^{2}}\sum_{i=1}^{N}\int_{0}^{t}(1-\rho_{s}^{2})\sigma^{2}(s,X_{s}^{i})(\phi^{\prime}(X_{s}^{i}))^{2}ds.$

The boundedness of $\sigma^{2}$ and $\phi^{\prime}$ yields

$[I_{\cdot}^{N}(\phi)]_{t}=\left\|\phi^{\prime}\right\|_{\infty}^{2}O(N^{-1}),\quad\text{as
}N\to\infty.$

By Doob’s martingale inequality it follows that

$\mathbb{E}\left[\sup_{t\in[0,T]}\lvert
I_{t}^{N}(\phi)\rvert^{2}\right]\leqslant
4\mathbb{E}\left[I_{T}^{N}(\phi)^{2}\right]=4\mathbb{E}\left[[I_{\cdot}^{N}(\phi)]_{T}\right]\to
0\mbox{ as $N\to\infty$}.$

From [14, Theorem 2.5] we obtain

$\mathbb{E}\left[\sup_{t\in[0,T]}\lvert J_{t}^{N}(\phi)\rvert^{2}\right]\to
0,\qquad{\text{as $N\to\infty$}}.$

∎

###  4.2 Convergence to the Limit SPDE

The next step is now to show that limit points of the empirical measure
processes $\nu^{N}$ indeed solve our SPDE. First, we need to verify tightness
and conclude that there exist limit points. To do this we follow the approach
in [16], showing tightness in $D_{\mathcal{S}^{\prime}}$, the space of càdlàg
paths with values in the space of tempered distributions, equipped with the M1
topology and then verifying that the limit points are sub-probability measure-
valued processes.

######  Proposition 4.2.

For every $t\in[0,T]$ and $\eta>0$, the empirical loss process,
$\mathcal{L}^{N}$, obeys

$\lim_{\delta\to
0}\lim_{N\to\infty}\mathbb{P}\left(\mathcal{L}^{N}_{t+\delta}-\mathcal{L}_{t}^{N}\geqslant\eta\right)=0,\quad\mathcal{L}_{t}^{N}\coloneqq\frac{1}{N}\sum_{i=1}^{N}\mathbbm{1}_{\tau^{i}\leqslant
t}.$ (4.2)

###### Proof.

Using the triangle inequality we can estimate

$\displaystyle\mathbb{P}\left(\mathcal{L}^{N}_{t+\delta}-\mathcal{L}_{t}^{N}\geqslant\eta\right)\leqslant$
$\displaystyle\mathbb{P}\left(\left\lvert\mathcal{L}^{N}_{t+\delta}-\mathcal{L}_{t}^{N}-\frac{1}{N\kappa}\sum_{i=1}^{N}\left(L_{t+\delta\wedge\tau^{i}}^{i}-L_{t\wedge\tau^{i}}^{i}\right)\right\rvert\geqslant\frac{\eta}{2}\right)$
$\displaystyle+\mathbb{P}\left(\frac{1}{N\kappa}\sum_{i=1}^{N}\left(L_{t+\delta\wedge\tau^{i}}^{i}-L_{t\wedge\tau^{i}}^{i}\right)\geqslant\frac{\eta}{2}\right).$

Applying the convergence result for $J_{t}^{N}(\phi)$ as $N\to\infty$ in
Proposition 4.1, which holds for all
$\phi\in\mathcal{C}_{0}^{E,\kappa}(\mathbb{R})$, at the two time points $t$
and $t+\delta$ we see that the first probability on the right-hand side
vanishes as $N\to\infty$. Markov’s inequality can be applied to the second
term to obtain

$\mathbb{P}\left(\frac{1}{N\kappa}\sum_{i=1}^{N}\left(L_{t+\delta\wedge\tau^{i}}^{i}-L_{t\wedge\tau^{i}}^{i}\right)\geqslant\frac{\eta}{2}\right)\leqslant\frac{2}{\eta\kappa}\mathbb{E}\left[L_{t+\delta}^{i}-L_{t}^{i}\right].$

Since the particles $X^{i}$ are continuous, the local time processes $L^{i}$
are also continuous. Thus, we get convergence to $0$ as $\delta\to 0$ uniform
in $N\geqslant 1$. ∎

######  Proposition 4.3 (Tightness).

The sequence of empirical-measure processes $(\nu^{N})$ is tight on the space
$(D_{\mathcal{S}^{\prime}},M1)$ and the sequence $(\nu^{N},W^{0})$ is tight on
$(D_{\mathcal{S}^{\prime}},M1)\times(C_{\mathbb{R}},\left\|\cdot\right\|_{\infty})$.

###### Proof.

Using [25, Theorem 3.2] it suffices to show that, for arbitrary
$\phi\in\mathcal{S}$, the process $\langle\nu^{N},\phi\rangle$ is tight on
$D_{\mathbb{R}}$ in the M1 topology. For all $t\in[0,T]$, we decompose
$\langle\nu_{t}^{N},\phi\rangle$ as

$\langle\nu_{t}^{N},\phi\rangle=\langle\hat{\nu}^{N}_{t},\phi\rangle-\phi(0)\mathcal{L}_{t}^{N},\quad\hat{\nu}^{N}_{t}\coloneqq\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{t\wedge\tau^{i}}^{i}},$
(4.3)

with $\mathcal{L}^{N}_{t}$ as defined in (4.2). The term
$\phi(0)\mathcal{L}_{t}^{N}$ is monotone and does not contribute to the
M1-modulus of continuity. By [25, Propositions 4.1 and 4.2] and the arguments
in the proof of [16, Proposition 5.1] it is then sufficient to establish that
we have

$\mathbb{E}\left[\lvert\langle\hat{\nu}_{t}^{N},\phi\rangle-\langle\hat{\nu}_{s}^{N},\phi\rangle\rvert^{4}\right]=O(\lvert
t-s\rvert^{2})\quad\text{as }\lvert t-s\rvert\to 0,$ (4.4)

and that for any $\varepsilon>0$

$\lim_{\delta\to
0}\lim_{N\to\infty}\mathbb{P}\left(\sup_{t\leqslant\delta}\lvert\langle\nu_{t}^{N}-\nu_{0}^{N},\phi\rangle\rvert+\sup_{t\in(T-\delta,T)}\lvert\langle\nu_{T}^{N}-\nu_{t}^{N},\phi\rangle\rvert>\varepsilon\right)=0.$
(4.5)

For any $s,t\in[0,T]$ it holds that

$\displaystyle\mathbb{E}\left[\lvert\langle\hat{\nu}_{t}^{N},\phi\rangle-\langle\hat{\nu}_{s}^{N},\phi\rangle\rvert^{4}\right]$
$\displaystyle\leqslant\frac{1}{N}\sum_{i=1}^{N}\mathbb{E}\left[\lvert\phi(X_{t}^{i})-\phi(X_{s}^{i})\rvert^{4}\right]\leqslant\left\|\phi\right\|_{Lip}^{4}\mathbb{E}\left[\lvert
X_{t}^{i}-X_{s}^{i}\rvert^{4}\right]$

where $\left\|\phi\right\|_{Lip}$ denotes the Lipschitz constant of the
function $\phi$. It remains to deal with the expectation on the right-hand
side. By using the inequality $(a+b+c)^{4}\leqslant 27(a^{4}+b^{4}+c^{4})$ we
obtain

$\displaystyle\mathbb{E}\left[\lvert X_{t}^{i}-X_{s}^{i}\rvert^{4}\right]$
$\displaystyle\leqslant
27\Big{(}\mathbb{E}\left[\left\lvert\int_{s}^{t}\mu(u,X_{u}^{i})du\right\rvert^{4}\right]+\mathbb{E}\left[\left\lvert\int_{s}^{t}\sigma(u,X_{u}^{i})dB_{u}^{i}\right\rvert^{4}\right]+\mathbb{E}\left[\left\lvert
L_{t}^{i}-L_{s}^{i}\right\rvert^{4}\right]\Big{)}.$

Since the coefficient $\sigma$ is bounded, the stochastic integral is a
martingale and we can apply the Burkholder–Davis–Gundy inequality [31, Theorem
IV.42.1] to find that

$\mathbb{E}\left[\left\lvert\int_{s}^{t}\sigma(u,X_{u}^{i})dB_{u}^{i}\right\rvert^{4}\right]=O(\lvert
t-s\rvert^{2}),\quad\text{as }\lvert t-s\rvert\to 0.$

The boundedness assumption in Assumption 2.2 $(i)$ for the drift term $\mu$
yields

$\mathbb{E}\left[\left\lvert\int_{s}^{t}\mu(u,X_{u}^{i})du\right\rvert^{4}\right]=O(\lvert
t-s\rvert^{4}),\quad\text{as $\lvert t-s\rvert\to 0$}.$

The estimate $\mathbb{E}\left[\left\lvert
L_{t}^{i}-L_{s}^{i}\right\rvert^{4}\right]=O(\lvert t-s\rvert^{2})$, as
$\lvert t-s\rvert\to 0$, follows using the local time representation from the
Skorokhod problem. Hence, we have $\mathbb{E}[\lvert
X_{t}^{i}-X_{s}^{i}\rvert^{4}]=O(\lvert t-s\rvert^{2})$ as $\lvert
t-s\rvert\to 0$, and so we obtain (4.4).

To establish (4.5) it is enough to consider the small time interval
$(0,\delta)$ as the result for the interval $(T-\delta,T)$ follows similarly.
We can apply (4.3) to (4.5) to get

$\mathbb{P}\left(\sup_{t\leqslant\delta}\lvert\langle\nu_{t}^{N}-\nu_{0}^{N},\phi\rangle\rvert>\varepsilon\right)\leqslant\mathbb{P}\left(\sup_{t\leqslant\delta}\lvert\langle\hat{\nu}_{t}^{N}-\hat{\nu}_{0}^{N},\phi\rangle\rvert\geqslant\frac{\varepsilon}{2}\right)+\mathbb{P}\left(\lvert\phi(0)\rvert\mathcal{L}^{N}_{\delta}\geqslant\frac{\varepsilon}{2}\right).$

We can then again use the estimates in the poof of [16, Proposition 5.1]
combined with the arguments for (4.4) and obtain that the first term converges
to $0$ as $\delta\to 0$, uniformly in $N\geqslant 1$. Proposition 4.2 provides
the same result for the second term and the tightness follows. ∎

At this stage we have only established that there exists a subsequence such
that $(\nu^{N_{k}},W^{0})\to(\nu^{*},W^{0})$ weakly in the space of
$\mathcal{S}^{\prime}$-valued càdlàg processes equipped with the M1 topology.
However, we can deduce that the limiting processes actually take values in the
space of sub-probability measures supported on $[0,\infty)$ by the same
arguments as [16, Proposition 5.4].

######  Proposition 4.4.

Let $(\nu^{*},W^{0})$ realise the limiting law of $(\nu^{N_{k}},W^{0})$. Then
each $\nu_{t}^{*}$ is a sub-probability measure supported on $[0,\infty)$ for
all $t\in[0,T]$, with probability 1.

Next, we confirm that there is a suitable filtration for the limit points.

######  Lemma 4.5.

Let $(\nu^{*},W^{0})$ be the limit of a subsequence $(\nu^{N_{k}},W^{0})$.
Then there is a filtration $\mathcal{F}^{\nu^{*},W^{0}}$ so that $\nu^{*}$ is
adapted and $W^{0}$ is an $\mathcal{F}^{\nu^{*},W^{0}}$-Brownian motion.

###### Proof.

On the sample space $\Omega=D_{\mathcal{S}^{\prime}}\times C_{\mathbb{R}}$ we
define the filtration

$\mathcal{F}_{t}^{\nu^{*},W^{0}}\coloneqq\sigma\left(\\{(\nu^{*},W^{0})\mapsto(\langle\nu^{*}_{s},\phi\rangle,W^{0}_{s}):s<t,\phi\in\mathcal{S}\\}\right).$
(4.6)

This is a sensible filtration because the projections
$\pi^{\phi^{1},\cdot,\phi^{n}}_{t_{1},\cdot,t_{n}}$ given by

$\pi^{\phi^{1},\cdot,\phi^{n}}_{t_{1},\cdot,t_{n}}:D_{\mathcal{S}^{\prime}}\to\mathbb{R}^{n},\pi^{\phi^{1},\cdot,\phi^{n}}_{t_{1},\cdot,t_{n}}(\xi)=(\xi_{t_{1}}(\phi_{1}),\ldots,\xi_{t_{n}}(\phi_{n})),$

generate the Borel sets on $D_{\mathcal{S}^{\prime}}$ as shown in [25]. It is
clear that $\nu^{*}$ is adapted to $\mathcal{F}^{\nu^{*},W^{0}}$. It remains
to show that $W^{0}$ is a Brownian motion in this filtration. This follows
with the same arguments as in [26, Section 4.1]. ∎

To show that limit points of the empirical-measure process $\nu^{N}$ solve the
SPDE we will prove that solutions of the finite evolution equation of
Proposition 4.1 converge to solutions of our SPDE. This will be done using the
martingale approach of [16] and [17]. This approach easily allows for a higher
level a generality that also covers an extension considered in Section 7. The
martingale components are given by

$\displaystyle\begin{split}M^{\phi}(\xi)(t)\coloneqq&\langle\xi_{t},\phi\rangle-\langle\xi_{0},\phi\rangle-\int_{0}^{t}\langle\xi_{s},\mu(s,\cdot)\phi^{\prime}\rangle
ds\\\
&-\frac{1}{2}\int_{0}^{t}\langle\xi_{s},\sigma^{2}(s,\cdot)\phi^{\prime\prime}\rangle
ds,\\\
S^{\phi}(\xi)(t)\coloneqq&M^{\phi}(\xi)(t)^{2}-\int_{0}^{t}\langle\xi_{s},\sigma(s,\cdot)\rho(s)\phi^{\prime}\rangle^{2}ds,\\\
C^{\phi}(\xi,w)(t)\coloneqq&M^{\phi}(\xi)(t)\cdot
w(t)-\int_{0}^{t}\langle\xi_{s},\sigma(s,\cdot)\rho(s)\phi^{\prime}\rangle
ds\end{split}$ (4.7)

for $\xi\in D_{\mathcal{S}^{\prime}}$, $w\in C_{\mathbb{R}}$ and
$\phi\in\mathcal{C}^{E,\kappa}_{0}(\mathbb{R})$. Using the results from [16,
Section 5] we obtain the weak convergences

$\displaystyle M^{\phi}(\nu^{N_{k}})(t)$ $\displaystyle\to
M^{\phi}(\nu^{*})(t),$ $\displaystyle S^{\phi}(\nu^{N_{k}})(t)$
$\displaystyle\to S^{\phi}(\nu^{*})(t),$ $\displaystyle
C^{\phi}(\nu^{N_{k}},W^{0})(t)$ $\displaystyle\to
C^{\phi}(\nu^{*},W^{0})(t)\qquad{\text{in }\mathbb{R}}.$

on a deterministic cocountable subset of $[0,T]$ for all
$\phi\in\mathcal{C}^{E,\kappa}_{0}(\mathbb{R})$. Then, the convergence of the
solutions follows as in [16, Proposition 5.11].

######  Proposition 4.6 (Evolution equation).

Let $(N_{k})$ be a subsequence such that the weak convergence
$(\nu^{N_{k}},W^{0})\to(\nu^{*},W^{0})$ holds. Then, for every
$\phi\in\mathcal{C}^{E,\kappa}_{0}(\mathbb{R})$ the process
$M^{\phi}(\nu^{*}),S^{\phi}(\nu^{*})$ and $C^{\phi}(\nu^{*},W)$ are
martingales. As a consequence, $\nu^{*}$ and $W^{0}$ satisfy the limit SPDE
from Theorem 2.4.

###  4.3 Regularity of the Limit Solution

We now show that the solutions we constructed as limits from the particle
system satisfy the regularity conditions of Definition 2.3 using the estimates
from Section 3.

######  Proposition 4.7.

Let $(\nu^{*},W^{0})$ be a limit point of the sequence of empirical-measure
processes. Then the process $(\nu_{t}^{*})_{t\in[0,T]}$ is in the class
$\Lambda$.

###### Proof.

In Proposition 4.4 we already showed that $\nu^{*}$ takes values in the space
of sub-probability measures supported on $[0,\infty)$. It remains to prove the
regularity conditions on the sub-probability measures.

Consider a finite open interval $I=(x,y)\subseteq\mathbb{R}$. Take $\delta>0$
and any $\phi_{\delta}\in\mathcal{S}$ such that $\phi_{\delta}=1$ on I,
$\phi_{\delta}=0$ on $(-\infty,x-\delta)\cup(y+\delta,\infty)$ and
$\phi_{\delta}\in(0,1)$ otherwise. From $(\nu^{N_{k}},W)\to(\nu^{*},W)$ weakly
we get that $\int_{0}^{t}\langle\nu_{s}^{N_{k}},\phi_{\delta}\rangle
ds\to\int_{0}^{t}\langle\nu_{s}^{*},\phi_{\delta}\rangle$ds weakly by [34,
Theorem 11.5.1]. We have

$\mathbb{E}\left[\int_{0}^{T}\nu_{t}^{*}(I)dt\right]\leqslant\mathbb{E}\left[\int_{0}^{T}\langle\nu_{t}^{*},\phi_{\delta}\rangle
dt\right]=\lim_{k\to\infty}\mathbb{E}\left[\int_{0}^{T}\langle\nu_{t}^{N_{k}},\phi_{\delta}\rangle
dt\right].$

Considering the bound in Lemma 3.3 and taking the limit $\delta\to 0$ yields
(iv) in the definition of the class $\Lambda$.

Using the same approach we obtain that for $\varepsilon>0$

$\displaystyle\mathbb{E}\left[\nu_{t}^{*}(0,\varepsilon)^{2}\right]$
$\displaystyle\leqslant\liminf_{N\to\infty}\frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{j\neq
i,j=1}^{N}\mathbbm{P}(0<X_{t}^{i}<\varepsilon+\delta,0<X_{t}^{j}<\varepsilon+\delta).$

By employing the estimate in Proposition 3.5 and taking $\delta$ to zero we
obtain (iii) because $q>1$ ensures the necessary integrability.

For the exponential-tails condition consider $I=(\lambda,\eta)$ for $\eta>0$,
$\lambda<\eta$. The approach for (iv) yields

$\displaystyle\mathbb{E}\left[\int_{0}^{T}\nu_{t}^{*}(\lambda,\eta)dt\right]$
$\displaystyle\leqslant\liminf_{k\to\infty}\mathbb{E}\left[\int_{0}^{T}\nu_{t}^{N_{k}}(\lambda-\delta,\eta+\delta)dt\right]$
$\displaystyle\leqslant\liminf_{k\to\infty}\mathbb{E}\left[\int_{0}^{T}\nu_{t}^{N_{k}}(\lambda-\delta,\infty)dt\right]=o(e^{-\alpha(\lambda-\delta)}),$

by using Proposition 3.6. Taking the limit $\delta\to 0$ and then
$\eta\to\infty$ yields (ii). ∎

Combining the results from Proposition 4.6 and Proposition 4.7 we can conclude
our first main result Theorem 2.4.

##  5 Uniqueness of the SPDE

In this section we prove uniqueness of solutions to the SPDE in the class
$\Lambda$. Before presenting the uniqueness proof we show how we can derive an
alternative weak boundary formulation for the SPDE (2.1) that is helpful in
dealing with boundary terms appearing in the uniqueness proof.

To overcome the difficulties of working with solutions of low regularity, we
regularize the weak solutions to our SPDE through convolution with a nice
mollifier. The elastic heat kernel $G_{\varepsilon}^{E,\kappa}(x,y)$, defined
in (B.2), is a natural candidate for this, since it is an element of the space
of test functions $\mathcal{C}^{E,\kappa}_{0}(\mathbb{R})$ and since its
explicit form makes it easy to perform manipulations. This leads us to
consider the convolution operator $T_{\varepsilon}^{E,\kappa}$ given by
$(T_{\varepsilon}^{E,\kappa}\zeta)(x)=\langle\zeta,G_{\varepsilon}^{E,\kappa}(x,\cdot)\rangle$.
While we use a different operator, we take the same approach as in [16, 17]
and work in the space $H^{-1}$, the dual of the first Sobolev space. This
approach now rests on the following crucial inequality, connecting $L^{2}$
estimates for the regularized process with an $H^{-1}$ estimate for the non-
regularized process:

$\left\|\zeta\right\|_{-1}\leqslant C\liminf_{\varepsilon\to
0}\left(\left\|\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\zeta\right\|_{L^{2}(0,\infty)}+\left\lvert\int_{0}^{\infty}T_{\varepsilon}^{E,\kappa}\zeta(y)dy\right\rvert\right),$
(5.1)

for any $\zeta\in H^{-1}$, where $\partial^{-1}_{x}$ denotes the anti-
derivative as defined in (B.3). The proof of this can be found in Lemma B.3
and further details are reserved for Appendix B.

Differently from [16, 17], here we work with $H^{-1}$ as the dual of
$H^{1}(0,\infty)$ rather than $H^{1}_{0}(0,\infty)$. In our proof of
uniqueness, we will show that two different solutions to the SPDE, say $\nu$
and $\tilde{\nu}$, must satisfy
$\mathbb{E}[\left\|\nu_{t}-\tilde{\nu}_{t}\right\|_{-1}]=0$ for all
$t\in[0,T]$. In doing so, the first term on the right-hand side of (5.1) is
only concerned with the difference $\zeta=\nu_{t}-\tilde{\nu}_{t}$ restricted
to $(0,\infty)$, while forcing the second term to zero ensures that potential
point masses of $\nu_{t}$ and $\tilde{\nu}_{t}$ at the origin must be of the
same value (recall the solutions are sub-probability measures, which a priori
need not have the same mass at any given time). As discussed in Section 2, the
unique solution turns out to have a density, but we are not making any such
assumptions on our class of solutions at this point.

Before proceeding to show uniqueness, we first give two auxiliary results that
will help us deal with the behaviour of our estimates near the boundary.

######  Lemma 5.1.

Let $\nu$ be a process in the class $\Lambda$ and $p_{\varepsilon}$ the
Gaussian heat kernel defined in (A.1). Then we have

$\mathbb{E}\left[\int_{0}^{T}\int_{0}^{\infty}\left(\int_{0}^{\infty}p_{\varepsilon}(x+y)\nu_{t}(dy)\right)^{2}dxdt\right]\to
0,\qquad{\text{ as }\varepsilon\to 0}.$

###### Proof.

Arguing as in [16, Lemma 7.6] we have

$\displaystyle\lvert\langle\nu_{t},p_{\varepsilon}(x+\cdot)\rangle\rvert$
$\displaystyle\leqslant
e^{-x^{2}/\varepsilon}\int_{0}^{\infty}p_{\varepsilon}(y)\nu_{t}(dy)$
$\displaystyle\leqslant
c_{1}e^{-x^{2}/\varepsilon}\varepsilon^{-1/2}[\nu_{t}(0,\varepsilon^{\eta})+\exp(-\varepsilon^{2\eta-1}/2)],$

for some $\eta\in(0,\frac{1}{2})$ and $c_{1}>0$. Squaring and integrating over
$x>0$ gives

$\int_{0}^{\infty}\lvert\langle\nu_{t},p_{\varepsilon}(x+\cdot)\rangle\rvert^{2}dx\leqslant
c_{2}\varepsilon^{-1/2}[\nu_{t}(0,\varepsilon^{\eta})^{2}+\exp(-\varepsilon^{2\eta-1})],$

for some $c_{2}>0$. Since $\nu$ is in the class $\Lambda$, the boundary decay
yields a $\gamma>0$ so that

$\mathbbm{E}\left[\int_{0}^{T}\int_{0}^{\infty}\lvert\langle\nu_{t},p_{\varepsilon}(x+\cdot)\rangle\rvert^{2}dx\right]=O(\varepsilon^{\eta(1+\gamma)-\frac{1}{2}})+O(\varepsilon^{-\frac{1}{2}}\exp(-\varepsilon^{2\eta-1})).$

Therefore, we get the desired estimate by choosing $\eta$ such that
$(1+\gamma)^{-1}<2\eta<1$. ∎

######  Lemma 5.2.

Let $\nu$ be a process in the class $\Lambda$ and let
$g_{\varepsilon}^{E,\kappa}$ denote the correction term in the definition of
the elastic heat kernel (A.3). Then we have that

$\mathbb{E}\left[\int_{0}^{T}\int_{0}^{\infty}\left(\int_{0}^{\infty}g_{\varepsilon}^{E,\kappa}(x,y)\nu_{t}(dy)\right)^{2}dxdt\right]\to
0,\qquad{\text{for }\varepsilon\to 0}.$

###### Proof.

We start by looking at the inner integral. Using Lemma A.3 we get

$\displaystyle\int_{0}^{\infty}g_{\varepsilon}^{E,\kappa}(x,y)\nu_{t}(dy)\leqslant\kappa
e^{-\frac{x^{2}}{2\varepsilon}}\nu_{t}(0,\infty)$
$\displaystyle\leqslant\kappa e^{-\frac{x^{2}}{2\varepsilon}}.$

Now, considering the integral in $x$ we have

$\displaystyle\int_{0}^{\infty}\left(\int_{0}^{\infty}g_{\varepsilon}^{E,\kappa}(x,y)\nu_{t}(dy)\right)^{2}dx$
$\displaystyle\leqslant\kappa^{2}\int_{0}^{\infty}e^{-\frac{x^{2}}{\varepsilon}}dx=\frac{1}{2}\kappa^{2}\sqrt{\pi}\varepsilon^{1/2}.$

We then obtain

$\mathbb{E}\left[\int_{0}^{T}\int_{0}^{\infty}\left(\int_{0}^{\infty}g_{\varepsilon}^{E,\kappa}(x,y)\nu_{t}(dy)\right)^{2}dxdt\right]=O(\varepsilon^{1/2})$
(5.2)

and the result follows. ∎

###  5.1 Weak Boundary Condition and Elastic Boundary Terms

As discussed in Section 1.2, we need a particular weak formulation of the
elastic boundary condition. This plays a critical role in dealing with the
boundary terms appearing in (5.14) in the final uniqueness proof, presented in
the next subsection. With $\phi^{E,\kappa}_{\varepsilon}$ as in (1.7), we note
that this gives a smooth function approximating 1 on $[0,\infty)$ and
satisfying

$\displaystyle\kappa\phi^{E,\kappa}_{\varepsilon}(0)=\partial_{x}\phi^{E,\kappa}_{\varepsilon}(0),$
(5.3)

which it inherits from the elastic kernel $G^{E,\kappa}_{\varepsilon}$. We
then obtain the following.

######  Proposition 5.3 (Weak boundary condition).

Let $(\nu,W^{0})$ be a solution to (2.1) with $\nu$ in the class $\Lambda$ and
let $\phi^{E,\kappa}_{\varepsilon}$ be as in (1.7). Then (1.8) holds for all
$\varepsilon>0$.

###### Proof.

By (5.3), we see that the convolution
$\phi^{E,\kappa}_{\varepsilon}(x)=\int_{0}^{\infty}G^{E,\kappa}_{\varepsilon}(x,y)dy$
results in a valid test function in $\mathcal{C}^{E,\kappa}_{0}(\mathbb{R})$.
Using also Fubini’s theorem to change the order of integration between $d\nu$
and $dx$, the assumption that $\nu$ is a solution to (2.1) therefore gives

$\displaystyle\langle\nu_{t},\phi^{E,\kappa}_{\varepsilon}\rangle-\langle\nu_{0},\phi^{E,\kappa}_{\varepsilon}\rangle=\int_{0}^{\infty}\langle\nu_{t},G^{E,\kappa}_{\varepsilon}(x,\cdot)\rangle-\langle\nu_{0},G^{E,\kappa}_{\varepsilon}(x,\cdot)\rangle
dx$
$\displaystyle\;=\int_{0}^{\infty}\Big{(}\int_{0}^{t}\langle\nu_{s},\mu_{s}\partial_{y}G_{\varepsilon}^{E,\kappa}(x,\cdot)\rangle
ds$
$\displaystyle\qquad\qquad+\int_{0}^{t}\langle\nu_{s},\frac{\sigma_{s}^{2}}{2}\partial_{yy}G_{\varepsilon}^{E,\kappa}(x,\cdot)\rangle
ds+\int_{0}^{t}\rho_{s}\langle\nu_{s},\sigma_{s}\partial_{y}G_{\varepsilon}^{E,\kappa}(x,\cdot)\rangle
dW_{s}^{0}\Big{)}dx.$

Next, we employ Lemma A.2 to switch the derivatives from $y$ to $x$. This
yields

$\displaystyle\langle\nu_{t},$
$\displaystyle\phi^{E,\kappa}_{\varepsilon}\rangle-\langle\nu_{0},\phi^{E,\kappa}_{\varepsilon}\rangle$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\Big{(}\int_{0}^{t}-\partial_{x}\langle\nu_{s},\mu_{s}G_{\varepsilon}^{E}(x,\cdot)\rangle
ds+\int_{0}^{t}\partial_{xx}\langle\nu_{s},\frac{\sigma_{s}^{2}}{2}G_{\varepsilon}^{E}(x,\cdot)\rangle
ds$
$\displaystyle-\int_{0}^{t}\rho_{s}\partial_{x}\langle\nu_{s},\sigma_{s}G_{\varepsilon}^{E}(x,\cdot)\rangle
dW_{s}^{0}\Big{)}dx$
$\displaystyle+\int_{0}^{\infty}\left(2\int_{0}^{t}\partial_{x}\langle\nu_{s},\mu_{s}p_{\varepsilon}(x+\cdot)\rangle
ds+2\int_{0}^{t}\rho_{s}\partial_{x}\langle\nu_{s},\sigma_{s}p_{\varepsilon}(x+\cdot)\rangle
dW_{s}^{0}\right)dx$
$\displaystyle-\int_{0}^{\infty}\left(2\int_{0}^{t}\partial_{x}\langle\nu_{s},\mu_{s}g_{\varepsilon}^{E,\kappa}(x,\cdot)\rangle
ds-2\int_{0}^{t}\rho_{s}\partial_{x}\langle\nu_{s},\sigma_{s}g_{\varepsilon}^{E,\kappa}(x,\cdot)\rangle
dW_{s}^{0}\right)dx.$

with $g_{\varepsilon}^{E,\kappa}$ as defined in (A.3) and $p_{\varepsilon}$
the Gaussian heat kernel given in (A.1). We can now interchange the order of
integration again and compute the integrals in $x$ to obtain

$\displaystyle\langle\nu_{t},\phi^{E,\kappa}_{\varepsilon}\rangle-\langle\nu_{0},\phi^{E,\kappa}_{\varepsilon}\rangle=$
$\displaystyle\int_{0}^{t}\langle\nu_{s},\mu_{s}G_{\varepsilon}^{E,\kappa}(0,\cdot)\rangle
ds-\int_{0}^{t}\partial_{x}\langle\nu_{s},\frac{\sigma^{2}}{2}G_{\varepsilon}^{E,\kappa}(0,\cdot)\rangle
ds$
$\displaystyle+\int_{0}^{t}\rho_{s}\langle\nu_{s},\sigma_{s}G_{\varepsilon}^{E,\kappa}(0,\cdot)\rangle
dW_{s}^{0}$
$\displaystyle-2\int_{0}^{t}\langle\nu_{s},\mu_{s}p_{\varepsilon}(\cdot)\rangle
ds-2\int_{0}^{t}\rho_{s}\langle\nu_{s},\sigma_{s}p_{\varepsilon}(\cdot)\rangle
dW_{s}^{0}$
$\displaystyle+2\int_{0}^{t}\langle\nu_{s},\mu_{s}g_{\varepsilon}^{E,\kappa}(0,\cdot)\rangle
ds+2\int_{0}^{t}\rho_{s}\langle\nu_{s},\sigma_{s}g_{\varepsilon}^{E,\kappa}(0,\cdot)\rangle
dW_{s}^{0}.$

Using the specific form of the elastic heat kernel we have that

$\displaystyle\int_{0}^{t}\langle\nu_{s},\mu_{s}G_{\varepsilon}^{E,\kappa}(0,\cdot)\rangle
ds=$ $\displaystyle
2\int_{0}^{t}\langle\nu_{s},\mu_{s}p_{\varepsilon}(\cdot)\rangle
ds-\int_{0}^{t}\langle\nu_{s},g_{\varepsilon}^{E,\kappa}(0,\cdot)\rangle ds$
$\displaystyle\int_{0}^{t}\rho_{s}\langle\nu_{s},\sigma_{s}G_{\varepsilon}^{E,\kappa}(0,\cdot)\rangle
dW_{s}^{0}=$ $\displaystyle
2\int_{0}^{t}\rho_{s}\langle\nu_{s},\sigma_{s}p_{\varepsilon}(\cdot)\rangle
dW_{s}^{0}-\int_{0}^{t}\rho_{s}\langle\nu_{s},\sigma_{s}g_{\varepsilon}^{E,\kappa}(0,\cdot)\rangle
dW_{s}^{0}.$

By using these relations the equation above simplifies to

$\displaystyle\langle\nu_{t},\phi^{E,\kappa}_{\varepsilon}\rangle-\langle\nu_{0},\phi^{E,\kappa}_{\varepsilon}\rangle=$
$\displaystyle-\int_{0}^{t}\partial_{x}\langle\nu_{s},\frac{\sigma_{s}^{2}}{2}G_{\varepsilon}^{E,\kappa}(0,\cdot)\rangle
ds$
$\displaystyle+\int_{0}^{t}\langle\nu_{s},\mu_{s}g_{\varepsilon}^{E,\kappa}(0,\cdot)\rangle
ds+\int_{0}^{t}\rho\langle\nu_{s},\sigma_{s}g_{\varepsilon}^{E,\kappa}(0,\cdot)\rangle
dW_{s}^{0}.$

Finally, we can then use the boundary condition of the elastic heat kernel to
obtain the relaxed boundary condition (1.8). ∎

######  Lemma 5.4 (Elastic Boundary Term).

Let $\nu$ and $\tilde{\nu}$ be two solutions to the SPDE (2.1) in the class
$\Lambda$ and denote their difference by $\Delta$, i.e.
$\Delta\coloneqq\nu-\tilde{\nu}$. Then we have

$\displaystyle\mathbb{E}\Bigg{[}\int_{0}^{t}-\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{s}(0)\sigma_{s}^{2}(0)T_{\varepsilon}^{E,\kappa}\Delta_{s}(0)$
$\displaystyle-\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{s}(0)\left(\mathcal{E}_{s,\varepsilon}^{\sigma^{2}}-\tilde{\mathcal{E}}_{s,\varepsilon}^{\sigma^{2}}\right)(0)ds\Bigg{]}$
$\displaystyle=-\frac{1}{\kappa}\mathbb{E}\left[(\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{t}(0))^{2}\right]+o(1).$

with
$\mathcal{E}^{\sigma^{2}}_{s,\varepsilon},\tilde{\mathcal{E}}^{\sigma^{2}}_{s,\varepsilon}$
error terms as defined in (5.10) for measures $\nu_{s}$ and $\tilde{\nu_{s}}$,
respectively.

###### Proof.

Combining Proposition 5.3 with the definition of the operator
$T_{\varepsilon}^{E,\kappa}$, the anti-derivative and the error terms defined
in (5.10) we obtain

$\displaystyle\begin{split}&\int_{0}^{s}\sigma_{u}^{2}(0)T_{\varepsilon}^{E,\kappa}\Delta_{u}(0)du=\frac{2}{\kappa}\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{s}(0)-\int_{0}^{s}(\mathcal{E}_{u,\varepsilon}^{\sigma^{2}}-\mathcal{E}_{u,\varepsilon}^{\tilde{\sigma}^{2}})(0)du\\\
&\qquad+\frac{2}{\kappa}\int_{0}^{s}\langle\Delta_{u},\mu_{u}g_{\varepsilon}^{E,\kappa}(0,\cdot)\rangle
du+\frac{2}{\kappa}\int_{0}^{s}\rho_{u}\langle\Delta_{u},\sigma_{u}g_{\varepsilon}^{E,\kappa}(0,\cdot)\rangle
dW_{u}^{0}.\end{split}$ (5.4)

With this relation we obtain

$\displaystyle-\mathbb{E}\Bigg{[}\int_{0}^{t}\sigma_{s}(0)^{2}\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}$
$\displaystyle\Delta_{s}(0)T_{\varepsilon}^{E,\kappa}\Delta_{s}(0)ds\Bigg{]}$
$\displaystyle=$
$\displaystyle-\mathbb{E}\left[\int_{0}^{t}\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{s}(0)d(\int_{0}^{s}\sigma_{u}^{2}(0)T_{\varepsilon}^{E,\kappa}\Delta_{u}(0)du)\right]$
$\displaystyle=$
$\displaystyle-\frac{1}{\kappa}\mathbb{E}\left[2\int_{0}^{t}\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{s}(0)d(\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{s}(0))\right]$
$\displaystyle+\mathbb{E}\left[\int_{0}^{t}\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{s}(0)(\mathcal{E}_{s,\varepsilon}^{\sigma^{2}}-\tilde{\mathcal{E}}_{s,\varepsilon}^{\sigma^{2}})(0)ds\right]$
$\displaystyle-\frac{2}{\kappa}\mathbb{E}\left[\int_{0}^{t}\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{s}(0)\langle\Delta_{s},\mu_{s}g_{\varepsilon}^{E,\kappa}(0,\cdot)\rangle
ds\right]$
$\displaystyle-\frac{2}{\kappa}\mathbb{E}\left[\int_{0}^{t}\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{s}(0)\rho_{s}\langle\Delta_{s},\sigma_{s}g_{\varepsilon}^{E,\kappa}(0,\cdot)\rangle
dW_{s}^{0}\right].$

By combining the boundedness of $\sigma$ with Lemma A.3 we get that the
stochastic integral in the last line is a martingale and thus the term
vanishes. For the first term, using the fact that
$\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{\cdot}(0)$ is a continuous
semimartingale, we have

$(\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{t}(0))^{2}=2\int_{0}^{t}\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{s}(0)d(\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{s}(0))+[\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{\cdot}(0)]_{t}.$

Using equation (5.4) we obtain that

$[\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{\cdot}(0)]_{t}=\int_{0}^{t}\rho_{s}^{2}(\langle\Delta_{s},\sigma_{s}g_{\varepsilon}^{E,\kappa}(0,\cdot)\rangle)^{2}ds.$

This yields

$\displaystyle-\mathbb{E}\Bigg{[}\int_{0}^{t}\sigma_{s}(0)^{2}\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{s}(0)T_{\varepsilon}^{E,\kappa}\Delta_{s}(0)ds\Bigg{]}$
(5.5)
$\displaystyle=-\frac{1}{\kappa}\mathbb{E}\left[(\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{t}(0))^{2}\right]+\mathbb{E}\left[\int_{0}^{t}\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{s}(0)(\mathcal{E}_{s,\varepsilon}^{\sigma^{2}}-\tilde{\mathcal{E}}_{s,\varepsilon}^{\sigma^{2}})(0)ds\right]+J_{1}+J_{2}$
(5.6)

with the two terms $J_{1}$ and $J_{2}$ defined as

$\displaystyle J_{1}$
$\displaystyle\coloneqq\frac{1}{\kappa}\mathbb{E}\left[\int_{0}^{t}\rho_{s}^{2}(\langle\Delta_{s},\sigma_{s}g_{\varepsilon}^{E,\kappa}(0,\cdot)\rangle)^{2}ds\right]$
$\displaystyle J_{2}$
$\displaystyle\coloneqq-\frac{2}{\kappa}\mathbb{E}\left[\int_{0}^{t}\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{s}(0)\langle\Delta_{s},\mu_{s}g_{\varepsilon}^{E,\kappa}(0,\cdot)\rangle
ds\right].$

From Lemma A.3 we have the estimate

$g_{\varepsilon}^{E,\kappa}(0,y)\leqslant\kappa
e^{-\frac{y^{2}}{2\varepsilon}}.$

For a process $\xi$ in the class $\Lambda$ we get

$\displaystyle\begin{split}\int_{0}^{\infty}g_{\varepsilon}^{E,\kappa}(0,y)\xi_{s}(dy)&\leqslant\kappa\int_{0}^{\varepsilon^{1/4}}e^{-\frac{y^{2}}{2\varepsilon}}\xi_{s}(dy)+\kappa\int_{\varepsilon^{1/4}}^{\infty}e^{-\frac{y^{2}}{2\varepsilon}}\xi_{s}(dy)\\\
&\leqslant\kappa\xi_{s}(0,\varepsilon^{1/4})+\kappa
e^{-\frac{\varepsilon^{-1/2}}{2}}\end{split}$ (5.7)

The coefficients $\mu,\sigma,\rho$ are bounded by assumption and

$\lvert\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{s}(0)\rvert\leqslant\lvert\Delta_{s}(0,\infty)\rvert\leqslant
2.$

We can combine this with (5.7) to estimate

$\displaystyle J_{1}\leqslant
C\left(\int_{0}^{t}\mathbb{E}\left[\nu_{s}(0,\varepsilon^{1/4})^{2}\right]ds+\int_{0}^{t}\mathbb{E}\left[\tilde{\nu}_{s}(0,\varepsilon^{1/4})^{2}\right]ds\right)+O(e^{-\varepsilon^{-1/2}})\to
0,$ (5.8)

as $\varepsilon\to 0$, because $\nu$ and $\tilde{\nu}$ are in the class
$\Lambda$. For $J_{2}$ we similarly get

$\displaystyle\begin{split}\lvert J_{2}\rvert&\leqslant
C\mathbb{E}\left[\int_{0}^{t}\langle\nu_{s},g_{\varepsilon}^{E,\kappa}(0,\cdot)+\langle\tilde{\nu}_{s},g_{\varepsilon}^{E,\kappa}(0,\cdot)\rangle\rangle
ds\right]\\\
&\leqslant\int_{0}^{t}\mathbb{E}\left[\nu_{s}(0,\varepsilon^{1/4})\right]+\mathbb{E}\left[\tilde{\nu}_{s}(0,\varepsilon^{1/4})\right]ds+O(e^{-\varepsilon^{-1/2}/2})\to
0,\quad\text{as $\varepsilon\to 0$},\end{split}$ (5.9)

again, by the properties of the class $\Lambda$. Combining (5.5), (5.8) and
(5.9) we obtain

$\displaystyle-\mathbb{E}\Bigg{[}\int_{0}^{t}\sigma_{s}(0)^{2}\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{s}(0)T_{\varepsilon}^{E,\kappa}\Delta_{s}(0)ds\Bigg{]}$
$\displaystyle=-\frac{1}{\kappa}\mathbb{E}\left[(\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{t}(0))^{2}\right]+\mathbb{E}\left[\int_{0}^{t}\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{s}(0)(\mathcal{E}_{s,\varepsilon}^{\sigma^{2}}-\tilde{\mathcal{E}}_{s,\varepsilon}^{\sigma^{2}})(0)ds\right]+o(1)$

as $\varepsilon\to 0$, and so the result follows. ∎

###  5.2 Proof of uniqueness for solutions to the SPDE

We can now give the proof of Theorem 2.5. Lemma 5.4 from the previous section
plays an important role in our arguments. It is used in (5.14) to deal with
the boundary terms that arise when integrating by parts.

###### Proof of Theorem 2.5.

Let $(\nu,W^{0})$ be a solution to the SPDE (2.1) in the class $\Lambda$. Take
the elastic heat kernel function $y\mapsto G_{\varepsilon}^{E,\kappa}(x,y)$ as
a test function in the SPDE and apply the switching of derivatives from $y$ to
$x$ using Lemma A.2 to obtain the following dynamics for the smoothed measure
$T^{E,\kappa}_{\varepsilon}\nu_{t}$

$\displaystyle dT_{\varepsilon}^{E,\kappa}\nu_{t}(x)=$
$\displaystyle-\partial_{x}\langle\nu_{t},\mu_{t}G_{\varepsilon}^{E,\kappa}(x,\cdot)\rangle
dt+\frac{1}{2}\partial_{xx}\langle\nu_{t},\sigma_{t}^{2}G_{\varepsilon}^{E,\kappa}(x,\cdot)\rangle
dt$
$\displaystyle-\rho_{t}\partial_{x}\langle\nu_{t},\sigma_{t}G_{\varepsilon}^{E,\kappa}(x,\cdot)\rangle
dW_{t}^{0}$
$\displaystyle+2\partial_{x}\langle\nu_{t},\mu_{t}p_{\varepsilon}(x+\cdot)\rangle
dt+2\rho_{t}\partial_{x}\langle\nu_{t},\sigma_{t}p_{\varepsilon}(x+\cdot)\rangle
dW_{t}^{0}$
$\displaystyle-2\partial_{x}\langle\nu_{t},\mu_{t}g_{\varepsilon}^{E,\kappa}(x,\cdot)\rangle
dt-2\rho_{t}\partial_{x}\langle\nu_{t},\sigma_{t}g_{\varepsilon}^{E,\kappa}(x,\cdot)\rangle
dW_{t}^{0}.$

Next, we integrate to introduce the anti-derivative defined in (B.3)

$\displaystyle d\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\nu_{t}(x)=$
$\displaystyle-\langle\nu_{t},\mu_{t}G_{\varepsilon}^{E,\kappa}(x,\cdot)\rangle
dt+\frac{1}{2}\partial_{x}\langle\nu_{t},\sigma_{t}^{2}G_{\varepsilon}^{E,\kappa}(x,\cdot)\rangle
dt$
$\displaystyle-\rho_{t}\langle\nu_{t},\sigma_{t}G_{\varepsilon}^{E,\kappa}(x,\cdot)\rangle
dW_{t}^{0}$
$\displaystyle+2\langle\nu_{t},\mu_{t}p_{\varepsilon}(x+\cdot)\rangle
dt+2\rho_{t}\langle\nu_{t},\sigma_{t}p_{\varepsilon}(x+\cdot)\rangle
dW_{t}^{0}$
$\displaystyle-2\langle\nu_{t},\mu_{t}g_{\varepsilon}^{E,\kappa}(x,\cdot)\rangle
dt-2\rho_{t}\langle\nu_{t},\sigma_{t}g_{\varepsilon}^{E,\kappa}(x,\cdot)\rangle
dW_{t}^{0}.$

We can take the coefficients $\mu,\sigma$ and $\sigma^{2}$ outside of the
integration against the measure $\nu_{t}$ by introducing error terms
$\mathcal{E}_{t,\varepsilon}^{h}$ defined by

$\mathcal{E}_{t,\varepsilon}^{h}\coloneqq\langle\nu_{t},h_{t}(\cdot)G^{E,\kappa}_{\varepsilon}(x,\cdot)\rangle-
h_{t}(x)T_{\varepsilon}^{E,\kappa}\nu_{t}(x).$ (5.10)

for $h_{t}(x)=h(t,x)$ representing $\mu,\sigma$ or $\sigma^{2}$. The equation
then becomes

$\displaystyle d\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\nu_{t}=$
$\displaystyle-(\mu_{t}T_{\varepsilon}^{E,\kappa}\nu_{t}+\mathcal{E}_{t,\varepsilon}^{\mu})dt+\frac{1}{2}\partial_{x}(\sigma^{2}_{t}T_{\varepsilon}^{E,\kappa}\nu_{t}+\mathcal{E}_{t,\varepsilon}^{\sigma^{2}})dt$
$\displaystyle-\rho_{t}(\sigma_{t}T_{\varepsilon}^{E,\kappa}\nu_{t}+\mathcal{E}_{t,\varepsilon}^{\sigma})dW_{t}^{0}$
$\displaystyle+2\langle\nu_{t},\mu_{t}p_{\varepsilon}(x+\cdot)\rangle
dt+2\rho_{t}\langle\nu_{t},\sigma_{t}p_{\varepsilon}(x+\cdot)\rangle
dW_{t}^{0}$
$\displaystyle-2\langle\nu_{t},\mu_{t}g_{\varepsilon}^{E,\kappa}(x+\cdot)\rangle
dt-2\rho_{t}\langle\nu_{t},\sigma_{t}g_{\varepsilon}^{E,\kappa}(x+\cdot)\rangle
dW_{t}^{0}.$

To simplify the notation, we denote by $o_{sq}(1)$ any family of functions
$\\{(f_{t,\varepsilon})_{t\in[0,T]}\\}_{\varepsilon>0}$ such that

$\mathbb{E}\int_{0}^{T}\left\|f_{t,\varepsilon}\right\|^{2}_{L^{2}(0,\infty)}dt\to
0,\quad\text{as }\varepsilon\to 0.$ (5.11)

By using the limit behavior from Lemma 5.1 and Lemma 5.2, as well as the
results for $\mathcal{E}^{h}_{t,\varepsilon}$ from [16, Lemma 8.1], which also
hold in the elastic case because of the regularity of the class $\Lambda$, we
have

$\displaystyle\begin{split}d\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\nu_{t}=&-\mu_{t}T_{\varepsilon}^{E,\kappa}\nu_{t}dt+\frac{1}{2}\partial_{x}(\sigma^{2}_{t}T_{\varepsilon}^{E,\kappa}\nu_{t}+\mathcal{E}_{t,\varepsilon}^{\sigma^{2}})dt\\\
&-\rho_{t}\sigma_{t}T_{\varepsilon}^{E,\kappa}\nu_{t}dW_{t}^{0}+o_{sq}(1)dt+o_{sq}(1)dW_{t}^{0}.\end{split}$
(5.12)

Let $(\tilde{\nu},W^{0})$ be another solution to the linear SPDE with the same
Brownian motion $W^{0}$ and define the difference of the two solutions as
$\Delta\coloneqq\nu-\tilde{\nu}$. The dynamics of this are

$\displaystyle d\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{t}=$
$\displaystyle-\mu_{t}T_{\varepsilon}^{E,\kappa}\Delta_{t}dt+\frac{1}{2}\partial_{x}(\sigma_{t}^{2}T_{\varepsilon}^{E,\kappa}\Delta_{t}+\mathcal{E}_{t,\varepsilon}^{\sigma^{2}}-\tilde{\mathcal{E}}_{t,\varepsilon}^{\sigma^{2}})dt$
$\displaystyle-\rho_{t}\sigma_{t}T_{\varepsilon}^{E,\kappa}\Delta_{t}dW_{t}^{0}+o_{sq}(1)dt+o_{sq}(1)dW_{t}^{0}.$

Next we use Itô’s formula to find an equation for the square of the anti-
derivative

$\displaystyle d(\partial_{x}^{-1}T^{E,\kappa}_{\varepsilon}\Delta_{t})^{2}=$
$\displaystyle-2\mu_{t}(\partial_{x}^{-1}T^{E,\kappa}_{\varepsilon}\Delta_{t})T_{\varepsilon}^{E,\kappa}\Delta_{t}dt$
$\displaystyle+(\partial_{x}^{-1}T^{E,\kappa}_{\varepsilon}\Delta_{t})\partial_{x}(\sigma_{t}^{2}T_{\varepsilon}^{E,\kappa}\Delta_{t}+\mathcal{E}_{t,\varepsilon}^{\sigma^{2}}-\tilde{\mathcal{E}}_{t,\varepsilon}^{\sigma^{2}})dt$
$\displaystyle+\sigma^{2}_{t}\rho_{t}^{2}(T^{E,\kappa}_{\varepsilon}\Delta_{t})^{2}dt$
$\displaystyle-2(\partial_{x}^{-1}T^{E,\kappa}_{\varepsilon}\Delta_{t})\sigma_{t}\rho_{t}T^{E,\kappa}_{\varepsilon}\Delta_{t}dW_{t}^{0}$
$\displaystyle+\rho_{t}\sigma_{t}T_{\varepsilon}^{E,\kappa}\Delta_{t}o_{sq}(1)dt$
$\displaystyle+(\partial_{x}^{-1}T^{E,\kappa}_{\varepsilon}\Delta_{t})o_{sq}(1)dt$
$\displaystyle+(\partial_{x}^{-1}T^{E,\kappa}_{\varepsilon}\Delta_{t})o_{sq}(1)dW_{t}+o_{sq}(1)^{2}dt.$

We are interested in estimates for
$\mathbb{E}[\|\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{t}\|_{2}^{2}]$
to use the $H^{-1}$-estimate (B.4). We obtain these by integrating over $x>0$
and taking expectation. We deal with the terms on the right-hand side
individually. Combining the subexponential-tails property of $\nu_{t}$ and
$\tilde{\nu}_{t}$ with [16, Lemma 8.3 and Lemma 8.7], which are easily adapted
to the elastic heat kernel, the stochastic integral terms are martingales and
thus vanish when we take expectation. For the first term we use the
generalized Young’s inequality with free parameter $\eta>0$ to obtain

$\displaystyle\begin{split}\mathbb{E}\Bigg{[}-2&\int_{0}^{t}\int_{0}^{\infty}\mu_{s}(\partial_{x}^{-1}T^{E,\kappa}_{\varepsilon}\Delta_{s})T_{\varepsilon}^{E,\kappa}\Delta_{s}dxds\Bigg{]}\\\
&\leqslant
c_{\eta}\int_{0}^{t}\mathbb{E}\left[\left\|\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{s}\right\|_{2}^{2}\right]ds+\eta\mathbb{E}\left[\int_{0}^{t}\int_{0}^{\infty}(\mu_{s})^{2}\lvert
T_{\varepsilon}^{E,\kappa}\Delta_{s}\rvert^{2}dxds\right].\end{split}$ (5.13)

We employ integration by parts followed by the generalized Young’s inequality
for the second term. Lemma 5.4 from Section 5.1 allows us to handle the
boundary term that appears when integrating by parts. Thus, we have

$\displaystyle\begin{split}\mathbb{E}&\Bigg{[}\int_{0}^{t}\int_{0}^{\infty}(\partial_{x}^{-1}T^{E,\kappa}_{\varepsilon}\Delta_{s})\partial_{x}(\sigma_{s}^{2}T^{E,\kappa}_{\varepsilon}\Delta_{s}+\mathcal{E}_{s,\varepsilon}^{\sigma^{2}}-\tilde{\mathcal{E}}_{s,\varepsilon}^{\sigma^{2}})dxds\Bigg{]}\\\
&\leqslant-\frac{1}{\kappa}\mathbb{E}\left[(\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{s}(0))^{2}\right]-\mathbb{E}\left[\int_{0}^{t}\int_{0}^{\infty}\sigma_{s}^{2}\lvert
T^{E,\kappa}_{\varepsilon}\Delta_{s}\rvert^{2}dxds\right]\\\
&\qquad+\eta\mathbb{E}\left[\int_{0}^{t}\int_{0}^{\infty}\lvert
T_{\varepsilon}^{E,\kappa}\Delta_{s}\rvert^{2}dxds\right]+o(1).\end{split}$
(5.14)

We leave the third term as it is. For the remaining terms we proceed to use
the generalized Young’s inequality as in (5.13). Putting everything back
together we obtain

$\displaystyle\mathbb{E}\Bigg{[}\left\|\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{t}\right\|^{2}_{2}\Bigg{]}+\frac{1}{\kappa}\mathbb{E}\left[(\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{t}(0))^{2}\right]\leqslant
c_{\eta}\mathbb{E}\left[\int_{0}^{t}\left\|\partial_{x}^{-1}T^{E,\kappa}_{\varepsilon}\Delta\right\|_{2}^{2}ds\right]$
$\displaystyle\qquad-\mathbb{E}\left[\int_{0}^{t}\int_{0}^{\infty}(\sigma_{s}^{2}-\sigma_{s}^{2}\rho_{s}^{2}(1+\eta)-\mu_{s}^{2}\eta-\eta)\lvert
T^{E,\kappa}_{\varepsilon}\Delta_{s}\rvert^{2}dxds\right]+o(1).$

Since $\rho_{s}^{2}<1$ we can choose the free parameter $\eta$ small enough
such that

$\sigma_{s}^{2}-\sigma_{s}^{2}\rho_{s}^{2}(1+\eta)-\mu_{s}^{2}\eta-\eta\geqslant
c_{0}$ (5.15)

for all $x$ and $s$ and for some constant $c_{0}>0$. This yields

$\displaystyle\begin{split}&\mathbb{E}\left[\left\|\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta\right\|^{2}_{2}\right]+\frac{1}{\kappa}\mathbb{E}\left[(\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{t}(0))^{2}\right]\\\
&\qquad\leqslant
c_{\eta}\int_{0}^{t}\mathbb{E}\left[\left\|\partial_{x}^{-1}T^{E,\kappa}_{\varepsilon}\Delta_{s}\right\|_{2}^{2}\right]ds-
c_{0}\mathbb{E}\left[\int_{0}^{t}\left\|T_{\varepsilon}^{E,\kappa}\Delta_{s}\right\|_{2}^{2}ds\right]+o(1).\end{split}$
(5.16)

We then get the estimate

$\displaystyle\begin{split}&\mathbb{E}\left[\left\|\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta\right\|^{2}_{2}\right]+\frac{1}{\kappa}\mathbb{E}\left[(\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{t}(0))^{2}\right]\\\
&\qquad\leqslant
c_{\eta}\int_{0}^{t}\mathbb{E}\left[\left\|\partial_{x}^{-1}T^{E,\kappa}_{\varepsilon}\Delta_{s}\right\|_{2}^{2}\right]ds+c_{\eta}\int_{0}^{t}\frac{1}{\kappa}\mathbb{E}\left[(\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{s}(0))^{2}\right]ds+o(1).\end{split}$

By Gronwall’s lemma it follows that

$\lim_{\varepsilon\to
0}\left(\mathbb{E}\left[\left\|\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{t}\right\|^{2}_{2}\right]+\frac{1}{\kappa}\mathbb{E}\left[(\partial_{x}^{-1}T_{\varepsilon}^{E,\kappa}\Delta_{t}(0))^{2}\right]\right)=0.$
(5.17)

Now, recall the $H^{-1}$-estimate (B.4). We can find another constant $C_{1}$
such that

$\left\|\Delta_{t}\right\|_{-1}\leqslant C_{1}\liminf_{\varepsilon\to
0}\left(\left\|\partial_{x}^{-1}T^{E,\kappa}_{\varepsilon}\Delta_{t}\right\|_{L^{2}(0,\infty)}+\frac{1}{\sqrt{\kappa}}\left\lvert\int_{0}^{\infty}T^{E,\kappa}_{\varepsilon}\Delta_{t}(y)dy\right\rvert\right).$
(5.18)

Using $(a+b)^{2}\leqslant 2(a^{2}+b^{2})$ and Fatou’s lemma, we obtain from
(5.17) that

$\displaystyle\mathbb{E}\left[\liminf_{\varepsilon\to
0}\left(\left\|\partial_{x}^{-1}T^{E}_{\varepsilon}\Delta_{t}\right\|_{L^{2}(0,\infty)}+\frac{1}{\sqrt{\kappa}}\left\lvert\int_{0}^{\infty}T^{E}_{\varepsilon}\Delta_{t}(y)dy\right\rvert\right)^{2}\right]=0$

and so, as a consequence of (5.18),
$\mathbb{E}\left[\left\|\Delta_{t}\right\|_{-1}\right]=0$. In turn, we can
deduce the equality $\nu_{t}=\tilde{\nu}_{t}$ for all $t\in[0,T]$, with
probability 1, which concludes the proof. ∎

##  6 Existence of a density and its integrability

In this section we establish the existence and regularity results for the
densities of the sub-probability measures $\nu_{t},t\in[0,T]$, for a solution
to the SPDE (2.1). We use an approach based on energy estimates in $L^{2}$,
similar to the arguments of Section 5, to establish existence of a density of
$\nu_{t}$ restricted $(0,\infty)$. Unfortunately, these arguments do not allow
us to rule out the presence of an atom at the boundary. However, as we see in
Section 6.1, this can be achieved by a result of [6] for reflected Brownian
motion.

######  Lemma 6.1.

Let $(\nu,W^{0})$ be the unique solution to the SPDE (2.1) with $\nu$ in the
class $\Lambda$. Then we have

$\lim_{\varepsilon\to
0}\mathbb{E}\left[\int_{0}^{t}\left\|T_{\varepsilon}^{E,\kappa}\nu_{s}\right\|_{2}^{2}ds\right]<\infty\qquad\text{for
all}\quad t\in[0,T].$

###### Proof.

The proof follows by performing the same estimates as in the proof of Theorem
2.5 but for $\nu$ instead of the difference of two solutions. Instead of
disregarding the negative term on the right-hand side involving $c_{0}$ as in
(5.16), we move it over to the left-hand side and suitably adjust the
application of Gronwall’s lemma. ∎

Using this result we deduce that there is an $L^{2}$-density for a.e.
$(\omega,t)\in\Omega\times[0,T]$.

######  Proposition 6.2.

Let $(\nu,W^{0})$ be the unique solution to the SPDE (2.1) with $\nu$ in the
class $\Lambda$. Then for almost every $(\omega,t)\in\Omega\times[0,T]$ the
measure $\nu_{t}(\omega)$ has a density $V_{t}(\omega)$ on $[0,\infty)$ and we
have the integrated estimate

$\mathbb{E}\left[\int_{0}^{t}\left\|V_{s}\right\|_{2}^{2}ds\right]<\infty,\qquad\text{for
all}\quad t\in[0,T].$ (6.1)

###### Proof.

Fatou’s lemma applied to the result of Lemma 6.1 yields a bounded sequence
$(T_{\varepsilon_{n}}^{E,\kappa}\nu_{t})_{n}$ in $L^{2}$ for almost every
$(\omega,t)\in\Omega\times[0,T]$. Combining the weak compactness of bounded
sequences in a Hilbert space with the weak convergence of the smoothed
measure, along with Schwartz functions $\mathcal{S}$ being dense in $L^{2}$,
gives the result. ∎

Fixing $\omega$ (in an almost sure set), the above proposition only guarantees
that $V_{t}(\omega)$ exists in $L^{2}(0,\infty)$ for almost every $t\in[0,T]$.
Restricting each measure $\nu_{t}$ to $(0,\infty)$, we obtain instead the
existence of a density for all $t\in[0,T]$, with probability 1, but we only
show $L^{2}$ integrability away from the origin, which of course leaves the
possibility of a Dirac mass at zero. Before verifying this, we first show
several auxiliary results.

Take sets $U\Subset W\Subset(0,\infty)$, where $\Subset$ denotes compact
containment, and let $\psi$ be a smooth cut-off function such that $\psi=1$ on
$U$, $\psi\in(0,1)$ on $W\setminus U$ and $\psi=0$ otherwise. We have

$\lvert\partial_{x}\psi\rvert+\lvert\partial_{xx}\psi\rvert\leqslant
C1_{W\setminus U}.$

where the constant $C$ only depends on $U$ and $W$. Moreover, we define the
error term

$\bar{\mathcal{E}}_{t,\varepsilon}^{h}(x)=\partial_{x}\nu_{t}(h_{t}G_{\varepsilon}^{E,\kappa}(x,\cdot))-h_{t}(x)\partial_{x}T_{\varepsilon}^{E,\kappa}\nu_{t}(x)+\partial_{x}h_{t}(x)\mathcal{H}_{t,\varepsilon}(x),$
(6.2)

where

$\mathcal{H}_{t,\varepsilon}(x)=\nu_{t}((x-y)\partial_{x}G_{\varepsilon}^{E,\kappa}(x,\cdot)),$
(6.3)

for a function $h$ representing $\mu,\sigma^{2}$ or $\sigma$.

######  Lemma 6.3.

Let $(\nu,W^{0})$ be the unique solution to the SPDE (2.1) with $\nu$ in the
class $\Lambda$. Then we have the convergence

$\mathbb{E}\left[\int_{0}^{t}\int_{0}^{\infty}\psi^{2}(x)\left(\int_{0}^{\infty}\partial_{x}p_{\varepsilon}(x+y)\nu_{s}(dy)\right)^{2}dxds\right]\to
0,\quad\text{as $\varepsilon\to 0$}$

with the cut-off function $\psi$ specified above.

###### Proof.

Splitting the derivative of the Gaussian heat kernel into two parts we get

$\displaystyle\int_{0}^{\infty}\psi^{2}(x)\left(\int_{0}^{\infty}\frac{1}{\sqrt{2\pi\varepsilon}}\frac{x+y}{\varepsilon}e^{-\frac{(x+y)^{2}}{2\varepsilon}}\nu_{s}(dy)\right)^{2}dx$
$\displaystyle\leqslant
2\int_{0}^{\infty}\psi^{2}(x)e^{-x^{2}/\varepsilon}\left(\int_{0}^{\infty}\frac{1}{\sqrt{2\pi\varepsilon}}\frac{y}{\varepsilon}e^{-y^{2}/2\varepsilon}\nu_{s}(dy)\right)^{2}dx$
$\displaystyle+2\int_{0}^{\infty}\psi^{2}(x)e^{-x^{2}/\varepsilon}\frac{x^{2}}{\varepsilon^{2}}\left(\int_{0}^{\infty}\frac{1}{\sqrt{2\pi\varepsilon}}e^{-y^{2}/2\varepsilon}\nu_{s}(dy)\right)^{2}dx.$

For the first term on the right-hand side we have

$\displaystyle\int_{0}^{\infty}\psi^{2}(x)e^{-x^{2}/\varepsilon}\frac{1}{2\pi\varepsilon^{3}}\left(\int_{0}^{\infty}ye^{-y^{2}/2\varepsilon}\nu_{s}(dy)\right)^{2}dx$
$\displaystyle\leqslant\int_{0}^{\infty}\psi^{2}(x)e^{-x^{2}/\varepsilon}\frac{1}{2\pi\varepsilon^{2}}dx=O(\varepsilon^{-3/2}e^{-w^{2}/2\varepsilon})$

where $w=\inf W$. Then we obtain

$\displaystyle\mathbb{E}\left[\int_{0}^{t}\int_{0}^{\infty}\psi^{2}(x)e^{-x^{2}/\varepsilon}\left(\int_{0}^{\infty}\frac{1}{\sqrt{2\pi\varepsilon}}\frac{y}{\varepsilon}e^{-y^{2}/2\varepsilon}\nu_{s}(dy)\right)^{2}dxds\right]=O(\varepsilon^{-3/2}e^{-w^{2}/2\varepsilon}).$

We can argue similarly for the second term and hence obtain the desired
convergence. ∎

A similar estimate holds for the elastic correction term
$g_{\varepsilon}^{E,\kappa}$.

######  Lemma 6.4.

Let $(\nu,W^{0})$ be the unique solution to the SPDE (2.1) with $\nu$ in the
class $\Lambda$. Then we have the convergence

$\mathbb{E}\left[\int_{0}^{t}\int_{0}^{\infty}\psi^{2}(x)\left(\int_{0}^{\infty}\partial_{x}g_{\varepsilon}(x,y)\nu_{s}(dy)\right)^{2}dxds\right]\to
0,\quad\text{as $\varepsilon\to 0$}$

with the cut-off function $\psi$ specified above.

###### Proof.

The derivative of the elastic correction term $g_{\varepsilon}^{E,\kappa}$ is
given by

$\displaystyle\partial_{x}g_{\varepsilon}(x,y)=$
$\displaystyle\partial_{x}\left(\kappa
e^{\kappa(x+y)}e^{\frac{\kappa^{2}\varepsilon}{2}}\left(1-\mbox{Erf}\left(\frac{x+y+\kappa\varepsilon}{\sqrt{2\varepsilon}}\right)\right)\right)$
$\displaystyle=$ $\displaystyle
2\kappa^{2}e^{\kappa(x+y)}e^{\frac{\kappa^{2}\varepsilon}{2}}\frac{1}{\sqrt{2\pi\varepsilon}}\int_{0}^{\infty}\exp\left(-\frac{(z+x+y+\kappa\varepsilon)^{2}}{2\varepsilon}\right)dz$
$\displaystyle-2\kappa
e^{\kappa(x+y)}e^{\frac{\kappa^{2}\varepsilon}{2}}\frac{1}{\sqrt{2\pi\varepsilon}}\exp\left(-\frac{(y+x+\kappa\varepsilon)^{2}}{2\varepsilon}\right).$

We can then estimate

$\displaystyle
2\kappa^{2}e^{\kappa(x+y)}e^{\frac{\kappa^{2}\varepsilon}{2}}\frac{1}{\sqrt{2\pi\varepsilon}}\int_{0}^{\infty}\exp\left(-\frac{(z+x+y+\kappa\varepsilon)^{2}}{2\varepsilon}\right)dz\leqslant\kappa^{2}e^{-\frac{(x+y)^{2}}{2\varepsilon}}$

by Lemma A.3. For the other term we get

$2\kappa
e^{\kappa(x+y)}e^{\frac{\kappa^{2}\varepsilon}{2}}\frac{1}{\sqrt{2\pi\varepsilon}}\exp\left(-\frac{(y+x+\kappa\varepsilon)^{2}}{2\varepsilon}\right)\leqslant\frac{2\kappa}{\sqrt{2\pi\varepsilon}}e^{-\frac{(x+y)^{2}}{2\varepsilon}}.$

Thus, we get

$\displaystyle\mathbb{E}\int_{0}^{t}\int_{0}^{\infty}\psi^{2}(x)\left(\int_{0}^{\infty}\partial_{x}g_{\varepsilon}(x,y)\nu_{s}(dy)\right)^{2}dxds$
$\displaystyle\leqslant\mathbb{E}\int_{0}^{t}\int_{0}^{\infty}\psi^{2}(x)\kappa^{4}\left(\int_{0}^{\infty}e^{-\frac{(x+y)^{2}}{2\varepsilon}}\nu_{s}(dy)\right)^{2}dxds$
$\displaystyle\quad+\mathbb{E}\int_{0}^{t}\int_{0}^{\infty}\psi^{2}(x)4\kappa^{2}\left(\int_{0}^{\infty}\frac{1}{\sqrt{2\pi\varepsilon}}e^{-\frac{(x+y)^{2}}{2\varepsilon}}\nu_{s}(dy)\right)^{2}dxds$
$\displaystyle\leqslant\mathbb{E}\int_{0}^{t}\int_{0}^{\infty}\psi^{2}(x)\kappa^{4}e^{-\frac{x^{2}}{\varepsilon}}dxds+\mathbb{E}\int_{0}^{t}\int_{0}^{\infty}\psi^{2}(x)\frac{e^{-x^{2}/\varepsilon}}{2\pi\varepsilon}dxds=O(\varepsilon^{-1/2}e^{-w^{2}/2\varepsilon})$

with $w=\inf W$. ∎

The necessary controls on the error term
$\bar{\mathcal{E}}^{h}_{t,\varepsilon}$ are proved in the following lemmas.

######  Lemma 6.5.

Let $\bar{\mathcal{E}}^{h}_{s,\varepsilon}$ be the error term defined in (6.2)
and $\psi$ the cut-off function specified above. Then we have the convergence

$\mathbb{E}\left[\int_{0}^{t}\left\|\psi\bar{\mathcal{E}}^{h}_{s,\varepsilon}\right\|_{2}^{2}ds\right]\to
0\quad\text{as $\varepsilon\to 0$}.$

###### Proof.

We can combine the estimates in [16, Lemma 8.2] with the definition of the
elastic heat kernel to get

$\displaystyle\lvert\bar{\mathcal{E}}_{s,\varepsilon}^{h}\rvert$
$\displaystyle\leqslant
C\left\|\partial_{xx}h\right\|_{\infty}\int_{0}^{\infty}\lvert
x-y\rvert^{2}\lvert\partial_{x}G_{\varepsilon}^{E,\kappa}\rvert\nu_{s}(dy)$
$\displaystyle\leqslant
C\left\|\partial_{xx}h\right\|_{\infty}\int_{0}^{\infty}\lvert
x-y\rvert^{3}\varepsilon^{-1}p_{\varepsilon}(x-y)\nu_{s}(dy)$
$\displaystyle\quad+C\left\|\partial_{xx}h\right\|_{\infty}\int_{0}^{\infty}\lvert
x-y\rvert^{2}\lvert\partial_{x}p_{\varepsilon}(x+y)\rvert\nu_{s}(dy)$
$\displaystyle\quad+C\left\|\partial_{xx}h\right\|_{\infty}\int_{0}^{\infty}\lvert
x-y\rvert^{2}\lvert\partial_{x}g_{\varepsilon}^{E,\kappa}(x,y)\rvert\nu_{s}(dy).$

The result follows for the first term by the results from the absorbing case
in [16, Lemma 8.2] . Similar to the calculations in Lemma 6.3 and Lemma 6.4 we
can show that

$\mathbb{E}\left[\int_{0}^{t}\int_{0}^{\infty}\psi^{2}(x)\left(\int_{0}^{\infty}\lvert
x-y\rvert^{2}\lvert\partial_{x}g_{\varepsilon}(x,y)^{E,\kappa}\rvert\nu_{s}(dy)\right)^{2}dxds\right]=O(\varepsilon^{-1/2}e^{-w^{2}/2\varepsilon})$

and

$\mathbb{E}\left[\int_{0}^{t}\int_{0}^{\infty}\psi^{2}(x)\left(\int_{0}^{\infty}\lvert
x-y\rvert^{2}\lvert\partial_{x}p_{\varepsilon}(x+y)\rvert\nu_{s}(dy)\right)^{2}dxds\right]=O(\varepsilon^{-5/2}e^{-w^{2}/2\varepsilon}).$

∎

To simplify the notation, we denote by $o^{\psi}_{sq}(1)$ any family of
functions $\\{(f_{t,\varepsilon})_{t\in[0,T]}\\}_{\varepsilon>0}$ such that

$\mathbb{E}\left[\int_{0}^{T}\left\|\psi
f_{t,\varepsilon}\right\|^{2}_{2}dt\right]\to 0,\qquad\text{as $\varepsilon\to
0$}.$

The final auxiliary result is a control on the other error term
$\mathcal{H}_{t,\varepsilon}$.

######  Lemma 6.6.

Let $\mathcal{H}_{t,\varepsilon}$ be the error term defined in (6.3). Then
there is a constant $c_{\mathcal{H}}$ such that

$\lvert\mathcal{H}_{t,\varepsilon}\rvert\leqslant c_{\mathcal{H}}\lvert
T_{2\varepsilon}^{E,\kappa}\nu_{t}\rvert+o_{sq}^{\psi}(1)$

for $\varepsilon>0$ small enough.

###### Proof.

We can use the definition of the elastic heat kernel
$G_{\varepsilon}^{E,\kappa}$ to estimate

$\lvert\mathcal{H}_{t,\varepsilon}\rvert\leqslant\lvert\nu_{t}((x-y)\partial_{x}p_{\varepsilon}(x-y))\rvert+\lvert\nu_{t}((x-y)\partial_{x}p_{\varepsilon}(x+y))\rvert+\lvert\nu_{t}((x-y)\partial_{x}g_{\varepsilon}^{E,\kappa}(x,y))\rvert.$

For the first term there is a constant $c$ such that

$\lvert\nu_{t}((x-y)\partial_{x}p_{\varepsilon}(x-y))\rvert\leqslant
c\nu_{t}(p_{2\varepsilon}(x-y))$

by [16, Lemma 8.2]. Moreover, we have the estimate

$g_{\varepsilon}^{E,\kappa}(x,y)\leqslant\kappa
e^{-\frac{(x+y)^{2}}{2\varepsilon}}\leqslant p_{\varepsilon}(x+y),$

for $\varepsilon$ small enough. Thus there is a constant $C$ such that

$\lvert\nu_{t}((x-y)\partial_{x}p_{\varepsilon}(x-y))\rvert\leqslant
CT_{2\varepsilon}^{E,\kappa}\nu_{t}.$

The estimates for the other two terms follows as in Lemma 6.3 and Lemma 6.4. ∎

Now we have all the tools to prove the existence of a density in the interior.

######  Proposition 6.7.

Let $(\nu,W^{0})$ be the unique solution to the SPDE (2.1) with $\nu$ in the
class $\Lambda$. The measure $\nu_{t}$ restricted to $(0,\infty)$ has a
density $V_{t}$ for all $t\in[0,T]$ with probability 1. The density is square
integrable on $(\delta,\infty)$ for every $\delta>0$.

###### Proof.

We proceed similarly to the uniqueness proof in Section 5. We use the test
function $y\mapsto G_{\varepsilon}^{E,\kappa}(x,y)$ in the SPDE (2.1), switch
derivatives from $y$ to $x$ and take the coefficients outside the integration
against the measure. This creates the error terms defined in (6.2) and (6.3).
Applying Itô’s formula we can then derive the dynamics

$\displaystyle d(T_{\varepsilon}\nu_{t})^{2}=$
$\displaystyle-2T_{\varepsilon}^{E,\kappa}\nu_{t}\left(\mu_{t}\partial_{x}T_{\varepsilon}^{E,\kappa}\nu_{t}-\partial_{x}\mu_{t}\mathcal{H}_{t,\varepsilon}+\bar{\mathcal{E}}_{t,\varepsilon}^{\mu}\right)dt$
$\displaystyle+T_{\varepsilon}^{E,\kappa}\nu_{t}\partial_{x}\left(\sigma_{t}^{2}\partial_{x}T_{\varepsilon}^{E,\kappa}\nu_{t}-\partial_{x}\sigma_{t}^{2}\mathcal{H}_{t,\varepsilon}+\bar{\mathcal{E}}_{t,\varepsilon}^{\sigma^{2}}\right)dt$
$\displaystyle-2T_{\varepsilon}^{E,\kappa}\nu_{t}\rho_{t}\left(\sigma_{t}\partial_{x}T_{\varepsilon}^{E,\kappa}\nu_{t}-\partial_{x}\sigma_{t}\mathcal{H}_{t,\varepsilon}+\bar{\mathcal{E}}_{t,\varepsilon}^{\sigma}\right)dW_{t}^{0}$
$\displaystyle+4T_{\varepsilon}^{E,\kappa}\partial_{x}\nu_{t}(\mu_{t}p_{\varepsilon}(x+\cdot))dt+4T_{\varepsilon}^{E,\kappa}\nu_{t}\partial_{x}\nu_{t}(\rho_{t}\sigma_{t}p_{\varepsilon}(x+\cdot))dW_{t}^{0}$
$\displaystyle-4T_{\varepsilon}^{E,\kappa}\nu_{t}\partial_{x}\nu_{t}(\mu_{t}g_{\varepsilon}^{E,\kappa}(x,\cdot))dt-4T_{\varepsilon}^{E,\kappa}\nu_{t}\partial_{x}\nu_{t}(\rho_{t}\sigma_{t}g_{\varepsilon}^{E,\kappa}(x,\cdot))dW_{t}^{0}$
$\displaystyle+\left(\rho_{t}\left(\sigma_{t}\partial_{x}T_{\varepsilon}^{E,\kappa}\nu_{t}-\partial_{x}\sigma_{t}\mathcal{H}_{t,\varepsilon}+\bar{\mathcal{E}}_{t,\varepsilon}^{\sigma}\right)+2\partial_{x}\nu_{t}(\rho_{t}\sigma_{t}p_{\varepsilon}(x+\cdot))-2\partial_{x}\nu_{t}(\rho_{t}\sigma_{t}g_{\varepsilon}(x,\cdot))\right)^{2}dt.$

Next, we multiply the equation by $\psi^{2}$ and integrate over $x$. We then
estimate all the terms on the right-hand side individually using the results
of Lemma 6.3, Lemma 6.4, Lemma 6.5 and Lemma 6.6 together with the generalized
Young’s inequality. For the first term we have

$\displaystyle-2\int_{0}^{t}\int_{0}^{\infty}\psi^{2}T_{\varepsilon}^{E,\kappa}\nu_{s}\left(\mu_{s}\partial_{x}T_{\varepsilon}^{E,\kappa}\nu_{s}-\partial_{x}\mu_{s}\mathcal{H}_{s,\varepsilon}+\bar{\mathcal{E}}_{s,\varepsilon}^{\mu}\right)dxds$
$\displaystyle\leqslant c_{\eta}\int_{0}^{t}\left\|\psi
T_{\varepsilon}^{E,\kappa}\nu_{s}\right\|_{2}^{2}ds+c_{\eta}\int_{0}^{t}\left\|\psi
T_{2\varepsilon}^{E,\kappa}\nu_{s}\right\|_{2}^{2}ds+\eta\int_{0}^{t}\int_{0}^{\infty}\mu_{s}^{2}\psi^{2}\lvert\partial_{x}T_{\varepsilon}^{E,\kappa}\nu_{s}\rvert^{2}dxds+o(1).$

For the term in the second line we apply integration by parts and obtain

$\displaystyle\int_{0}^{t}\int_{0}^{\infty}\psi^{2}T_{\varepsilon}^{E,\kappa}\nu_{s}\partial_{x}\left(\sigma^{2}\partial_{x}T_{\varepsilon}^{E,\kappa}\nu_{s}-\partial_{x}\sigma^{2}_{s}\mathcal{H}_{s,\varepsilon}+\bar{\mathcal{E}}_{s,\varepsilon}^{\sigma^{2}}\right)dxds$
$\displaystyle=-\int_{0}^{t}\int_{0}^{\infty}\psi^{2}\sigma^{2}\lvert\partial_{x}T_{\varepsilon}^{E,\kappa}\nu_{s}\rvert^{2}dxds-2\int_{0}^{t}\int_{0}^{\infty}\psi\psi^{\prime}T_{\varepsilon}^{E,\kappa}\nu_{s}\partial_{x}T_{\varepsilon}^{E,\kappa}\nu_{s}\sigma^{2}_{s}dxds$
$\displaystyle\quad+\int_{0}^{t}\int_{0}^{\infty}\psi^{2}\partial_{x}T_{\varepsilon}^{E,\kappa}\nu_{s}\partial_{x}\sigma_{s}^{2}\mathcal{H}_{s,\varepsilon}dxds+2\int_{0}^{t}\int_{0}^{\infty}\psi\psi^{\prime}T_{\varepsilon}^{E,\kappa}\nu_{s}\partial_{x}\sigma_{s}^{2}\mathcal{H}_{s,\varepsilon}dxds$
$\displaystyle\quad-\int_{0}^{t}\int_{0}^{\infty}\psi^{2}\partial_{x}T_{\varepsilon}^{E,\kappa}\nu_{s}\bar{\mathcal{E}}_{s,\varepsilon}^{\sigma^{2}}-2\int_{0}^{t}\int_{0}^{\infty}\psi\psi^{\prime}T_{\varepsilon}^{E,\kappa}\nu_{s}\bar{\mathcal{E}}_{s,\varepsilon}^{\sigma^{2}}dxds$
$\displaystyle\leqslant-\int_{0}^{t}\int_{0}^{\infty}\psi^{2}\sigma^{2}\lvert\partial_{x}T_{\varepsilon}^{E,\kappa}\nu_{s}\rvert^{2}dxds+\eta\int_{0}^{t}\int_{0}^{\infty}\psi^{2}\lvert\partial_{x}T_{\varepsilon}^{E,\kappa}\nu_{s}\rvert^{2}dxds$
$\displaystyle\quad+c_{\eta}\int_{0}^{t}\int_{0}^{\infty}\lvert
T_{\varepsilon}^{E,\kappa}\nu_{s}\rvert^{2}dxds+c_{\eta}\int_{0}^{t}\int_{0}^{\infty}\lvert
T_{2\varepsilon}^{E,\kappa}\nu_{s}\rvert^{2}dxds+o(1).$

For now, we leave the terms involving stochastic integrals unchanged. The
remaining terms in the fourth and fifth line can be estimated via

$\displaystyle
4\int_{0}^{t}\int_{0}^{\infty}\psi^{2}T_{\varepsilon}^{E,\kappa}\nu_{s}\partial_{x}\nu_{s}(\mu_{s}p_{\varepsilon}(x+\cdot))dxds+4\int_{0}^{t}\int_{0}^{\infty}\psi^{2}T_{\varepsilon}^{E,\kappa}\nu_{s}\partial_{x}\nu_{s}(\mu_{s}g_{\varepsilon}^{E,\kappa}(x,\cdot))dxds$
$\displaystyle\leqslant c_{\eta}\int_{0}^{t}\left\|\psi
T_{\varepsilon}^{E,\kappa}\nu_{s}\right\|_{2}^{2}ds+o(1).$

We can write the last term as

$\displaystyle\int_{0}^{t}\int_{0}^{\infty}\psi^{2}\Big{(}\rho_{s}\left(\sigma_{s}\partial_{x}T_{\varepsilon}^{E,\kappa}\nu_{s}-\partial_{x}\sigma_{s}\mathcal{H}_{s,\varepsilon}+\bar{\mathcal{E}}_{s,\varepsilon}^{\sigma}\right)+o_{sq}^{\psi}(1)\Big{)}^{2}dxds$
$\displaystyle=\int_{0}^{t}\int_{0}^{\infty}\psi^{2}\rho_{s}^{2}\left(\sigma_{s}\partial_{x}T_{\varepsilon}^{E,\kappa}\nu_{s}-\partial_{x}\sigma_{s}\mathcal{H}_{s,\varepsilon}+\bar{\mathcal{E}}_{s,\varepsilon}^{\sigma}\right)^{2}dxds$
$\displaystyle\quad+\int_{0}^{t}\int_{0}^{\infty}\psi^{2}\rho_{s}\left(\sigma_{s}\partial_{x}T_{\varepsilon}^{E,\kappa}\nu_{s}-\partial_{x}\sigma_{s}\mathcal{H}_{s,\varepsilon}+\bar{\mathcal{E}}_{s,\varepsilon}^{\sigma}\right)o_{sq}^{\psi}(1)dxds+o(1)$
$\displaystyle\leqslant(1+\eta)\int_{0}^{t}\int_{0}^{\infty}\psi^{2}\rho_{s}^{2}\sigma_{s}^{2}\lvert\partial_{x}T_{\varepsilon}^{E,\kappa}\nu_{s}\rvert^{2}dxds$
$\displaystyle\quad-(1+\eta)\int_{0}^{t}\int_{0}^{\infty}\psi^{2}\rho_{s}^{2}\sigma_{s}\partial_{x}T_{\varepsilon}^{E,\kappa}\nu_{s}\left(\partial_{x}\sigma_{s}\mathcal{H}_{s,\varepsilon}-\bar{\mathcal{E}}^{\sigma}_{s,\varepsilon}\right)dxds$
$\displaystyle\quad+(1+\eta)\int_{0}^{t}\int_{0}^{\infty}\psi^{2}\rho^{2}\left(\partial_{x}\sigma\mathcal{H}_{s,\varepsilon}-\bar{\mathcal{E}}^{\sigma}_{s,\varepsilon}\right)^{2}dxds+o(1).$

We obtain

$\displaystyle\int_{0}^{t}\int_{0}^{\infty}\psi^{2}\Big{(}\rho_{s}\left(\sigma_{s}\partial_{x}T_{\varepsilon}^{E,\kappa}\nu_{s}-\partial_{x}\sigma_{s}\mathcal{H}_{s,\varepsilon}+\bar{\mathcal{E}}_{s,\varepsilon}^{\sigma}\right)+o_{sq}^{\psi}(1)\Big{)}^{2}dxds$
$\displaystyle\leqslant(1+2\eta+\eta^{2})\int_{0}^{t}\int_{0}^{\infty}\psi^{2}\rho_{s}^{2}\sigma_{s}^{2}\lvert\partial_{x}T_{\varepsilon}^{E,\kappa}\nu_{s}\rvert^{2}dxds+c_{\eta}\int_{0}^{t}\left\|\psi
T_{2\varepsilon}^{E,\kappa}\nu_{s}\right\|_{2}^{2}ds+o(1).$

Putting everything back together and choosing $\eta>0$ small enough such that

$\sigma_{s}^{2}-\rho_{s}^{2}(1+2\eta+\eta^{2})\sigma^{2}_{s}-\eta\mu_{s}^{2}-\eta\geqslant
c_{0}$

for some $c_{0}>0$ yields

$\displaystyle\left\|\psi
T_{\varepsilon}^{E,\kappa}\nu_{t}\right\|_{2}^{2}\leqslant$
$\displaystyle\left\|T_{\varepsilon}^{E,\kappa}\nu_{0}\right\|_{2}^{2}+c_{\eta}\int_{0}^{t}\left\|T_{\varepsilon}^{E,\kappa}\nu_{s}\right\|_{2}^{2}ds+c_{\eta}\int_{0}^{t}\left\|T_{2\varepsilon}^{E,\kappa}\nu_{s}\right\|_{2}^{2}ds+o(1)$
$\displaystyle-2\int_{0}^{t}\int_{0}^{\infty}\psi^{2}T_{\varepsilon}^{E,\kappa}\nu_{s}\rho_{s}\left(\sigma_{s}\partial_{x}T_{\varepsilon}^{E,\kappa}\nu_{s}-\partial_{x}\sigma_{t}\mathcal{H}_{t,\varepsilon}+\bar{\mathcal{E}}_{t,\varepsilon}^{\sigma}\right)dxdW_{s}^{0}$
$\displaystyle+4\int_{0}^{t}\int_{0}^{\infty}\psi^{2}T_{\varepsilon}^{E,\kappa}\nu_{s}\partial_{x}\nu_{s}(\rho_{s}\sigma_{s}p_{\varepsilon}(x+\cdot))dxdW_{s}^{0}$
$\displaystyle-4\int_{0}^{t}\int_{0}^{\infty}\psi^{2}T_{\varepsilon}^{E,\kappa}\nu_{s}\partial_{x}\nu_{s}(\rho_{s}\sigma_{s}g_{\varepsilon}^{E,\kappa}(x,\cdot))dxdW_{s}^{0}.$

Next, we take supremum over $t$ and then expectation. Using [16, Lemma 8.5] to
estimate the stochastic integrals, which can easily be adapted to this case,
we get

$\displaystyle\mathbb{E}\left[\sup_{s\in[0,t]}\left\|\psi
T_{\varepsilon}^{E,\kappa}\nu_{s}\right\|_{2}^{2}\right]\leqslant$
$\displaystyle
c^{\prime}\mathbb{E}\left[\left\|T_{\varepsilon}^{E,\kappa}\nu_{0}\right\|_{2}^{2}\right]+c^{\prime}\mathbb{E}\left[\int_{0}^{t}\left\|T_{\varepsilon}^{E,\kappa}\nu_{s}\right\|_{2}^{2}ds\right]$
$\displaystyle+c^{\prime}\mathbb{E}\left[\int_{0}^{t}\left\|T_{2\varepsilon}^{E,\kappa}\nu_{s}\right\|_{2}^{2}ds\right]+o(1)$

with a constant $c^{\prime}>0$. Taking the limit $\varepsilon\to 0$ and using
Proposition 6.1, Assumption 2.1 and Lemma B.1 we obtain

$\lim_{\varepsilon\to
0}\mathbb{E}\left[\sup_{s\in[0,t]}\left\|T_{\varepsilon}^{E,\kappa}\nu_{s}\right\|_{L^{2}(U)}^{2}\right]\leqslant\lim_{\varepsilon\to
0}\mathbb{E}\left[\sup_{s\in[0,t]}\left\|\psi
T_{\varepsilon}^{E,\kappa}\nu_{s}\right\|_{2}^{2}\right]<\infty.$ (6.4)

Thus, we have the existence of an $L^{2}$-density in the set $U$ using the
same arguments as in the proof of Proposition 6.2 and since $U$ was arbitrary
the result follows. ∎

###  6.1 Ruling out a Dirac mass at the origin

As a special case of Theorem 7.3 in the next section, we can characterize the
unique solution $(\nu,W^{0})$ to the SPDE (2.1) as the conditional law

$\nu_{t}(dx)=\mathbb{P}\left(X_{t}\in
dx,\,t<\tau\right|\mathcal{F}_{t}^{W^{0}})$ (6.5)

of a given reflected diffusion

$dX_{t}=\mu(t,X_{t})dt+\sigma(t,X_{t})\rho(t)W_{t}^{0}+\sigma(t,X_{t})(1-\rho(t)^{2})^{\frac{1}{2}}dW_{t}+dL_{t},$
(6.6)

where $W$ is a Brownian motion independent of $W^{0}$ and $\tau$ is the
elastic killing time defined by a standard exponential random variable
independent of $X$. This point of view makes it easy to deduce from [6] that
there cannot be an atom at the origin.

###### Proof of Proposition 2.7.

Express the unique solution $\nu$ as (6.5). We can then apply the scale
transformation $\zeta$ defined in Lemma 3.1 to (6.6) in order to obtain

$\nu_{t}(A)\leqslant\mathbb{P}\left(Z_{0}+\rho
W_{t}^{0}+\sqrt{1-\rho^{2}}W_{t}^{1}+\int_{0}^{t}\tilde{\mu}_{s}ds+L_{t}\in\zeta(t,A)\big{|}\mathcal{F}_{t}^{W^{0}}\right),$

for any Borel set $A$ in $[0,\infty)$. By Lemma 3.1, we have that

$Z_{0}+\rho
W_{t}^{0}+\sqrt{1-\rho^{2}}W_{t}^{1}+\int_{0}^{t}\tilde{\mu}_{s}ds\geqslant
Z_{0}+\rho W_{t}^{0}+\sqrt{1-\rho^{2}}W_{t}^{1}-Ct$ (6.7)

for all $t\in[0,T]$ almost surely with a constant $C>0$. Note that the
processes in (6.7) only differ by the drift term. Moreover the drift $-Ct$ is
always decreasing by more than the drift $\int_{0}^{t}\tilde{\mu}_{s}ds$. Thus
we have

$\displaystyle\left\\{Z_{0}+\rho
W_{t}^{0}+\sqrt{1-\rho^{2}}W_{t}^{1}+\int_{0}^{t}\tilde{\mu}_{s}ds+L_{t}=0\right\\}$
$\displaystyle=\left\\{Z_{0}+\rho
W_{t}^{0}+\sqrt{1-\rho^{2}}W_{t}^{1}+\int_{0}^{t}\tilde{\mu}_{s}ds=\inf_{u\leqslant
t}\left\\{Z_{0}+\rho
W_{u}^{0}+\sqrt{1-\rho^{2}}W_{u}^{1}+\int_{0}^{u}\tilde{\mu}_{s}ds\right\\}\right\\}$
$\displaystyle\subseteq\left\\{Z_{0}+\rho
W_{t}^{0}+\sqrt{1-\rho^{2}}W_{t}^{1}-Ct=\inf_{u\leqslant t}\left\\{Z_{0}+\rho
W_{u}^{0}+\sqrt{1-\rho^{2}}W_{u}^{1}-uC\right\\}\right\\}$
$\displaystyle=\left\\{Z_{0}+\rho
W_{t}^{0}+\sqrt{1-\rho^{2}}W_{t}^{1}-Ct+\tilde{L}_{t}=0\right\\},$

where $\tilde{L}$ is the local time of the process in the last line. As a
result we obtain

$0\leqslant\nu_{t}(\\{0\\})\leqslant\mathbb{P}\left(Z_{0}+\rho
W_{t}^{0}+\sqrt{1-\rho^{2}}W_{t}^{1}-Ct+\tilde{L}_{t}=0\Big{|}\mathcal{F}_{t}^{W^{0}}\right),$

for all $t\in[0,T]$ almost surely. With $f(t):=Ct$, we obviously have that
$f(t)/\sqrt{t}$ is non-decreasing, so $f$ is not in the upper class of
Brownian motion by [19, p.144]. Thus, [6, Theorem 2.2 and Theorem 2.5] gives
that the probability on the right-hand side is zero for all $t\in[0,T]$ almost
surely, and hence $\nu_{t}$ cannot have an atom at zero for any $t$. Combining
this with Proposition 6.7 we can conclude that we have a density everywhere on
the positive half-line. ∎

While the previous theorem establishes existence of a density for the entire
positive half-line, we do not get the square integrability all the way up to
the boundary, beyond a set of times of full measure as guaranteed by Theorem
2.6. It is unclear to us how to extend this result to include all $t\in[0,T]$
and we actually conjecture it to be false for all $t\in[0,T]$. To see where
the problem lies we consider the simpler case of a reflecting boundary and set
the drift $\mu$ to zero and the volatility $\sigma$ to one. The solution to
the SPDE then takes the form

$\nu_{t}(dx)=\mathbb{P}\left(\rho
W_{t}^{0}+\sqrt{1-\rho^{2}}W_{t}^{1}+L_{t}\in
dx\,|\,\mathcal{F}^{W^{0}}_{t}\right).$

We can interpret this conditional distribution as the distribution of the
scaled Brownian motion $\sqrt{1-\rho^{2}}W^{1}$ reflected on a given fixed
Brownian path $-\rho W^{0}$. This is a particular instance of a Brownian
motion reflected in a time-dependent domain, which, together with the related
heat equation, have been studied in a series of papers [5, 6, 7, 8]. In [6]
the authors show that, at the boundary of such domains, both singularities of
the density as well as atoms are possible. While they establish that there are
no atoms when the boundary is a path of a Brownian motion, in [8] it is shown
that in this case the density exhibits blow-up at the boundary on a dense
subset of times $t$. These effects arise when the boundary sharply moves into
the domain and the diffusion cannot transport the heat away fast enough. The
authors show that, for any boundary that locally moves in sharper than
Brownian motion (i.e., is in the upper class of Brownian motion), an atom
occurs. This leads us to suspect that the blow-up is too strong to allow for
square integrability of the density at the origin.

##  7 Extension to nonlinear interactions in the drift

In this section we discuss some of the results we can obtain and some of the
difficulties associated to a natural extension of the particle system which
allows for interaction through a dependence on the empirical measure
$\nu^{N}_{t}$ in the drift coefficient $\mu$. The particle system then becomes

$\displaystyle\begin{split}X_{t}^{i,N}&=X_{0}^{i}+\int_{0}^{t}\mu(s,X_{s}^{i,N},\nu_{s}^{N})ds+\int_{0}^{t}\sigma(s,X_{s}^{i,N})\rho(s)dW_{s}^{0}\\\
&+\int_{0}^{t}\sigma(s,X_{s}^{i,N})(1-\rho(s)^{2})^{\frac{1}{2}}dW_{s}^{i}+L_{t}^{i,N},\end{split}$
(7.1)

together with the elastic stopping times $\tau^{i,N}$. In this case we need
additional assumptions on the regularity of $\mu$ with respect to the measure
variable.

######  Assumption 7.1 (Regularity in the Measure Variable).

The drift coefficient $\mu$ is Lipschitz continuous in the measure variable
$\nu$ on the space of sub-probability measures $\mathbf{M}_{\leqslant
1}(\mathbb{R})$ with respect to the bounded Lipschitz distance $d_{0}$ which
is given by

$d_{0}(\nu,\tilde{\nu})\coloneqq\sup\\{\lvert\langle\psi,\nu-\tilde{\nu}\rangle\rvert:\left\|\psi\right\|_{Lip}\leqslant
1,\left\|\psi\right\|_{\infty}\leqslant 1\\},$ (7.2)

i.e., there exists a fixed constant $c>0$ such that
$\lvert\mu(t,x,\nu)-\mu(t,x,\tilde{\nu})\rvert\leqslant
cd_{0}(\nu,\tilde{\nu})$.

Through the empirical measure in the drift coefficient the elastic stopping
times $\tau^{i,N}$ become part of the equations for the particles. Thus, well-
posedness of the particle system needs additional arguments compared to the
system (1.1). The existence of such a particle system can be established by
adjusting the arguments in [14, Theorem 2.3] to this case. We need to add a
common noise in the filtrations defined therein. Another difference is that in
our case the interactions are through the empirical measure not the loss
function. An important step in the construction in [14, Section 3] is, for a
given particle $i$, to consider the system without particle $i$ given that
$t<\tau^{i,N}$. In our case such a system will, in contrast to the system in
[14], not be independent of the particle $i$ because of the empirical-measure
interaction. However, given the condition $t<\tau^{i,N}$ we can consider this
system with the empirical measure

$\nu_{t}^{N,-i}(dx)=\frac{1}{N}\left(\sum_{j=1,j\neq
i}^{N}\delta_{X_{t}^{j,N}}(dx)\mathbbm{1}_{t<\tau^{j}}+\delta_{X_{t}^{i,N}}(dx)\right).$

Thus, the system without particle $i$ depends on $W^{i}$ and $X^{i}_{0}$ but
not $\chi^{i}$. Considering this fact, the construction in [14] carries
through in our case. Using Assumptions 2.1,2.2 and 7.1 a standard argument
relying on the Lipschitz continuity and Gronwall’s lemma can be used to
establish uniqueness.

Following the same arguments as in Section 4 we obtain the existence result in
this case. The methods we used in the proofs in Section 4 can easily be
generalised to allow for interaction in the drift. In particular the
martingale approach can be extended in a straight forward way to $\mu$
depending also on $\xi$ in (4.7).

######  Theorem 7.2.

Let $(\nu^{N},W^{0})$ be a sequence with $\nu^{N}$ being empirical-measure
processes corresponding to the particle system (7.1) that satisfy Assumptions
2.1, 2.2 and 7.1. Then $(\nu^{N},W^{0})$ possesses converging subsequences in
$(D_{\mathcal{S}^{\prime}},M1)\times(C_{\mathbb{R}},\left\|\right\|_{\infty})$.
Moreover, for any limit point $(\nu,W^{0})$ the process $\nu$ is in the class
$\Lambda$ and $(\nu,W^{0})$ satisfies the SPDE

$\displaystyle\begin{split}\langle\nu_{t},\phi\rangle=\langle\nu_{0},\phi\rangle&+\int_{0}^{t}\langle\nu_{s},\mu(s,\cdot,\nu_{s})\phi^{\prime}\rangle
ds+\frac{1}{2}\int_{0}^{t}\langle\nu_{s},\sigma^{2}(s,\cdot)\phi^{\prime\prime}\rangle
ds\\\ &+\int_{0}^{t}\langle\nu_{s},\rho(s)\sigma(s,\cdot)\phi^{\prime}\rangle
dW_{s}^{0}\end{split}$ (7.3)

for all times $t\in[0,T]$ and all test functions
$\phi\in\mathcal{C}^{E,\kappa}_{0}(\mathbb{R})$, where

$\mathcal{C}^{E,\kappa}_{0}(\mathbb{R})=\\{\phi\in\mathcal{S}:\partial_{x}\phi(0)=\kappa\phi(0)\\}.$

Note that in this case the limit equation becomes a nonlinear SPDE with a
nonlinearity in the drift term $\mu$. A convenient way of obtaining regularity
for solutions to this nonlinear SPDE is to use the following probabilistic
representation.

######  Theorem 7.3.

Let $(\nu,W^{0})$ be a solution to the SPDE (7.3) with $\nu$ in the class
$\Lambda$. Then, for all $t\in[0,T]$ we have the following representation of
$\nu_{t}$

$\nu_{t}=\mathbb{P}\left(X_{t}\in\cdot,t<\tau\right|\mathcal{F}_{t}^{\nu,W^{0}}),$
(7.4)

where $X$ is a reflecting particle with dynamics given by

$dX_{t}=\mu(t,X_{t},\nu_{t})dt+\sigma(t,X_{t})\rho_{t}W_{t}^{0}+\sigma(t,X_{t})(1-\rho_{t}^{2})^{\frac{1}{2}}dW_{t}+dL_{t},$

with $W^{0},W$ independent Brownian motions and $X_{0}$ being distributed
according to $\nu_{0}$. The stopping time $\tau$ is the elastic stopping time
associated with $X$ and $(\mathcal{F}^{\nu,W^{0}}_{t})$ the filtration
generated by $\nu$ and $W^{0}$.

###### Proof.

The proof is based on a decoupling argument and the linear uniqueness result.
Let $\nu$ be a weak solution to the nonlinear SPDE which exists by Theorem
7.2. We then define a stochastic process $X$ as the solution to the SDE

$dX_{t}=\mu_{t}^{\nu}dt+\sigma(t,X_{t})(\rho_{t}dW_{t}^{0}+(1-\rho_{t}^{2})^{1/2}dW_{t}^{1})+dL_{t},$
(7.5)

where the drift term $\mu^{\nu}$ is given by

$\mu_{t}^{\nu}\coloneqq\mu(t,X_{t},\nu_{t})$

and $W^{0},W^{1}$ are independent Brownian motions. The associated elastic
stopping time is $\tau^{lin}\coloneqq\inf\\{t>0:L_{t}>\xi\\}$ for an
independent exponential random variable $\xi\sim\mbox{Exp}(\kappa)$. Define a
process $(\nu_{t}^{lin})$ taking values in the space of sub-probability
measures as

$\langle\nu_{t}^{lin},\phi\rangle\coloneqq\mathbb{E}\left[\phi(X_{t})\mathbbm{1}_{t<\tau^{lin}}|\mathcal{F}_{t}^{\nu,W^{0}}\right]$
(7.6)

for $\phi\in\mathcal{S}$. Take a test function
$\phi\in\mathcal{C}^{E,\kappa}_{0}(\mathbb{R})$, apply Itô’s formula,
rearrange and take conditional expectation to find that $\nu_{t}^{lin}$ solves
the linear SPDE

$\displaystyle\begin{split}d\langle\nu_{t}^{lin},\phi\rangle=&\langle\nu_{t}^{lin},\mu_{t}^{\nu}\phi^{\prime}\rangle
dt+\frac{1}{2}\langle\nu_{t}^{lin},\sigma^{2}_{t}\phi^{\prime\prime}\rangle
dt\\\ &+\langle\nu_{t}^{lin},\rho_{t}\sigma_{t}\phi^{\prime}\rangle
dW_{t}^{0}\end{split}$ (7.7)

for all $\phi\in\mathcal{C}^{E,\kappa}_{0}(\mathbb{R})$. By the same arguments
as for the particle system in Section 3, one easily checks that the solution
is in the class $\Lambda$. Since $\nu$ is a solution to the nonlinear SPDE, it
also solves the SPDE (7.7). By Therorem 2.5 solutions to (7.7) are unique in
the class $\Lambda$, so we have $\nu_{t}=\nu_{t}^{lin}$. This yields the
desired result. ∎

Based on this representation the same regularity results as in Theorem 2.6 and
Proposition 2.7 can be shown using the methods in Section 6. The question of
uniqueness of solutions to the nonlinear SPDE is more subtle. In the case of
an absorbing boundary, uniform $L^{2}$ regularity is used to deal with the
nonlinearity by employing an argument involving a sequence of stopping times
(see [16, 17]). As discussed in Section 6 we do not expect such regularity to
hold in the elastic case. This makes the extension to the nonlinear case
appear significantly more difficult than in the absorbing case and an entirely
different approach may be needed.

##  8 Absorption and Reflection as Limiting Cases

Intuitively, an elastic boundary condition acts as a mixture of an absorbing
boundary and a reflecting boundary – with the positive parameter $\kappa$
controlling the balance of the two. This can be seen from the elastic
condition

$\partial_{x}\phi(0)=\kappa\phi(0),$ (8.1)

if we take the limits $\kappa\to 0$ and $\kappa\to\infty$. When taking
$\kappa$ to $\infty$ we obtain the absorbing boundary condition $\phi(0)=0$
and taking $\kappa$ to $0$ yields the reflecting boundary condition
$\partial_{x}\phi(0)=0$. This shows that the absorbing and reflecting cases
can be obtained as limits of the elastic case. Moreover, we observe the same
when analysing the elastic stopping times

$\tau=\inf\\{t>0:L_{t}>\chi^{\kappa}\\},\quad\chi^{\kappa}\sim\mbox{Exp}(\kappa).$

In the limit $\kappa\to\infty$ we get $\chi^{\kappa}\to 0$ and the stopping
time $\tau$ becomes the first time the process $X$ hits the boundary at 0,
$i.e.$ the absorbing stopping time. For $\kappa\to 0$ we have the convergence
$\chi^{\kappa}\to\infty$ and the elastic killing time $\tau^{i}$ converges to
$\infty$. As a result the limit process becomes purely reflecting. The goal of
this section is to show that we also have this convergence at the level of the
measure-valued processes as we let $\kappa$ go to $0$ or $\infty$. We will
write $\nu^{\kappa}$ for the solution to the elastic SPDE to emphasis the
dependence on the elastic-killing parameter $\kappa$.

SPDEs of the type (2.1) with an absorbing boundary are studied in [16] and in
[17]. Existence and uniqueness of solutions is proved in these articles. The
absorbing nature of the boundary is defined through the space of test
functions

$\mathcal{C}^{A}_{0}(\mathbb{R})\coloneqq\\{\phi\in\mathcal{S}(\mathbb{R}):\phi(0)=0\\}.$
(8.2)

By using the space of test functions

$\mathcal{C}^{R}_{0}(\mathbb{R})\coloneqq\\{\phi\in\mathcal{S}(\mathbb{R}):\partial_{x}\phi(0)=0\\}$
(8.3)

in the formulation of the SPDE (2.1) we have instead a reflecting boundary.
With a few adjustments to the proof for the elastic case, we can also prove
Theorem 2.8.

###### Proof of Theorem 2.8.

The existence and necessary regularity of the weak solution follows through a
particle approximation as done in Section 3 and Section 4. Note that the
required estimates for the reflecting system were established alongside the
estimates for the elastic system in Section 3.

The kernel smoothing method with the reflecting heat kernel
$G_{\varepsilon}^{R}$ can be employed to prove uniqueness. In the reflecting
case the weak formulation of the boundary simplifies because the process takes
values in the space of probability measures. Denoting by $T_{\varepsilon}^{R}$
the convolution operator with $G_{\varepsilon}^{R}$ and by $\nu^{0}$ a
solution to the reflecting SPDE we see that for the anti-derivative at the
boundary, an application of Fubini’s theorem and the fact that
$G^{R}_{\varepsilon}$ integrates to 1 gives
$\partial_{x}^{-1}T_{\varepsilon}^{R}\nu^{0}_{t}(0)=-\nu^{0}_{t}[0,\infty)$.
The rest of the proof now follows with the same arguments as in the elastic
case. ∎

Now that we have existence and uniqueness for the three boundary cases, we can
prove Theorem 2.9 on the weak convergence of the solution to the elastic
equation to the absorbing and reflecting counter parts in the following way

  * •

verify tightness of $(\nu^{\kappa})$ to establish existence of limit points

  * •

show that limit points solve the absorbing or reflecting SPDEs, respectively,

  * •

deduce weak convergence from uniqueness of solutions.

The first step concerns tightness. As in the case of the particle
approximations we consider tightness on the space of càdlàg processes with
values in the space of tempered distributions.

######  Lemma 8.1.

Let $(\nu^{\kappa},W^{0})$ be a sequence of solutions with $\nu^{\kappa}$ in
the class $\Lambda$ to the SPDE (2.1) with elastic boundary condition for
parameter $\kappa>0$. Then, the sequence $(\nu^{\kappa},W^{0})$ is tight on
$(D_{\mathcal{S}^{\prime}},M1)\times(C_{\mathbb{R}},\left\|\cdot\right\|_{\infty})$
for both cases, $\kappa\to\infty$ and $\kappa\to 0$ .

###### Proof.

We follow the same approach as in Proposition 4.3. By the probabilistic
representation from Proposition 7.3 we know that for $\phi\in\mathcal{S}$ we
have

$\langle\nu_{t}^{\kappa},\phi\rangle=\mathbb{E}\left[\phi(X_{t})\mathbbm{1}_{t<\tau^{\kappa}}|\mathcal{F}_{t}^{\nu^{\kappa},W^{0}}\right].$
(8.4)

We can then again apply the decomposition of [25, Proposition 4.2] and obtain
that the first condition we need to show is

$\displaystyle\begin{split}\mathbb{E}\left[\lvert\langle\hat{\nu}_{t}^{\kappa},\phi\rangle-\langle\hat{\nu}_{s}^{\kappa},\phi\rangle\rvert^{4}\right]=O(\lvert
t-s\rvert^{2}),\qquad{\text{as }\lvert t-s\rvert\to 0}\end{split}$ (8.5)

for

$\langle\hat{\nu}_{t}^{\kappa},\phi\rangle=\mathbb{E}\left[\phi(X_{t\wedge\tau^{\kappa}})|\mathcal{F}_{t}^{\nu^{\kappa},W^{0}}\right].$

The definition of $X$ is independent of $\kappa$ and we can conclude condition
(8.5) with the same methods as in Proposition 4.3. To conclude tightness using
the approach of Proposition 4.3 we need to verify that for all $t\in[0,T]$ and
$\eta>0$

$\lim_{\delta\to 0}\limsup_{\kappa\to
0}\mathbb{P}(\mathcal{L}^{\kappa}_{t+\delta}-\mathcal{L}^{\kappa}_{t}\geqslant\eta)=0$
(8.6)

where

$\mathcal{L}_{t}^{\kappa}=\mathbb{P}\left(\tau^{\kappa}\leqslant
t|\mathcal{F}_{t}^{\nu^{\kappa},W^{0}}\right)$

to obtain tightness in the case $\kappa\to 0$ and the same condition with
$\limsup_{\kappa\to\infty}$ for the other case. Markov’s inequality yields

$\mathbb{P}(\mathcal{L}^{\kappa}_{t+\delta}-\mathcal{L}^{\kappa}_{t}\geqslant\eta)\leqslant\eta^{-1}\mathbb{P}(t<\tau^{\kappa}\leqslant
t+\delta)$

We consider a process $\tilde{Y}$ given by

$\tilde{Y}_{t}^{\kappa}=X_{0}^{\kappa}+\int_{0}^{t}\mu_{s}ds+\int_{0}^{t}\sigma_{s}dW_{s}$

where $dW_{s}=\rho_{s}dW_{s}^{0}+\sqrt{1-\rho^{2}_{s}}dW_{s}^{1}$. The
reflecting particle $X$ is then given by

$X_{t}=\tilde{Y}_{t}+L_{t}$

using the Skorokhod problem. Moreover, we set

$Y_{t}^{\kappa}=\tilde{Y}_{t}^{\kappa}+\chi^{\kappa}.$

We can apply this together with the definition of the stopping time
$\tau^{\kappa}$ to obtain

$\displaystyle\mathbb{P}\left(\tau^{\kappa}\leqslant t\right)$
$\displaystyle=\mathbb{P}\left(L_{t}\geqslant\chi^{\kappa}\right)=\mathbb{P}\left(\inf_{s\in[0,t]}\tilde{Y}_{s}\leqslant-\chi^{\kappa}\right)=\mathbb{P}\left(\inf_{s\in[0,t]}Y_{s}^{\kappa}\leqslant
0\right).$

Taking $\varepsilon>0$ we have

$\displaystyle\mathbb{P}\left(t<\tau^{\kappa}\leqslant t+\delta\right)=$
$\displaystyle\mathbb{P}\left(t<\tau^{\kappa}\leqslant
t+\delta,Y_{t}^{\kappa}\geqslant\varepsilon\right)+\mathbb{P}\left(t<\tau^{\kappa}\leqslant
t+\delta,Y_{t}^{\kappa}\in(0,\varepsilon)\right)$
$\displaystyle+\mathbb{P}\left(t<\tau^{\kappa}\leqslant
t+\delta,Y_{t}^{\kappa}\leqslant 0\right).$

The last term is zero because $Y_{t}^{\kappa}\leqslant 0$ implies
$\tau^{\kappa}\leqslant t$. This gives the estimate

$\mathbb{P}\left(t<\tau^{\kappa}\leqslant
t+\delta\right)\leqslant\mathbb{P}\left(t<\tau^{\kappa}\leqslant
t+\delta,Y_{t}^{\kappa}\geqslant\varepsilon\right)+\mathbb{P}\left(Y_{t}^{\kappa}\in(0,\varepsilon)\right).$
(8.7)

To deal with the second term on the right-hand side we apply the scale
transformation and change of measure discussed in Section 3 to transform
$Y^{\kappa}$ into a Brownian motion. The only dependence on $\kappa$ is then
left in the initial condition. However, note that $\chi^{\kappa}$ converges to
zero or infinity almost surely if we take the limit $\kappa\to\infty$ or
$\kappa\to 0$, respectively. Since the function $\zeta$ is continuous in the
$x$ variable and the measure $\mathbb{Q}$ equivalent to $\mathbb{P}$ we also
get $\mathbb{Q}$-almost sure convergences to zero or infinity for
$\zeta(t,\chi^{\kappa})$. Using the properties of Brownian motion we then have

$\limsup_{\kappa\to\infty}\mathbb{P}(Y_{t}^{\kappa}\in(0,\varepsilon))\leqslant
C_{q}\mathbb{Q}\left(B_{t}\in(0,C_{\mu,\sigma}\varepsilon)\right)^{1/q}=o(1),\quad\text{as
$\varepsilon\to 0$},$

for $q>1$, a constant $C_{q}>0$ depending on $q$ and $B$ a Brownian motion
under $\mathbb{Q}$. In the case $\kappa\to 0$ we get

$\limsup_{\kappa\to 0}\mathbb{P}(Y_{t}^{\kappa}\in(0,\varepsilon))=0.$

We follow the structure in the proof of [16, Proposition 4.7 ] to estimate the
other term in (8.7). We get

$\displaystyle\mathbb{P}\left(t<\tau^{\kappa}\leqslant
t+\delta,Y_{t}^{\kappa}\geqslant\varepsilon\right)$
$\displaystyle\leqslant\mathbb{P}\left(\inf_{s\in[t,t+\delta]}Y_{s}^{\kappa}\leqslant
0,Y_{t}^{\kappa}\geqslant\varepsilon\right)$
$\displaystyle\leqslant\mathbb{P}\left(\inf_{s\in[t,t+\delta]}(Y_{s}^{\kappa}-Y_{t}^{\kappa})\leqslant-\varepsilon\right).$

Using the scale transformation $\zeta$ from Section 3 we define
$U_{s}^{\kappa}\coloneqq\zeta(t+s,Y_{t+s}^{\kappa})-\zeta(t+s,Y_{t}^{\kappa})$.
The dynamics of $U^{\kappa}$ are given by

$dU_{s}^{\kappa}=u_{s}^{\kappa}ds+dW_{s}$

where $u^{\kappa}$ is a uniformly bounded drift coefficient. This means we can
find a constant $c_{1}>0$ such that

$\displaystyle\mathbb{P}\left(\inf_{s\in[t,t+\delta]}(Y_{s}^{\kappa}-Y_{t}^{\kappa})\leqslant-\varepsilon\right)$
$\displaystyle=\mathbb{P}\left(\inf_{s\in[0,\delta]}U_{s}^{\kappa}\leqslant-\kappa\right)$
$\displaystyle\leqslant\mathbb{P}\left(\inf_{s\in[0,\delta]}W_{s}\leqslant-
c_{1}(\varepsilon-\delta)\right)=\Phi(-c_{1}\delta^{-1/2}(\varepsilon-\delta))$

where $\Phi$ is the normal c.d.f. We need that $\varepsilon-\delta>0$ and the
convergences

$\delta^{-1/2}(\varepsilon(\delta)-\delta)\to\infty,\quad\varepsilon(\delta)\to
0.$

Choosing $\varepsilon(\delta)=\delta^{1/2}\log(1/\delta)$ gives the desired
convergences and we get (8.6). This completes the tightness proof. ∎

To show weak convergence, it is now sufficient to prove that the limit points
solve the respective SPDE for the reflecting and absorbing cases. To do this,
we again rely on the martingale approach described in Section 4.2. Consider
the maps $M^{\phi}(\nu^{\kappa}),S^{\phi}(\nu^{\kappa})$ and
$C^{\phi}(\nu^{\kappa})$ defined in (4.7). We need to show that, as we take
the limit $\kappa\to 0/\infty$, we obtain the corresponding quantities for the
reflecting and absorbing cases, respectively.

######  Proposition 8.2.

Fix $t\leqslant T$ and define $\Psi_{h}:\\{\xi\in
D_{\mathcal{S}^{\prime}}:\xi_{s}\in\mathbf{M}_{\leqslant
1}(\mathbb{R})\\}\times\mathcal{S}\to\mathbb{R}$ by

$\Psi_{h}(\xi,\phi)\coloneqq\int_{0}^{t}\langle\xi_{s},h(s,\cdot)\phi\rangle
ds$ (8.8)

with $h$ denoting a placeholder for $\mu,\sigma^{2}$ or $\rho\sigma$. If
$\nu^{\kappa}\to\nu^{*}$ weakly on $(D_{\mathcal{S}^{\prime}},M1)$ and
$\phi^{\kappa}\to\phi^{*}$ in $\mathcal{S}$, then we also have the convergence
$\Psi^{\kappa}_{h}\to\Psi^{*}_{h}$ weakly on $\mathbb{R}$.

###### Proof.

We show the result for the case $\kappa\to 0$. The other case follows using
the same line of argument. We fix a bounded Lipschitz function $f\in
Lip(\mathbb{R})$. By Lemma 8.1 and the resulting relative compactness we can
find a weakly convergent subsequence, again denoted by
$(\nu^{\kappa},\phi^{\kappa})$. By Skorokhod representation we can assume
almost sure convergence and use the triangle inequality and the Lipschitz
continuity of $f$ to obtain

$\displaystyle\left\lvert\mathbb{E}\left[f(\Psi^{\kappa}_{h})\right]-\mathbb{E}\left[f(\Psi^{*}_{h})\right]\right\rvert\leqslant$
$\displaystyle
C\Bigg{(}\mathbb{E}\left[\left\lvert\int_{0}^{t}\langle\nu_{s}^{\kappa}-\nu_{s}^{*},h(s,\cdot)\phi^{*}\rangle
ds\right\rvert\right]$
$\displaystyle+\mathbb{E}\left[\left\lvert\int_{0}^{t}\langle\nu_{s}^{\kappa},h(s,\cdot)\phi^{*}-h(s,\cdot)\phi^{\kappa}\rangle
ds\right\rvert\right]\Bigg{)}$ $\displaystyle\eqqcolon$ $\displaystyle
C\left(\mathbb{E}[I_{1}^{\kappa}]+\mathbb{E}[I_{2}^{\kappa}]\right)$

for some $C>0$. We can treat the first term as in [17, Proposition 4.7]. For
the second term involving $I_{2}^{\kappa}$, using that $h$ is bounded and