# Solving the hyperbolic Anderson model 1:
Skorohod setting

Xia Chen  Department of Mathematics, University of Tennessee, Knoxville
<EMAIL_ADDRESS>,  Aurélien Deya  Institut Elie Cartan, University of
Lorraine<EMAIL_ADDRESS>,  Jian Song  Research Center for
Mathematics and Interdisciplinary Sciences, Shandong University; School of
Mathematics, Shandong University<EMAIL_ADDRESS>and  Samy Tindel
Department of Mathematics, Purdue University<EMAIL_ADDRESS>
###### Abstract.

This paper is concerned with a wave equation in dimension $d\in\\{1,2,3\\}$,
with a multiplicative space-time Gaussian noise which is fractional in time
and homogeneous in space. We provide necessary and sufficient conditions on
the space-time covariance of the Gaussian noise, allowing the existence and
uniqueness of a mild Skorohod solution.

###### Key words and phrases:

Stochastic wave equation, Malliavin calculus, Skorohod equation

###### 2010 Mathematics Subject Classification:

60H15, 60H07, 60F10

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 The Gaussian noise and Malliavin calculus
    2. 2.2 Mild solution for the Skorohod equation
  3. 3 Existence and uniqueness in the Skorohod setting
    1. 3.1 On the necessary condition
    2. 3.2 Bound for the Laplace transform of $n$-th chaos contributions
    3. 3.3 On the sufficiency of the condition
    4. 3.4 Proof of Corollary 1.3

##  1\. Introduction

In the series of articles [CDOT1, CDOT2], we started a line of research aiming
at a comparative study between the Skorohod and Stratonovich settings for the
parabolic Anderson model in very rough environments. At the core of our
project in the aforementioned papers lies the following observation: while the
Stratonovich solution might be seen as more physically relevant, the Skorohod
solution often offers more possibilities in terms of quantitative analysis
(moments, asymptotics, see for instance [Balan12, chen19, hhnt]). In [CDOT1,
CDOT2], we were thus able to transfer some nontrivial information about
moments of the stochastic heat equation from the Skorohod to the Stratonovich
equation.

The current article can be seen as a new chapter in this global picture.
Indeed, the stochastic wave equation is another canonical model of random
evolution which deserves a thorough quantitative study, just like for the heat
equation. In addition, the toolbox allowing to handle basic issues for the
wave equation is necessarily different in nature from the parabolic case. It
thus seems natural to explore connections between the Stratonovich and the
Skorohod worlds in a hyperbolic setting. We start this long term program here
by an in-depth study of existence-uniqueness results in the Skorohod realm.

To be more specific, consider the following stochastic wave equation on
$\mathbb{R}^{d}$ with $d$ in the set $\\{1,2,3\\}$,

$\begin{cases}\dfrac{\partial^{2}u}{\partial t^{2}}(t,x)=\Delta
u(t,x)+u\,\dot{W}(t,x),~{}~{}t>0,x\in\mathbb{R}^{d},\vspace{0.2cm}\\\
u(0,x)=u_{0}(x),\quad\dfrac{\partial u}{\partial t}(0,x)=u_{1}(x).\end{cases}$
(1.1)

In equation (1.1), $\Delta$ stands for the usual Laplace operator in
$\mathbb{R}^{d}$, and $u_{0},u_{1}$ are initial conditions satisfying some
appropriate upper bounds (see Hypotheses (2.14)-(2.15) below). As far as the
forcing $\dot{W}$ above is concerned, we consider a centered Gaussian noise
whose covariance is given by

$\mathbb{E}[\dot{W}(s,x)\dot{W}(t,y)]=|s-t|^{-\alpha_{0}}\gamma(x-y).$ (1.2)

The parameter $\alpha_{0}$ in (1.2) is an arbitrary number in $[0,1)$, which
means that the noise $\dot{W}$ we consider here is either fractional in time
($\alpha_{0}\in(0,1)$) or independent of time ($\alpha_{0}=0$). The spatial
covariance of $\dot{W}$ is encoded by a (possibly singular) non-negative and
non-negative definite function $\gamma$ whose spectral measure (i.e. the
Fourier transform of $\gamma$) is denoted by $\mu$. For instance if the noise
$\dot{W}$ is white in space, its spatial covariance function $\gamma(x)$ is
the Dirac delta function $\delta(x)$ with $\mu(d\xi)=d\xi$. Another example of
interest for this class of functions is the Riesz kernel
$\gamma(x)=|x|^{-\alpha}$ with $\alpha\in(0,d)$, for which we have
$\mu(d\xi)=C_{\alpha}|\xi|^{\alpha-d}d\xi$.

With this set of assumptions in hand, we can state our main result in a
slightly informal way (see Proposition 3.3 and Theorem 3.8 for more rigorous
versions).

######  Theorem 1.1.

Let $\dot{W}$ be a Gaussian noise with covariance function given by (1.2),
where $\alpha_{0}\in[0,1)$ and $\gamma$ admits a spectral measure $\mu$. Then
under appropriate regularity conditions on $u_{0}$ and $u_{1}$,

$\int_{\mathbb{R}^{d}}\Big{(}{1\over 1+|\xi|^{2}}\Big{)}^{3-\alpha_{0}\over
2}\mu(d\xi)<\infty,$ (1.3)

is a necessary and sufficient condition on $\mu$ so that equation (1.1) admits
a unique Skorohod solution when $\alpha_{0}\in(0,1)$, and a sufficient
condition when $\alpha_{0}=0$.

######  Remark 1.2.

We would like to stress that our condition (1.3) encompasses a wide variety of
spatial covariance functions for the Gaussian noise $\dot{W}$, besides the
typical examples such as the above-mentioned Dirac delta function and Riesz
kernels. For instance, one may consider a periodic spatial covariance function
$\gamma(x)$ for which $\mu$ becomes a discrete measure supported on the
lattice space $a{\mathbb{Z}}^{d}$ for some constant $a>0$. In fact, one of the
appeal of Theorem 1.1 is the generality of our framework.

If the Gaussian noise $\dot{W}$ possesses spatial homogeneity properties, then
Theorem 1.1 generates the following convenient wellposedness criterion (see
Section 3.4 for a proof of this corollary).

######  Corollary 1.3.

Assume that the spatial covariance $\gamma$ is non-negative, non-negative
definite, and that there exists $\alpha>0$ for which

$\gamma(cx)=c^{-\alpha}\gamma(x)\quad\text{for all}\ c>0,\
x\in\mathbb{R}^{d}.$ (1.4)

Then the condition (1.3) holds true (or equivalently, the wave equation (1.1)
has a unique Skorohod solution) if and only if $\alpha_{0}+\alpha<3$.

Let us complete the above statement with a few remarks.

######  Remark 1.4.

It is readily checked that the spectral measure of a homogeneous measure of
order $\nu\in\mathbb{R}$ is a homogeneous measure of order $d-\nu$. Therefore,
our non-negative definiteness condition on $\gamma$ actually rules out the
case of $\alpha$-homogeneity with $\alpha>d$, since a homogeneous measure of
negative order is identically zero (this can be seen by letting $c$ tends to
$0$ in relation (1.4)). As the only homogeneous measures of order 0 on
$\mathbb{R}^{d}$ are the constant multiples of the Lebesgue meaure, the only
case with $\alpha=d$ is when $\gamma(\cdot)$ is a constant multiple of Dirac
function (i.e., $\dot{W}$ is a spatial white noise). In particular, as
$(\alpha_{0},\alpha)\in[0,1)\times(0,d]$, the condition $\alpha_{0}+\alpha<3$
is automatically verified for $d=1,2$.

######  Remark 1.5.

Our Corollary 1.3 encompasses the Riesz kernel case $\gamma(x)=|x|^{-\alpha}$
with $\alpha\in(0,d)$, which obviously satisfies the homogeneous property
(1.4). Another similar example comes from fractional Brownian sheets with
Hurst parameters $H_{j}\in(\frac{1}{2},1)$ for $j=1,\dots,d$. In this case we
have $\gamma(x)=\prod_{j=1}^{d}|x_{j}|^{-(2-2H_{j})}$, and the coefficient
$\alpha$ in (1.4) is given by $\alpha=\sum_{j=1}^{d}(2-2H_{j})$. In both the
Riesz kernel and fractional sheet situation, the condition
$\alpha_{0}+\alpha<3$ is necessary and sufficient in order to solve (1.1). For
the fractional Brownian sheet situation, this condition can be recast as
$H_{0}+\sum_{j=1}^{d}H_{j}>d-1/2$.

######  Remark 1.6.

The range of application of Corollary 1.3 also includes the spatial white
noise. In this case we have $\gamma(x)=\delta(x)$, and we get $\alpha=d$. In
particular one can solve equation (1.1) driven by a purely spatial white noise
($\alpha_{0}=0$, $\alpha=d$) in dimensions $d=1,2$ but not in dimension $d=3$.

Theorem 1.1 and Corollary 1.3 are the first results giving necessary and
sufficient conditions in order to solve equation (1.1) in the Skorohod setting
for a general space-time fractional Gaussian noise. However, the stochastic
wave equation driven by multiplicative Gaussian noise (also known as
hyperbolic Anderson model) has been extensively studied in recent years. We
now briefly recall some literature related to the problem of existence and
uniqueness of the mild Skorohod solution.

  1. (i)

In [walsh86], Walsh developed an Itô-type stochastic calculus for martingale
measures and used it to study stochastic partial differential equations
(SPDEs). In particular, the stochastic wave equation in dimension one was
considered therein. Then adapting some results by Peszat and Zabczyk [PZ] to
the random field setting in [dalang99], Dalang extended Walsh’s definition of
stochastic integral with respect to martingale measures. This allowed him to
solve SPDEs whose Green function is a Schwartz distribution rather than a
classical function. As an application, assuming that the Gaussian noise
$\dot{W}$ in (1.1) is white in time (i.e., replacing the temporal covariance
$|s-t|^{-\alpha_{0}}$ in (1.2) by the Dirac delta function $\delta(s-t)$), the
following so-called Dalang’s condition

$\int_{\mathbb{R}^{d}}\frac{1}{1+|\xi|^{2}}\mu(d\xi)<\infty,$ (1.5)

was proved in [dalang99] (see Remark 14 (a) therein) to be a necessary and
sufficient condition for the existence and uniqueness of mild Itô (equivalent
to Skorohod in this context) solution for $d\in\\{1,2,3\\}$. Walsh’s theory
was further extended in [cd08], where stochastic wave equations in any
dimension were studied. Observe that condition (1.5) coincides with our own
assumption (1.3) in the temporal white noise case which corresponds to
$\alpha_{0}=1$.

  2. (ii)

When $\dot{W}$ has a covariance given by (1.2), the Gaussian noise is colored
in time and thus the stochastic calculus for martingale measures used in
[walsh86, dalang99, cd08] does not apply in this situation. Balan [Balan12]
employed Malliavin calculus (see e.g. [nualart]) to obtain the existence and
uniqueness of a mild Skorohod solution to (1.1) for $d\in\\{1,2,3\\}$. She
worked with a space-time colored noise with $\alpha_{0}\in(0,1)$ and under
Dalang’s condition (1.5). This result was extended to any dimension $d$ in
[bs17]. Our result goes beyond the assumptions of [Balan12], since the
hypothesis (1.3) is weaker than Dalang’s condition (1.5) whenever
$\alpha_{0}<1$.

  3. (iii)

In the special case $d=1$, a study of the fractional space-time noise was
carried out in [ssx20]. More specifically [ssx20] handled the case of a
fractional noise in time with index $\alpha_{0}\in(0,1)$, while $\gamma$ was
rougher than in [Balan12, bs17]. Namely in [ssx20] the spatial component of
the noise is assumed to be the distributional second derivative of the
function $x\mapsto|x|^{2H}$ with $H<1/2$. The condition obtained therein was
$\alpha_{0}\in[0,1)$ and $\alpha\in(1,3/2)$. Notice that one cannot really
compare our current paper with [ssx20], since our positivity assumptions rule
out the possibility of considering a very rough noise in space.

  4. (iv)

In the recent paper [BCC21], for (1.1) with time-independent homogeneous
Gaussian noise (i.e., $\alpha_{0}=0$ in (1.2) and $0<\alpha\leq d$ in (1.4)),
the existence and uniqueness of the mild Skorohod solution was obtained under
the conditions $0<\alpha<d\leq 3$ and $0<\alpha=d\leq 2$ respectively. This is
indeed consistent with our assumptions in Theorem 1.1 (see also Corollary 1.3
and the subsequent remarks). In the sequel we will highlight this point by
preforming several separate computations for the specific time-independent
case. It should be observed that even in the time independent case, our
setting is more general than [BCC21]. Indeed, our contribution encompasses
cases with no density for the measure $\mu$, as well as no convolution
decomposition ($\gamma=K*K$) and no homogeneity for $\gamma$.

As one can see from this review, our main Theorem 1.1 gives a general
framework allowing to solve hyperbolic Anderson models in dimension $d\leq 3$.
It includes and goes beyond most of the aforementioned references. One should
also mention the recent efforts [De, OO] in order to properly define wave
equations with additive noise and polynomial nonlinear terms, in the rough
paths sense. Further comments on those contributions will be made in our
forthcoming paper [CDST] on Stratonovich solutions.

We now summarize the methodology employed in order to prove Theorem 1.1,
focusing on the sufficient condition. With respect to the heat equation
situation, one of the main obstacle is that one cannot appeal to Feynman-Kac
type formulae in order to analyze the equation. Therefore, we shall only rely
on a proper control of the chaos expansion for a candidate solution $u$ to
equation (1.1). As we will see in (2.12), this chaos expansion takes the form

$u(t,x)=\sum_{n=0}^{\infty}I_{n}(f_{n}(\cdot,t,x)),$ (1.6)

for a sequence of functions $f_{n}$ based on products of the wave kernel. Our
main task is thus reduced to a sharp control of some weighted norms of the
functions $f_{n}$. Some important tools towards this aim are the following:

  * •

Second moment computations in (1.6) in terms some $L^{2}$-norms of functions
involving the noise covariance and the wave kernel.

  * •

Poissonization (or Laplace transform) methods in order to be reduced to
$L^{1}$ (as opposed to $L^{2}$) norms and products of $1$-d integrals.

Some of the ingredients described above are already contained in [BCC21].
However, the presence of a nontrivial time covariance induces a more technical
and challenging situation. In order to proceed with the main steps described
above, a delicate study based on Fourier analysis is needed.

Let us finally emphasize a striking phenomenon revealed by a close examination
of the subsequent strategy and arguments. Namely, under suitable initial
conditions and referring to the chaos expansion (1.6), proving the main
convergence result

$\sum_{n=0}^{\infty}n!\lVert I_{n}(f_{n}(.,t,x))\rVert_{\mathcal{H}^{\otimes
n}}^{2}<\infty,\quad\text{for all}\ t>0$

is in fact equivalent to proving the (much) weaker property

$\lVert I_{1}(f_{1}(.,t,x))\rVert_{\mathcal{H}}<\infty,\quad\text{for some}\
t>0.$

This explains in particular how condition (1.3) becomes necessary in the
statement of Theorem 1.1.

This paper is organized as follows. In Section 2, we provide some
preliminaries on Gaussian analysis related to the noise $\dot{W}$ and equation
(1.1). In Section 3, we first show that (1.3) is actually a necessary
condition for the second moment of the first chaos of the mild Skorohod
solution to be finite. This yields the necessity of condition (1.3). Then in
Sections 3.2-3.3 we show that under our assumption (1.3), the chaos expansion
of the solution does converge in $L^{2}(\Omega)$ by estimating the Laplace
transform of the second moment of each chaos. This corresponds to the
sufficient part in Theorem 1.1.

Throughout the paper, we use the symbol $C$ for a generic positive constant
which may be different in different places.

##  2\. Preliminaries

This section is devoted to a better grasp on the Gaussian analysis related to
the noise $\dot{W}$ defined by (1.2). Then we shall state a rigorous version
of the mild formulation for equation (1.1).

###  2.1. The Gaussian noise and Malliavin calculus

In this subsection we collect some preliminaries on Malliavin calculus for the
Gaussian noise $\dot{W}$ with covariance given by (1.2). For more details, we
refer to [nualart]. Let us first label our running assumptions on the noise
coefficients.

######  Hypothesis 2.1.

Recall that the covariance function of $\dot{W}$ is formally given by (1.2).
Then we assume that $\alpha_{0}\in[0,1)$. Moreover, the function $\gamma$ is
supposed to be non-negative and non-negative definite with spectral measure
$\mu$.

With Hypothesis 2.1 in mind, we define a set of functions encoding the
covariance structure of $\dot{W}$. Namely, let $\mathcal{H}$ be the Hilbert
space which is the completion of the Schwartz space
$\mathcal{S}(\mathbb{R}_{+}\times\mathbb{R})$ under the inner product

$\displaystyle\langle\varphi,\phi\rangle_{\mathcal{H}}=$
$\displaystyle\int_{\mathbb{R}_{+}^{2}}\int_{\mathbb{R}^{2d}}|r-s|^{-\alpha_{0}}\varphi(r,x)\phi(s,y)\gamma(x-y)\,dxdydrds.$
(2.1)

One can also write this inner product in spatial Fourier mode as follows:

$\displaystyle\langle\varphi,\phi\rangle_{\mathcal{H}}=$
$\displaystyle\int_{\mathbb{R}_{+}^{2}}\int_{\mathbb{R}^{d}}|r-s|^{-\alpha_{0}}\widehat{\varphi}(r,\xi)\overline{\widehat{\phi}(s,\xi)}\,\mu(d\xi)drds,$
(2.2)

where we recall from Hypothesis 2.1 that $\alpha_{0}\in(0,1)$, and $\gamma$ is
a non-negative definite function with spectral measure $\mu$.

We can now introduce $W$ as an isonormal Gaussian process. Specifically, on a
complete probability space $(\Omega,\mathcal{F},P)$, let
$W=\\{W(\varphi),\varphi\in\mathcal{H}\\}$ be a Gaussian family with
covariance given by

$\mathbb{E}[W(\varphi)W(\phi)]=\langle\varphi,\phi\rangle_{\mathcal{H}}.$
(2.3)

Then $W(\varphi)$ for $\varphi\in\mathcal{H}$ is called the Wiener integral of
$\varphi$ with respect to $W$ and we also denote
$\int_{\mathbb{R}_{+}}\int_{\mathbb{R}^{d}}\varphi(t,x)W(dt,dx):=W(\varphi).$

Let $h(x_{1},\dots,x_{n})$ be a smooth function such that its partial
derivatives have at most polynomial growth. Then for smooth and cylindrical
random variables of the form $F=h(W(\varphi_{1}),\dots,W(\varphi_{n}))$, one
can define the Malliavin derivative $DF$ as the $\mathcal{H}$-valued random
variable

$DF:=\sum_{k=1}^{n}\frac{\partial h}{\partial
x_{k}}(W(\varphi_{1}),\dots,W(\varphi_{n}))\,\varphi_{k}.$

One can verify that $D:L^{2}(\Omega)\to L^{2}(\Omega;\mathcal{H})$ is a
closable operator, and then we define the Sobolev space $\mathbb{D}^{1,2}$ as
the closure of the space of the smooth and cylindrical random variables under
the norm

$\|F\|_{1,2}=\sqrt{\mathbb{E}[F^{2}]+\mathbb{E}[\|DF\|_{\mathcal{H}}^{2}]}.$

Denote by $\text{Dom }\delta$ the domain of the divergence operator $\delta$,
which is the set of $u\in L^{2}(\Omega;\mathcal{H})$ such that $|E[\langle
DF,u\rangle_{\mathcal{H}}]|\leq c_{F}\|F\|_{2}$ with some constant $c_{F}$
depending on $F$, for all $F\in\mathbb{D}^{1,2}$. The divergence operator
$\delta$ (also known as the Skorohod integral) is the adjoint of the Malliavin
derivative operator $D$ defined by the duality

$\mathbb{E}[F\,\delta(u)]=E[\langle DF,u\rangle_{\mathcal{H}}],~{}\text{ for
all }F\in\mathbb{D}^{1,2}\text{ and }u\in\text{Dom }\delta.$ (2.4)

Thus, for $u\in\text{Dom }\delta$, we have $\delta(u)\in L^{2}(\Omega)$. It is
also readily checked from (2.4) with $F\equiv 1$ that
$\mathbb{E}[\delta(u)]=0.$ Note that we will use the following notation

$\delta(u)=\int_{\mathbb{R}_{+}}\int_{\mathbb{R}}u(t,x)W(dt,dx),\quad\text{for
all}\quad u\in\text{Dom }\delta.$ (2.5)

To end this subsection, we recall the Wiener chaos expansion of a random
variable $F\in L^{2}(\Omega)$. Let $\mathbf{H}_{0}=\mathbb{R}$, and for
integers $n\geq 1$, let $\mathbf{H}_{n}$ be the closed linear subspace of
$L^{2}(\Omega)$ containing the set of random variables
$\left\\{H_{n}(W(\varphi)),\varphi\in\mathcal{H},\|\varphi\|_{\mathcal{H}}=1\right\\}$,
where $H_{n}$ is the $n$-th Hermite polynomial (i.e.,
$H_{n}(x)=(-1)^{n}e^{x^{2}}\frac{d^{n}}{dx^{n}}(e^{-x^{2}})$). Then
$\mathbf{H}_{n}$ is called the $n$-th Wiener chaos of $W$. Assuming
$\mathcal{F}$ is the $\sigma$-field generated by
$\\{W(\varphi),\varphi\in\mathcal{H}\\}$, we have the following Wiener chaos
decomposition

$L^{2}(\Omega,\mathcal{F},P)=\bigoplus_{n=0}^{\infty}\mathbf{H}_{n}.$

For $n\geq 1$, let $\mathcal{H}^{\otimes n}$ be the $n$-th tensor product of
$\mathcal{H}$ and $\widetilde{\mathcal{H}}^{\otimes n}$ be the symmetrization
of $\mathcal{H}^{\otimes n}$. Then the mapping $I_{n}(h^{\otimes
n})=H_{n}(W(h))$ for $h\in\mathcal{H}$ can be extended to a linear isometry
between $\widetilde{\mathcal{H}}^{\otimes n}$ and the $n$-th Wiener chaos
$\mathbf{H}_{n}$. Thus, for any random variable $F\in
L^{2}(\Omega,\mathcal{F},P)$, the following unique Wiener chaos expansion in
$L^{2}(\Omega)$ holds true,

$F=\mathbb{E}[F]+\sum_{n=1}^{\infty}I_{n}(f_{n}),\quad\text{with
}f_{n}\in\widetilde{\mathcal{H}}^{\otimes n}.$

Furthermore, noting that

$\mathbb{E}\left[\left|I_{n}(f_{n})\right|^{2}\right]=n!\|f_{n}\|^{2}_{\mathcal{H}^{\otimes
n}},$

we have

$\mathbb{E}[|F|^{2}]=\left(\mathbb{E}[F]\right)^{2}+\sum_{n=1}^{\infty}\mathbb{E}\left[\left|I_{n}(f_{n})\right|^{2}\right]=\left(\mathbb{E}[F]\right)^{2}+\sum_{n=1}^{\infty}n!\|f_{n}\|^{2}_{\mathcal{H}^{\otimes
n}}.$ (2.6)

###  2.2. Mild solution for the Skorohod equation

In this subsection, we define the mild Skorohod solution to (1.1) and derive
its Wiener chaos expansion. We start by introducing some more notation about
filtrations related to our problem.

######  Notation 2.2.

In the sequel we write $\\{\mathcal{F}_{t}\\}_{t\geq 0}$ for the filtration
generated by the time increments of $\dot{W}$. That is we set

$\mathcal{F}_{t}=\sigma\\{W(\mathbf{1}_{[0,s]}\varphi);\,0\leq s\leq
t,\,\varphi\in\mathcal{S}(\mathbb{R})\\}\vee\mathcal{N},$

where $\mathcal{N}$ is the collection of null sets.

We now label some notation concerning the wave kernel.

######  Notation 2.3.

We denote by $G_{t}(x)$ the fundamental solution of the wave equation, and
recall that we consider here the dimensions $d=1,2,3$. The generalized
function $G_{t}$ is characterized by its spatial Fourier transform

$\widehat{G}_{t}(\xi)=\frac{\sin(t|\xi|)}{|\xi|}.$ (2.7)

For two functions $u_{0},u_{1}$ of the variable $x$ (whose regularity will be
specified below), we also set

$w(t,x)=\frac{\partial}{\partial
t}\left(G_{t}*u_{0}\right)(x)+\left(G_{t}*u_{1}\right)(x),$ (2.8)

where $*$ denotes the convolution in space.

The expression (2.7) of the fundamental solution $\widehat{G}_{t}$ in Fourier
modes will prove to be crucial in our computations below. However, we will
also resort to properties of $G_{t}$ in direct modes. To this aim, we recall
that for $d\in\\{1,2,3\\}$, the fundamental solution $G_{t}(x)$ has explicit
expressions:

$G_{t}(x)=\left\\{\begin{array}[]{lll}\displaystyle\frac{1}{2}\mathbf{1}_{[|x|<t]}&\text{
if }d=1,\vspace{0.2cm}\\\
\displaystyle\frac{1}{2\pi}\frac{1}{\sqrt{t^{2}-|x|^{2}}}\mathbf{1}_{[|x|<t]}&\text{
if }d=2,\vspace{0.2cm}\\\ \displaystyle\frac{1}{4\pi t}\sigma_{t}&\text{ if
}d=3,\end{array}\right.$

where $\sigma_{t}$ is the uniform measure on the sphere
$\\{x\in\mathbb{R}^{3}:|x|=t\\}$.

With this notation in hand, we are ready to state a rigorous definition of the
Skorohod solution to equation (1.1).

######  Definition 2.4.

An $\\{\mathcal{F}_{t}\\}_{t\geq 0}$-adapted random field $u=\\{u(t,x),t\geq
0,x\in\mathbb{R}^{d}\\}$ is called a mild Skorohod solution to (1.1) if
$\mathbb{E}[u^{2}(t,x)]<\infty$ for each
$(t,x)\in\mathbb{R}_{+}\times\mathbb{R}^{d}$ and if it satisfies the following
integral equation,

$u(t,x)=w(t,x)+\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)u(s,y)W(ds,dy),$
(2.9)

where $w(t,x)$ is given in (2.8) and the stochastic integral on the right-hand
side is a Skorohod integral like in (2.5). In particular, it is implicitly
assumed that for each $t\geq 0,x\in\mathbb{R}$, the process
$v_{t,x}(s,y)=G_{t-s}(x-y)u(s,y)\mathbf{1}_{[0,t]}(s)$ lies in Dom $\delta$
(see (2.4)).

Let us say a few words about the chaos decomposition for the mild solution
(2.9). First for $n\in\mathbb{N}$ we denote

$f^{w}_{n}(s_{1},x_{1},\dots,s_{n},x_{n},t,x)=\frac{1}{n!}\sum_{\sigma\in\Sigma_{n}}G_{t-s_{\sigma(n)}}(x-x_{\sigma(n)})\cdots
G_{s_{\sigma(2)}-s_{\sigma(1)}}(x_{\sigma(2)}-x_{\sigma(1)})\\\ \times
w(s_{\sigma(1)},x_{\sigma(1)})\mathbf{1}_{[0<s_{\sigma(1)}<\dots<s_{\sigma(n)}<t]},$
(2.10)

where $\Sigma_{n}$ is the set of permutations on $\\{1,2,\dots,n\\}$, $G_{t}$
is the wave kernel defined by (2.7) and $w$ is the convolution (2.8). Then the
following result can be found, for instance, in [Balan12].

######  Proposition 2.5.

There exists a unique mild Skorohod solution to (1.1) if and only if the
function $w(t,x)$ given by (2.8) is well-defined and the series
$\sum_{n=1}^{\infty}I_{n}(f_{n}(\cdot,t,x))$ converges in $L^{2}(\Omega)$ for
all $t>0$, i.e.,

$\sum_{n=1}^{\infty}n!\|f^{w}_{n}(\cdot,t,x)\|^{2}_{\mathcal{H}^{\otimes
n}}<\infty,~{}\text{ for all }t>0\text{ and }x\in\mathbb{R}^{d}.$ (2.11)

Whenever (2.11) is met, we have the following Wiener chaos expansion for the
solution $u$ to equation (1.1):

$u(t,x)=w(t,x)+\sum_{n=1}^{\infty}I_{n}(f_{n}^{w}(\cdot,t,x)).$ (2.12)

In order to state some necessary and sufficient conditions allowing to solve
(1.1), we have to introduce another piece of notation. Namely we will call
$f_{n}$ the function $f^{w}$ in (2.10) obtained when $w\equiv 1$. More
specifically we have

$\displaystyle
f_{n}(s_{1},x_{1},\dots,s_{n},x_{n},t,x)=\frac{1}{n!}\sum_{\sigma\in\Sigma_{n}}$
$\displaystyle G_{t-s_{\sigma(n)}}(x-x_{\sigma(n)})\cdots
G_{s_{\sigma(2)}-s_{\sigma(1)}}(x_{\sigma(2)}-x_{\sigma(1)})$
$\displaystyle\times\mathbf{1}_{[0<s_{\sigma(1)}<\dots<s_{\sigma(n)}<t]}.$
(2.13)

We can now state our bound on the function $w$.

######  Proposition 2.6.

Let $u_{0}$ and $u_{1}$ be the initial conditions in (1.1), and recall that
$w$ is defined by (2.8). Let us assume that

$\sup_{x\in\mathbb{R}^{d}}|u_{1}(x)|<\infty,$ (2.14)

and for $u_{0}$ we suppose

$\sup_{x\in\mathbb{R}^{d}}|u_{0}(x)|<\infty~{}\text{ if
}~{}d=1,\quad\text{and}\quad\int_{\mathbb{R}^{d}}|\widehat{u}_{0}(\xi)|d\xi<\infty~{}\text{
if }~{}d=2,3.$ (2.15)

Then the function $w$ satisfies

$\sup_{t\in[0,T],x\in\mathbb{R}^{d}}|w(t,x)|<\infty.$ (2.16)

###### Proof.

We resort to expression (2.8) and will bound the two terms therein.

First, in order to bound the term $G_{t}*u_{1}$ in (2.8), note that
(regardless of the dimension $d=1,2,3$) we have
$\int_{\mathbb{R}^{d}}G_{t}(x)dx=t$ for all $t>0$. Hence if
$\|u_{1}\|_{\infty}=M<\infty$, for all $x\in\mathbb{R}^{d}$ we get

$|G_{t}*u_{1}(x)|\leq\|u_{1}\|_{\infty}\int_{\mathbb{R}^{d}}G_{t}(y)dy=Mt.$
(2.17)

Regarding the term $\frac{\partial}{\partial t}(G_{t}*u_{0})(x)$ in (2.8), we
separate the study according to the dimension. That is for $d=1$ it is readily
checked that

$\frac{\partial}{\partial
t}(G_{t}*u_{0})(x)=\frac{1}{2}\left(u_{0}(x+t)+u_{0}(x-t)\right).$

Therefore, if $\|u_{0}\|_{\infty}=M$, we clearly have

$\left|\frac{\partial}{\partial t}(G_{t}*u_{0})(x)\right|\leq M,$ (2.18)

uniformly in $x\in\mathbb{R}$. For $d=2,3$, we express the convolution in
Fourier modes. Namely using
$\widehat{G_{t}*u_{0}}=\widehat{G}_{t}\widehat{u}_{0}$ and appealing to
expression (2.7) for the Fourier transform of $G_{t}$, one can write

$\frac{\partial}{\partial
t}\left(G_{t}*u_{0}\right)(x)=\frac{\partial}{\partial
t}\left(\int_{\mathbb{R}^{d}}\frac{\sin(t|\xi|)}{|\xi|}\widehat{u}_{0}(\xi)e^{i\xi\cdot
x}d\xi\right)=\int_{\mathbb{R}^{d}}\cos(t|\xi|)\widehat{u}_{0}(\xi)e^{i\xi\cdot
x}d\xi.$

Hence if one assumes $\|\widehat{u}_{0}\|_{L^{1}(\mathbb{R}^{d})}=M$, we get

$\left|\frac{\partial}{\partial
t}(G_{t}*u_{0})(x)\right|\leq\int_{\mathbb{R}^{d}}|\widehat{u}_{0}(\xi)|d\xi=M,$
(2.19)

uniformly in $x\in\mathbb{R}^{d}$.

We can now conclude our claim: we simply gather (2.17), (2.18) and (2.19) into
expression (2.8). This yields our estimate (2.16). ∎

##  3\. Existence and uniqueness in the Skorohod setting

We now turn to the existence and uniqueness problem for the Skorohod equation
(2.9), starting with the statement of our main theorem.

######  Theorem 3.1.

Consider equation (2.9) driven by a centered Gaussian noise $\dot{W}$. The
covariance function of $\dot{W}$ is given by (1.2), with $\alpha_{0}\in(0,1)$
and $\gamma$ satisfying Hypothesis 2.1. Then the following holds true.

  1. (i)

If (2.14) and (2.15) are satisfied, we have that equation (2.9) admits a
unique solution as soon as $\gamma$ satisfies (1.3), that is

$\int_{\mathbb{R}^{d}}\left(\frac{1}{1+|\xi|^{2}}\right)^{\frac{3-\alpha_{0}}{2}}\mu(d\xi)<\infty.$

  2. (ii)

Recall that $w$ is defined by (2.8). If in addition we have $w(x)\geq a_{0}$
with a strictly positive constant $a_{0}$, then (1.3) is also a necessary
condition in order to get a unique solution to (2.9).

######  Remark 3.2.

A simple example of initial condition $(u_{0},u_{1})$ in (1.1) fulfilling
(2.14)-(2.15) and giving rise to a strictly positive $w$ is $u_{0}\equiv 0$
and $u_{1}\equiv 1$.

The remainder of the section is devoted to a proof Theorem 3.1. We start with
the necessary condition.

###  3.1. On the necessary condition

Recall that the covariance function of $\dot{W}$ is given by (1.2), with a
non-negative constant $\alpha_{0}$ and a non-negative and non-negative
definite function $\gamma(x)$ whose Fourier transform is $\mu$. In Proposition
3.3 below, we provide a necessary condition (1.3) for the existence and
uniqueness of the Skorohod solution to (1.1).

######  Proposition 3.3.

Let $W$ be a noise whose covariance function is given by (1.2). We assume that
Hypothesis 2.1 holds true with $\alpha_{0}\in(0,1)$ and a function $\gamma$.
Assume that the function $w$ given by (2.8) is lower bounded by a constant
$a_{0}>0$. Then a necessary condition in order to solve equation (1.1) is
(1.3).

###### Proof.

Recall that the function $f_{1}^{w}$ corresponding to the first chaos in our
decomposition (2.12) is defined by

$f_{1}^{w}(s,y;t,x)=G_{t-s}(x-y)w(s,y)\mathbf{1}_{[0,t]}(s).$ (3.1)

Since we have assumed $w(x)\geq a_{0}>0$ uniformly in $x\in\mathbb{R}^{d}$, it
suffices to show that (1.3) is a necessary condition for
$\|f_{1}(\cdot,t,x)\|^{2}_{\mathcal{H}}$ to be finite, where we recall that we
have set $f_{1}=f_{1}^{w}$ for the initial condition $w(s,y)\equiv 1$.
Furthermore, thanks to (2.2) we have the following expression for $f_{1}$:

$\displaystyle\|f_{1}(\cdot,t,x)\|_{\mathcal{H}}^{2}=$
$\displaystyle\int_{0}^{t}\int_{0}^{t}\int_{\mathbb{R}^{d}}|r-s|^{-\alpha_{0}}\frac{\sin(r|\xi|)}{|\xi|}\frac{\sin(s|\xi|)}{|\xi|}\mu(d\xi)drds.$

Using the fact that the Fourier transform of $|t|^{-\alpha_{0}}$ is
proportional to $|\lambda|^{\alpha_{0}-1}$, we thus obtain

$\displaystyle\|f_{1}(\cdot,t,x)\|_{\mathcal{H}}^{2}=c_{\alpha_{0}}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}}|\lambda|^{\alpha_{0}-1}\left|\int_{0}^{t}e^{i\lambda
s}\frac{\sin(s|\xi|)}{|\xi|}ds\right|^{2}d\lambda\,\mu(d\xi).$ (3.2)

We now lower bound the square in the right-hand side of (3.2). First,
separating real and imaginary parts, we easily get

$\displaystyle\left|\int_{0}^{t}e^{i\lambda
s}\frac{\sin(s|\xi|)}{|\xi|}ds\right|^{2}\geq$
$\displaystyle\frac{1}{|\xi|^{2}}\Big{|}Q_{t}(\lambda,\xi)\Big{|}^{2},$ (3.3)

where we have set

$Q_{t}(\lambda,\xi)=\int_{0}^{t}\cos(\lambda s)\sin(|\xi|s)ds.$

Then notice that $Q_{t}(\lambda,\xi)$ is an integral which can be computed
explicitly thanks to elementary methods. We get

$\displaystyle Q_{t}(\lambda,\xi)=$
$\displaystyle\frac{1}{2}\int_{0}^{t}\Big{(}\sin\big{(}(\lambda+|\xi|)s\big{)}-\sin\big{(}(\lambda-|\xi|)s\big{)}\Big{)}ds$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{1-\cos\big{(}(\lambda+|\xi|)t\big{)}}{\lambda+|\xi|}+\frac{1-\cos\big{(}(|\xi|-\lambda)t\big{)}}{|\xi|-\lambda}\right).$
(3.4)

Let us focus our attention on the region $0\leq\lambda\leq|\xi|$ in the right-
hand side above. In this case, both terms in (3.1) are non-negative and as a
consequence,

$\Big{|}Q_{t}(\lambda,\xi)\Big{|}^{2}\geq\frac{1}{4}\left(\frac{1-\cos\big{(}(|\xi|-\lambda)t\big{)}}{|\xi|-\lambda}\right)^{2}.$

Therefore, for $|\xi|>0$, we get the following lower bound:

$\displaystyle\int_{\mathbb{R}}|\lambda|^{\alpha_{0}-1}\Big{|}Q_{t}(\lambda,\xi)\Big{|}^{2}d\lambda\geq\int_{0}^{|\xi|}|\lambda|^{\alpha_{0}-1}\Big{|}Q_{t}(\lambda,\xi)\Big{|}^{2}d\lambda$
$\displaystyle\geq\frac{1}{4}\int_{0}^{|\xi|}|\lambda|^{\alpha_{0}-1}\left(\frac{1-\cos\big{(}(|\xi|-\lambda)t\big{)}}{|\xi|-\lambda}\right)^{2}d\lambda\geq\frac{1}{4}|\xi|^{\alpha_{0}-1}\int_{0}^{|\xi|}\left(\frac{1-\cos\big{(}(|\xi|-\lambda)t\big{)}}{|\xi|-\lambda}\right)^{2}d\lambda.$

Thanks to the elementary change of variable $|\xi|-\lambda=\tau$, we thus end
up with

$\displaystyle\int_{\mathbb{R}}|\lambda|^{\alpha_{0}-1}\Big{|}Q_{t}(\lambda,\xi)\Big{|}^{2}d\lambda\geq\frac{1}{4}|\xi|^{\alpha_{0}-1}\int_{0}^{|\xi|}\left(\frac{1-\cos(\tau
t)}{\tau}\right)^{2}d\tau.$ (3.5)

Let us summarize our computations so far. Plugging (3.5) into (3.3) and then
(3.2), we have obtained

$\|f_{1}(\cdot,t,x)\|_{\mathcal{H}}^{2}\geq
C_{\alpha_{0}}\int_{\mathbb{R}^{d}}|\xi|^{\alpha_{0}-3}\int_{0}^{|\xi|}\left(\frac{1-\cos(\tau
t)}{\tau}\right)^{2}\,d\tau\,\mu(d\xi),$

where we recall that $C$ designates a constant whose exact value can change
from line to line. Hence, lower bounding the integral in $\xi$ over
$\mathbb{R}^{d}$ by an integral over $\\{|\xi|\geq 1\\}$ we get

$\displaystyle\|f_{1}(\cdot,t,x)\|_{\mathcal{H}}^{2}$ $\displaystyle\geq$
$\displaystyle C_{\alpha_{0}}\int_{\\{|\xi|\geq
1\\}}|\xi|^{\alpha_{0}-3}\int_{0}^{|\xi|}\left(\frac{1-\cos(\tau
t)}{\tau}\right)^{2}d\tau\,\mu(d\xi)$ (3.6) $\displaystyle\geq$ $\displaystyle
C_{\alpha_{0}}\left(\int_{0}^{1}\left(\frac{1-\cos(\tau
t)}{\tau}\right)^{2}d\tau\right)\int_{\\{|\xi|\geq
1\\}}|\xi|^{\alpha_{0}-3}\mu(d\xi).$

Observe that the integral over $\tau$ above is just a strictly positive
universal constant. Hence, one can recast (3.6) as

$\|f_{1}(\cdot,t,x)\|_{\mathcal{H}}^{2}\geq C\int_{\\{|\xi|\geq
1\\}}|\xi|^{\alpha_{0}-3}\mu(d\xi),$

where $C$ is a finite positive constant. Now it is readily checked that the
right-hand side above is finite if and only iff (1.3) is fulfilled. We have
thus found that (1.3) is a necessary condition to have
$\|f_{1}(\cdot,t,x)\|_{\mathcal{H}}^{2}<\infty,$ which concludes the proof. ∎

######  Remark 3.4.

If the noise is independent of time, namely if $\alpha_{0}=0$, it is readily
checked that (3.2) becomes

$\displaystyle\|f_{1}(\cdot,t,x)\|_{\mathcal{H}}^{2}=$
$\displaystyle\int_{\mathbb{R}^{d}}\left(\int_{0}^{t}\frac{\sin(s|\xi|)}{|\xi|}ds\right)^{2}\mu(d\xi)=\int_{\mathbb{R}^{d}}\frac{\left(1-\cos(t|\xi|)\right)^{2}}{|\xi|^{4}}\mu(d\xi).$

Therefore a sufficient condition (also necessary if the noise is spatially
homogeneous, see [BCC21]) for $\|f_{1}(\cdot,t,x)\|_{\mathcal{H}}^{2}$ to be
finite is

$\int_{\mathbb{R}^{d}}\left(\frac{1}{1+|\xi|^{2}}\right)^{2}\mu(d\xi)<\infty.$
(3.7)

Note that if we plug the value $\alpha_{0}=0$ into our condition (1.3), we get
something stronger than (3.7). This means that (1.3) is not an optimal
condition for $\|f_{1}(\cdot,t,x)\|_{\mathcal{H}}^{2}<\infty$ when
$\alpha_{0}=0$. However, our condition (1.3) with $\alpha_{0}=0$ is still a
sufficient condition for (1.1) with time-independent noise to have a unique
Skorohod solution as stated in Theorem 1.1. Hence we are not able (at this
moment) to close the gap between necessary and sufficient condition for the
time independent case $\alpha_{0}=0$. At a technical level, this is mainly due
to the considerations leading to relations (3.47)-(3.48) below.

###  3.2. Bound for the Laplace transform of $n$-th chaos contributions

Recall that the function $f_{n}^{w}$ is given by (2.10), and features in
(2.11) as the kernel for the $n$-th chaos contribution in the decomposition of
$u(t,x)$. As a first step toward (2.11), and accordingly toward existence and
uniqueness of a solution for (1.1), let us prove the following Proposition,
which provides an upper bound for the Laplace transform of the 2nd moment
$\mathbb{E}[|I_{n}(f_{n}(\cdot,t,x))|^{2}]$.

######  Proposition 3.5.

Assume that Hypothesis 2.1 holds true and that $u_{0},u_{1}$ satisfy the
conditions of Proposition 2.6. Let $p\geq 1$ be a fixed parameter. Then, there
exists a positive constant $C_{\alpha_{0}}$ depending only on $\alpha_{0}$
such that

$2\int_{0}^{\infty}e^{-2pt}\|f_{n}(\cdot,t,x)\|_{{\mathcal{H}}^{\otimes
n}}^{2}dt\leq\begin{cases}\displaystyle\left(C_{\alpha_{0}}\int_{\mathbb{R}^{d+1}}\frac{1}{|\tau|^{1-\alpha_{0}}\Phi_{p}(\tau,\eta)}\,d\tau\mu(d\eta)\right)^{n},&\alpha_{0}>0,\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\left(\int_{\mathbb{R}^{d}}\frac{1}{\Phi_{p}(0,\eta)}\,\mu(d\eta)\right)^{n},&\alpha_{0}=0,\end{cases}$
(3.8)

where $\Phi_{p}$ is the function defined on $\mathbb{R}^{d+1}$ by

$\Phi_{p}(\tau,\eta):=\Big{|}(p-i\tau)^{2}+|\eta|^{2}\Big{|}^{2}=4p^{2}\tau^{2}+(p^{2}+|\eta|^{2}-\tau^{2})^{2}.$
(3.9)

###### Proof.

This somewhat lengthy proof will be split onto several steps for the reader’s
convenience. We will focus on the case $\alpha_{0}>0$, the case $\alpha_{0}=0$
being proved in a similar way.

Step 1: Reduction to a constant initial condition. Let us write
$f^{w}_{n}=\frac{1}{n!}\sum_{\sigma\in\Sigma_{n}}f_{n}^{w,\sigma}$, with

$\displaystyle f_{n}^{w,\sigma}(s_{1},x_{1},\dots,s_{n},x_{n},t,x)$
$\displaystyle:=G_{t-s_{\sigma(n)}}(x-x_{\sigma(n)})\cdots
G_{s_{\sigma(2)}-s_{\sigma(1)}}(x_{\sigma(2)}-x_{\sigma(1)})\times
w(s_{\sigma(1)},x_{\sigma(1)})\mathbf{1}_{[0<s_{\sigma(1)}<\dots<s_{\sigma(n)}<t]}.$

It is easily seen, thanks to Jensen’s inequality applied to the uniform
measure on $\Sigma_{n}$, that

$\|f^{w}_{n}(\cdot,t,x)\|_{{\mathcal{H}}^{\otimes
n}}^{2}\leq\frac{1}{n!}\sum_{\sigma\in\Sigma_{n}}\|f_{n}^{w,\sigma}(\cdot,t,x)\|_{{\mathcal{H}}^{\otimes
n}}^{2}.$ (3.10)

Now, using first the expression (2.1) and then the result of Proposition 2.6
(which guarantees that $w\in L^{\infty}([0,T]\times\mathbb{R}^{d})$), we
derive that

$\displaystyle\|f^{w,\sigma}_{n}(\cdot,t,x)\|_{{\mathcal{H}}^{\otimes n}}^{2}$
$\displaystyle=\int_{(\mathbb{R}_{+}^{n})^{2}}d{\bf s}d{\bf
s^{\prime}}\int_{(\mathbb{R}^{d})^{2}}d{\bf x}d{\bf
x^{\prime}}\,\bigg{(}\prod_{k=1}^{n}|s_{k}-s_{k}^{\prime}|^{-\alpha_{0}}\gamma(x_{k}-x_{k}^{\prime})\bigg{)}$
$\displaystyle\qquad G_{t-s_{\sigma(n)}}(x-x_{\sigma(n)})\cdots
G_{s_{\sigma(2)}-s_{\sigma(1)}}(x_{\sigma(2)}-x_{\sigma(1)})\times
w(s_{\sigma(1)},x_{\sigma(1)})\mathbf{1}_{[0<s_{\sigma(1)}<\dots<s_{\sigma(n)}<t]}$
$\displaystyle\qquad\quad
G_{t-s^{\prime}_{\sigma(n)}}(x-x^{\prime}_{\sigma(n)})\cdots
G_{s^{\prime}_{\sigma(2)}-s^{\prime}_{\sigma(1)}}(x^{\prime}_{\sigma(2)}-x^{\prime}_{\sigma(1)})\times
w(s^{\prime}_{\sigma(1)},x^{\prime}_{\sigma(1)})\mathbf{1}_{[0<s^{\prime}_{\sigma(1)}<\dots<s^{\prime}_{\sigma(n)}<t]}$
$\displaystyle\leq\|w\|_{L^{\infty}([0,T]\times\mathbb{R}^{d})}^{2}\int_{([0,t]_{<}^{n})^{2}}\int_{(\mathbb{R}^{d})^{2}}\bigg{(}\prod_{k=1}^{n}|s_{k}-s_{k}^{\prime}|^{-\alpha_{0}}\gamma(x_{k}-x_{k}^{\prime})\bigg{)}$
$\displaystyle\hskip
113.81102pt\times\bigg{(}\prod_{k=1}^{n}G_{s_{k}-s_{k-1}}(x_{k}-x_{k-1})G_{s_{k}^{\prime}-s_{k-1}^{\prime}}(x_{k}^{\prime}-x_{k-1}^{\prime})\bigg{)}\,d{\bf
x}d{\bf x^{\prime}}d{\bf s}d{\bf s^{\prime}},$

where we use the convention $s_{0}=0$ and $x_{0}=x^{\prime}_{0}=x$, and the
notation $d{\bf s}=ds_{1}\cdots ds_{n}$ (we use similar conventions for $d{\bf
x},d{\bf x^{\prime}}$ and $d{\bf s^{\prime}}$). Note also that we use
$[0,t]^{n}_{<}$ to denote the $n$-th order simplex
$[0<s_{1}<s_{2}<\dots<s_{n}<t]$ on $[0,t]$. Injecting the latter estimate into
(3.10), and with expression (2.1) in mind, we get that

$\|f^{w}_{n}(\cdot,t,x)\|_{{\mathcal{H}}^{\otimes
n}}\leq\|w\|_{L^{\infty}([0,T]\times\mathbb{R}^{d})}\|g_{n}(\cdot,t,x)\|_{{\mathcal{H}}^{\otimes
n}},$

where $g_{n}$ is defined by

$g_{n}(s_{1},x_{1},\dots,s_{n},x_{n},t,x):=G_{t-s_{n}}(x-x_{n})\cdots
G_{s_{2}-s_{1}}(x_{2}-x_{1})\mathbf{1}_{[0,t]_{<}^{n}}(s_{1},\dots,s_{n}).$
(3.11)

In the sequel, we shall upper bound the terms involving $g_{n}$.

Step 2: Expression for the Laplace transform. One has

$\displaystyle\|g_{n}(\cdot,t,x)\|_{{\mathcal{H}}^{\otimes n}}^{2}$
$\displaystyle=\int_{([0,t]_{<}^{n})^{2}}\int_{(\mathbb{R}^{d})^{2}}\bigg{(}\prod_{k=1}^{n}|s_{k}-s_{k}^{\prime}|^{-\alpha_{0}}\gamma(x_{k}-x_{k}^{\prime})\bigg{)}$
$\displaystyle\qquad\times\bigg{(}\prod_{k=1}^{n}G_{s_{k}-s_{k-1}}(x_{k}-x_{k-1})G_{s_{k}^{\prime}-s_{k-1}^{\prime}}(x_{k}^{\prime}-x_{k-1}^{\prime})\bigg{)}d{\bf
x}d{\bf x^{\prime}}d{\bf s}d{\bf s^{\prime}}.$ (3.12)

In order to ease our arguments below, let us recast (3.12) in a slightly more
complicated way by doubling the $t$ variable. That is, write

$\|g_{n}(\cdot,t,x)\|^{2}_{\mathcal{H}^{\otimes n}}=F_{n}(t,t),$ (3.13)

where the function $F:\mathbb{R}^{2}\to\mathbb{R}_{+}$ is defined by

$F_{n}(t,\bar{t})=\int_{[0,t]_{<}^{n}\times[0,\bar{t}]_{<}^{n}}\int_{(\mathbb{R}^{d})^{2}}\bigg{(}\prod_{k=1}^{n}|s_{k}-s_{k}^{\prime}|^{-\alpha_{0}}\gamma(x_{k}-x_{k}^{\prime})\bigg{)}\\\
\times\bigg{(}\prod_{k=1}^{n}G_{s_{k}-s_{k-1}}(x_{k}-x_{k-1})G_{s_{k}^{\prime}-s_{k-1}^{\prime}}(x_{k}^{\prime}-x_{k-1}^{\prime})\bigg{)}d{\bf
x}d{\bf x^{\prime}}d{\bf s}d{\bf s^{\prime}}.$ (3.14)

Also recall that we are interested in the Laplace transform of
$\|g_{n}(\cdot,t,x)\|_{{\mathcal{H}}^{\otimes n}}^{2}$, for which we introduce
a notation:

$\Lambda_{n}(p)\equiv\int_{0}^{\infty}e^{-2pt}\|g_{n}(t,x,\cdot)\|_{{\mathcal{H}}^{\otimes
n}}^{2}dt=\int_{0}^{\infty}e^{-2pt}F_{n}(t,t)\,dt.$ (3.15)

Step 3: Reverse $L^{2}$ bounds. We will now use monotonicity properties of
exponential random variables in order to reduce the $L^{2}$-type norm (3.15)
to a product of $L^{1}$-type norms. The argument is inspired by page 954 in
[chen19] and goes as follows:

  1. (i)

The integral in (3.15) can be seen as an integral with respect to an
exponential variable $T$ with parameter $2p$. Namely

$\Lambda_{n}(p)=\frac{1}{2p}\mathbb{E}[F_{n}(T,T)],\quad\text{ with }\quad
T\sim\text{exp}(2p).$

  2. (ii)

It is elementary to prove the identity in law
$T\overset{(d)}{=}\tau\wedge\bar{\tau}$, with $\tau,\bar{\tau}$ i.i.d. with
common law exp$(p)$. Hence,

$\Lambda_{n}(p)=\frac{1}{2p}\mathbb{E}[F_{n}(\tau\wedge\bar{\tau},\tau\wedge\bar{\tau})].$
(3.16)

  3. (iii)

Due to the positivity of $\gamma$ and $G$, it is readily checked that both
$t\mapsto F_{n}(t,\bar{t})$ and $\bar{t}\mapsto F_{n}(t,\bar{t})$ are non-
decreasing functions. Plugging this information into (3.16) we obtain

$\Lambda_{n}(p)\leq\frac{1}{2p}\mathbb{E}[F_{n}(\tau,\bar{\tau})].$ (3.17)

Let us now recall the expression (3.14) for $F_{n}$ and write the expected
value with respect to $\tau,\bar{\tau}$ in (3.17) explicitly. Summarizing our
considerations (i)-(iii) above, we have obtained the following relation:

$\Lambda_{n}(p)\leq\frac{1}{2p}\int_{\mathbb{R}_{+}^{2}}dtd\bar{t}~{}p\,e^{-p(t+\bar{t})}\times\int_{[0,t]_{<}^{n}\times[0,\bar{t}]_{<}^{n}}\int_{(\mathbb{R}^{d})^{2}}\bigg{(}\prod_{k=1}^{n}|s_{k}-s_{k}^{\prime}|^{-\alpha_{0}}\gamma(x_{k}-x_{k}^{\prime})\bigg{)}\\\
\times\bigg{(}\prod_{k=1}^{n}G_{s_{k}-s_{k-1}}(x_{k}-x_{k-1})G_{s_{k}^{\prime}-s_{k-1}^{\prime}}(x_{k}^{\prime}-x_{k-1}^{\prime})\bigg{)}d{\bf
x}d{\bf x^{\prime}}d{\bf s}d{\bf s^{\prime}}.$ (3.18)

We now rearrange terms above thanks to Fubini’s theorem, in order to obtain a
more readable version of (3.18). To this aim, let us introduce an additional
notation. Namely, for $s_{1},\dots,s_{n}\in[0,t]$ and
$x_{1},\dots,x_{n}\in\mathbb{R}^{d}$, we set

$H_{p}(s_{1},x_{1}\dots,s_{n},x_{n})=\int_{0}^{\infty}e^{-pt}\bigg{(}\prod_{k=1}^{n}G_{s_{k}-s_{k-1}}(x_{k}-x_{k-1})\bigg{)}\mathbf{1}_{[0<s_{1}<\dots<s_{n}<t]}dt.$

Then (3.18) can be expressed as

$\Lambda_{n}(p)\leq\frac{p}{2}\int_{(\mathbb{R}^{d+1})^{2n}}H_{p}(s_{1},x_{1}\dots,s_{n},x_{n})H_{p}(s_{1}^{\prime},x_{1}^{\prime},\dots,s_{n}^{\prime},x_{n}^{\prime})\\\
\times\bigg{(}\prod_{k=1}^{n}|s_{k}-s_{k}^{\prime}|^{-\alpha_{0}}\gamma(x_{k}-x_{k}^{\prime})\bigg{)}d{\bf
x}d{\bf x^{\prime}}d{\bf s}d{\bf s^{\prime}}.$ (3.19)

Step 4: Some algebraic manipulations. Notice that the right-hand side of
(3.19) can be written in terms of a convolution product. That is, we have

$\Lambda_{n}(p)\leq\frac{p}{2}\int_{(\mathbb{R}^{d+1})^{n}}H_{p}(s_{1},x_{1},\dots,s_{n},x_{n})\phi_{p}(s_{1},x_{1},\dots,s_{n},x_{n})dyds,$
(3.20)

where

$\phi_{p}(s_{1},x_{1},\dots,s_{n},x_{n})=H_{p}*\left[\prod_{k=1}^{n}|s_{k}|^{-\alpha_{0}}\gamma(x_{k})\right].$

Therefore, one can express (3.20) in Fourier modes thanks to Plancherel’s
identity. This yields, for some positive constant $C_{\alpha_{0}}$,

$\Lambda_{n}(p)\leq\frac{p}{2}C^{n}_{\alpha_{0}}\int_{(\mathbb{R}^{d+1})^{n}}\bigg{(}\prod_{k=1}^{n}{1\over|\lambda_{k}|^{1-\alpha_{0}}}\bigg{)}\\\
\times\left|\int_{\mathbb{R}^{n}}\exp\left(i\sum_{k=1}^{n}\lambda_{k}s_{k}\right)\widehat{H}_{p}(s_{1},\xi_{1},\dots,s_{n},\xi_{n})d{\bf
s}\right|^{2}d\lambda\mu(d\xi),$ (3.21)

where we abuse the notation $d\lambda=d\lambda_{1}\cdots d\lambda_{n}$ and
$\mu(d\xi)=\mu(d\xi_{1})\cdots\mu(d\xi_{n}),$ and $\widehat{H}_{p}$ means the
Fourier transform of $H_{p}$ in its space variable.

Let us now say a few words about the computation of $\widehat{H}_{p}$ in
(3.21). Denoting without any further mention the spatial Fourier transform by
$\mathcal{F}g$ or $\widehat{g}$, we have

$\widehat{H}_{p}(s_{1},\xi_{1},\dots,s_{n},\xi_{n})\\\
=\int_{0}^{\infty}e^{-pt}\mathcal{F}\left(\prod_{k=1}^{n}G_{s_{k}-s_{k-1}}(y_{k}-y_{k-1})\right)(\xi_{1},\dots,\xi_{n})\,\mathbf{1}_{[0,t]^{n}_{<}}(s_{s},\dots,s_{n})\,dt.$
(3.22)

Furthermore, performing the change of variables $z_{k}=y_{k}-y_{k-1}$ and
rearranging the Fourier variables thanks to some easy algebraic manipulations,
it is readily checked that

$\mathcal{F}\left(\prod_{k=1}^{n}G_{s_{k}-s_{k-1}}(y_{k}-y_{k-1})\right)(\xi_{1},\dots,\xi_{n})=\prod_{k=1}^{n}\widehat{G}_{s_{k}-s_{k-1}}\left(\sum_{j=k}^{n}\xi_{j}\right).$

Plugging this identity in (3.22) and then in (3.21), we end up with

$\Lambda_{n}(p)\leq\frac{p}{2}C^{n}_{\alpha_{0}}\int_{(\mathbb{R}^{d+1})^{n}}\left|\Psi_{p}(\lambda,\xi)\right|^{2}\,\prod_{k=1}^{n}{1\over|\lambda_{k}|^{1-\alpha_{0}}}\,d\lambda\mu(d\xi),$
(3.23)

where we have set

$\Psi_{p}(\lambda,\xi)=\int_{0}^{\infty}e^{-pt}\int_{[0,t]_{<}^{n}}\exp\bigg{(}i\sum_{k=1}^{n}\lambda_{k}s_{k}\bigg{)}\prod_{k=1}^{n}\widehat{G}_{s_{k}-s_{k-1}}\Big{(}\sum_{j=k}^{n}\xi_{j}\Big{)}d{\bf
s}dt.$ (3.24)

Step 5: Some Fourier computations. In this step, we write a more explicit
version of (3.23) by computing some Fourier transforms. Namely write
$s_{k}-s_{k-1}=r_{k}$ for $k=1,\dots,n+1$, where we have set $s_{0}=0$ and
$s_{n+1}=t$. It is easily seen that the function $\Psi_{p}$ in (3.24) becomes

$\Psi_{p}(\lambda,\xi):=\int_{\mathbb{R}_{+}^{n+1}}\exp\left(-p\sum_{k=1}^{n+1}r_{k}\right)\prod_{k=1}^{n}\exp\left(ir_{k}\sum_{j=k}^{n}\lambda_{j}\right)\widehat{G}_{r_{k}}\Big{(}\sum_{j=k}^{n}\xi_{j}\Big{)}~{}d{\bf
r}.$

Integrating over the variable $r_{n+1}$, we are left with a product of $n$
integrals:

$\Psi_{p}(\lambda,\xi)=\frac{1}{p}\prod_{k=1}^{n}\int_{\mathbb{R}_{+}}\exp\left(\Big{(}-p+i\sum_{j=k}^{n}\lambda_{j}\Big{)}r\right)\widehat{G}_{r}\Big{(}\sum_{j=k}^{n}\xi_{j}\Big{)}dr.$
(3.25)

We now give a more explicit expression for $\Psi_{p}$ thanks to the expression
(2.7) for $\widehat{G}$. Namely, for any $\beta\in\mathbb{C}$ with positive
real part, it is well known that

$\int_{0}^{\infty}e^{-\beta
r}\frac{\sin(r|\eta|)}{|\eta|}dr=\frac{1}{\beta^{2}+|\eta|^{2}}.$ (3.26)

Reporting this expression into (3.25), we get

$\Psi_{p}(\lambda,\xi)=\frac{1}{p}\prod_{k=1}^{n}\bigg{[}\Big{(}p-i\sum_{j=k}^{n}\lambda_{j}\Big{)}^{2}+\Big{|}\sum_{j=k}^{n}\xi_{j}\Big{|}^{2}\bigg{]}^{-1}.$

Going back to (3.23), we have thus obtained

$\displaystyle\Lambda_{n}(p)$
$\displaystyle\leq\frac{C^{n}_{\alpha_{0}}}{2}\int_{(\mathbb{R}^{d+1})^{n}}\bigg{(}\prod_{k=1}^{n}{1\over|\lambda_{k}|^{1-\alpha_{0}}}\bigg{)}\prod_{k=1}^{n}\bigg{|}\Big{(}p-i\sum_{j=k}^{n}\lambda_{j}\Big{)}^{2}+\Big{|}\sum_{j=k}^{n}\xi_{j}\Big{|}^{2}\bigg{|}^{-2}d\lambda\,\mu(d\xi).$
(3.27)

Moreover, recalling the expression (3.9) for $\Phi_{p}$, (3.27) can be
expressed as

$\Lambda_{n}(p)\leq\frac{C^{n}_{\alpha_{0}}}{2}\int_{(\mathbb{R}^{d+1})^{n}}\bigg{(}\prod_{k=1}^{n}{1\over|\lambda_{k}|^{1-\alpha_{0}}}\bigg{)}\prod_{k=1}^{n}\left(\Phi_{p}\left(\sum_{j=k}^{n}\lambda_{j},\sum_{j=k}^{n}\xi_{j}\right)\right)^{-1}d\lambda\,\mu(d\xi).$
(3.28)

Step 6: Reduction to $1$-d integrals. Our next aim is to transform the right-
hand side of (3.28) into a product of integrals in $\mathbb{R}^{d+1}$. In
order to achieve this we invoke the following lemma, which is an elaboration
of [chen19, Lemma 3.1]:

######  Lemma 3.6.

Let $f$ be a function and $\nu$ be a measure, both defined on $\mathbb{R}^{m}$
and both positive. We assume that the Fourier transforms $\widehat{f}$ and
$\widehat{\nu}$ of $f$ and $\nu$ are positive functions. Then for all
$\eta\in\mathbb{R}^{m}$ we have

$\int_{\mathbb{R}^{m}}f(\xi-\eta)\nu(d\xi)\leq\int_{\mathbb{R}^{m}}f(\xi)\nu(d\xi).$
(3.29)

###### Proof.

A basic application of Parseval’s identity shows that

$\int_{\mathbb{R}^{m}}f(\xi-\eta)\nu(d\xi)=\int_{\mathbb{R}^{m}}e^{ix\cdot\eta}\widehat{f}(x)\widehat{\nu}(x)dx.$

We now trivially bound the oscillating exponential term by $1$ in the right
hand side above, and apply Parseval’s identity again, in order to get our
claim (3.29). ∎

Lemma 3.6 is applied successively to the $\lambda,\xi$ variables in (3.28). We
now provide some details about the $(\lambda_{1},\xi_{1})$-integrals. That is,
in the right-hand side of (3.28), the $(\lambda_{1},\xi_{1})$ variables appear
in the integral

$\Lambda_{n,1}(p)=\int_{\mathbb{R}^{d+1}}\left|\psi_{p}(\lambda_{1}+l,\xi_{1}+x)\right|^{2}\nu(d\lambda_{1},d\xi_{1}),$
(3.30)

where $l=\sum_{j=2}^{n}\lambda_{j},x=\sum_{j=2}^{n}\xi_{j}$ and where the
function $\psi_{p}$ and the measure $\nu$ are respectively defined by

$\psi_{p}(\tau,\eta)=\left(\Phi_{p}(\tau,\eta)\right)^{-1/2}=\Big{(}(p-i\tau)^{2}+|\eta|^{2}\Big{)}^{-1},\quad\nu(d\lambda_{1},d\xi_{1})=\frac{1}{|\lambda_{1}|^{1-\alpha_{0}}}d\lambda_{1}\mu(d\xi_{1}).$
(3.31)

If we wish to apply Lemma 3.6 to the integral in (3.30), we thus have to prove
that the space-time Fourier transforms of $|\psi_{p}|^{2}$ and $\nu$ are both
positive functions. This is argued below:

  1. (a)

Owing to (3.26) and (2.7), we have

$\displaystyle\psi_{p}(\tau,\eta)=\int_{\mathbb{R}^{d+1}}\exp\bigg{(}i(\tau
t+\eta\cdot x)\bigg{)}\mathbf{1}_{[0,\infty)}(t)e^{-pt}G_{t}(x)dxdt.$

Otherwise stated, $\psi_{p}$ is the Fourier transform of a non-negative
function $g_{p}$ defined on $\mathbb{R}^{d+1}$ by
$g_{p}(t,x)=\mathbf{1}_{[0,\infty)}(t)e^{-pt}G_{t}(x)$. Therefore
$\mathcal{F}\psi_{p}$ is a positive function. Since
$\mathcal{F}|\psi_{p}|^{2}=(\mathcal{F}\psi_{p})*(\mathcal{F}\bar{\psi}_{p})$,
we also get that $\mathcal{F}|\psi_{p}|^{2}$ is a positive function.

  2. (b)

According to (3.31), the space-time Fourier transform of the measure $\nu$ is
given by $\mathcal{F}\nu(t,x)=C_{d}|t|^{-\alpha_{0}}\gamma(x)$, which is a
positive function.

We are thus in a position to apply Lemma 3.6 to the integral in (3.30), which
yields

$\Lambda_{n,1}(p)\leq\int_{\mathbb{R}^{d+1}}\left|\psi_{p}(\lambda_{1},\xi_{1})\right|^{2}\nu(d\lambda_{1},d\xi_{1})=\int_{\mathbb{R}^{d+1}}\left(\Phi_{p}(\lambda_{1},\xi_{1})\right)^{-1}\nu(d\lambda_{1},d\xi_{1}),$
(3.32)

where we observe that the right-hand side above does not depend on the
variables $\lambda_{j},\xi_{j}$ with $j=2,\dots,n$ anymore, recalling that
$\Phi_{p}$ is given by (3.9).

As mentioned above, we now just have to iterate (3.32) into (3.27). This
allows to reduce $\Lambda_{n}(p)$ to a product of $n$ integrals in
$\mathbb{R}^{d+1}$:

$\displaystyle\Lambda_{n}(p)\leq\frac{1}{2}\bigg{(}C_{\alpha_{0}}\int_{\mathbb{R}^{d+1}}|\tau|^{\alpha_{0}-1}\left(\Phi_{p}(\tau,\eta)\right)^{-1}d\tau\mu(d\eta)\bigg{)}^{n}.$
(3.33)

Step 7: Conclusion. Recall that according to (3.15), $\Lambda_{n}(p)$ is the
Laplace transform of $t\mapsto\|g_{n}(t,x,\cdot)\|^{2}_{\mathcal{H}^{\otimes
n}}$. Moreover, we have reduced our computations for $f_{n}$ to those of
$g_{n}$ in Step 1. Hence (3.33) easily yields (3.8). This finishes our proof.
∎

###  3.3. On the sufficiency of the condition

In this subsection, we aim to show that the necessary condition (1.3) obtained
in Section 3.1 is also a sufficient condition for the existence and uniqueness
of the mild Skorohod solution to (1.1). We first give the following
preparatory result.

######  Lemma 3.7.

Assume that Hypothesis 2.1 holds true, and set

$L_{\alpha_{0},n}:=\begin{cases}\displaystyle\int_{\mathbb{R}^{d+1}}\frac{1}{|\lambda|^{1-\alpha_{0}}\,\Phi_{n}(\lambda,\xi)}~{}\mu(d\xi)d\lambda,&\alpha_{0}>0,\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\int_{\mathbb{R}^{d}}\frac{1}{\Phi_{n}(0,\xi)}~{}\mu(d\xi),&\alpha_{0}=0,\end{cases}$
(3.34)

where $\Phi_{n}$ is defined in (3.9). Then under the condition (1.3), we have

$L_{\alpha_{0},n}<\infty\ \text{ for every}\ n\geq
1,\qquad\text{and}\qquad\displaystyle\lim_{n\to\infty}nL_{\alpha_{0},n}=0\,.$
(3.35)

###### Proof.

Case 1: $\alpha_{0}>0$. Starting from the right hand side of (3.9), some
elementary manipulations show that

$4n^{2}\lambda^{2}+(n^{2}+|\xi|^{2}-\lambda^{2})^{2}=4n^{2}|\xi|^{2}+(n^{2}+\lambda^{2}-|\xi|^{2})^{2}.$

Therefore one can symmetrize $\Phi_{n}$ in the following way:

$\displaystyle\Phi_{n}(\lambda,\xi)$ $\displaystyle={1\over
2}\Big{\\{}4n^{2}\lambda^{2}+(n^{2}+|\xi|^{2}-\lambda^{2})^{2}+4n^{2}|\xi|^{2}+(n^{2}+\lambda^{2}-|\xi|^{2})^{2}\Big{\\}}$
$\displaystyle=n^{4}+(|\xi|+|\lambda|)^{2}(|\xi|-|\lambda|)^{2}+2n^{2}(|\xi|^{2}+\lambda^{2}).$

Next some elementary algebraic manipulations yield

$\displaystyle\Phi_{n}(\lambda,\xi)$ $\displaystyle\geq
n^{4}+(|\xi|+|\lambda|)^{2}(|\xi|-|\lambda|)^{2}+n^{2}(|\xi|+|\lambda|)^{2}$
$\displaystyle=n^{4}+(|\xi|+|\lambda|)^{2}\big{(}n^{2}+(|\xi|-|\lambda|)^{2}\big{)}.$

Hence to obtain the desired assertion (3.35), it suffices to prove that

$K_{n}<\infty\ (\text{for all}\ n\geq
1)\quad\text{and}\lim_{n\to\infty}nK_{n}=0,\quad\text{with}\quad
K_{n}:=\int_{\mathbb{R}^{d+1}}\tilde{\Phi}_{n}(\lambda,\xi)\,\mu(d\xi)d\lambda,$
(3.36)

and where the function $\tilde{\Phi}_{n}$ is defined by

$\tilde{\Phi}_{n}(\lambda,\xi):={1\over|\lambda|^{1-\alpha_{0}}\Big{(}n^{4}+(|\xi|+|\lambda|)^{2}\big{(}n^{2}+(|\xi|-|\lambda|)^{2}\big{)}\Big{)}}\,.$
(3.37)

The remainder of the proof is devoted to show (3.36).

In order to bound $K_{n}$ with a suitable (finite) quantity, let us fix a
large constant $a>0$. Then we decompose $K_{n}$ as

$K_{n}=K_{n}(a)+\bar{K}_{n}(a),$ (3.38)

where $K_{n}(a)$ and $\bar{K}_{n}(a)$ are respectively defined by

$K_{n}(a):=\int_{\mathbb{R}\times\\{|\xi|\leq
a\\}}\tilde{\Phi}_{n}(\lambda,\xi)\,\mu(d\xi)d\lambda,\qquad\bar{K}_{n}(a):=\int_{\mathbb{R}\times\\{|\xi|>a\\}}\tilde{\Phi}_{n}(\lambda,\xi)\,\mu(d\xi)d\lambda.$
(3.39)

We now bound those two terms separately.

In order to estimate $K_{n}(a)$ in (3.38), we simply write

$\tilde{\Phi}_{n}(\lambda,\xi)\leq\frac{1}{|\lambda|^{1-\alpha_{0}}(n^{4}+n^{2}\lambda^{2})},$

which yields

$nK_{n}(a)\leq\mu(|\xi|\leq
a)\int_{\mathbb{R}}\frac{d\lambda}{|\lambda|^{1-\alpha_{0}}n(n^{2}+\lambda^{2})}\\\
\leq\frac{\mu(|\xi|\leq
a)}{n}\int_{\mathbb{R}}\frac{d\lambda}{|\lambda|^{1-\alpha_{0}}(1+\lambda^{2})}.$

Since $\alpha_{0}\in(0,1)$, the latter estimate readily entails that

$K_{n}(a)<\infty\quad\text{and}\quad\lim_{n\to\infty}nK_{n}(a)=0.$ (3.40)

We now analyze the term $\bar{K}_{n}(a)$ in (3.39). For this we decompose the
integral and write

$\bar{K}_{n}(a)=\bar{K}_{n}^{1}(a)+\bar{K}_{n}^{2}(a),$ (3.41)

where $\bar{K}_{n}^{1}(a),\bar{K}_{n}^{2}(a)$ are given by

$\bar{K}_{n}^{1}(a):=\int_{\\{|\lambda|\leq|\xi|/2,\,|\xi|>a\\}}\tilde{\Phi}_{n}(\lambda,\xi)\,\mu(d\xi)d\lambda,\quad\bar{K}_{n}^{2}(a):=\int_{\\{|\lambda|>|\xi|/2\\},\,|\xi|>a\\}}\tilde{\Phi}_{n}(\lambda,\xi)\,\mu(d\xi)d\lambda.$

Now we estimate $\bar{K}_{n}^{1}(a)$. If $|\lambda|\leq\frac{1}{2}|\xi|$ and
recalling the expression (3.37) for $\tilde{\Phi}_{n}$, we have

$\tilde{\Phi}_{n}(\lambda,\xi)\leq\frac{1}{|\lambda|^{1-\alpha_{0}}\big{(}n^{4}+|\xi|^{2}(n^{2}+\frac{1}{4}|\xi|^{2})\big{)}}.$

Hence integrating first with respect to $\lambda$, we get

$\bar{K}_{n}^{1}(a)\leq\int_{\mathbb{R}^{d}}\frac{1}{n^{4}+|\xi|^{2}\big{(}n^{2}+\frac{1}{4}|\xi|^{2}\big{)}}\left(\int_{-{1\over
2}|\xi|}^{{1\over
2}|\xi|}{1\over|\lambda|^{1-\alpha_{0}}}d\lambda\right)\mu(d\xi).$

In addition, the elementary inequality $a^{2}+b^{2}\geq 2ab$ implies that
$n^{2}+\frac{1}{4}|\xi|^{2}\geq\frac{1}{2}n|\xi|$. We thus get

$\bar{K}_{n}^{1}(a)\leq\frac{C_{\alpha_{0}}}{n}\int_{\mathbb{R}^{d}}\frac{|\xi|^{\alpha_{0}}}{n^{3}+|\xi|^{3}}\mu(d\xi).$

Taking into account our assumption (1.3), we obtain that
$\bar{K}_{n}^{1}(a)<\infty$ for every fixed $n\geq 1$, and a classical
dominated convergence argument allows to conclude

$\lim_{n\to\infty}n\bar{K}_{n}^{1}(a)=0.$ (3.42)

As far as $\bar{K}_{n}^{2}(a)$ in (3.41) is concerned, on the set
$|\lambda|>\frac{1}{2}|\xi|$, there exists a constant $C>0$ such that

$\tilde{\Phi}_{n}(\lambda,\xi)\leq\frac{C}{|\xi|^{3-\alpha_{0}}\big{(}n^{2}+(|\xi|-|\lambda|)^{2}\big{)}}.$
(3.43)

Furthermore, one can argue that

$\int_{\\{|\lambda>|\xi|/2\\}}{1\over
n^{2}+(|\xi|-|\lambda|)^{2}}d\lambda\leq\int_{\mathbb{R}}{1\over
n^{2}+(|\xi|-|\lambda|)^{2}}d\lambda\\\ =2\int_{0}^{\infty}{1\over
n^{2}+(\lambda-|\xi|)^{2}}d\lambda\leq 2\int_{\mathbb{R}}{1\over
n^{2}+\lambda^{2}}d\lambda={2\pi\over n}.$ (3.44)

Plugging (3.43) and (3.44) into the definition of $\bar{K}_{n}^{2}(a)$, we
thus get

$nK_{n}^{2}(a)\leq
C_{\alpha_{0}}\int_{\\{|\xi|>a\\}}|\xi|^{-(3-\alpha_{0})}\mu(d\xi),$ (3.45)

which, due to assumption (1.3), already guarantees that $K_{n}^{2}(a)$ is
finite for every fixed $n\geq 1$.

Summarizing our considerations so far, we report (3.42) and (3.45) into the
decomposition (3.41) of $\bar{K}_{n}(a)$. Taking also into account (3.40) and
the decomposition (3.38), we end up with

$K_{n}<\infty\ \text{for every}\ n\geq
1\quad\text{and}\quad\limsup_{n\to\infty}nK_{n}\leq
C_{\alpha_{0}}\int_{\\{|\xi|>a\\}}|\xi|^{-(3-\alpha_{0})}\mu(d\xi).$ (3.46)

Eventually recall that we are working under (1.3). Moreover, the parameter $a$
in the right-hand side of (3.46) is arbitrarily large. Hence the right-hand
side of (3.46) is arbitrarily small. We thus get

$\lim_{n\to\infty}nK_{n}=0,$

which means that (3.36) is shown. As argued in the beginning of the proof,
this is sufficient to ensure that (3.35) holds true.

Case 2: $\alpha_{0}=0$. Noting that $\Phi_{n}(0,\xi)=(n^{2}+|\xi|^{2})^{2}$,
it suffices to prove

$\int_{\mathbb{R}^{d}}\frac{1}{n^{4}+|\xi|^{4}}\mu(d\xi)<\infty\ \text{for
every fixed}\ n\geq
1\quad\text{and}\quad\lim_{n\to\infty}\int_{\mathbb{R}^{d}}\frac{n}{n^{4}+|\xi|^{4}}\mu(d\xi)=0.$
(3.47)

Observe first that the condition (1.3) with $\alpha_{0}=0$ becomes

$\int_{\mathbb{R}^{d}}{1\over 1+|\xi|^{3}}\mu(d\xi)<\infty,$ (3.48)

and so the fact that
$\int_{\mathbb{R}^{d}}\frac{1}{n^{4}+|\xi|^{4}}\mu(d\xi)<\infty$ (for $n\geq
1$) is obvious. Besides, one has clearly

$\displaystyle\int_{\mathbb{R}^{d}}\frac{n}{n^{4}+|\xi|^{4}}\mu(d\xi)=\int_{\\{|\xi|\leq
n\\}}\frac{n}{n^{4}+|\xi|^{4}}\mu(d\xi)+\int_{\\{|\xi|>n\\}}\frac{n}{n^{4}+|\xi|^{4}}\mu(d\xi)$
(3.49)

The second term of (3.49) satisfies

$\int_{\\{|\xi|>n\\}}\frac{n}{n^{4}+|\xi|^{4}}\mu(d\xi)\leq\int_{\\{|\xi|>n\\}}\frac{n}{n^{4}+n|\xi|^{3}}\mu(d\xi)\leq\int_{\mathbb{R}^{d}}\frac{1}{n^{3}+|\xi|^{3}}\mu(d\xi)$

and hence, by (3.48) this second term converges to 0 as $n\to\infty$.
Regarding the first term of (3.49), we have

$\int_{\\{|\xi|\leq
n\\}}\frac{n}{n^{4}+|\xi|^{4}}\mu(d\xi)\leq\frac{1}{n^{3}}\mu(\\{|\xi|\leq
n\\}).$

In addition, observe that

$\frac{1}{n^{3}}\mu(\\{|\xi|\leq n\\})\leq 2\int_{\\{|\xi|\leq
n\\}}\frac{1}{n^{3}+|\xi|^{3}}\mu(d\xi)\leq
2\int_{\mathbb{R}^{d}}\frac{1}{n^{3}+|\xi|^{3}}\mu(d\xi),$

and the right hand side above also goes to 0 as $n\to 0$ thanks to dominated
convergence arguments, similarly to what we did for (3.42). This shows (3.47)
and our proof is complete. ∎

Lemma 3.7 was our last preliminary result before proving our existence and
uniqueness theorem. We now state and prove this result, which has to be
regarded as the main contribution of the current paper.

######  Theorem 3.8.

Recall that the function $f_{n}$ is given by (2.10). Assume that Hypothesis
2.1 holds true and assume the same conditions as in Proposition 2.6. Then we
have

$\sum_{n=0}^{\infty}n!\|f_{n}(\cdot,t,x)\|_{{\mathcal{H}}^{\otimes
n}}^{2}<\infty.$ (3.50)

Hence owing to Proposition 2.5, there is a unique mild Skorohod solution to
(1.1).

###### Proof.

We start by upper bounding the Laplace transform of the function
$t\mapsto\|f_{n}(\cdot,t,x)\|^{2}_{\mathcal{H}^{\otimes n}}$. To this aim,
apply directly (3.8) with $p=n$. We get

$\displaystyle
2\int_{0}^{\infty}e^{-2nt}\|f_{n}(\cdot,t,x)\|_{{\mathcal{H}}^{\otimes
n}}^{2}dt\leq\left(C_{\alpha_{0}}L_{\alpha_{0},n}\right)^{n},$ (3.51)

where $L_{\alpha_{0},n}$ is the (finite) quantity introduced in (3.34).

According to Lemma 3.7, it holds that
$\lim\limits_{n\to\infty}nL_{\alpha_{0},n}=0$, and hence some elementary
considerations yield

$\displaystyle\lim_{n\to\infty}\frac{1}{n}\log\big{(}n^{n}(C_{\alpha_{0}}L_{\alpha_{0},n})^{n}\big{)}=\lim_{n\to\infty}\big{(}\log
C_{\alpha_{0}}+\log(nL_{\alpha_{0},n})\big{)}=-\infty.$

This together with (3.51) lead to the following relation:

$\lim_{n\to\infty}{1\over n}\log
n^{n}\left(2\int_{0}^{\infty}e^{-2nt}\|f_{n}(\cdot,t,x)\|_{{\mathcal{H}}^{\otimes
n}}^{2}dt\right)=-\infty.$ (3.52)

This identity can be related to a single value of
$\|f_{n}(\cdot,t,x)\|_{\mathcal{H}^{\otimes n}}$ in the following way: for a
given constant $t>0$, write

$2\int_{0}^{\infty}e^{-2ns}\|f_{n}(\cdot,s,x)\|_{{\mathcal{H}}^{\otimes
n}}^{2}ds\geq
2\int_{t}^{\infty}e^{-2ns}\|f_{n}(\cdot,s,x)\|_{{\mathcal{H}}^{\otimes
n}}^{2}ds.$

Moreover, going back to expression (3.12) and taking into account the fact
that the kernels $G$ and $\gamma$ are positive, it is clear that
$s\mapsto\|f_{n}(\cdot,s,x)\|_{{\mathcal{H}}^{\otimes n}}$ is nondecreasing.
Therefore we get

$2\int_{0}^{\infty}e^{-2ns}\|f_{n}(\cdot,s,x)\|_{{\mathcal{H}}^{\otimes
n}}^{2}ds\geq\frac{1}{n}e^{-2nt}\|f_{n}(\cdot,t,x)\|_{{\mathcal{H}}^{\otimes
n}}^{2}.$ (3.53)

Combining (3.52) and (3.53), we thus get

$\lim_{n\to\infty}{1\over n}\log
n^{n}\|f_{n}(\cdot,t,x)\|_{{\mathcal{H}}^{\otimes n}}^{2}=-\infty.$ (3.54)

We are now ready to conclude the proof of (3.50). Indeed, since $n!\leq n^{n}$
for all $n\geq 0$ (with the convention $0^{0}=1$), we have

$\sum_{n=0}^{\infty}n!\|f_{n}(\cdot,t,x)\|^{2}_{\mathcal{H}^{\otimes
n}}\leq\sum_{n=0}^{\infty}n^{n}\|f_{n}(\cdot,t,x)\|^{2}_{\mathcal{H}^{\otimes
n}}.$ (3.55)

Moreover, relation (3.54) asserts the existence of $n_{0}\geq 1$ such that for
$n\geq n_{0}$ we have $n^{n}\|f_{n}(\cdot,t,x)\|_{\mathcal{H}^{\otimes n}}\leq
e^{-n}$. Plugging this information into (3.55), we trivially get (3.50). This
finishes our proof. ∎

######  Remark 3.9.

For the special case stated in Corollary 1.3, we can prove Theorem 3.8 by just
gathering Proposition 3.5 and the following scaling property valid for the
functions $g_{n}$ defined by (3.11):

$\|g_{n}(\cdot,t,x)\|_{{\mathcal{H}}^{\otimes
n}}^{2}=t^{(4-\alpha-\alpha_{0})n}\|g_{n}(\cdot,1,x)\|_{{\mathcal{H}}^{\otimes
n}}^{2}.$ (3.56)

In particular, we do not need to invoke the behavior of $L_{\alpha_{0},n}$ as
$n\to\infty$. More specifically, let us recall that (3.8) is also valid for
the functions $g_{n}(\cdot,t,x)$ defined by (3.11). Hence

$\int_{0}^{\infty}e^{-2t}\|g_{n}(\cdot,t,x)\|^{2}_{\mathcal{H}^{\otimes
n}}dt\leq\big{(}C_{\alpha_{0}}L_{\alpha_{0},1}\big{)}^{n},$ (3.57)

where $L_{\alpha_{0},1}<\infty$. Moreover, owing to (3.56), the left-hand side
of (3.57) can be recast as

$\|g_{n}(\cdot,1,x)\|_{\mathcal{H}^{\otimes
n}}^{2}\int_{0}^{\infty}e^{-2t}t^{(4-\alpha-\alpha_{0})n}dt.$ (3.58)

Let us also recall from Remark 1.3 that in the homogeneous case our condition
(1.3) reads $\alpha_{0}+\alpha<3$. Thus gathering (3.57) and (3.58), and
reporting to elementary properties of Gamma functions, we end up with

$\|g_{n}(\cdot,1,x)\|^{2}_{\mathcal{H}^{\otimes
n}}\leq\frac{C^{n}}{(n!)^{4-\alpha-\alpha_{0}}},$

for a constant $C>0.$ We can now invoke (3.56) again in order to get

$\|g_{n}(\cdot,t,x)\|^{2}_{\mathcal{H}^{\otimes
n}}\leq\frac{(Ct^{4-\alpha-\alpha_{0}})^{n}}{(n!)^{4-\alpha-\alpha_{0}}}.$

Since $4-\alpha-\alpha_{0}>1$, this is enough to ensure (3.50).

###  3.4. Proof of Corollary 1.3

It is readily checked that the Fourier transform of a $\alpha$-homogeneous
measure is homogeneous of order $d-\alpha$. In other words, condition (1.4)
can be easily recast as follows: for all bounded function
$\varphi:\mathbb{R}^{d}\to\mathbb{R}$ with compact support and all $c>0$,

$\int_{\mathbb{R}^{d}}\varphi(cx)\,\mu(dx)=c^{-\alpha}\int_{\mathbb{R}^{d}}\varphi(x)\,\mu(dx)\,.$

Applying this formula with $\varphi(\xi):=\mathbf{1}_{\\{|\xi|\leq 1\\}}$ and
$c=r^{-1}$, we obtain that

$\mu(\mathcal{B}(0,r))=r^{\alpha}\mu(\mathcal{B}(0,1))\quad\text{for all}\
r>0,$ (3.59)

where $\mathcal{B}(0,r):=\\{\xi\in\mathbb{R}^{d}:\ |\xi|\leq 1\\}$.

With relation (3.59) in mind, applying (3.61) in Lemma 3.10 below with
$\nu=\mu$ and $f(r)=(1+r^{2})^{(\alpha_{0}-3)/2}$ will enable us to establish
the following identity:

$\int_{\mathbb{R}^{d}}\bigg{(}\frac{1}{1+|\xi|^{2}}\bigg{)}^{\frac{3-\alpha_{0}}{2}}\mu(d\xi)=\alpha\,\mu(\mathcal{B}(0,1))\int_{0}^{\infty}\frac{dr}{r^{1-\alpha}}\bigg{(}\frac{1}{1+r^{2}}\bigg{)}^{\frac{3-\alpha_{0}}{2}}\,,$
(3.60)

from which we immediately derive the conclusion of Corollary 1.3 (recall that
$\alpha>0$).

######  Lemma 3.10.

Let $\nu$ be a Randon measure on $\mathbb{R}^{d}$ and denote
$g(r)=\nu(\mathcal{B}(0,r))$ for $r\geq 0.$ Then for any continuous function
$f:[0,\infty)\to[0,\infty)$, we have

$\int_{\mathbb{R}^{d}}f(|\xi|)\nu(d\xi)=\int_{0}^{\infty}f(r)dg(r).$ (3.61)

###### Proof.

It suffices to prove the following equality for all $R>0$:

$\int_{\mathcal{B}(0,R)}f(|\xi|)\nu(d\xi)=\int_{0}^{R}f(r)dg(r),$ (3.62)

where the integral on the right-hand side is a well-defined Riemann-Stieltjes
integral noting that $f$ is continuous and $g$ is increasing.

For a fixed positive number $R$, let $0=r_{0}<r_{1}<\dots<r_{n}=R$ be a
partition of the interval $[0,R]$. Denoting
$E_{k}=\mathcal{B}(0,r_{k})\backslash\mathcal{B}(0,r_{k-1})$ for
$k=1,\dots,n$, clearly we have

$\int_{\mathcal{B}(0,R)}f(|\xi|)\nu(d\xi)=\sum_{k=1}^{n}\int_{E_{k}}f(|\xi|)\nu(d\xi).$

By the continuity of $f$, we have that for each $k$,

$\int_{E_{k}}f(|\xi|)\nu(d\xi)=f(r_{k}^{*})\nu(E_{k})=f(r_{k}^{*})[g(r_{k})-g(r_{k-1})],$

for some $r_{k}^{*}\in[r_{k-1},r_{k}]$. Thus, we have

$\int_{\mathcal{B}(0,R)}f(|\xi|)\nu(d\xi)=\sum_{i=1}^{n}f(r_{k}^{*})[g(r_{k})-g(r_{k-1})],$

and letting $n\to\infty$ yields the desired (3.62). ∎

#### Acknowledgment.

X. Chen is partially supported by the Simons foundation grant 585506. J. Song
is partially supported by Shandong University grant 11140089963041 and
National Natural Science Foundation of China grant 12071256. S. Tindel is
partially supported by the NSF grant DMS-1952966.

## References


# Reproducing the Metric-Based Evaluation of
a Set of Controllable Text Generation Techniques

###### Abstract

Rerunning a metric-based evaluation should be more straightforward, and
results should be closer, than in a human-based evaluation, especially where
code and model checkpoints are made available by the original authors. As this
report of our efforts to rerun a metric-based evaluation of a set of single-
attribute and multiple-attribute controllable text generation (CTG) techniques
shows however, such reruns of evaluations do not always produce results that
are the same as the original results, and can reveal errors in the reporting
of the original work.

Keywords: Reproduction, metric evaluation, controllable text generation.

Reproducing the Metric-Based Evaluation of
a Set of Controllable Text Generation Techniques

Michela Lorandi, Anya Belz
---
ADAPT Centre, Dublin City University, Ireland
{michela.lorandi<EMAIL_ADDRESS>

Abstract content

## 1\. Introduction

Over the past few years, the fields of natural language processing (NLP) and
machine learning (ML) have seen an increase in interest in reproducibility
Sinha et al. (2020); Branco et al. (2020); Belz et al. (2021); Belz and
Thomson (2023). Initially, efforts focussed on promoting and encouraging
sharing of all resources needed to rerun experiments, but increasingly it
became clear that exact reproduction of results is rarely the outcome even
where metric evaluation is concerned. The question is what can be concluded in
such situations beyond binary reproduced vs. not reproduced findings.

Belz et al. (2022; 2023) proposed QRA++, an approach to measuring how close
results from two evaluations are, and how reproducible evaluation measures
are, in order to facilitate comparison in terms of degree of reproducibility
between different methods of evaluation. This approach enables comparable,
quantified reproducibility results to be produced.

In this short report, we present our work rerunning the metric-based
evaluation of a set of single and multiple-attribute controllable text
generation techniques Gu et al. (2022, 2023). In the case of all except one
pair of scores from the original and reproduction evaluations, the two scores
are not the same, and we apply QRA++ to quantify the differences.

We start with a summary of the QRA++ measures we use (Section 2), followed by
a description of the specific original experiments we repeated in this
reproduction study (Section 3). We then describe how we went about repeating
the work (Section 4), before presenting the side-by-side results from the
original work and our reproduction along with the QRA++ measures of their
similarity (Section 5). We finish with some discussion and conclusions
(Section 6).

## 2\. QRA++ Measures

QRA++ distinguishes four types of results commonly reported in NLP and ML
papers:

  1. 1.

Type I results: single numerical scores, e.g. mean quality rating, error
count, etc.

  2. 2.

Type II results: sets of related numerical scores, e.g. set of Type I results
.

  3. 3.

Type III results: categorical labels attached to text spans of any length.

  4. 4.

Type IV results: Qualitative findings stated explicitly or implied by
quantitative results in the original paper.

The above are quantitatively assessed as follows:

  1. 1.

Type I results: Small-sample coefficient of variation CV* Belz (2022).

  2. 2.

Type II results: Pearson’s r, Spearman’s $\rho$.

  3. 3.

Type III results: Multi-rater: Fleiss’s $\kappa$; Multi-rater, multi-label:
Krippendorff’s $\alpha$.

  4. 4.

Type IV results: Proportion of findings that are / are not confirmed by the
repeat experiment. To obtain comparable results we restrict ourselves to
pairwise system ranks as findings.

In the work reported in this paper we have Type I, II and IV results, and
therefore apply the corresponding quantitative measures above.

## 3\. Original Work Being Repeated

In the present reproduction study, we carried out repeat evaluations of the
main new systems presented by Gu et al. (2022) and Gu et al. (2023). The
authors provide the code on
GitHub111https://github.com/HappyGu0524/MultiControl and the model checkpoints
on Google
Drive.222https://drive.google.com/drive/folders/14XHSG4IAGlAL9t-SYoTUKnAs5ARqHd5f

More precisely, the experimental grid we reproduced looks as follows:
{PriorCTG} x {Topic (World, Sports, Business, and Technology), Sentiment
(Positive and Negative), Toxicity (Toxic and Non-Toxic)} x {no extension,
extension} + {PriorCTG} x {Multi attribute (Topic, Sentiment and Non-Toxic)} x
{no optim, optim} + {MultiCTG} x {Multi attribute (Topic, Sentiment and Non-
Toxic)}. The individual systems (MultiCTG, PriorCTG +/- extend/optim) are
described in the next section.

### 3.1. Systems included in reproduction

We included the results for the four main new systems from the original work
Gu et al. (2022, 2023) in our reproduction study; we abbreviate system names
as follows: MultiCTG, PriorCTG, PriorCTG+extend, and PriorCTG+optim.

MultiCTG: This is the core new CTG approach proposed by Gu et al. (2022) which
directly searches for the intersection areas of multiple attribute
distributions to achieve control over multiple control attributes. The
attribute space is first estimated with an autoencoder structure, then the
intersections are iteratively approached via joint minimisation of distances
to points representing the controlled attributes.

PriorCTG: This is the core new CTG approach proposed by Gu et al. (2023),
which utilises a form of latent-space control, more specifically an invertible
transformation function, the Normalizing Flow, that maps the complex
distributions in latent space to simple Gaussian distributions in prior space.

PriorCTG+extend: The extend control strategy additionally achieves opposite
control, as in contrastive learning, by using negative weights when
interpolating.

PriorCTG+optim: The optim control strategy additionally optimises the
intersection of the single attribute representations in prior space to achieve
multiple-attribute control.

All systems are trained on the IMDb movie reviews dataset Maas et al. (2011),
the AGNews dataset Zhang et al. (2015), and the Jigsaw Toxic Comment
Classification Challenge Dataset cjadams (2017), respectively, for control of
sentiment, topic and detoxification attributes. Note that we did not include
any of the baseline systems in the reproduction.

Methods |  Sentiment$\uparrow$ (%) |  Topic$\uparrow$ (%) | Detox.$\uparrow$ | PPL.$\downarrow$ | Dist.-1/2/3$\uparrow$ (%)
---|---|---|---|---|---
Avg. | Pos. | Neg. | Avg. | W. | S. | B. | T. | (%)
PriorCTG | 97.1 | 99.9 | 94.3 | 95.9 | 95.5 | 99.3 | 90.2 | 98.7 | 90.7 | 61 | 42.0 / 79.7 / 88.4
PriorCTG Repro | 98.2 | 99.9 | 96.6 | 94.8 | 93.4 | 97.8 | 88.5 | 99.5 | 96.9 | 59.7 | 41.9 / 79.5 / 88.4
PriorCTG+extend | 99.7 | 99.9 | 99.5 | 97.8 | 97.9 | 99.4 | 94.0 | 99.8 | 95.7 | 61.6 | 42.4 / 79.4 / 88.1
PriorCTG+extend Repro | 99.3 | 99.9 | 98.7 | 98.2 | 98.2 | 99.5 | 95.5 | 99.8 | 99.9 | 60.8 | 42.3 / 79.2 / 88.1
Table 1: Side-by-side metric results from original work Gu et al. (2023) and reproduction study for single-attribute control (last two rows in Table 1 in the original paper). The results of the last two columns are obtained using our own implementation. For PriorCTG and PriorCTG+extend systems (see Section 3). Repro=Reproduction results. Methods |  Average$\uparrow$ (%) |  Sentiment$\uparrow$ (%) |  Topic$\uparrow$ (%) |  Detoxification$\uparrow$ (%) | PPL.$\downarrow$ |  Dist.$\uparrow$ (%)
---|---|---|---|---|---|---
MultiCTG |  87.4 $\pm$ 10.9 |  86.7 $\pm$ 10.5 |  84.8 $\pm$ 14.2 |  90.7 $\pm$ 7.4 | 31.3 | 59.0
MultiCTG Repro |  88.4 $\pm$ 8.3 |  84.9 $\pm$ 11.5 |  84.5 $\pm$ 14.4 |  95.9 $\pm$ 5.5 | 31.5 | 59.2
PriorCTG |  89.9 $\pm$ 8.7 |  88.0 $\pm$ 10.6 |  87.4 $\pm$ 8.5 |  94.3 $\pm$ 3.2 | 38.9 | 65.3
PriorCTG Repro |  91.1 $\pm$ 6.7 |  88.0 $\pm$ 10.2 |  87.1 $\pm$ 11.2 |  98.3 $\pm$ 1.6 | 38.3 | 65.2
PriorCTG+optim |  92.2 $\pm$ 8.6 |  92.5 $\pm$ 8.5 |  89.3 $\pm$ 11.0 |  94.9 $\pm$ 3.4 | 33.0 | 61.7
PriorCTG+optim Repro |  93.2 $\pm$ 7.2 |  91.8 $\pm$ 9.7 |  89.3 $\pm$ 12.4 |  98.6 $\pm$ 1.1 | 32.5 | 62
Table 2: Side-by-side metric results from original work Gu et al. (2022, 2023)
and reproduction study for multiple-attribute control. Results for MultiCTG
are from the third to last row in Gu et al. (2022). Original results for the
other two systems are from the last two rows in Table 3 in Gu et al. (2023).
The results of the last two columns are obtained using our own implementation.
For system and metrics descriptions see Section 3). Repro=Reproduction
results.

### 3.2. Evaluation metrics

The metrics in this section are all described in detail in Gu et al. (2022).
The main set of metrics assesses single-attribute control performance (called
‘attribute relevance’ in the original papers), computed as the percentage of
outputs that are classified as having the given intended control attribute
value by a specific classifier.

For Sentiment control performance, the classifier is DeBERTa He et al. (2020)
finetuned on the Yelp dataset Zhang et al. (2015).

For Topic control performance, the classifier is DeBERTa finetuned on the
AGNews dataset Zhang et al. (2015) utilizing the portion of dataset not used
during the model’s training.

For Toxicity control performance, there is a discrepancy between what the
paper says and what is in the evaluation script shared on GitHub. According to
the former, toxicity is measured with the Google Perspective
API.333https://www.perspectiveapi.com/ However, the script uses a toxicity
classifier obtained by finetuning DeBERTa on the Jigsaw Toxic Comment
Classification Challenge Dataset,444https://www.kaggle.com/c/jigsaw-toxic-
comment-classification-challenge/ analogous to control performance assessment
for the other control attributes. We ran the evaluation both with Perspective
and with the DeBERTa classifier, and found that scores obtained with the
latter were closer to the original scores, so those are what we used.

Multiple-attribute control performance is computed as the average of the
single-attribute control performance scores for the three attributes being
controlled.

Perplexity is calculated by GPT2-large following the Contrastive Prefix method
Qian et al. (2022). Note that we used our own implementation as no code was
shared for this.

Distinctness Li et al. (2016) is computed as the percentage of distinct
n-grams in the continuations generated from a given set of prefixes. System-
level 1-gram, 2-gram, and 3-gram distinctness scores are obtained by averaging
over prefix-level distinctness scores. In multi-control setting, the average
of system-level Distinct-1, 2 and 3 is computed. Here too we used our own
implementation based on Yu et al. (2021) implementation, because the code was
not shared either by Li et al. or by Gu et al.

This gives us six main types of metrics (the three classifier-based metrics,
their average (for multiple-attribute control), perplexity, and distinctness).
In Table 1 we additionally give the average over the individual control
performance scores (Avg. columns) for sentiment, topic and toxicity.

## 4\. Reproduction Work

Our first step was to download the code and model checkpoints from the
authors’ Github and Drive repositories, and recreate the environments on our
machine with a GPU RTXA6000 with 48GB RAM.

We then re-executed the inference phase of the experiments involving PriorCTG
from Gu et al. (2023), first those with single-attribute control, i.e. where
Topic, Sentiment or Toxicity are being controlled individually, and then those
with multiple-attribute control, where Topic, Sentiment and Toxicity are being
controlled at the same time.

For multiple-attribute control we also re-executed the inference phase of the
experiments involving MultiCTG from Gu et al. (2022). This gave us sets of 35
$\times$ 5 $=$ 175 outputs (35 inputs from the PPLM Prompts test set $\times$
5 repetitions of prompting and collecting the outputs) for each
system/attribute combination.

Note that as in the original work, outputs are generated for all values of all
controlled attributes (single-attribute case) or for all combinations of
controlled attribute values (multiple-attribute case), results for all of
which except Toxicity$=$toxic (‘Detox(ification)’ in the tables) are reported
in the results tables. In the multiple-attribute case, the average over
different attribute value combinations, along with the corresponding standard
deviation, is reported.

For the evaluation, we computed the metrics listed in Section 3. Recall from
Section 3.2 that we used the script provided by the authors for Sentiment,
Topic and Toxicity control performance assessment. However, we coded our own
scripts to compute Perplexity and Distinct-n, as scripts are not provided for
these. We also use our own code for the standard deviations in the multiple-
attribute table. For all scripts we use parameters as provided by the authors.

Note that as a result of some of the evaluation scripts not being shared, we
have two distint reproduction situations (which in QRA++ is reflected in the
measurement conditions): (a) for the classifier-based control-performance
measures, we use our outputs (regenerated by us using the original authors’
code) and evaluate them with the original authors’ scripts; and (b) for
perplexity and distinctness, we use our outputs and our evaluation scripts. In
the former case differences in scores can only be due to differences in
executing the original authors’ code, whereas in the latter case, differences
can be due to both execution and differences in the evaluation code.

In order to avoid this dual possible source of difference for perplexity and
distinctness scores, we decided to re-evaluate the original authors’ outputs
with our own script. This means that the scores in our tables are not the same
as in the two original papers for these two metrics. But it means CV* scores
and other reproducibility measures are comparable across all metrics.

## 5\. Side-by-Side Results and QRA++ Assessment

Tables 1 and 2 present side-by-side evaluation results for the original and
reproduction work, for each of the six metrics from Section 3, plus, in Table
1 only, averages over individual control performance scores (Avg. columns).
Reall that we reevaluated the original authors’ outputs in terms of Perplexity
and Distinctness (see preceding section).

System | CV* between original and reproduction scores for each evaluation measure
---|---
Sent | Sent | Sent | Topic | Topic | Topic | Topic | Topic | Detox | PPL | Dist-1 | Dist-2 | Dist-3
avg | pos | neg | avg | W | S | B | T |  |  |  |  |
Prior-CTG | 1.12 | 0 | 2.4 | 1.15 | 2.22 | 1.52 | 1.9 | 0.8 | 6.59 | 2.15 | 0.24 | 0.25 | 0
Prior-CTG+ext | 0.4 | 0 | 0.8 | 0.41 | 0.31 | 0.1 | 1.58 | 0 | 4.28 | 1.3 | 0.24 | 0.25 | 0
Average | 0.76 | 0 | 1.6 | 0.78 | 1.27 | 0.81 | 1.74 | 0.4 | 5.44 | 1.725 | 0.24 | 0.25 | 0
Table 3: CV* for each pair of original and reproduction metric scores, for the Prior-CTG and Prior-CTG+extend systems, and the average over both systems. System |  CV* between original and reproduction scores for each evaluation measure
---|---
Avg |  Sentiment |  Topic |  Detox |  PPL |  Distinct-n
Multi-CTG  |  1.13 |  2.09 |  0.35 |  5.56 |  0.64 |  0.34
Prior-CTG  |  1.32 |  0.0 |  0.34 |  4.14 |  1.52 |  0.15
Prior-CTG+optim  |  1.08 |  0.76 |  0.0 |  3.81 |  1.52 |  0.48
Average  |  1.18 |  0.95 |  0.23 |  4.5 |  1.23 |  0.32
Table 4: CV* for each pair of original and reproduction metric scores, for the
Multi-CTG, Prior-CTG and Prior-CTG+optim systems, and the average over all
three.

### 5.1. Type IV results

Regarding Type IV results (findings), here we are assessing relative
performance between systems, such that each pairwise ranking counts as one
finding. Note that statistical significance was not computed in the original
work.

For single-attribute control (Table 1), in the original work, Prior CTG+extend
has higher scores than PriorCTG according to all metrics except for Perplexity
and 2-gram and 3-gram Distinctness where PriorCTG scores are very slightly
higher. For Sentiment/Pos, scores are identical. In our reproduction
evaluations, these two systems are ranked the same way in all cases, giving us
a perfect proportion of 13/13 findings upheld for this table.

For multiple-attribute control (scores in Table 2), the same type of analysis
gives us a proportion of 18/18 findings upheld (pairwise ranks confirmed).

### 5.2. Type I results

For Type I results, we computed CV* values for all individual system/metric
level original and reproduction scores. We report the individual scores, as
well as the mean per metric.

For single-attribute control (scores in Table 1), Table 3 shows CV* scores for
each pair of original and reproduction metric scores, for the Prior-CTG and
Prior-CTG+extend systems, and the average over both systems (last row).

One clear tendency is that the Prior-CTG system has better reproducibility
scores across the board than Prior-CTG+extend (except for distinctness metrics
where the two systems are tied).

Looking at metric-level differences (‘Average’ row), we can see that
Perplexity and (by a smaller margin) Detoxification Control have lower
reproducibility than the other metrics.

For multiple-attribute control (scores in Table 2), Table 4 shows CV* scores
for each pair of original and reproduction metric scores, for the Multi-CTG,
Prior-CTG and Prior-CTG+optim systems, and the average over all three (last
row). We can see that here too, the Perplexity and Detoxification Control
metrics have the poorest reproducibility.

We can also see a slight tendency for the classifier scores for the Prior-
CTG+optim system to have better reproducibility than the other two systems
(but not for PPL and Distinct-n), but the picture is more mixed than for the
single-attribute control systems.

### 5.3. Type II results

For Type II results we compute Pearson’s correlation coefficients between sets
of metric scores in two ways, (i) for each metric (i.e. how do all the scores
for each metric correlate between original and reproduction), and (ii) for
each system (i.e. how do all the scores for each system correlate).

For single-attribute control (scores in Table 1), system-level Pearson’s
between all metric results in the original and reproduction runs is above 0.99
for both Prior-CTG and Prior-CTG+extend. Mean metric-level Pearson’s is
perfect (but note that we have only two score pairs all of which are ranked
identically).

For multiple-attribute control (scores in Table 2), system-level Pearson’s
between all metric results in the original and reproduction runs is above 0.99
for all three systems. Metric-level Pearson’s is above 0.99 for all metrics
except the sentiment-classifier metric which at $r=$ 0.969 is slightly lower
than the other metrics. Mean metric-level $r$ is 0.994.

## 6\. Discussion and Conclusion

The main challenges in carrying out our reproduction study were (i) lack of
clarity in the paper with respect to what the averages and standard deviations
in results tables were computed over, and (ii) discrepancies between the
shared code and what the paper said, e.g. the paper says toxicity was assessed
with Perspective, whereas the shared evaluation script has a toxicity
classifier.

Our quantified reproducibility assessments revealed a high degree of
reproducibility at the study level for Type II and Type IV results. For Type I
results, study-level CV* (computed as the mean of metric-level means) was
1.154 for single-attribute control, and 1.402 for multiple-attribute control.
While this compares well to reproducibility results in human evaluations which
very rarely achieve study-level CV* below 5 in pairwise comparisons of
original study and one reproduction, it does confirm once again that even with
identical code, we cannot necessarily expect to get the same results.

In terms of metric-level CV*, the Detoxification control metric had notably
worse reproducibility than the others which may be partly but not entirely
explainable by the fact that only Toxicity=nontoxic was taken into account
here.

In terms of the results that tend to be considered as most important, Type IV
results or findings upheld, reproducibility was perfect with all pairwise
rankings being identical in the original and reproduction experiments.

## Bibliographical References

##

  * Belz (2022) Anya Belz. 2022.  A metrological perspective on reproducibility in nlp.  _Computational Linguistics_ , 48(4):1125–1135.
  * Belz et al. (2021) Anya Belz, Shubham Agarwal, Anastasia Shimorina, and Ehud Reiter. 2021.  A systematic review of reproducibility research in natural language processing.  In _Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume_ , pages 381–393.
  * Belz and Thomson (2023) Anya Belz and Craig Thomson. 2023.  The 2023 repronlp shared task on reproducibility of evaluations in nlp: Overview and results.  In _Proceedings of the 3rd Workshop on Human Evaluation of NLP Systems_ , pages 35–48.
  * Branco et al. (2020) António Branco, Nicoletta Calzolari, Piek Vossen, Gertjan Van Noord, Dieter Van Uytvanck, Joao Silva, Luís Gomes, André Moreira, and Willem Elbers. 2020.  A shared task of a new, collaborative type to foster reproducibility: A first exercise in the area of language science and technology with reprolang2020.  In _Proceedings of The 12th Language Resources and Evaluation Conference_ , pages 5539–5545. European Language Resources Association (ELRA).
  * cjadams (2017) Julia Elliott Lucas Dixon Mark McDonald nithum Will Cukierski cjadams, Jeffrey Sorensen. 2017.  Toxic comment classification challenge.
  * Gu et al. (2022) Yuxuan Gu, Xiaocheng Feng, Sicheng Ma, Lingyuan Zhang, Heng Gong, and Bing Qin. 2022.  A distributional lens for multi-aspect controllable text generation.  In _Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing_ , pages 1023–1043, Abu Dhabi, United Arab Emirates. Association for Computational Linguistics.
  * Gu et al. (2023) Yuxuan Gu, Xiaocheng Feng, Sicheng Ma, Lingyuan Zhang, Heng Gong, Weihong Zhong, and Bing Qin. 2023.  Controllable text generation via probability density estimation in the latent space.  In _Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 12590–12616, Toronto, Canada. Association for Computational Linguistics.
  * He et al. (2020) Pengcheng He, Xiaodong Liu, Jianfeng Gao, and Weizhu Chen. 2020.  Deberta: Decoding-enhanced bert with disentangled attention.  In _International Conference on Learning Representations_.
  * Li et al. (2016) Jiwei Li, Michel Galley, Chris Brockett, Jianfeng Gao, and Bill Dolan. 2016.  A diversity-promoting objective function for neural conversation models.  In _Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 110–119, San Diego, California. Association for Computational Linguistics.
  * Maas et al. (2011) Andrew L. Maas, Raymond E. Daly, Peter T. Pham, Dan Huang, Andrew Y. Ng, and Christopher Potts. 2011.  Learning word vectors for sentiment analysis.  In _Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies_ , pages 142–150, Portland, Oregon, USA. Association for Computational Linguistics.
  * Qian et al. (2022) Jing Qian, Li Dong, Yelong Shen, Furu Wei, and Weizhu Chen. 2022.  Controllable natural language generation with contrastive prefixes.  In _Findings of the Association for Computational Linguistics: ACL 2022_ , pages 2912–2924, Dublin, Ireland. Association for Computational Linguistics.
  * Sinha et al. (2020) Koustuv Sinha, Joelle Pineau, Jessica Forde, Rosemary Nan Ke, and Hugo Larochelle. 2020.  Neurips 2019 reproducibility challenge.  6\.
  * Whitaker (2017) Kirstie Whitaker. 2017.  The MT Reproducibility Checklist.  {https://www.cs.mcgill.ca/~jpineau/ReproducibilityChecklist.pdf}.
  * Yu et al. (2021) Dian Yu, Zhou Yu, and Kenji Sagae. 2021.  Attribute alignment: Controlling text generation from pre-trained language models.  In _Findings of the Association for Computational Linguistics: EMNLP 2021_ , pages 2251–2268, Punta Cana, Dominican Republic. Association for Computational Linguistics.
  * Zhang et al. (2015) Xiang Zhang, Junbo Zhao, and Yann LeCun. 2015.  Character-level convolutional networks for text classification.  _Advances in neural information processing systems_ , 28.

##  Appendix A Perplexity and Distinct-n implementation

No code was shared to compute perplexity and Distinct-n, hence we used our own
implementation. Perplexity is calculated using the evaluate library of
HuggingFace555https://huggingface.co/docs/evaluate/en/index using GPT-2 Large.

System-level Distinct-n (n=1, 2, 3) is the average Distinct-n at prefix-level,
which is computed as the number of unique n-grams in the set of generated
outputs with the same prefix over the total amount of tokens. GPT-2 is used to
tokenise the texts.

Table 5 and 6 show Perplexity and Distinct-n results reported in the original
work, the results of the original study computed using our implementation and
the reproduction using our implementation.

Methods |  PPL. (%) $\downarrow$ |  Dist.-1/2/3$\uparrow$ (%)
---|---|---
PriorCTG | 54.3 | 29.1 / 70.1 / 86.9
PriorCTG using our implementation | 61 | 42.0 / 79.7 / 88.4
PriorCTG Repro | 59.7 | 41.9 / 79.5 / 88.4
PriorCTG+extend | 54.6 | 29.8 / 70.5 / 86.8
PriorCTG+extend using our implementation | 61.6 | 42.4 / 79.4 / 88.1
PriorCTG+extend Repro | 60.8 | 42.3 / 79.2 / 88.1
Table 5: Side-by-side metric results from original work Gu et al. (2023), original work Gu et al. (2023) computed using our own implementation and reproduction study using our own implementation for single-attribute control (last two rows in Table 1 in the original paper). For PriorCTG and PriorCTG+extend systems (see Section 3). Repro=Reproduction results. Methods | PPL.$\downarrow$ |  Dist.$\uparrow$ (%)
---|---|---
MultiCTG | 28.4 | 49.5
MultiCTG using our implementation | 31.3 | 59.0
MultiCTG Repro | 31.5 | 59.2
PriorCTG | 34.7 | 55.5
PriorCTG using our implementation | 38.9 | 65.3
PriorCTG Repro | 38.3 | 65.2
PriorCTG+optim | 29.6 | 51.6
PriorCTG+optim using our implementation | 33.0 | 61.7
PriorCTG+optim Repro | 32.5 | 62
Table 6: Side-by-side metric results from original work Gu et al. (2022,
2023), original work Gu et al. (2022, 2023) computed using our own
implementation and reproduction study using our own implementation for
multiple-attribute control. Results for MultiCTG are from the third to last
row in Gu et al. (2022). Original results for the other two systems are from
the last two rows in Table 3 in Gu et al. (2023). For system and metrics
descriptions see Section 3). Repro=Reproduction results.