# Polynomial rates via deconvolution for nonparametric estimation in McKean-
Vlasov SDEs

Chiara Amorino∗, Denis Belomestny‡, Vytautė Pilipauskaitė†,
Mark Podolskij∗, Shi-Yuan Zhou∗

###### Abstract

This paper investigates the estimation of the interaction function for a class
of McKean-Vlasov stochastic differential equations. The estimation is based on
observations of the associated particle system at time $T$, considering the
scenario where both the time horizon $T$ and the number of particles $N$ tend
to infinity. Our proposed method recovers polynomial rates of convergence for
the resulting estimator. This is achieved under the assumption of
exponentially decaying tails for the interaction function. Additionally, we
conduct a thorough analysis of the transform of the associated invariant
density as a complex function, providing essential insights for our main
results.

Keywords: convergence rates, deconvolution, interacting particle system,
McKean-Vlasov SDEs, mean-field models, propagation of chaos

AMS 2010 subject classifications: 62G20, 62M05, 60G07, 60H10

 The authors gratefully acknowledge financial support of ERC Consolidator
Grant 815703 “STAMFORD: Statistical Methods for High Dimensional Diffusions”.
VP acknowledges support of the Independent Research Fund Denmark - Natural
Sciences, grant DFF-10.46540/2032-00005B. ∗Unité de Recherche en
Mathématiques, Université du Luxembourg†Department of Mathematical Sciences,
Aalborg University, Denmark ‡ Faculty of Mathematics, University of Duisburg-
Essen

###### Contents

  1. 1 Introduction
  2. 2 Model and assumptions
    1. 2.1 Probabilistic Results
  3. 3 Statistical Framework and Main Results
    1. 3.1 The Estimation Procedure
    2. 3.2 Main Results: Convergence Rates
  4. 4 Proof of the Main Results
    1. 4.1 Proof of Theorem 2.3
    2. 4.2 Proof of Proposition 3.1
    3. 4.3 Proof of Theorem 3.2
    4. 4.4 Proof of Corollary 3.3
    5. 4.5 Proof of Theorem 3.5
  5. 5 On the Fourier transforms
    1. 5.1 Properties of $\mathcal{F}(\pi)$
  6. 6 Proofs of technical results
    1. 6.1 Proof of Lemma 2.1
    2. 6.2 Proof of Proposition 2.5
    3. 6.3 Proof of Lemma 4.1

##  1 Introduction

The foundation of stochastic systems involving interacting particles and the
development of nonlinear Markov processes, initially introduced by McKean in
the 1960s [39], can be traced back to their roots in statistical physics,
particularly within the domain of plasma physics. Over subsequent decades, the
significance of these systems in probability theory has steadily grown. This
area has witnessed the development of fundamental probabilistic tools,
including propagation of chaos, geometric inequalities, and concentration
inequalities. Pioneering contributions from researchers such as Méléard [40],
Malrieu [37], Cattiaux et al. [12], and Sznitman [8] have played a crucial
role in shaping this field.

However, formulating a modern statistical inference program for these systems
remained challenging until the early 2000s, with few exceptions, such as
Kasonga’s early paper [34]. Several factors contributed to this challenge.
Firstly, the advanced probabilistic tools required for estimation were still
under development. Secondly, the microscopic particle systems originating from
statistical physics were not naturally observable, making the motivation for
statistical inference less apparent. This situation began to change around the
2010s with the widespread adoption of these models in various fields where
data became observable and collectable. Applications expanded into diverse
fields, including the social sciences (e.g., opinion dynamics [13] and
cooperative behaviors [9]), mathematical biology (e.g., structured models in
population dynamics [41] and neuroscience [3]), and finance (e.g., the study
of systemic risk [21] and smile calibration [27]). Mean-field games have
emerged as a new frontier for statistical developments, as evident in
references [11, 18, 26]. This transition has led to a growing need for a
systematic statistical inference program, which constitutes the primary focus
of this paper. Recently, this interest has manifested in two primary
directions. On one front, statistical investigations are rooted in the direct
observation of large interacting particle systems, as evidenced in works [2,
14, 15, 16, 17, 44]. On the other front, statistical inference revolves around
the observation of the mean-field limit, the McKean-Vlasov process, as
exemplified in [22, 23, 48]. Concerning stationary McKean-Vlasov SDEs, the
literature is relatively sparse. To the best of our knowledge, only a handful
of references exist, including [45] and [24], which focus on the special
McKean-Vlasov model without a potential term. In [25], a more general model is
explored.

This paper focuses on statistical inference for an interacting particle system
described by the following stochastic differential equation:

${}X_{t}^{i,N}=X_{0}^{i}+B_{t}^{i}-\int_{0}^{t}V^{\prime}(X_{s}^{i,N})ds-\frac{1}{2N}\sum_{j=1}^{N}\int_{0}^{t}W^{\prime}(X_{s}^{i,N}-X_{s}^{j,N})ds,\qquad
1\leq i\leq N,$ (1)

where the processes $B^{i}:=(B_{t}^{i})_{t\geq 0}$ are independent standard
Brownian motions with unitary diffusion coefficient and $X_{0}^{i}$ are i.i.d.
random variables with distribution $\mu_{0}(dx)$. The function $V$ is referred
to as the confinement potential, while $W$ (or $W^{\prime}$) is the
interaction potential (or interaction function respectively). Our primary
objective is to estimate the interaction function $W^{\prime}$ based on
observations $X^{1,N}_{T},...,X^{N,N}_{T}$ of the particles, which are
solutions of the system (1). Our approach hinges on the analysis of the
associated inverse problem concerning the underlying stationary Fokker-Planck
equation, relying on various results related to the probabilistic properties
of the model.

Our research is closely connected to a recent study [5] that discusses
estimation of the interaction function with a specific semiparametric
structure. The authors in [5] develop an estimation procedure, demonstrating
convergence rates that critically depend on the tail behavior of the
nonparametric part of the interaction function $W$. Specifically, assuming a
polynomial decay of the tails, they establish logarithmic convergence rates,
proven to be optimal in that context. This naturally raises the question of
whether polynomial rates can be achieved under a different set of conditions.
Our paper aims to address this question, and our key finding is that, by
assuming exponential decay of the interaction function $W$, we can introduce
an estimator that achieves polynomial convergence rates, as demonstrated in
Theorem 3.5.

Compared to the framework proposed in [5], our model features some
distinctions. The smoothness of the confinement potential $V$ emerges as a
crucial factor influencing the achieved convergence rate. Additionally, the
regularity of the invariant density $\pi$ of the associated McKean-Vlasov
equation and the analysis of its Fourier transform $\mathcal{F}(\pi)$ are
vital for establishing the asymptotic properties of our estimator. Notably, a
lower bound on $\mathcal{F}(\pi)$ is required, presenting one of the primary
challenges in our paper. We address this challenge using Hadamard
factorization, leading to the desired lower bound under mild assumptions on
the model. Furthermore, we provide an example demonstrating how a non-smooth
confinement potential $V$ results in the Fourier transform $\mathcal{F}(\pi)$
exhibiting a polynomial decay. As another interesting situation, we consider
the case of non-smooth potential $W$ and show that even in this case the
Fourier transform of $\pi$ decays exponentially fast. Further technical tools
include the extension of the Kantorovich-Rubinstein dual theorem to functions
lacking Lipschitz continuity, presented in Theorem 2.3, and an extension of
the uniform propagation of chaos in the $L^{2p}$-norm without convexity
assumption for $W$, as demonstrated in Proposition 2.5.

The structure of the paper is as follows. In Section 2, we introduce the model
assumptions. This section also offers a concise overview of interacting
particle systems, and the relevant tools about Fourier and Laplace transforms.
Crucially, we present key probabilistic results that lay the groundwork for
our main findings. Section 3 is dedicated to formulating our primary
statistical problem and the associated estimation procedure. Additionally, we
establish upper bounds on the $L^{2}$ risk of the proposed drift estimator.
The proof of our main results is provided in Section 4, where we delve into
the detailed verification of our key findings. Finally, in Section 5, we
explore the sufficient conditions necessary to ensure that the transforms
satisfy the requirements for our main results. All remaining proofs are
collected in Section 6.

#### Notation

All random variables and stochastic processes are defined on a filtered
probability space $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq
0},\mathbb{P})$. Throughout the paper, we use the symbol $c$ to represent
positive constants, although these constants may vary from one line to
another. For any function $f:\mathbb{R}\rightarrow\mathbb{R}$, we denote its
supremum as $\|f\|_{\infty}:=\sup_{y\in\mathbb{R}}|f(y)|$. The notation
$x_{n}\lesssim y_{n}$ signifies the existence of a constant $c>0$, independent
of $n$, such that $x_{n}\leq cy_{n}$. The derivatives of a function $f$ are
denoted as $f^{\prime},f^{\prime\prime},\ldots,$ or $f^{(k)}$, $k\geq 1$. We
use $C^{k}(\mathbb{R})$ to denote the space of $k$ times continuously
differentiable functions. For a complex number $z\in\mathbb{C}$, we denote its
complex conjugate, real part, and imaginary part as $\overline{z}$,
$\operatorname{Re}(z)$, and $\operatorname{Im}(z)$, respectively. For
$a\in\mathbb{R}$, we use the notation

$\mathcal{L}_{a}:=\\{y+ia:y\in\mathbb{R}\\}.$

##  2 Model and assumptions

We start our analysis by introducing a set of assumptions on the confinement
potential $V$ and the interaction potential $W$. It will become evident that
the smoothness of $V$ plays a pivotal role in our asymptotic analysis. Our
investigation encompasses two distinct scenarios: (a) The confinement
potential of infinite smoothness takes the form $V(x)=\alpha x^{2}/2$ for a
positive constant $\alpha$, (b) The confinement potential is expressed as
$V(x)=\alpha x^{2}/2+\widetilde{V}(x)$, where $\alpha>0$ and $\widetilde{V}$
is a known function characterized by non-smooth features, as elucidated below.
We assume that $V$ and $W$ satisfy the following hypothesis:

######  Assumption 1.

The potentials $W\colon\mathbb{R}\to\mathbb{R}$ and
$V\colon\mathbb{R}\to\mathbb{R}$ are such that

  * •

The interaction potential $W\in C^{2}(\mathbb{R})$ is even with bounded
derivatives $W^{\prime}\in L^{1}(\mathbb{R})$ and $W^{\prime\prime}$ such that
$\inf_{x\in\mathbb{R}}W^{\prime\prime}(x)=-C_{W}$ for some $C_{W}>0$.

  * •

The confinement potential $V$ is given by
$V(x)=\frac{\alpha}{2}x^{2}+\widetilde{V}(x)$, with
$\alpha+\inf_{x\in\mathbb{R}}\widetilde{V}^{\prime\prime}(x)=C_{V}>0$, where
$C_{V}$ and $C_{W}$ satisfy the relation $C_{V}-C_{W}>0$. $\widetilde{V}$ is
given by either

    1. A1.

$\widetilde{V}=0$, or

    2. A2.

$\widetilde{V}$ is even, and there exists $J\in\mathbb{N},\ J\geq 2$ such that
$\widetilde{V}\in C^{J}(\mathbb{R})$ and $\widetilde{V}\not\in
C^{J+1}(\mathbb{R})$. Furthermore, for each $2\leq j\leq J$,
$\sup_{x\in\mathbb{R}}|\widetilde{V}^{(j)}(x)|=\tilde{c}_{j}<\infty$.

Additionally, the initial distribution admits a density
$\mu_{0}(dx)=\mu_{0}(x)dx$ which satisfies

$\int_{\mathbb{R}}\exp(cx)\mu_{0}(x)dx<\infty,\quad\forall
c\in\mathbb{R},\qquad\int_{\mathbb{R}}\log(\mu_{0}(x))\mu_{0}(x)dx<\infty.$
(2)

##### Discussion

We would like to emphasize that we do not assume convexity of the interaction
potential $W$, as is commonly done in most works. Instead, our assumption is
that $W^{\prime\prime}$ is bounded below by a constant $-C_{W}<0$. While the
geometric convergence of the distribution of the system (1) to the invariant
distribution for $t\to\infty$ now follows from [10, Theorem 2.1], Proposition
2.5 establishes a uniform (in time) propagation of chaos under the above
assumptions. Note also that A1 can be seen as a special case of A2, where
$J=\infty$.

###  2.1 Probabilistic Results

The mean field equation associated to the interacting particle system
introduced in (1) is given by the $1$-dimensional McKean-Vlasov equation

${}\overline{X}_{t}=\overline{X}_{0}+B_{t}-\int_{0}^{t}V^{\prime}(\overline{X}_{s})ds-\frac{1}{2}\int_{0}^{t}(W^{\prime}*\mu_{s})(\overline{X}_{s})ds,\qquad
t\geq 0,$ (3)

where $\mu_{t}(dx)=\mathbb{P}(\overline{X}_{t}\in dx)$ and

$(W^{\prime}*\mu_{t})(x)=\int_{\mathbb{R}}W^{\prime}(x-y)\mu_{t}(dy),\qquad
x\in\mathbb{R},\qquad t\geq 0.$

Under Assumption 1, existence and uniqueness of strong solutions of Eq. 1 and
Eq. 3 follow as in [6, 10, 38]. Additionally, the measure $\mu_{t}$ possesses
a smooth Lebesgue density and the McKean-Vlasov equation admits an invariant
density $\pi$ solving the stationary Fokker-Planck equation

$\frac{1}{2}\pi^{\prime\prime}=-\frac{\mathrm{d}}{\mathrm{d}x}\left(\left(V^{\prime}+\frac{1}{2}W^{\prime}*\pi\right)\pi\right),$
(4)

which means that $\pi$ is given by

${}\pi(x)=\frac{1}{Z_{\pi}}\exp\left(-2V(x)-W*\pi(x)\right),\quad
x\in\mathbb{R},$ (5)

with a normalizing constant

$Z_{\pi}=\int_{\mathbb{R}}\exp\left(-2V(x)-W*\pi(x)\right)dx<\infty.$

The invariant density can be upper and lower bounded under our assumptions,
according to the following lemma. Its proof can be found in Section 6.

######  Lemma 2.1.

Suppose that Assumption 1 holds true. Then, for any $x\in\mathbb{R}$, there
exist two constants $c_{1},\,c_{2}>0$ such that

$c_{1}\exp(-\widetilde{C}x^{2})\leq\pi(x)\leq c_{2}\exp(-C_{V}x^{2}),$

with $\widetilde{C}:=\alpha+\tilde{c}_{2}$ and $C_{{V}}$ and $\tilde{c}_{2}$
are as in Assumption 1. Furthermore, it holds that

$|\pi^{(n)}(x)|\leq c(1+|x|)^{n}\exp(-C_{V}x^{2}),\quad\leavevmode\nobreak\
0\leq n\leq J.$

A crucial tool in our estimation procedure consists in the result which
combines uniform in time propagation of chaos for (1) with convergence to the
equilibrium of Eq. 3. In order to state it we start by introducing the
Wasserstein metric. The Wasserstein $p$-distance between two measures $\mu$,
$\nu$ on $\mathbb{R}$ is defined by

$W_{p}(\mu,\nu):=\left(\inf_{X\sim\mu,Y\sim\nu}\mathbb{E}[|X-Y|^{p}]\right)^{\frac{1}{p}},$

where the infimum is taken over all the possible couplings $(X,Y)$ of random
variables $X$ and $Y$ with respective laws $\mu$ and $\nu$.

The following result combines Point (iv) of Theorem 2.1 in [10] with Theorem
5.1 in [38] or Theorem 3.1 in [12], adapted to the current framework, see also
our Proposition 2.5 below.

######  Theorem 2.2.

Let $\overline{X}^{i}$, $1\leq i\leq N$, be i.i.d. copies of the process
$\overline{X}$ defined in (3) so that every $\overline{X}^{i}$ is driven by
the same Brownian motion as the i-th particle of the system (1) and they are
equal at time $0$. Denote by

$\Pi_{N,T}=N^{-1}\sum_{i=1}^{N}\delta_{X_{T}^{i,N}}$

the empirical distribution of the particle system $X_{T}^{i,N}$ for $1\leq
i\leq N$, and by $\Pi$ the law associated to the invariant density $\pi$.
Under Assumption 1 there exist a constant $c>0$ independent of $N$ and $T$
such that

$\sup_{t\geq
0}\mathbb{E}\left[|X_{t}^{i,N}-\overline{X}_{t}^{i,N}|^{2}\right]\leq cN^{-1}$

and

$\mathbb{E}[W_{1}^{2}(\Pi_{N,T},\Pi)]\leq c\left(N^{-1}+\exp(-\lambda
T)\right)=:cN_{T}^{-1},$

where $\lambda=C_{V}-C_{W}>0$, and $C_{V}$ and $C_{W}$ have been introduced in
Assumption 1.

The aforementioned bound asserts that the invariant distribution $\Pi$ of the
mean field equation can be accurately approximated by the empirical measure
$\Pi_{N,T}$, while providing an associated error bound for this approximation.
In the upcoming estimation procedure all convergence rates will be measured in
terms of $N_{T}$. In the following, we will present and prove a similar
result, but from a dual perspective, involving the Laplace transforms of
$\Pi_{N,T}$ and $\Pi$.

We opt to replace the use of Fourier transform, as seen in [5], with the
Laplace transform. This choice is made with a similar intent as the authors of
[4], allowing for greater flexibility and the inclusion of diverse scenarios.

Before we proceed further, let us now introduce some notation and properties
concerning the Laplace transform. For any locally integrable function $\phi$,
we define the bilateral Laplace transform as follows:

${}\widehat{\phi}(z):=\int_{\mathbb{R}}\phi(t)\exp(-zt)dt.$ (6)

The Laplace transform $\widehat{\phi}(z)$ is an analytic function within the
convergence region $\Sigma_{\phi}$, which typically takes the form of a
vertical strip in the complex plane:

$\Sigma_{\phi}:=\left\\{z\in\mathbb{C}\,:\,x_{\phi}^{-}\leq\operatorname{Re}(z)\leq
x_{\phi}^{+}\right\\}$

for some $x_{\phi}^{-}$, $x_{\phi}^{+}$ such that $-\infty<x_{\phi}^{-}\leq
x_{\phi}^{+}<\infty$. The convergence region $\Sigma_{\phi}$ can degenerate to
a vertical line in the complex plane, in such case it is
$\Sigma_{\phi}:=\left\\{z\in\mathbb{C}\,:\,\operatorname{Re}(z)=x_{\phi}\right\\}$,
with $x_{\phi}\in\mathbb{R}$. If $\phi$ is a probability density, then the
imaginary axis always belongs to the convergence region $\Sigma_{\phi}$. In
this case the Fourier transform of $\phi$

$\widehat{\phi}(-iy)=\mathcal{F}(\phi)(y):=\int_{\mathbb{R}}\phi(t)\exp(iyt)dt,\qquad
y\in\mathbb{R},$

is the characteristic function of $\phi$. This degenerate case pertains to
distributions whose characteristic function lacks the ability to be
analytically extended to a strip surrounding the imaginary axis in the complex
plane.

We are now ready to state a propagation of chaos type result for the Fourier
transforms that is reminiscent of Theorem 2.2 by means of the Kantorovich-
Rubinstein theorem. In particular, we can demonstrate that the transform of
$\Pi$ can also be effectively approximated by the transform of the empirical
measure $\Pi_{N,T}$. The proof can be found in Section 4.

######  Theorem 2.3.

Under Assumption 1, there exists a constant $c>0$ such that, for any
$z\in\mathcal{L}_{\pm a}$,

$\mathbb{E}[|\mathcal{F}(\Pi)(z)-\mathcal{F}(\Pi_{N,T})(z)|^{2}]\leq
c|z|^{2}N_{T}^{-1}.$

######  Remark 2.4.

Note that for $z\in\mathbb{R}$, this theorem is a direct implication of the
Kantorovich-Rubinstein dual formulation, applicable to $1$-Lipschitz
functions. However, when dealing with $z\in\mathcal{L}_{\pm a}$, the function
$\exp(iz)$ no longer possesses Lipschitz continuity, thereby rendering the use
of the Kantorovich-Rubinstein dual formulation unfeasible. This motivates us
to establish this analogous formulation. ∎

Some crucial results will be instrumental in the derivation of the theorem
above. Specifically, we present an extension of the propagation of chaos
theorem, as stated in Proposition 2.5 below, in the $L^{2p}$ norm, for $p\geq
1$. For the detailed proof of this result, we refer to Section 6.

######  Proposition 2.5.

Under Assumption 1 for any $p\geq 1$ there exists a constant $c>0$ such that
uniformly in $N$ and $i$,

$\sup_{t\geq
0}\mathbb{E}\left[|X_{t}^{i,N}-\overline{X}_{t}^{i,N}|^{2p}\right]\leq
cN^{-p}.$

##  3 Statistical Framework and Main Results

###  3.1 The Estimation Procedure

Suppose we observe the data $X_{T}^{1,N},\ldots,X_{T}^{N,N}$ in the asymptotic
framework where both $N$ and $T$ tend towards infinity. Our goal is to
estimate the interaction function $W^{\prime}$. In particular, we will propose
an estimator $W^{\prime}_{N,T}$ and study its performance by considering the
associated mean integrated squared error, aiming to achieve polynomial
convergence rates.

The estimation procedure is semiparametric in the sense that it consists of
four different steps, involving both parameter estimation and nonparametric
estimation techniques.

  1. 1.

The first step consists in the estimation of the derivative of the log-density
which we denote as $l(y)$:

$l(y):=(\log\pi)^{\prime}(y)=\frac{\pi^{\prime}(y)}{\pi(y)},\qquad
y\in\mathbb{R}.$

This will be achieved by the introduction of kernel estimators for both $\pi$
and $\pi^{\prime}$. Let $K$ be a smooth kernel of order $m\geq 2$, that is

$\int_{\mathbb{R}}K(x)dx=1,\qquad\int_{\mathbb{R}}x^{j}K(x)dx=0\quad
j=0,...,m-1,\qquad\int_{\mathbb{R}}x^{m}K(x)dx\neq 0.$

It is worth noting the choice of the kernel order, denoted by $m$, is flexible
and can be determined by the statistician. As we will see later on, the choice
of $m$ is determined by the regularity of $V$: a smooth confinement potential
allows us to choose an arbitrary $m\in\mathbb{N}$. On the other hand, if $V$
is non-smooth as described in Assumption 1, A2, we need to additionally assume
that $m\leq J$. Indeed, this is a standard restriction on the order of the
kernel when estimating non-smooth functions. Let us also introduce the
bandwidths $h_{i}:=h_{i,N,T}$, $i=0,1$, which satisfy $h_{i}\to 0$ for $N,T$
going to $\infty$ and the notation $K_{h}(x)=\frac{1}{h}K(\frac{x}{h})$. Then,
we can define the kernel estimators $\pi_{N,T}$ and $\pi_{N,T}^{\prime}$ for
$\pi$ and $\pi^{\prime}$, respectively, as below:

$\pi_{N,T}(y):=\frac{1}{N}\sum_{i=1}^{N}K_{h_{0}}(y-X_{T}^{i,N}),\qquad\pi^{\prime}_{N,T}(y):=\frac{1}{Nh_{1}}\sum_{i=1}^{N}K_{h_{1}}(y-X_{T}^{i,N}),\quad
y\in\mathbb{R}.$

An estimator for the derivative of the log-density $l(y)$ is then given by

$l_{N,T}(y):=\frac{\pi^{\prime}_{N,T}(y)}{\pi_{N,T}(y)}1_{\\{\pi_{N,T}(y)>\delta\\}},$

where $\delta=\delta_{N,T}\to 0$ as $N,T\rightarrow\infty$.

  2. 2.

In the second step, we estimate the parameter $\alpha>0$ appearing in the
confinement potential. This is based on the identity

$l(y)=-2\alpha y-2\widetilde{V}^{\prime}(y)-W^{\prime}*\pi(y)$

and on a contrast function method. Indeed, since $W^{\prime}\in
L^{1}(\mathbb{R})$, we know that $\left|W^{\prime}*\pi(y)\right|\rightarrow 0$
as $|y|\rightarrow\infty$, which allows us to construct a minimal contrast
estimator for $\alpha$. In particular, for any $\epsilon\in(0,1)$ arbitrarily
small, we can introduce an integrable weight function $w$ with support on
$[\epsilon,1]$ and a parameter $U=U_{N,T}$, which satisfies $U\to\infty$ for
$N,T\rightarrow\infty$. Then, we can define the estimator $\alpha_{N,T}$ for
$\alpha$ as

$\alpha_{N,T}:=\underset{\alpha\in\mathbb{R}}{\operatorname{argmin}}\int_{\mathbb{R}}\left(l_{N,T}(y)+2\alpha
y+2\widetilde{V}^{\prime}(y)\right)^{2}w_{U}(y)dy.$

  3. 3.

Using the results in the previous step we can construct an estimator
$\Psi_{N,T}$ of $\Psi:=-W^{\prime}*\pi$. Indeed, given the estimators
$l_{N,T}$ and $\alpha_{N,T}$ constructed above, we can define

$\Psi_{N,T}(y):=\left(l_{N,T}(y)+2\alpha_{N,T}\,y+2\widetilde{V}^{\prime}(y)\right)1_{\\{|y|\leq\epsilon
U\\}},\qquad y\in\mathbb{R}.$

  4. 4.

The last step consists in applying the deconvolution and the inverse Laplace
transform to obtain an estimator for $W^{\prime}$. Note that because we want
to consider values of $\mathcal{F}(\Pi)$ in the strip of analyticity (e.g. to
escape zeros of $\mathcal{F}(\Pi)$), we can not use the standard
regularization techniques of deconvolution problems consisting of cutting off
the estimates outside bounded intervals on real line, see [50]. Instead, we
are going to employ Tikhonov-type regularization. More specifically, for some
$a\geq 0$, choose a sequence of entire functions $\rho_{N,T}$ such that for
$z\in\mathcal{L}_{\pm a}$,
$\left|\mathcal{F}\left(\Pi_{N,T}\right)(z)+\rho_{N,T}(z)\right|\geq\varepsilon_{N,T}>0$,
where $\varepsilon_{N,T}\rightarrow 0$ for $N,T\rightarrow\infty$. Then, we
define the estimator $W_{N,T}^{\prime}$ via

$\mathcal{F}\left(W_{N,T}^{\prime}\right)(z):=-\frac{\mathcal{F}\left(\Psi_{N,T}\right)(z)}{\mathcal{F}\left(\Pi_{N,T}\right)(z)+\rho_{N,T}(z)}.$

Observe that the right hand side is well-defined since $\Psi_{N,T}\in
L^{1}(\mathbb{R})\cap L^{2}(\mathbb{R})$. Additionally, for all
$z\in\mathcal{L}_{\pm a}$,
$|\mathcal{F}(W_{N,T}^{\prime})(z)|\leq\varepsilon_{N,T}^{-1}\left|\mathcal{F}(\Psi_{N,T})(z)\right|$,
such that $W_{N,T}^{\prime}$, as the inverse Fourier transform of the right
hand side, is well-defined.

It is natural to draw a comparison between our proposed estimation approach
with the one presented in [5], particularly in scenarios where the interacting
drift exhibits polynomial tails. Although the overall steps in the estimation
process share similarities, our context introduces several novel
considerations.

Our estimation procedure can be divided into two parts, depending on whether
we are addressing the first case with infinite smoothness, where
$\widetilde{V}=0$, or the less smooth case where $\widetilde{V}$ adheres to
condition A2. The model distinction arises from the absence of the potential
function $V(x)$ in [5]. Instead, the interaction potential in [5] comprises
two components: the potential, encompassing trigonometric and polynomial
functions, and the non-parametric component of $W$. Despite this difference,
it does not significantly impact the estimation procedure. The parametric
component in [5] plays a role similar to the confinement potential in our
context, and both are estimated through a contrast function, yielding
comparable results in Steps 1 and 2 for both cases.

The deviation becomes evident in Step 3, where the constraint on exponential
tails of $W^{\prime}$ results in polynomial convergence rates (see Theorem
3.2), a contrast to the logarithmic convergence rates for $\Psi$ in [5] due to
polynomial tails of $W^{\prime}$.

The divergence continues into the fourth and final step, where the estimation
procedure takes on entirely different forms. The primary challenge lies in
analyzing the joint decay of the transforms of $W^{\prime}$ and $\pi$,
introducing the condition
$\int_{\mathcal{L}_{a}}|\frac{\mathcal{F}(W^{\prime})(z)}{\mathcal{F}(\Pi)(z)}|^{2}dz<\infty$.
Analyzing such a condition proves to be a complex task. Notably, the analysis
hinges on studying the zeros of the transform of $\pi$, leading us to use the
Laplace transform instead of the Fourier transform. It is worth noting that
selecting $a=0$ in $\mathcal{L}_{a}$ allows obtaining the Fourier transform
from the Laplace transform. A comprehensive explanation regarding the
fulfillment of the mentioned constraint can be found in Section 5.

###  3.2 Main Results: Convergence Rates

Let us start with the first step, which consists in the estimation of $l$. We
remark again that the order of the kernel can be chosen arbitrarily in the
case of $\widetilde{V}=0$, whereas for $\widetilde{V}$ as in Assumption 1, A2,
we require the condition $m\leq J$. In the sequel the bandwidth $h_{0}$ for
$\pi_{N,T}$ is chosen as

${}h_{0}:=N_{T}^{-\frac{1}{2(m+1)}}.$ (7)

Similarly, for the estimation of $\pi’_{N,T}$ we choose

${}h_{1}:=N_{T}^{-\frac{1}{2(m+2)}}.$ (8)

Finally, the threshold parameter is chosen as

${}\delta:=\frac{c_{1}}{2}\exp(-\widetilde{C}U^{2}),$ (9)

where $c_{1}$, $\widetilde{C}$ are the constants appearing in Lemma 2.1. We
remark that the choice of $h_{0}$ and $h_{1}$ is the same as in [5] while the
choice of $\delta$ is due to our lower bound on $\pi$ as presented in Lemma
2.1.

######  Proposition 3.1.

Let $h_{0}$, $h_{1}$ and $\delta$ be as above and let $U\geq 1$. Assume that
Assumption 1 holds. Then it is

$\sup_{|x|\leq
U}\mathbb{E}\left[|l_{N,T}(x)-l(x)|^{2}\right]^{\frac{1}{2}}\lesssim\exp(\widetilde{C}U^{2})\left(N_{T}^{-\frac{m}{2(m+2)}}+UN_{T}^{-\frac{m}{2(m+1)}}\right).$

We now proceed to estimate $\alpha$ in Step 2, employing the estimator
$\alpha_{N,T}$. In our current context, this step is less challenging than it
was in the previous work cited as [5], thanks to the specific model under
consideration. In particular, the estimation of $\alpha$ essentially involves
simplifying Step 2 from [5] to the scalar case, where an additional potential
$\widetilde{V}$ has been introduced. However, this potential is already known,
and it can be chosen in a way such that it does not contribute to the
convergence rate at this stage of the analysis.

######  Theorem 3.2.

Let $U\geq 1$ and recall that $m$ is the order of the kernel $K$. If
Assumption 1 holds then, for any $\epsilon\in(0,1)$,

$\displaystyle{}\mathbb{E}\left[\int_{\mathbb{R}}|\Psi_{N,T}(y)-\Psi(y)|^{2}dy\right]^{\frac{1}{2}}$
$\displaystyle\lesssim\exp(\widetilde{C}U^{2})U^{\frac{1}{2}}\left(N_{T}^{-\frac{m}{2(m+2)}}+UN_{T}^{-\frac{m}{2(m+1)}}\right)$
(10) $\displaystyle+\frac{2\exp(-C_{V}(\epsilon\frac{U}{2})^{2})}{\epsilon
U}+\left(\int_{|y|>\frac{\epsilon
U}{2}}|W^{\prime}(y)|^{2}dy\right)^{\frac{1}{2}}.$

The dependence of the convergence rate for $\Psi_{N,T}$ on the tail behavior
of the function $W^{\prime}$ is evident in Theorem 3.2. In particular, when
the tails of $W^{\prime}$ exhibit exponential decay, the subsequent corollary,
which directly follows from the previous result, provides a precise bound.

######  Corollary 3.3.

In the setting of previous theorem, let $p>{C_{V}}$ and assume that

$\limsup_{x\to\infty}\exp(2px^{2})\int_{|y|>x}|W^{\prime}(y)|^{2}dy<\infty.$
(11)

Then, for any $\epsilon\in(0,1)$, choosing $U^{2}=c_{u}\log(N_{T})$, where

${}c_{u}=\frac{m}{2(m+2)}\frac{1}{(\widetilde{C}+C_{V}\frac{\epsilon^{2}}{4})}$
(12)

gives

${}\mathbb{E}\left[\int_{\mathbb{R}}|\Psi_{N,T}(y)-\Psi(y)|^{2}dy\right]^{\frac{1}{2}}\lesssim(\log
N_{T})^{\frac{1}{4}}N_{T}^{-\gamma},$ (13)

where

${}\gamma=\frac{m}{2(m+2)}\frac{1}{1+\frac{4\widetilde{C}}{\epsilon^{2}C_{V}}}.$
(14)

######  Remark 3.4.

One might question why the convergence rate above depends on the auxiliary
parameter $\epsilon\in(0,1)$ introduced in the second step of our estimation
procedure. By following the proof of Theorem 3.2, it is easy to verify that
when the value of $\alpha$ is known and does not require pre-estimation, the
results still hold with $\epsilon=1$. However, when estimating the parameter
$\alpha$, we lose the option of setting $\epsilon=1$ and can only use
$\epsilon\in(0,1)$, which slightly affects our convergence rate. The optimal
choice is to take $\epsilon$ as close to $1$ as possible, resulting in a
convergence rate $\gamma$ equal to

$\frac{m}{2(m+2)}\frac{1}{1+\frac{4\widetilde{C}}{C_{V}}}-\tilde{\epsilon}$

for any arbitrarily small $\tilde{\epsilon}$. ∎

The estimation of $\Psi$ leads us to the estimation of $W^{\prime}$ as stated
in the following theorem. In this context, an assumption regarding the decay
of the transforms of $\pi$ and $W^{\prime}$ will be crucial.

######  Assumption 2.

Recall that $\mathcal{L}_{a}=\left\\{y+ia:y\in\mathbb{R}\right\\}$ for some
$a\geq 0$. There exists an $\overline{a}\geq 0$ such that

$\int_{\mathcal{L}_{\pm\overline{a}}}\left|\frac{\mathcal{F}\left(W^{\prime}\right)(z)}{\mathcal{F}(\Pi)(z)}\right|^{2}dz<\infty.$
(15)

We will further analyze this assumption in Section 5. In such section, we will
employ tools from complex analysis to establish a lower bound on
$\mathcal{F}(\Pi)(z)$, allowing us to verify that Assumption 2 holds true in
specific situations. Additionally, at the conclusion of Section 5, we provide
an example illustrating how a non-smooth confinement potential implies
polynomial decay in the transform of the invariant density. This example
assists us in verifying the validity of Assumption 2 as mentioned earlier.

In the sequel, $\varepsilon_{N,T}$ is chosen as

$\varepsilon_{N,T}=\exp\left(\frac{a\epsilon}{2}(c_{u}\log
N_{T})^{\frac{1}{2}}\right)(\log
N_{T})^{\frac{1}{4}}N_{T}^{-\frac{\gamma}{2}}.$

From a quick look at Theorem 3.5, it is easy to see that it provides the final
convergence rate for the estimation of $W^{\prime}$.

######  Theorem 3.5.

Assume that Assumption 1 and Assumption 2 hold true for some $a\geq 0$. Then

$\displaystyle\mathbb{E}\left[\int_{\mathbb{R}}|W^{\prime}_{N,T}(y)-W^{\prime}(y)|^{2}dy\right]^{\frac{1}{2}}\leq
c\exp\left(\frac{a\epsilon}{2}(c_{u}\log N_{T})^{\frac{1}{2}}\right)(\log
N_{T})^{\frac{1}{4}}N_{T}^{-\frac{\gamma}{2}},$

where $c_{u}$ is as in (12) and $\gamma$ as in (14).

######  Remark 3.6.

Observe that when Assumption 2 holds for $a=0$ the convergence rate found in
Theorem 3.5 is $(\log N_{T})^{\frac{1}{4}}N_{T}^{-\frac{\gamma}{2}}$. Thanks
to Hadamard representation we will see in Section 5 we can obtain such result
if $\mathcal{F}(\Pi)(z)$ does not have zeros on the real line (see Theorem 5.4
below). In Theorem 3.5, we consider a more general case, where we allow the
function $\mathcal{F}(\Pi)(z)$ to have real zeros. The crucial condition is
indeed that there exists at least one line parallel to the real axis on which
such function does not have any zeros. Let us stress that this condition is
much weaker then one requiring no zeros of $\mathcal{F}(\Pi)(z)$ on the real
axis ($a=0$) since there is a large class of densities with Fourier transforms
vanishing only on the real line, see, e.g. [7]. We refer to Section 5 for more
details. ∎

######  Remark 3.7.

Even when Assumption 2 holds for $a\neq 0$, the observed convergence rate, as
determined in the aforementioned theorem, remains polynomial. This outcome
arises from the dominance of the polynomial term over the exponential term in
Theorem 3.5. ∎

######  Remark 3.8.

It is important to highlight that the convergence rate outlined in Theorem 3.5
is contingent upon the smoothness of the confinement potential
$\widetilde{V}$. Specifically, in the scenario of smooth potentials, we have
the flexibility to set the parameter $m$ to be arbitrarily large.
Additionally, it’s noteworthy to mention that $\epsilon\in(0,1)$ with the
optimal choice being in close proximity to $1$. This choice results in
${\gamma}$ being close to $1/(2(1+4\widetilde{C}/C_{V}))$.

In contrast, when dealing with a confinement potential of smoothness $J$, we
encounter the constraint $m\leq J$. In this case, the optimal choice is to set
$m=J$, leading to ${\gamma}$ being close to
$J/(2(J+2)(1+4\widetilde{C}/C_{V}))$. ∎

##  4 Proof of the Main Results

Let us introduce some notation and properties of the Laplace transform, as
defined in (6), that will be useful in the following section. The inverse
Laplace transform is given by the following formula

$\phi(t)=\frac{1}{2\pi
i}\int_{x-i\infty}^{x+i\infty}\widehat{\phi}(z)\exp(zt)dz=\frac{1}{2\pi}\int_{-\infty}^{\infty}\widehat{\phi}(x+iy)\exp((x+iy)t)dy,$

with $x\in(x_{\phi}^{-},x_{\phi}^{+})$. The bilateral Laplace transform is
unique in the sense that if $\widehat{\phi}_{1}$ and $\widehat{\phi}_{2}$ are
such that $\widehat{\phi}_{1}(z)=\widehat{\phi}_{2}(z)$ in a common strip of
convergence
$\operatorname{Re}(z)\in(x_{\phi_{1}}^{-},x_{\phi_{1}}^{+})\cap(x_{\phi_{2}}^{-},x_{\phi_{2}}^{+})$,
then $\phi_{1}(t)=\phi_{2}(t)$ for almost all $t\in\mathbb{R}$ (see Theorem 6b
in [49]).

Moreover, the following identities will prove to be useful. Let $\phi$ be a
locally integrable function and $a>0$. By Parseval’s theorem, (see Theorem
31.7 in [19])

$\displaystyle\|\phi\|_{2}^{2}$ $\displaystyle=\frac{1}{2\pi
i}\int_{a-i\infty}^{a+i\infty}\overline{\mathcal{F}(\phi)(-i\overline{s})}\mathcal{F}(\phi)(is)ds$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}\overline{\mathcal{F}(\phi)(-ia-y)}\mathcal{F}(\phi)(ia-y)dy.$

Then,

$\displaystyle\|\phi\|^{2}_{2}$
$\displaystyle\lesssim\int_{-\infty}^{\infty}|\overline{\mathcal{F}(\phi)(-ia-y)}\mathcal{F}(\phi)(ia-y)|dy\lesssim\Delta_{-}+\Delta_{+},$

where

$\displaystyle\Delta_{-}$
$\displaystyle=\int_{-\infty}^{\infty}|\overline{\mathcal{F}(\phi)(-ia-y)}|^{2}dy=\int_{\mathcal{L}_{-a}}|\mathcal{F}(\phi)(z)|^{2}dz,$
$\displaystyle\Delta_{+}$
$\displaystyle=\int_{-\infty}^{\infty}|\mathcal{F}(\phi)(ia-y)|^{2}dy=\int_{\mathcal{L}_{a}}|\mathcal{F}(\phi)(z)|^{2}dz,$

and $\mathcal{L}_{a}=\\{y+ia:y\in\mathbb{R}\\}$.

###  4.1 Proof of Theorem 2.3

As it will be helpful for the proof of Theorem 2.3, we now state a lemma
establishing that finite exponential moments of
$\overline{X}^{1}_{0}=X^{1,N}_{0}=X^{i}_{0}$ imply finite exponential moments
of $\overline{X}^{1}_{t}$ and $X^{1,N}_{t}$ for all $t\geq 0$. Its proof can
be found in Section 6.3.
Remark that the sub-Gaussianity implies the required finite exponential
moments (2). Indeed, a centered random variable $Z$ is sub-Gaussian if there
exists a $K>0$ such that $\mathbb{E}[\exp(tZ)]\leq\exp(K^{2}t^{2}/2)$ for all
$t\in\mathbb{R}$. For sufficient conditions ensuring sub-Gaussianity see
Definition 30 in [16] and references therein.

######  Lemma 4.1.

Under Assumption 1, we have

$\sup_{t\geq 0}\mathbb{E}\left[\exp(\pm c\overline{X}_{t})+\exp(\pm
cX^{i,N}_{t})\right]<\infty$

uniformly in $i$ and $N$. Additionally, for any $k\geq 1$,

${}\sup_{t\geq
0}\mathbb{E}\left[|\overline{X}_{t}|^{k}+|X^{i,N}_{t}|^{k}\right]<\infty$ (16)

uniformly in $i$ and $N$.

We start proving Theorem 2.3 by introducing some notation. Let
$\overline{\Pi}_{t}$ be the law of $\overline{X}_{t}$, $\Pi$ the law with
density $\pi$ and $\Pi^{N}_{T}=\Pi_{N,T}$ with

$\displaystyle\mathcal{F}(\Pi^{N}_{T})(z)$
$\displaystyle=\frac{1}{N}\sum_{j=1}^{N}\exp(izX_{T}^{j,N}),\qquad\mathcal{F}(\overline{\Pi}^{N}_{T})(z)=\frac{1}{N}\sum_{j=1}^{N}\exp(iz\overline{X}_{T}^{j}),$
$\displaystyle\mathcal{F}(\overline{\Pi}_{T})(z)$
$\displaystyle=\mathbb{E}[\exp(iz\overline{X}_{T})],\qquad\mathcal{F}(\Pi)(z)=\mathcal{F}(\pi)(z)=\int_{-\infty}^{\infty}\exp(izx)\pi(x)dx.$

In order to prove the result we now consider the following decomposition:

$\mathbb{E}[|\mathcal{F}(\Pi)(z)-\mathcal{F}(\Pi^{N}_{T})(z)|^{2}]\lesssim\sum_{i=1}^{3}\delta_{i}(z),$

where

$\displaystyle\delta_{1}(z)=|\mathcal{F}(\Pi)(z)-\mathcal{F}(\overline{\Pi}_{T})(z)|^{2},\qquad\delta_{2}(z)=\mathbb{E}\left[|\mathcal{F}(\overline{\Pi}_{T})(z)-\mathcal{F}(\overline{\Pi}^{N}_{T})(z)|^{2}\right],$

and

$\displaystyle\delta_{3}(z)$
$\displaystyle=\mathbb{E}\left[|\mathcal{F}(\overline{\Pi}^{N}_{T})(z)-\mathcal{F}(\Pi^{N}_{T})(z)|^{2}\right]$
$\displaystyle\leq\left(\frac{1}{N}\sum_{j=1}^{N}\mathbb{E}\left[|\exp(iz\overline{X}_{T}^{j})-\exp(izX_{T}^{j,N})|^{2}\right]^{\frac{1}{2}}\right)^{2}$
$\displaystyle=\mathbb{E}\left[|\exp(iz\overline{X}^{1}_{T})-\exp(izX_{T}^{1,N})|^{2}\right].$

For $x=x_{1}+ix_{2}$ and $y=y_{1}+iy_{2}$, we have that

$\displaystyle{}\left|\exp(x+y)-\exp(y)\right|$
$\displaystyle=\left|\exp(x)-1\right|\exp(y_{1})$
$\displaystyle\leq\left(\left|\exp(ix_{2})-1\right|\exp(x_{1})+\left|\exp(x_{1})-1\right|\right)\exp(y_{1})$
$\displaystyle\leq(|x_{2}|\exp(x_{1})+|x_{1}|(\exp(x_{1})+1))\exp(y_{1})$
$\displaystyle\leq\sqrt{2}|x|(\exp(x_{1}+y_{1})+\exp(y_{1}))$ (17)

which implies, for any $z\in\mathcal{L}_{a}$,

$\delta_{3}(z)\leq
2|z|^{2}\mathbb{E}\left[|\overline{X}^{1}_{T}-X^{1,N}_{T}|^{2}(\exp(-a\overline{X}^{1}_{T})+\exp(-aX^{1,N}_{T}))^{2}\right].$

Using the Cauchy-Schwarz inequality, we get

$\delta_{3}(z)\lesssim|z|^{2}\mathbb{E}\left[|\overline{X}^{1}_{T}-X^{1,N}_{T}|^{4}\right]^{\frac{1}{2}},$

because exponential moments of $\overline{X}^{1}_{T}$ and $X^{1,N}_{T}$ are
uniformly in $T,N$ finite according to Lemma 4.1. By Proposition 2.5,

$\mathbb{E}\left[|\overline{X}^{1}_{T}-X^{1,N}_{T}|^{4}\right]\lesssim\frac{1}{N^{2}},$

which yields

${}\delta_{3}(z)\lesssim\frac{|z|^{2}}{N}.$ (18)

Next, consider $\delta_{2}(z)$. Since $\overline{\Pi}^{N}_{T}$ is based on
i.i.d. $\overline{X}^{1}_{T},\dots,\overline{X}^{N}_{T}$ with common law
$\overline{\Pi}_{T}$, we have that

${}\delta_{2}(z)=\frac{1}{N}\mathbb{E}\left[|\mathbb{E}[\exp(iz\overline{X}^{1}_{T})]-\exp(iz\overline{X}^{1}_{T})|^{2}\right]\lesssim\frac{1}{N},$
(19)

where exponential moments of $\overline{X}^{1}_{T}$ are uniformly in $T$
finite by Lemma 4.1. Finally, let us deal with $\delta_{1}(z)$. We introduce a
random vector $(\overline{X}_{T},X)$ with
$\mathcal{L}(\overline{X}_{T})=\overline{\Pi}_{T}$, $\mathcal{L}(X)=\Pi$ which
attains
$\mathbb{E}[|\overline{X}_{T}-X|^{2}]=W_{2}^{2}(\overline{\Pi}_{T},\Pi)$.
Then, using again (4.1), it is

$\displaystyle\delta_{1}(z)^{\frac{1}{2}}$
$\displaystyle=|\mathbb{E}[\exp(iz\overline{X}_{T})]-\mathbb{E}[\exp(izX)]|$
$\displaystyle\leq\mathbb{E}[|\exp(iz\overline{X}_{T})-\exp(izX)|]$
$\displaystyle\leq\sqrt{2}|z|\mathbb{E}[|\overline{X}_{T}-X|(\exp(-a\overline{X}_{T})+\exp(-aX))]$

Using the Cauchy-Schwarz inequality we obtain

$\displaystyle\mathbb{E}[|\overline{X}_{T}-\overline{X}|(\exp(-a\overline{X}_{T})+\exp(-aX))]\lesssim
W_{2}(\overline{\Pi}_{T},\Pi)$

since the moments and the exponential moments of $\overline{X}_{T}$, $X$ are
uniformly in $T$ finite by Lemma 4.1 and the fact that $\mathcal{L}(X)=\Pi$ is
the invariant law of the McKean-Vlasov stochastic differential equation.
Theorem 1.4 in [38] ensures that there exist two constants $c$ and
$\lambda:=C_{V}-C_{W}>0$ such that

$W_{2}(\overline{\Pi}_{T},\Pi)\leq c\exp(-\lambda T).$

It yields that

${}\delta_{1}(z)\lesssim|z|^{2}\exp(-2\lambda T).$ (20)

From (18), (19) and (20) we get, for any $z\in\mathcal{L}_{a}$,

$\mathbb{E}[|\mathcal{F}(\Pi)(z)-\mathcal{F}(\Pi^{N}_{T})(z)|^{2}]\lesssim|z|^{2}\left(\frac{1}{N}+\exp(-\lambda
T)\right)=\frac{|z|^{2}}{N_{T}}.$

Our reasoning easily extends to $z\in\mathcal{L}_{-a}$, thereby concluding the
proof of the theorem.

###  4.2 Proof of Proposition 3.1

The proof of Proposition 3.1 follows closely the proof of Proposition 4.1 in
[5]. Recall that $l_{N,T}(x)$ is a kernel estimator of

$l(x)=\frac{\pi^{\prime}(x)}{\pi(x)}=-2V^{\prime}(x)-W^{\prime}\ast\pi(x).$
(21)

Let us decompose its error into the sum

$|l_{N,T}(x)-l(x)|=|l(x)|\mathds{1}_{\\{\pi_{N,T}(x)\leq\delta\\}}+r(x)$ (22)

where

$\displaystyle r(x)$
$\displaystyle:=|l_{N,T}(x)-l(x)|\mathds{1}_{\\{\pi_{N,T}(x)>\delta\\}}$
$\displaystyle\leq\delta^{-1}\left(|\pi^{\prime}_{N,T}(x)-\pi^{\prime}(x)|+|l(x)||\pi_{N,T}(x)-\pi(x)|\right).$
(23)

Under our assumptions, we have that $|V^{\prime}(x)|\leq
c(1+|x|+|\widetilde{V}^{\prime}(x)|)\leq c(1+|x|)$, where we have used that
$\widetilde{V}^{\prime}(0)=0$ and $\widetilde{V}^{\prime\prime}$ is bounded.
Moreover,
$|W^{\prime}\ast\pi(x)|\leq\|W^{\prime}\|_{\infty}\|\pi\|_{1}=\|W^{\prime}\|_{\infty}<\infty$,
which imply

$\sup_{|x|\leq U}|l(x)|\lesssim U.$

Next, Lemma 2.1 gives the lower bound $\pi(x)\geq
c_{1}\exp(-\widetilde{C}|x|^{2})$, which in turn implies $\pi(x)\geq 2\delta$
for all $|x|\leq U$. It follows that for all $|x|\leq U$,

$\displaystyle\mathbb{P}(\pi_{N,T}(x)\leq\delta)$
$\displaystyle=\mathbb{P}(\pi(x)-\pi_{N,T}(x)\geq\pi(x)-\delta)$
$\displaystyle\leq\mathbb{P}(\|\pi-\pi_{N,T}\|_{\infty}\geq\delta)\leq\delta^{-2}\mathbb{E}\left[\|\pi-\pi_{N,T}\|^{2}_{\infty}\right].$

Finally, consider

$\pi_{N,T}(x)-\pi(x)=r_{0}(x)+r_{1}(x)$

with

$r_{0}(x)=K_{h_{0}}\ast(\Pi_{N,T}(x)-\Pi)(x),\quad
r_{1}(x)=K_{h_{0}}\ast\Pi(x)-\pi(x),$

where recall $K_{h_{0}}(x)=h_{0}^{-1}K(h_{0}^{-1}x)$ is a scaled kernel. We
get

$|r_{0}(x)|\leq ch_{0}^{-1}W_{1}(\Pi_{N,T},\Pi)$

by applying the Kantorovich-Rubinstein theorem, moreover,
$\mathbb{E}[W_{1}^{2}(\Pi_{N,T},\Pi)]\leq cN_{T}^{-1}$ because of Theorem 2.2.
After substitution, we have that

$r_{1}(x)=\int_{\mathbb{R}}(\pi(x+h_{0}y)-\pi(x))K(y)dy,$

where by the Taylor theorem

$\pi(x+yh_{0})=\pi(x)+\sum_{i=1}^{m-1}\frac{\pi^{(k)}(x)}{k!}(yh_{0})^{k}+\frac{\pi^{(m)}(x+\tau
yh_{0})}{m!}(yh_{0})^{m}$

for some $0\leq\tau\leq 1$. Recall that $K$ has order $m$. Moreover, the bound
in Lemma 2.1 ensures that $|\pi^{(m)}(x)|\leq c$, hence,

$|r_{1}(x)|\leq ch_{0}^{m}$

uniformly in $x\in\mathbb{R}$. Our choice of $h_{0}$ yields

$\mathbb{E}\left[\|\pi_{N,T}-\pi\|_{\infty}^{2}\right]\lesssim
N_{T}^{-\frac{m}{m+1}}$

and similarly, that of $h_{1}$ yields

$\mathbb{E}\left[\|\pi^{\prime}_{N,T}-\pi^{\prime}\|_{\infty}^{2}\right]\lesssim
N_{T}^{-\frac{m}{m+2}}.$

Using these bounds in (22), (23), we obtain

$\sup_{|x|\leq
U}\mathbb{E}\left[|l_{N,T}(x)-l(x)|^{2}\right]^{\frac{1}{2}}\lesssim\exp(\widetilde{C}U^{2})\left(N_{T}^{-\frac{m}{2(m+2)}}+UN_{T}^{-\frac{m}{2(m+1)}}\right),$

which concludes the proof.

###  4.3 Proof of Theorem 3.2

Recall that

$\displaystyle\Psi_{N,T}(x)$
$\displaystyle=\left(l_{N,T}(x)+2\alpha_{N,T}x+2\widetilde{V}^{\prime}(x)\right)1_{\\{|x|\leq\epsilon
U\\}},$ $\displaystyle\Psi(x)$ $\displaystyle=l(x)+2\alpha
x+2\widetilde{V}^{\prime}(x)=-W^{\prime}\ast\pi(x).$

We decompose the mean integrated squared error of $\Psi_{N,T}$ into

${}\mathbb{E}\left[\int_{\mathbb{R}}\left(\Psi_{N,T}(x)-\Psi(x)\right)^{2}dx\right]^{\frac{1}{2}}=\left(\int_{|x|>\epsilon
U}(\Psi(x))^{2}dx\right)^{\frac{1}{2}}+I$ (24)

where

$I:=\left(\int_{|x|\leq\epsilon
U}\mathbb{E}\left[(\Psi_{N,T}(x)-\Psi(x))^{2}\right]dx\right)^{\frac{1}{2}}.$

Applying Minkowski’s inequality, we get $I\leq I_{1}+I_{2}$, where

$\displaystyle I_{1}$ $\displaystyle:=\left(\int_{|x|\leq\epsilon
U}\mathbb{E}\left[(l_{N,T}(x)-l(x))^{2}\right]dx\right)^{\frac{1}{2}}\lesssim
U^{\frac{1}{2}}\sup_{|x|\leq
U}\mathbb{E}\left[(l_{N,T}(x)-l(x))^{2}\right]^{\frac{1}{2}},$ $\displaystyle
I_{2}$ $\displaystyle:=\left(\int_{|x|\leq\epsilon
U}\mathbb{E}\left[(2\alpha_{N,T}x-2\alpha
x)^{2}\right]dx\right)^{\frac{1}{2}}\lesssim
U^{\frac{3}{2}}\mathbb{E}\left[(\alpha_{N,T}-\alpha)^{2}\right]^{\frac{1}{2}}.$

Next, consider the mean squared error of

$\alpha_{N,T}:=\operatorname{arg}\min_{\alpha\in\mathbb{R}}\int_{\mathbb{R}}\left(l_{N,T}(x)+2\alpha
x+2\widetilde{V}^{\prime}(x)\right)^{2}w_{U}(x)dx$

where $w_{U}(\cdot):=w(\cdot/U)/U$. The estimator $\alpha_{N,T}$ and $\alpha$
can be computed explicitly via

$\displaystyle\alpha_{N,T}$
$\displaystyle=-\frac{1}{2C_{2}U^{2}}\int_{\mathbb{R}}\left(l_{N,T}(x)+2\widetilde{V}^{\prime}(x)\right)xw_{U}(x)dx,$
$\displaystyle\alpha$
$\displaystyle=-\frac{1}{2C_{2}U^{2}}\int_{\mathbb{R}}\left(l(x)+2\widetilde{V}^{\prime}(x)+W^{\prime}\ast\pi(x)\right)xw_{U}(x)dx,$

where $C_{2}:=\int_{\mathbb{R}}x^{2}w(x)dx$. Since the support of $w_{U}$ is
$[\epsilon U,U]$, Jensen’s inequality can be applied to get

$\displaystyle(\alpha_{N,T}-\alpha)^{2}\leq\frac{1}{4C^{2}_{2}U^{4}}\int_{\mathbb{R}}\left(l_{N,T}(x)-l(x)-W^{\prime}\ast\pi(x)\right)^{2}x^{2}w_{U}(x)dx.$

By Minkowski’s inequality, we obtain

$\mathbb{E}\left[(\alpha_{N,T}-\alpha)^{2}\right]^{\frac{1}{2}}\leq\frac{1}{2C_{2}U^{2}}(J_{1}+J_{2}),$

where

$\displaystyle J_{1}$
$\displaystyle:=\left(\int_{\mathbb{R}}\mathbb{E}\left[(l_{N,T}(x)-l(x))^{2}\right]x^{2}w_{U}(x)dx\right)^{\frac{1}{2}}\leq
C_{2}^{\frac{1}{2}}U\sup_{|x|\leq
U}\mathbb{E}\left[(l_{N,T}(x)-l(x))^{2}\right]^{\frac{1}{2}},$ $\displaystyle
J_{2}$
$\displaystyle:=\left(\int_{\mathbb{R}}(W^{\prime}\ast\pi(x))^{2}x^{2}w_{U}(x)dx\right)^{\frac{1}{2}}\leq\frac{C_{\infty}U^{\frac{1}{2}}}{2}\left(\int_{|x|>\epsilon
U}(W^{\prime}\ast\pi(x))^{2}dx\right)^{\frac{1}{2}}$

with $C_{\infty}:=\sup_{x\in\mathbb{R}}x^{2}w(x).$ Using the above bounds in
(24) we conclude that

$\displaystyle\mathbb{E}\left[\int_{\mathbb{R}}(\Psi_{N,T}(x)-\Psi(x))^{2}dx\right]^{\frac{1}{2}}$
$\displaystyle\lesssim U^{\frac{1}{2}}\sup_{|x|\leq
U}\mathbb{E}\left[(l_{N,T}(x)-l(x))^{2}\right]^{\frac{1}{2}}$
$\displaystyle+\left(\int_{|x|>\epsilon
U}(W^{\prime}\ast\pi(x))^{2}dx\right)^{\frac{1}{2}}.$

The first term on the right hand side has been studied in Proposition 3.1, and
we are therefore left to study the second term. It is

$\left(\int_{|x|>\epsilon
U}(W^{\prime}\ast\pi(x))^{2}dx\right)^{\frac{1}{2}}=\left(\int_{|x|>\epsilon
U}\left(\int_{\mathbb{R}}W^{\prime}(x-y)\pi(y)dy\right)^{2}dx\right)^{\frac{1}{2}}.$

By Minkowski’s inequality this is bounded by

$\displaystyle{}\int_{|y|\leq\frac{\epsilon}{2}U}\left(\int_{|x|>\epsilon
U}|W^{\prime}(x-y)|^{2}dx\right)^{\frac{1}{2}}\pi(y)dy+\int_{|y|>\frac{\epsilon}{2}U}\left(\int_{|x|>\epsilon
U}|W^{\prime}(x-y)|^{2}dx\right)^{\frac{1}{2}}\pi(y)dy.$ (25)

Then, we apply a change of variables $x-y:=\widetilde{x}$, observing that
$|x|>\epsilon U$ and $|y|\leq\epsilon U/2$ imply $|x-y|>\epsilon U/2$. For the
second integral we enlarge the domain of integration to $\mathbb{R}$. It
follows that (25) is upper bounded by

$\left(\int_{|x|>\frac{\epsilon}{2}U}|W^{\prime}(x)|^{2}dx\right)^{\frac{1}{2}}\int_{\mathbb{R}}\pi(y)dy+\left(\int_{|y|>\frac{\epsilon}{2}U}\pi(y)dy\right)\left(\int_{\mathbb{R}}|W^{\prime}(x)|^{2}dx\right)^{\frac{1}{2}}.$

Thanks to Lemma 2.1, we know that

$\int_{\frac{\epsilon}{2}U}^{\infty}\pi(x)dx\lesssim\int_{\frac{\epsilon}{2}U}^{\infty}\overline{\pi}(x)dx,$

with $\overline{\pi}(x)=c_{2}\exp(-C_{V}x^{2})$ satisfying
$\overline{\pi}^{\prime}(x)=-2c_{2}C_{V}x\overline{\pi}(x)$. Hence, we can
write

$\int_{u}^{\infty}\overline{\pi}(x)dx\leq\frac{1}{u}\int_{u}^{\infty}x\overline{\pi}(x)dx=\frac{\overline{\pi}(u)}{2c_{2}C_{V}u}.$

It implies that

${}\int_{\epsilon\frac{U}{2}}^{\infty}\pi(x)dx\lesssim\frac{2\exp(-C_{V}(\frac{\epsilon}{2}U)^{2})}{\epsilon
U}$ (26)

Then, the boundedness of $\int_{\mathbb{R}}\pi(y)dy$ is straightforward, while
$\int_{\mathbb{R}}|W^{\prime}(x)|^{2}dx$ is bounded as $W^{\prime}\in
L^{1}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$. It follows

${}\left(\int_{|x|>\epsilon
U}(W^{\prime}\ast\pi(x))^{2}dx\right)^{\frac{1}{2}}\lesssim\left(\int_{|x|>\frac{\epsilon}{2}U}|W^{\prime}(x)|^{2}dx\right)^{\frac{1}{2}}+\frac{2\exp(-C_{V}(\frac{\epsilon}{2}U)^{2})}{\epsilon
U},$ (27)

as we wanted.

###  4.4 Proof of Corollary 3.3

Corollary 3.3 is a consequence of Theorem 3.2 and of the exponential decay of
the tails of $W^{\prime}$. The choice of the threshold

${}U^{2}=c_{u}\log(N_{T})$ (28)

gives

$\left(\int_{|y|>\frac{\epsilon}{2}U}|W^{\prime}(y)|^{2}dy\right)^{\frac{1}{2}}\lesssim\exp\left(-\frac{p\epsilon^{2}U^{2}}{4}\right)=N_{T}^{-\frac{p\epsilon^{2}c_{u}}{4}}.$

Together with Theorem 3.2 it implies that

$\displaystyle\mathbb{E}\left[\int_{\mathbb{R}}|\Psi_{N,T}(y)-\Psi(y)|^{2}dy\right]^{\frac{1}{2}}$
$\displaystyle\lesssim(\log
N_{T})^{\frac{1}{4}}N_{T}^{\widetilde{C}c_{u}}\left(N_{T}^{-\frac{m}{2(m+2)}}+(\log
N_{T})^{\frac{1}{2}}N_{T}^{-\frac{m}{2(m+1)}}\right)$
$\displaystyle+N_{T}^{-C_{V}\frac{\epsilon^{2}}{4}c_{u}}+N_{T}^{-p\frac{\epsilon^{2}}{4}c_{u}}.$

Recall that $p>C_{V}$. Then, we can choose $c_{u}$ in order to obtain the
balance between the remaining two terms above:

${}c_{u}=\frac{m}{2(m+2)}\frac{1}{(\widetilde{C}+C_{V}\frac{\epsilon^{2}}{4})}.$
(29)

Then, the convergence rate is

$\displaystyle(\log
N_{T})^{\frac{1}{4}}N_{T}^{-\frac{m}{2(m+2)}\frac{C_{V}\frac{\epsilon^{2}}{4}}{(\widetilde{C}+C_{V}\frac{\epsilon^{2}}{4})}}$

as claimed.

###  4.5 Proof of Theorem 3.5

Let $\mathcal{L}_{a}:=\\{y+ia:y\in\mathbb{R}\\}$ for some $a\geq 0$. Assume
$\operatorname{Re}(\mathcal{F}(\Pi)(s))>0$ for all $s\in\mathcal{L}_{a}$.
Recall that we defined a sequence of entire functions
$\rho_{N,T}(s):=\mathcal{F}(\Pi)(s)-\mathcal{F}(\Pi_{N,T})(s)+\varepsilon_{N,T}$,
$s\in\mathbb{C}$, for some $\varepsilon_{N,T}>0$. Note that for all
$s\in\mathcal{L}_{a}$

$|\mathcal{F}(\Pi_{N,T})(s)+\rho_{N,T}(s)|=|\mathcal{F}(\Pi)(s)+\varepsilon_{N,T}|\geq\varepsilon_{N,T}>0$
(30)

and $W^{\prime}_{N,T}$ is defined via

$\mathcal{F}(W^{\prime}_{N,T})(s):=-\frac{\mathcal{F}(\Psi_{N,T})(s)}{\mathcal{F}(\Pi_{N,T})(s)+\rho_{N,T}(s)}.$

Let $\delta_{N,T}:=W^{\prime}_{N,T}-W^{\prime}$. Plancherel’s theorem gives

$\displaystyle\int_{-\infty}^{\infty}|\delta_{N,T}(x)|^{2}dx$
$\displaystyle=\frac{1}{2\pi
i}\int_{a-i\infty}^{a+i\infty}\overline{\mathcal{F}(\delta_{N,T})(i(-\overline{s}))}\mathcal{F}(\delta_{N,T})(is)ds$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}\overline{\mathcal{F}(\delta_{N,T})(y-ia)}\mathcal{F}(\delta_{N,T})(y+ia)dy\lesssim\Delta_{-a}+\Delta_{a},$
(31)

where

$\displaystyle\Delta_{\pm
a}:=\int_{-\infty}^{\infty}|\mathcal{F}(\delta_{N,T})(y\pm
ia)|^{2}dy=\int_{\mathcal{L}_{\pm a}}|\mathcal{F}(\delta_{N,T})(s)|^{2}ds.$

It suffices to consider $\Delta_{a}$, because the analysis of $\Delta_{-a}$
follows a similar route. Rewrite

$\mathcal{F}(\delta_{N,T})(s)=-\frac{\mathcal{F}(\Psi_{N,T})(s)-\mathcal{F}(\Psi)(s)+\varepsilon_{N,T}\mathcal{F}(W^{\prime})(s)}{\mathcal{F}(\Pi_{N,T})(s)+\rho_{N,T}(s)}.$

Let us deal with the denominator by using (30) and
$\operatorname{Re}(\mathcal{F}(\Pi)(s))+\varepsilon_{N,T}>\operatorname{Re}(\mathcal{F}(\Pi)(s))>0$
for all $s\in\mathcal{L}_{a}$. We get

$\Delta_{a}\lesssim\frac{1}{\varepsilon_{N,T}^{2}}\Delta_{a,1}+\varepsilon_{N,T}^{2}\int_{\mathcal{L}_{a}}\left|\frac{\mathcal{F}(W^{\prime})(s)}{\mathcal{F}(\Pi)(s)}\right|^{2}ds,$
(32)

where

$\Delta_{a,1}:=\int_{\mathcal{L}_{a}}|\mathcal{F}(\Psi_{N,T})(s)-\mathcal{F}(\Psi)(s)|^{2}ds.$

$\displaystyle{}\mathbb{E}[\Delta_{a,1}]^{\frac{1}{2}}=\left(\int_{-\infty}^{\infty}\exp(-2ax)\mathbb{E}\left[|\Psi_{N,T}(x)-\Psi(x)|^{2}\right]dx\right)^{\frac{1}{2}}\leq
D_{1}+D_{2},$ (33)

where

$\displaystyle D_{1}$ $\displaystyle:=\left(\int_{|x|>\epsilon
U}\exp(-2ax)|\Psi(x)|^{2}dx\right)^{\frac{1}{2}},$ $\displaystyle D_{2}$
$\displaystyle:=(2\epsilon U)^{\frac{1}{2}}\exp(a\epsilon
U)\sup_{|x|\leq\epsilon
U}\mathbb{E}\left[|\Psi_{N,T}(x)-\Psi(x)|^{2}\right]^{\frac{1}{2}}.$

We start handling $D_{1}$, while the analysis on $D_{2}$ heavily relies on the
bounds gathered in previous steps. We recall that

$\Psi(x)=-W^{\prime}\ast\pi(x)=-\int_{\mathbb{R}}W^{\prime}(x-y)\pi(y)dy.$

By the Minkowski inequality,

$\displaystyle D_{1}$
$\displaystyle\leq\int_{\mathbb{R}}\left(\int_{|x|>\epsilon
U}\exp(-2ax)W^{\prime}(x-y)^{2}dx\right)^{\frac{1}{2}}\pi(y)dy=I_{1}+I_{2},$

where after a change of variable the r.h.s. has been decomposed into

$\displaystyle I_{1}$
$\displaystyle:=\int_{|y+c_{0}|\leq\frac{\epsilon}{2}U}\left(\int_{|x+y|>\epsilon
U}\exp(-2ax)W^{\prime}(x)^{2}dx\right)^{\frac{1}{2}}\exp(-ay)\pi(y)dy,$
$\displaystyle I_{2}$
$\displaystyle:=\int_{|y+c_{0}|>\frac{\epsilon}{2}U}\left(\int_{|x+y|>\epsilon
U}\exp(-2ax)W^{\prime}(x)^{2}dx\right)^{\frac{1}{2}}\exp(-ay)\pi(y)dy$

with $c_{0}:=\frac{a}{2C_{V}}$. Note that $|y+c_{0}|\leq\frac{\epsilon}{2}U$
and $|x+y|>\epsilon U$ imply $|x-c_{0}|>\frac{\epsilon}{2}U$ in the inner
integral in $I_{1}$. Let us also enlarge the domain of integration to
$\mathbb{R}$ in the outer and inner integrals in $I_{1}$ and $I_{2}$
respectively. Then

$I_{1}+I_{2}\leq
J_{1}\int_{\mathbb{R}}\exp(-ay)\pi(y)dy+J_{2}\left(\int_{\mathbb{R}}\exp(-2ax)W^{\prime}(x)^{2}dx\right)^{\frac{1}{2}},$
(34)

where

$J_{1}:=\left(\int_{|x-c_{0}|>\frac{\epsilon}{2}U}\exp(-2ax)W^{\prime}(x)^{2}dx\right)^{\frac{1}{2}},\qquad
J_{2}:=\int_{|y+c_{0}|>\frac{\epsilon}{2}U}\exp(-ay)\pi(y)dy.$

The upper bound on $\pi$ in Lemma 2.1 implies that the first integral on the
r.h.s. of (34) is finite, furthermore,

$\displaystyle J_{2}$
$\displaystyle\lesssim\int_{|y+c_{0}|>\frac{\epsilon}{2}U}\exp(-ay-
C_{V}y^{2})dy$
$\displaystyle\eqsim\int_{|y|>\frac{\epsilon}{2}U}\exp(-C_{V}y^{2})dy\lesssim\frac{\exp(-C_{V}(\frac{\epsilon}{2}U)^{2})}{\frac{\epsilon}{2}U}.$

Now consider $J_{1}$ and the second integral on the r.h.s. of (34), where,
recall, $W^{\prime}\in L^{2}(\mathbb{R})$ is odd and satisfies the assumption
(11) for $p>C_{V}$. This means that there exists $c>0$ such that

$\int_{x}^{\infty}W^{\prime}(u)^{2}du\leq c\exp(-2px^{2})$

for all $x\geq 1$, that is

$\int_{1}^{\infty}\chi(u)W^{\prime}(u)^{2}du\leq
c\int_{1}^{\infty}\chi(u)dF(u),$

where $F(u):=1-\exp(-2pu^{2})$ and $\chi(u):=\mathds{1}(u\geq x)$,
$u\in\mathbb{R}$, for all $x\geq 1$. By monotone approximation the above
inequality remains valid for all non-negative non-decreasing functions
$\chi:[1,\infty)\to[0,\infty)$, for example, $\chi(u)=\exp(2au)$. We get that
the second integral of $W^{\prime}\in L^{2}(\mathbb{R})$ on the r.h.s. of (34)
is finite, moreover,

$\displaystyle(J_{1})^{2}$
$\displaystyle\lesssim\int^{\infty}_{\frac{\epsilon}{2}U-c_{0}}\exp(2ax)W^{\prime}(x)^{2}dx$
$\displaystyle\lesssim\int_{\frac{\epsilon}{2}U-c_{0}}^{\infty}\exp(2ax)dF(x)=\int_{\frac{\epsilon}{2}U-c_{0}}^{\infty}\exp(2ax-2px^{2})(4px)dx$
$\displaystyle\lesssim\exp\left(-2p\left(\frac{\epsilon}{2}U-c_{1}\right)^{2}\right),$

where $c_{1}=c_{0}+\frac{a}{2p}$. Since $p>C_{V}$, it follows that

$(J_{1})^{2}\lesssim\frac{\exp(-2C_{V}(\frac{\epsilon}{2}U)^{2})}{(\frac{\epsilon}{2}U)^{2}}.$

We conclude that

$D_{1}\lesssim\frac{\exp(-C_{V}(\frac{\epsilon}{2}U)^{2})}{\frac{\epsilon}{2}U}.$
(35)

Regarding $D_{2}$, we recall that from the definition of $\Psi_{N,T}(y)$ and
$\Psi(y)$ we deduce

$D_{2}\lesssim(2\epsilon U)^{\frac{1}{2}}\exp(a\epsilon U)(R_{1}+R_{2}),$

where

$\displaystyle R_{2}$ $\displaystyle:=\sup_{|y|\leq
U}\mathbb{E}\left[|l_{N,T}(y)-l(y)|^{2}\right]^{\frac{1}{2}},$ $\displaystyle
R_{1}$ $\displaystyle:=\epsilon
U\mathbb{E}\left[\left(\alpha_{N,T}-\alpha\right)^{2}\right]^{\frac{1}{2}}\lesssim
R_{2}+U^{-\frac{1}{2}}\left(\int_{|x|>\epsilon
U}\Psi(x)^{2}dx\right)^{\frac{1}{2}}.$

The integral in the upper bound on $R_{1}$ coincides with $D_{1}$ when $a=0$
hence satisfies (35), whereas the upper bound on $R_{2}$ follows from Lemma
3.1. We get the upper bound

$\displaystyle D_{2}\lesssim\exp\left(a\epsilon U\right)\left(\left(\epsilon
U\right)^{\frac{1}{2}}\exp(\widetilde{C}U^{2})\left(N_{T}^{-\frac{m}{2(m+2)}}+UN_{T}^{-\frac{m}{2(m+1)}}\right)+\frac{\exp(-C_{V}(\frac{\epsilon}{2}U)^{2})}{\frac{\epsilon}{2}U}\right),$

which also works for $\mathbb{E}[\Delta_{a,1}]^{\frac{1}{2}}$. The choice
$U^{2}=c_{u}\log N_{T}$ for $c_{u}$ as in (29) gives us

$\mathbb{E}[\Delta_{a,1}]^{\frac{1}{2}}\lesssim\exp\left(a\epsilon(c_{u}\log
N_{T})^{\frac{1}{2}}\right)(\log
N_{T})^{\frac{1}{4}}N_{T}^{-\gamma}=:\lambda_{N,T}$ (36)

for $\gamma$ as in (14). Finally, from (4.5), (32) and (36) it follows that

$\mathbb{E}\left[\int_{-\infty}^{\infty}\left|\delta_{N,T}(y)\right|^{2}dy\right]^{\frac{1}{2}}\lesssim\frac{\lambda_{N,T}}{\varepsilon_{N,T}}+\varepsilon_{N,T},$

where note that $\varepsilon_{N,T}$ can be chosen such that
$\varepsilon_{N,T}:=\lambda_{N,T}^{\frac{1}{2}}\rightarrow 0$ for
$N,T\rightarrow\infty$. It yields

$\mathbb{E}\left[\int_{-\infty}^{\infty}\left|\delta_{N,T}(y)\right|^{2}dy\right]^{\frac{1}{2}}\lesssim\lambda^{\frac{1}{2}}_{N,T}$

as required.

##  5 On the Fourier transforms

We have observed that Assumption 2 is crucial in order to obtain the
polynomial convergence rate stated in Theorem 3.5. However, this assumption
may appear somewhat unclear at first. The objective of this section is to
investigate sufficient conditions that guarantee the validity of Assumption 2.
Given our focus on the super-smooth case, it seems natural to require that the
transform of the function $W^{\prime}$ we aim to estimate decays exponentially
fast. Verifying the condition (15), however, entails seeking a lower bound for
the transform of $\pi$, which will be the main objective of this section. We
will thus commence by examining the properties of the Fourier transform of
$\pi$, with the clear intention of studying the set of zeros and finding a
lower bound outside that set. Remark that it is the same whether considering
the transform of the distribution $\Pi$ or the density $\pi$, and it is

$\mathcal{F}(\Pi)=\mathcal{F}(\pi)=\int_{\mathbb{R}}\exp(izx)\pi(x)dx.$

###  5.1 Properties of $\mathcal{F}(\pi)$

We will use tools from complex analysis to study $\mathcal{F}(\pi)$ as a
function defined on $\mathbb{C}$. More specifically, our aim is to represent
the entire function $\mathcal{F}(\pi)$ via the Hadamard factorisation theorem.
For definitions of the terminology used in this work, we refer to [30]. We
begin by deriving an upper bound of the order of $\mathcal{F}(\pi)$, which
stems from the fact that $\pi$ has Gaussian tails.

######  Theorem 5.1.

Let Assumption 1 hold and $\pi$ be as in Eq. 5. The map

$\mathcal{F}(\pi)\colon\mathbb{C}\to\mathbb{C},\quad
z\mapsto\int_{\mathbb{R}}\exp(izx)\pi(x)\textnormal{d}x$ (37)

is an entire function which coincides with the characteristic function of
$\pi$ on $\mathbb{R}$. Moreover, the order of $\mathcal{F}(\pi)$ does not
exceed 2.

###### Proof.

The integral in (37) exists and defines a continuous function at any $z=a+ib$
with $a,b\in\mathbb{R}$ since
$|\mathcal{F}(\pi)(a+ib)|\leq\int_{\mathbb{R}}\exp(-bx)\pi(x)dx<\infty$ by
using $\pi(x)\lesssim\exp(-C_{V}x^{2})$ and
$\|W^{\prime}\ast\pi\|_{\infty}\leq\|W^{\prime}\|_{\infty}<\infty$.
Furthermore, we have $\int_{C}\mathcal{F}(\pi)(z)\,dz=0$ for any closed
contour $C$ which follows from the Cauchy theorem applied to the function
$z\mapsto\exp(izx).$ By Morera’s theorem, $\mathcal{F}(\pi)$ is an entire
function. To determine the order of $\mathcal{F}(\pi)$, we use the inequality

$\left|zx\right|\leq\frac{1}{2}\left(\frac{1}{c}\left|z\right|^{2}+c\left|x\right|^{2}\right)\quad\forall
z,x\in\mathbb{C},\ c>0,$

such that, for any $z\in\mathbb{C}$,

$\displaystyle\left|\mathcal{F}(\pi)(z)\right|$
$\displaystyle\leq\int_{\mathbb{R}}\exp(|\operatorname{Im}(z)x|)\pi(x)dx$
$\displaystyle\leq\exp\left(\frac{\operatorname{Im}(z)^{2}}{2c}\right)\int_{\mathbb{R}}\exp\left(\frac{cx^{2}}{2}\right)\pi(x)dx.$

Recall that according to Lemma 2.1, $\pi(x)\lesssim\exp(-C_{V}x^{2})$. Hence,
the above integral is finite if $x\mapsto\exp(-(C_{V}-c/2)x^{2})$ is
integrable on $\mathbb{R}$, which is the case for $C_{V}>c/2$. As a result,
the order of $\mathcal{F}(\pi)$ does not exceed 2. ∎

Now that we have that $\mathcal{F}(\pi)$ is an entire function of finite
order, we will find an expression for the function using Hadamard’s
factorisation theorem. For this, we denote the zeros of $\mathcal{F}(\pi)$ as
$(a_{j})_{j\in\mathbb{N}}$ and order them by increasing modulus. Note that the
zeros are symmetric around the imaginary axis, in other words, if $a_{j}$ is a
zero of $\mathcal{F}(\pi)$, then so is its negative conjugate
$-\overline{a}_{j}$. Moreover, $\mathcal{F}(\pi)$ has no zeros on the
imaginary axis because $\mathcal{F}(\pi)(ia)>0$ for all $a\in\mathbb{R}$, see
[35, Corollary 1 to Theorem 2.3.2]. Let us introduce the critical exponent of
convergence $\rho_{1}$ of the sequence $(a_{j})_{j\in\mathbb{N}}$:

$\rho_{1}=\inf\left\\{r>0:\sum_{j\in\mathbb{N}}\frac{1}{|a_{j}|^{r}}<\infty\right\\}.$

We denote the order of $\mathcal{F}(\pi)$ as $\rho$ and make the following

######  Assumption 3.

$\mathcal{F}(\pi)$ satisfies either $\rho_{1}<\rho$ or $\rho_{1}=\rho<2$.

We will use the Hadamard canonical factors

$\displaystyle E_{d}(z)=\begin{cases}1-z,&d=0,\\\
(1-z)\exp(z),&d=1,\end{cases}$

defined for $z\in\mathbb{C}$, to study the infinite product representation of
$\mathcal{F}(\pi)$.

######  Theorem 5.2.

Let Assumptions 1 and 3 hold and $\pi$ be as in (5) and $\mathcal{F}(\pi)$ as
in (37). Then, there exist $p_{1}\in\mathbb{R}$ and $p_{2}\geq 0$ such that
for all $z\in\mathbb{C}$,

$\mathcal{F}(\pi)(z)=\exp(-p_{2}z^{2}+ip_{1}z)\prod_{j\in\mathbb{N}}E_{1}\left(\frac{z}{a_{j}}\right).$
(38)

###### Proof.

Firstly, we consider the case $\rho_{1}<\rho$. Then $\rho$ is either $2$ or
$1$ by [30, Lemma 4.10.1]. The representation in (38) follows from [35,
Remark, page 42]. We note that if $\mathcal{F}(\pi)$ is of order $\rho=2$,
then $p_{2}>0$ because $p_{2}=0$ would lead to the contradiction
$\rho\leq\max(1,\rho_{1})$ by [35, Theorems 1.2.5, 1.2.7, 1.2.8]. However, if
$\mathcal{F}(\pi)$ is of order $\rho=1$ then its representation may be reduced
to

$\mathcal{F}(\pi)(z)=\exp\left(i\widetilde{p}_{1}z\right)\prod_{j}E_{0}\left(\frac{z}{a_{j}}\right),\qquad\widetilde{p}_{1}:=p_{1}+\sum_{j}\operatorname{Im}\left(\frac{1}{a_{j}}\right).$
(39)

Next, we turn to the case $\rho_{1}=\rho<2$. According to Hadamard’s
factorization theorem,

$\mathcal{F}(\pi)(z)=\exp(q_{1}z+q_{0})\prod_{j}E_{1}\left(\frac{z}{a_{j}}\right)$

for some $q_{1},q_{0}\in\mathbb{C}$. We note that $q_{0}=0$ since
$\mathcal{F}(\pi)(0)=1$. In order to say something about $q_{1}$, we note that
$X$ with a probability density function $\pi$ satisfies
$\mathbb{E}[|X|]<\infty$. Moreover, $\mathbb{E}[X]=0$, whence

$\displaystyle
0=\operatorname{Re}\left(\frac{\mathcal{F}(\pi)^{\prime}(0)}{\mathcal{F}(\pi)(0)}\right)=\operatorname{Re}\left(\log\mathcal{F}(\pi)(z)\right)^{\prime}|_{z=0}=\left.\operatorname{Re}\left(q_{1}+\sum_{j}\frac{z}{a_{j}(z-a_{j})}\right)\right|_{z=0}=\operatorname{Re}(q_{1}).$

We conclude that (38) holds true with $p_{2}=0$,
$p_{1}=\operatorname{Im}(q_{1})$. ∎

######  Remark 5.3.

We have $p_{2}>0$ only if $\rho_{1}<\rho=2$. If $\mathcal{F}(\pi)$ has no
zeros then $\pi$ must be a density of a normal distribution with mean zero by
[35, Corollary to Theorem 2.5.1], since $\rho$ is finite. ∎

The following step consists in bounding the infinite product in (38). If
$\lfloor\rho_{1}\rfloor=0$, the term $\prod_{j\in\mathbb{N}}\exp(z/a_{j})$ is
well-defined, and the representation becomes

$\mathcal{F}(\pi)(z)=\exp(-p_{2}z^{2}+i\widetilde{p}_{1}z)\prod_{j\in\mathbb{N}}E_{0}\left(\frac{z}{a_{j}}\right),$

where $\widetilde{p}_{1}$ is the same as in (39). Let us now use [20, Lemma
4.12]. For all $z\in\mathbb{C}$ outside $\cup_{j}B_{\epsilon_{j}}(a_{j})$,
where $\epsilon_{j}=1/|a_{j}|^{\rho_{1}+\varepsilon}$ and $\varepsilon>0$,
$\rho_{1}+\varepsilon\leq 2$, we get

$\left|\prod_{j\in\mathbb{N}}E_{\lfloor\rho_{1}\rfloor}\left(\frac{z}{a_{j}}\right)\right|\gtrsim\exp(-c_{\pi}|z|^{\rho_{1}+\varepsilon}).$

It leads us to the following theorem.

######  Theorem 5.4.

Let Assumptions 1 and 3 hold. Then there exist $c_{\pi}>0$ and a family of
positive numbers $(\epsilon_{j})_{j\in\mathbb{N}}$ such that, for any
$z\in\mathbb{C}$ outside $\cup_{j\geq 1}B_{\epsilon_{j}}(a_{j})$, it holds
that

${}|\mathcal{F}(\pi)(z)|\gtrsim\exp(-c_{\pi}|z|^{2}).$ (40)

Thanks to Theorem 5.4, we can explicitly describe a scenario where condition
(15) is satisfied, assuming that we are in the super-smooth case where the
tails of the transform of the interaction function are exponential.

Let us introduce slowly varying functions $l(t)$, for $t\geq 0$ (see also
[47]). These are positive and measurable functions such that, for each
$\lambda>0$, the following holds as $t\rightarrow\infty$: $\frac{l(\lambda
t)}{l(t)}\rightarrow 1$.

######  Assumption 4.

Let $(a_{j})_{j\in\mathbb{N}}$, $c_{\pi}$, and
$(\epsilon_{j})_{j\in\mathbb{N}}$ be as in Theorem 5.4. Assume that
$\cup_{j\geq 1}B_{\epsilon_{j}}(a_{j})\cap\mathcal{L}_{0}=\emptyset$.
Furthermore, there is a slowly varying function $l$ such that
$\liminf_{z\rightarrow\infty}l(z)>c_{\pi}$ for which the following holds true
on $\mathcal{L}_{0}$:

$|\mathcal{F}(W^{\prime})(z)|=O(\exp(-|z|^{2}l(|z|)))\quad\text{as
}\left|z\right|\to\infty.$ (41)

Note that in the aforementioned assumption, we assert that the transform of
$W^{\prime}$ exhibits a decay that is almost Gaussian. This observation aligns
with our expectations, given the nature of the model under consideration,
where the confinement potential is driven by $x^{2}$.
It is noteworthy to observe that Theorem 5 in [46], specifically in the
scenario where $\bar{a}=0$, furnishes both necessary and sufficient conditions
for the existence of non-trivial functions exhibiting an almost Gaussian
nature, accompanied by an almost Gaussian Fourier transform as defined in
Equation (41). Furthermore, according to Theorem 1 in [28], if both
$W^{\prime}$ and its Fourier transform $\mathcal{F}(W^{\prime})(z)$ follow the
asymptotic behavior $O(|z|^{s}\exp(-\frac{1}{2}z^{2}))$ as $z\to\infty$, then
each can be expressed as a finite linear combination of Hermite functions.
This provides concrete examples that satisfy our assumptions.

######  Corollary 5.5.

Let Assumptions 1, 3 and 4 hold. Then, we have

$\int_{\mathcal{L}_{0}}\left|\frac{\mathcal{F}\left(W^{\prime}\right)(z)}{\mathcal{F}(\pi)(z)}\right|^{2}dz<\infty.$

###### Proof.

The corollary is a straightforward consequence of Theorem 5.4 and Assumption
4. We have indeed

$\int_{\mathcal{L}_{0}}\left|\frac{\mathcal{F}\left(W^{\prime}\right)(z)}{\mathcal{F}(\pi)(z)}\right|^{2}dz\leq\int_{\mathcal{L}_{0}}\exp(-|z|^{2}l(|z|)+c_{\pi}|z|^{2})dz,$

which is bounded due to Assumption 4. ∎

We can conclude by noting that, under the assumptions of Corollary 5.5,
Assumption 2 is clearly satisfied. This implies that we can achieve a
polynomial convergence rate for the estimation of $W^{\prime}$, as stated in
Theorem 3.5.

######  Example 5.6.

To conclude this section, we provide an example demonstrating how a non-smooth
confinement potential leads to an invariant density with a Fourier transform
exhibiting polynomial decay. We have that

$\pi(x)=\frac{1}{Z_{\pi}}\exp\left(-2V(x)-W*\pi(x)\right),$

where $V(x)=(\alpha/2)x^{2}+\widetilde{V}(x)$ and $W(x)$ satisfy Assumption 1.
In addition, assume that $W$, $\widetilde{V}$ are infinitely differentiable
respectively on $\mathbb{R}$, $\mathbb{R}\setminus\\{0\\}$ and there exist
$\widetilde{V}^{(J+1)}(0^{\pm}):=\lim_{x\to
0^{\pm}}\widetilde{V}^{(J+1)}(x)<\infty$ such that

$\widetilde{V}^{(J+1)}(0^{+})\neq\widetilde{V}^{(J+1)}(0^{-}).$

Furthermore, $\widetilde{V}^{(j)}$, $2\leq j\leq J+2$, are bounded on
$\mathbb{R}\setminus\\{0\\}$. It provides that $\pi$ is infinitely
differentiable on $\mathbb{R}\setminus\\{0\\}$ and there exist
$\pi^{(J+1)}(0^{+})\neq\pi^{(J+1)}(0^{-})$. Then, iteratively integrating by
parts we obtain

$\displaystyle\mathcal{F}(\pi)(z)=\frac{1}{(-iz)^{(J+1)}}(I_{-}(z)+I_{+}(z)),$

where

$\displaystyle I_{-}(z):=\int_{-\infty}^{0}\exp(izx)\pi^{(J+1)}(x)dx,\qquad
I_{+}(z):=\int_{0}^{\infty}\exp(izx)\pi^{(J+1)}(x)dx.$

If we integrate by parts once more, we obtain

$I_{+}(z)=\frac{1}{iz}\left(-\pi^{(J+1)}(0^{+})-\int_{0}^{\infty}\exp(izx)\pi^{(J+2)}(x)dx\right)$
(42)

since $\pi^{(J+1)}(x)\to 0$ as $x\to\infty$ and $\pi^{(J+2)}$ is integrable,
which in turn follow using the same arguments as in the proof of Lemma 2.1. By
Riemann’s lemma, the last integral in (42) tends to zero and so

$I_{+}(z):=-\frac{1}{iz}\pi^{(J+1)}(0^{+})+o\left(\frac{1}{z}\right)$

as $z\to\infty$. Clearly an analogous reasoning applies to $I_{-}(z)$. It
yields that for all large enough $z$,

$|\mathcal{F}(\pi)(z)|\geq\frac{c}{|z|^{J+2}}.$

Hence, the Fourier transform of $\pi$ has in this case a polynomial decay, as
claimed.

######  Example 5.7.

Consider an ODE of the form (Burgers equation)

$\pi_{xx}+(((W^{\prime}+x)\ast\pi)\pi)_{x}=0$

with $W^{\prime}(x)=\alpha\delta_{0}$ for some $\alpha\in(0,1).$ Even though
$W\not\in C^{2}(\mathbb{R})$ in this case, it is still instructive to study
this situation. In particular, this example shows that for non-smooth
potentials $W$, the Fourier transform of $\pi$ can decay rather fast. Let us
introduce $\varphi(x):=\alpha\delta_{0}+x$. Then we have

$((W^{\prime}+x)\ast\pi)(x)=\int_{\mathbb{R}}\varphi(x-y)\pi(y)\,dy=\alpha\pi(x)+(x-\mu)$

provided $\int_{\mathbb{R}}\pi(y)\,dy=1$ and
$\int_{\mathbb{R}}y\pi(y)\,dy=\mu.$ Then

$\pi_{x}+(x-\mu)\pi+\alpha\pi^{2}=0.$

Applying the transformation

$\pi=y^{\prime}/(\alpha y),$

we obtain

$(y^{\prime}/y)^{2}+(y^{\prime\prime}y-(y^{\prime})^{2})/y^{2}+(x-\mu)y^{\prime}/y=0$

or

$y^{\prime\prime}+(x-\mu)y^{\prime}=0.$

The general solution of this equation is given by

$y(x)=c_{1}+c_{2}\int_{-\infty}^{x}\exp(-(s-\mu)^{2}/2)\,ds,\quad
c_{1},c_{2}\in\mathbb{R}.$

Hence, we deduce the identity

$\pi(x)=\frac{\exp(-(x-\mu)^{2}/2)}{c+\alpha\int_{-\infty}^{x}\exp(-(s-\mu)^{2}/2)\,ds}$

with the constant $c>0$ chosen in such a way that
$\int_{\mathbb{R}}\pi(x)\,dx=1.$ The Fourier transform of $\pi$ can be written
as

$\displaystyle\mathcal{F}(\pi)(\omega)$
$\displaystyle=\int_{\mathbb{R}}\frac{\exp(i\omega
x-(x-\mu)^{2}/2)}{F(x-\mu)}\,dx=\exp(i\omega\mu)\int_{\mathbb{R}}\frac{\exp(S(\omega;z))}{F(z)}\,dz$

where $S(\omega;z)=i\omega z-z^{2}/2$ and

$\displaystyle F(z)$
$\displaystyle=c+\alpha\int_{-\infty}^{z}\exp(-s^{2}/2)\,ds$
$\displaystyle=c+\alpha\sqrt{\pi/2}+\alpha\int_{0}^{z}\exp(-s^{2}/2)\,ds,$

having used that
$\int_{-\infty}^{z}\exp(-s^{2}/2)\,ds+\int_{z}^{0}\exp(-s^{2}/2)\,ds=\sqrt{\pi/2}$.
We have the identity

$\displaystyle\int_{0}^{iw}\exp(-t^{2}/2)\,dt=i\sqrt{2}\exp(w^{2}/2)D_{+}(w/\sqrt{2})$

where $D_{+}$ is the Dawson function satisfying

$D_{+}(x)=\frac{1}{2x}+\frac{1}{4x^{3}}+O\left(\frac{1}{x^{5}}\right),\quad
x\to\infty.$

Hence, we conclude that

$\displaystyle\int_{0}^{iw}\exp(-t^{2}/2)\,dt=-\frac{\exp(w^{2}/2)}{iw}\left[1+O\left(\frac{1}{w^{2}}\right)\right],\quad
w\to\infty.$

The same asymptotic holds for $\mathrm{Re}(w)/\mathrm{Im}(w)\to\infty.$ We can
therefore write

$F(iw)=c+\alpha\sqrt{\frac{\pi}{2}}-\frac{\alpha}{iw}\exp(w^{2}/2)\left[1+O\left(\frac{1}{w^{2}}\right)\right],\quad
w\to\infty.$

Furthermore, note that the solution of the equation

$\frac{\exp(w^{2}/2)}{iw}=q$

has the form

$w=\sqrt{2\log(q)+2\log(i\sqrt{2\log(q)})}+O(\log^{-1}(q))$

for $q$ large enough. Then the function $F(z)$ has zero at the point
$z_{0}=iw_{0}$ with

$\displaystyle w_{0}$
$\displaystyle=\sqrt{2\log((c+\alpha\sqrt{\pi/2})/\alpha)+2\log(i\sqrt{2\log((c+\alpha\sqrt{\pi/2})/\alpha)})}$
$\displaystyle+O(\log^{-1}((c+\alpha\sqrt{\pi/2})/\alpha))$

for $\alpha$ small enough (note that $c\to\sqrt{2\pi}$ for $\alpha\to 0$).
Observe that we can write

${}\int_{\mathbb{R}}\frac{\exp(S(\omega;z))}{F(z)}\,dz=I(w)+\int_{2iw_{0}-\infty}^{2iw_{0}+\infty}\frac{\exp(S(\omega;z))}{F(z)}\,dz,$
(43)

where clearly

$I(w)=\int_{\mathbb{R}}\frac{\exp(S(\omega;z))}{F(z)}\,dz-\int_{2iw_{0}-\infty}^{2iw_{0}+\infty}\frac{\exp(S(\omega;z))}{F(z)}\,dz.$

On $I(w)$ we can use Cauchy integral theorem. Indeed, it can be considered as
an integral over closed contour of as $\exp(-z^{2}/2)\to 0$ as
$|\mathrm{Re}(z)|\to\infty.$ In the strip
$\\{0<\mathrm{Im}(z)<2\mathrm{Re}(w_{0})\\},$ the function
$\exp(S(\omega;z))/F(z)$ has a simple pole in $z_{0}=iw_{0}$ with residue

$\lim_{z\rightarrow
z_{0}}(z-z_{0})\frac{\exp(S(\omega;z))}{F(z)}=\lim_{z\rightarrow
z_{0}}\frac{\exp(S(\omega;z))}{F^{\prime}(z)}=\frac{\exp(-\omega
w_{0}+w_{0}^{2}/2)}{\alpha\exp(w_{0}^{2}/2)}=\alpha^{-1}\exp(-\omega w_{0}).$

Recall that, if a function $f$ is analytic inside a contour $C$ except at some
isolated singularities $a_{1},...,a_{n}$, then Cauchy’s residue theorem
ensures that $\int_{C}f(z)dz=2\pi i\sum_{k=1}^{n}Res(f,a_{k})$. Hence, we
obtain

$I(\omega)=2\pi i\alpha^{-1}\exp(-\omega w_{0}).$

Besides that, the contribution of the second integral in (43) is negligible as

$\int_{2iw_{0}-\infty}^{2iw_{0}+\infty}\frac{\exp(S(\omega;z))}{F(z)}\,dz=\int_{-\infty}^{\infty}\frac{\exp(S(\omega;z+2iw_{0}))}{F(z+2iw_{0})}\,dz=O(\exp(-2\omega
w_{0}))$

since $\inf_{z\in\mathbb{R}}|F(z+2iw_{0})|>0.$ Thus, we deduce that

$\mathcal{F}(\pi)(\omega)=\frac{2\pi i}{\alpha}\exp(-\omega
w_{0}+i\omega\mu)+O(\exp(-2\omega w_{0})).$

##  6 Proofs of technical results

This section is devoted to proofs of our technical lemmas. We start by
providing the proof of Lemma 2.1.

###  6.1 Proof of Lemma 2.1

We can write the invariant density $\pi(x)$ in the following equivalent ways:

$\displaystyle\pi(x)$
$\displaystyle=\frac{1}{Z_{\pi}}\exp(-2V(x)-W\ast\pi(x))$ (44)
$\displaystyle=\frac{1}{Z_{0}}\exp(-2V_{0}(x)-W_{0}\ast\pi(x)),$ (45)

where $Z_{\pi},Z_{0}$ are the normalizing constants and

$\displaystyle V_{0}(x)$
$\displaystyle:=\int_{0}^{x}V^{\prime}(u)du=V(x)-V(0),$ $\displaystyle
W_{0}(x)$ $\displaystyle:=\int_{0}^{x}W^{\prime}(u)du=W(x)-W(0),$

satisfy $W_{0}(0)=V_{0}(0)=0$ and $W_{0}^{\prime}(x)=W^{\prime}(x)$,
$V^{\prime}_{0}(x)=V^{\prime}(x)$ for all $x\in\mathbb{R}$. Note

$W_{0}\ast\pi(x)=W\ast\pi(x)-W(0)$

and

$W_{0}\ast\pi(x)-W_{0}\ast\pi(0)=\int_{0}^{x}W^{\prime}\ast\pi(u)du=W\ast\pi(x)-W\ast\pi(0).$

To obtain the upper and lower bounds on $\pi(x)$ let us use the representation
(45). According to Assumption 1 we have $\|W^{\prime}\|_{1}<\infty$. We deduce
that for all $x$,

$|W_{0}(x)|\leq\int_{0}^{|x|}|W^{\prime}(u)|du\leq\|W^{\prime}\|_{1},$

hence,

$|W_{0}\ast\pi(x)|\leq\|W_{0}\|_{\infty}\|\pi\|_{1}\leq\|W^{\prime}\|_{1}.$

We get that for all $x$,

$\pi(x)\leq\frac{1}{Z_{0}}\exp(-2V_{0}(x)+\|W^{\prime}\|_{1})\eqsim\exp(-2V_{0}(x))$
(46)

and

$\pi(x)\geq\frac{1}{Z_{0}}\exp(-2V_{0}(x)-\|W^{\prime}\|_{1})\eqsim\exp(-2V_{0}(x)).$

As we have assumed that there exists a $C_{V}>0$ such that
$V^{\prime\prime}\geq C_{V}$ we obtain, for all $x\geq 0$,

$\displaystyle V_{0}(x)$
$\displaystyle=\int_{0}^{x}(V^{\prime}(u)-V^{\prime}(0))du+V^{\prime}(0)x$
$\displaystyle\geq\frac{C_{V}}{2}x^{2}+V^{\prime}(0)x$

and

$\displaystyle V_{0}(-x)$ $\displaystyle=\int_{-x}^{0}V^{\prime}(u)du$
$\displaystyle=\int_{-x}^{0}(V^{\prime}(0)-V^{\prime}(u))du-V^{\prime}(0)x$
$\displaystyle\geq\frac{C_{V}}{2}x^{2}-V^{\prime}(0)x.$

In conclusion, for all $x\in\mathbb{R}$, we deduce that

$V_{0}(x)\geq\frac{C_{V}}{2}x^{2}+V^{\prime}(0)x.$ (47)

Recall that $V^{\prime}(0)=0$ under both A1 and A2 as $\widetilde{V}=0$ or
$\widetilde{V}\in C^{2}(\mathbb{R})$ is assumed to be even. The proof about
the upper bound of $\pi$ is therefore concluded.
Regarding the lower bound, we have that

${}\pi(x)\geq c\exp(-\alpha x^{2}-2\widetilde{V}(x)).$ (48)

Observe that, similarly as above, we can take advantage of the fact that
$\tilde{V}^{\prime\prime}\leq\tilde{c}_{2}$ to obtain, for all $x\geq 0$,

$\displaystyle\widetilde{V}(x)-\widetilde{V}(0)$
$\displaystyle=\int_{0}^{x}(\widetilde{V}^{\prime}(u)-\widetilde{V}^{\prime}(0))du+\widetilde{V}^{\prime}(0)x$
$\displaystyle\leq\tilde{c}_{2}\int_{0}^{x}udu=\frac{\tilde{c}_{2}}{2}x^{2},$

having also used that $\widetilde{V}^{\prime}(0)=0$. An analogous reasoning
holds true for $x\leq 0$. It implies that, for any $x\in\mathbb{R}$,
$\widetilde{V}(x)\leq\widetilde{V}(0)+\frac{\tilde{c}_{2}}{2}x^{2}$. Replacing
it in (48) we obtain

$\pi(x)\geq c\exp(-\alpha
x^{2}-\tilde{c}_{2}x^{2}-2\widetilde{V}(0))=c\exp(-\widetilde{C}x^{2})$

with $\widetilde{C}=\alpha+\tilde{c}_{2}$, as we wanted.

Let us move to the proof of the upper bound on the derivatives of $\pi$. More
specifically, we want to prove by induction that for every natural number
$n\leq J$ there exists $c>0$ such that

${}|\pi^{(n)}(x)|\leq c(1+|x|)^{n}\pi(x).$ (49)

Let us begin with the base case $n=1$. Then $\pi=-\varphi^{\prime}\pi$, where
$\varphi^{\prime}:=2V^{\prime}+W^{\prime}\ast\pi$. Note that
$|\varphi^{\prime}(x)|\leq c(|x|+1)$, where $|V^{\prime}(x)|\leq c|x|$ follows
from $\|V^{\prime\prime}\|_{\infty}<\infty$ and $V^{\prime}(0)=0$. Now assume
the claim (49) holds for all natural numbers up to and including $n<J$. Then
the $(n+1)$-th derivative of $\pi$ is

${}\pi^{(n+1)}=(\pi^{\prime})^{(n)}=-\sum_{k=0}^{n}\binom{n}{k}\varphi^{(k+1)}\pi^{(n-k)}.$
(50)

The inductive hypothesis ensures that $|\pi^{(n-k)}(x)|\leq
c(1+|x|)^{(n-k)}\pi(x)$. Moreover, we have that $|\varphi^{(k+1)}(x)|\leq
c(1+|x|)$ since
$\|W^{\prime}\ast\pi^{(k)}\|_{\infty}\leq\|W^{\prime}\|_{1}\|\pi^{(k)}\|_{\infty}<\infty$,
whereas $|V^{(k+1)}(x)|\leq c(1+|x|)$ follows from $\|V^{(j)}\|_{\infty}\leq
c$, $2\leq j\leq J$. We conclude by replacing it in (50), which yields

$\displaystyle|\pi^{(n+1)}(x)|$ $\displaystyle\leq
c\sum_{k=0}^{n}(1+|x|)(1+|x|)^{(n-k)}\pi(x)\leq c(1+|x|)^{n+1}\pi(x)$

as we wanted, for $n+1\leq J$.

###  6.2 Proof of Proposition 2.5

For convenience, we omit the dependency on $N$ in our notation $X^{i,N}_{t}$.
We will prove the claim by applying a Grönwall-type argument to the function

$y(t):=\mathbb{E}[|X^{i}_{t}-\overline{X}^{i}_{t}|^{2p}].$

We define

$\Pi^{N}_{t}:=\frac{1}{N}\sum_{i=1}^{N}\delta_{X^{i}_{t}},\qquad\overline{\Pi}^{N}_{t}:=\frac{1}{N}\sum_{i=1}^{N}\delta_{\overline{X}^{i}_{t}},\qquad\overline{\Pi}_{t}:=\mathcal{L}(\overline{X}_{t}).$

Since $(X^{i}_{t})_{t\geq 0}$ and $(\overline{X}^{i}_{t})_{t\geq 0}$ start at
$X^{i}_{0}=\overline{X}^{i}_{0}$ and are driven by the same Brownian motion
$(B^{i}_{t})_{t\geq 0}$, it holds that

$\displaystyle X^{i}_{t}-\overline{X}^{i}_{t}=$
$\displaystyle-\int_{0}^{t}(V^{\prime}(X^{i}_{s})-V^{\prime}(\overline{X}^{i}_{s})$
$\displaystyle+\frac{1}{2}W^{\prime}\ast\Pi^{N}_{s}(X^{i}_{s})-\frac{1}{2}W^{\prime}\ast\overline{\Pi}_{s}(\overline{X}^{i}_{s}))ds.$

Applying Itô’s formula, summing over $i=1,\dots,N$ and dividing by $N$, yields

$\displaystyle\frac{1}{N}\sum_{i=1}^{N}|X^{i}_{t}-\overline{X}^{i}_{t}|^{2p}=$
$\displaystyle-\frac{2p}{N}\sum_{i=1}^{N}\int_{0}^{t}\left(A_{i}(s)+\frac{1}{2}B_{i}(s)+\frac{1}{2}C_{i}(s)\right)ds,$

where

$\displaystyle A_{i}(s)$
$\displaystyle:=(X^{i}_{s}-\overline{X}^{i}_{s})^{2p-1}(V^{\prime}(X^{i}_{s})-V^{\prime}(\overline{X}^{i}_{s})),$
$\displaystyle B_{i}(s)$
$\displaystyle:=(X^{i}_{s}-\overline{X}^{i}_{s})^{2p-1}(W^{\prime}\ast\Pi^{N}_{s}(X^{i}_{s})-W^{\prime}\ast\overline{\Pi}^{N}_{s}(\overline{X}^{i}_{s})),$
$\displaystyle C_{i}(s)$
$\displaystyle:=(X^{i}_{s}-\overline{X}^{i}_{s})^{2p-1}(W^{\prime}\ast\overline{\Pi}^{N}_{s}(\overline{X}^{i}_{s})-W^{\prime}\ast\overline{\Pi}_{s}(\overline{X}^{i}_{s})).$

Taking the expectation and derivative gives

$y^{\prime}(t)=-\frac{p}{N}\sum_{i=1}^{N}\mathbb{E}[2A_{i}(t)+B_{i}(t)+C_{i}(t)].$

Using the assumption $V^{\prime\prime}\geq C_{V}>0$ and the mean value theorem
gives

$\displaystyle-\mathbb{E}[A_{i}(t)]\leq-
C_{V}\mathbb{E}[|X^{i}_{t}-\overline{X}^{i}_{t}|^{2p}].$

The analysis of $B_{i}(t)$ makes use of the symmetry of $W$ and the
exchangeability of $(X^{i}_{t},\overline{X}^{i}_{t})$, $i=1,\dots,N$. Indeed,
we obtain

$-\mathbb{E}[B_{i}(t)]=-\frac{1}{N}\sum_{j=1}^{N}\mathbb{E}[B_{ij}(t)],$

where

$\displaystyle\mathbb{E}[B_{ij}(t)]$
$\displaystyle=\mathbb{E}[(X^{i}_{t}-\overline{X}^{i}_{t})^{2p-1}(W^{\prime}(X^{i}_{t}-X^{j}_{t})-W^{\prime}(\overline{X}^{i}_{t}-\overline{X}^{j}_{t}))]$
$\displaystyle=\frac{1}{2}\mathbb{E}[((X^{i}_{t}-\overline{X}^{i}_{t})^{2p-1}-(X^{j}_{t}-\overline{X}^{j}_{t})^{2p-1})(W^{\prime}(X^{i}_{t}-X^{j}_{t})-W^{\prime}(\overline{X}^{i}_{t}-\overline{X}^{j}_{t}))].$

By the mean value theorem, the assumption $-W^{\prime\prime}\leq C_{W}$ gives

$\displaystyle-\mathbb{E}[B_{ij}(t)]$
$\displaystyle\leq\frac{C_{W}}{2}\mathbb{E}[((X^{i}_{t}-\overline{X}^{i}_{t})^{2p-1}-(X^{j}_{t}-\overline{X}^{j}_{t})^{2p-1})((X^{i}_{t}-X^{j}_{t})-(\overline{X}^{i}_{t}-\overline{X}^{j}_{t}))]$
$\displaystyle\leq 2C_{W}\mathbb{E}[|X^{i}_{t}-\overline{X}^{i}_{t}|^{2p}],$

hence,

$-\mathbb{E}[B_{i}(t)]\leq
2C_{W}\mathbb{E}[|X^{i}_{t}-\overline{X}^{i}_{t}|^{2p}].$

Hölder’s inequality for $C_{i}(t)$ implies

$-\mathbb{E}[C_{i}(t)]\leq\mathbb{E}[|X^{i}_{t}-\overline{X}^{i}_{t}|^{2p}]^{\frac{{2p-1}}{2p}}R_{i}(t)^{\frac{1}{2p}},$

where

$\displaystyle R_{i}(t):=$
$\displaystyle\mathbb{E}[|W^{\prime}\ast\overline{\Pi}^{N}_{t}(\overline{X}^{i}_{t})-W^{\prime}\ast\overline{\Pi}_{t}(\overline{X}^{i}_{t})|^{2p}]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\left|\frac{1}{N}\sum_{j=1}^{N}W^{\prime}(\overline{X}^{i}_{t}-\overline{X}^{j}_{t})-W^{\prime}\ast\overline{\Pi}_{t}(\overline{X}^{i}_{t})\right|^{2p}\right].$

Expanding this term, we deduce that

$\displaystyle
R_{i}(t)=\frac{1}{N^{2p}}\sum_{j_{1},\dots,j_{2p}=1}^{N}\mathbb{E}\left[\prod_{k=1}^{2p}\mathbb{E}[W^{\prime}(\overline{X}^{i}_{t}-\overline{X}^{j_{k}}_{t})-W^{\prime}\ast\overline{\Pi}_{t}(\overline{X}^{i}_{t})|\overline{X}^{i}_{t}]\right],$

because of the independence of $\overline{X}^{i}_{t}$, $i=1,\dots,N$. The key
remark to study this expectation is that for $i\neq j$,

$\mathbb{E}[W^{\prime}(\overline{X}^{i}_{t}-\overline{X}^{j}_{t})-W^{\prime}\ast\overline{\Pi}_{t}(\overline{X}^{i}_{t})|\overline{X}^{i}_{t}]=0$

since $\mathcal{L}(\overline{X}^{j}_{t})=\overline{\Pi}_{t}$. We observe that
if there exists $k$ such that $j_{k}\neq j_{\widetilde{k}}$
$\forall\widetilde{k}\in\\{1,...,2p\\}\setminus k$, then the $2p$-fold product
vanishes. In other words, in order for a term to contribute to the sum in
$R_{i}(t)$, for every index $j_{k}$ there must be another $j_{\widetilde{k}}$
such that $j_{k}=j_{\widetilde{k}}$. We can have at most $N^{p}$ of these
combinations of indices. We recall we have assumed
$\|W^{\prime}\|_{\infty}<\infty$, which implies
$\mathbb{E}[|W^{\prime}(\overline{X}^{i}_{t}-\overline{X}^{j}_{t})-W^{\prime}\ast\overline{\Pi}_{t}(\overline{X}^{i}_{t})|^{k}]<\infty$
for any $k\in\mathbb{N}$. Therefore, we have

$R_{i}(t)^{\frac{1}{2p}}\lesssim N^{-\frac{1}{2}}.$

In conclusion, by exchangeability,

$\displaystyle-\frac{1}{N}\sum_{i=1}^{N}\mathbb{E}[C_{i}(t)]\lesssim\frac{1}{\sqrt{N}}\left(\frac{1}{N}\sum_{i=1}^{N}\mathbb{E}[|X^{i}_{t}-\overline{X}^{i}_{t}|^{2p}]\right)^{\frac{1}{2p}}.$

Finally, we obtain

$\displaystyle y^{\prime}(t)\leq-2(C_{V}-C_{W})y(t)+\frac{c}{\sqrt{N}}$

and since $y(0)=0$ and $C_{V}-C_{W}>0$, the conclusion follows by integrating
this Grönwall-like differential inequality, which provides $y(t)\leq
c/\sqrt{N}$ uniformly in $t$.

###  6.3 Proof of Lemma 4.1

Observe that for $c>0$ and any random variable $X$,

$\displaystyle{}0<\frac{1}{2}\left(\mathbb{E}[\exp(cX)]+\mathbb{E}[\exp(-cX)]\right)$
$\displaystyle=1+\sum_{k=1}^{\infty}\frac{c^{2k}}{(2k)!}\mathbb{E}[X^{2k}],$
(51)

hence, it suffices to study the asymptotic growth rate of its even moments in
order to show $\mathbb{E}[\exp(\pm cX)]<\infty$.

Let us start by looking at $\mathbb{E}[\exp(\pm cX_{t}^{i,N})]$, which leads
us to study $\mathbb{E}[(X_{t}^{i,N})^{2k}]$ for $k\in\mathbb{N}$. For
convenience, we omit the dependency on $N$ in our notation and recall that the
particles follow the system of SDE’s:

$X^{i}_{t}=X^{i}_{0}+\int_{0}^{t}-V^{\prime}(X^{i}_{s})-\frac{1}{2N}\sum_{j=1}^{N}W^{\prime}(X^{i}_{s}-X^{j}_{s})ds+B^{i}_{t},\quad
i=1,\dots,N.$

Applying the Itô lemma gives

$\displaystyle(X^{i}_{t})^{2k}=(X^{i}_{0})^{2k}+\int_{0}^{t}$
$\displaystyle-2k(X^{i}_{s})^{2k-1}(V^{\prime}(X^{i}_{s})+\frac{1}{2N}\sum_{j=1}^{N}W^{\prime}(X^{i}_{s}-X^{j}_{s}))$
$\displaystyle+\frac{1}{2}(2k)(2k-1)(X^{i}_{s})^{2k-2}ds+2k\int_{0}^{t}(X^{i}_{s})^{2k-1}dB^{i}_{s}.$

Taking the expectation, and then differentiating,

$\displaystyle\frac{d}{dt}\mathbb{E}[(X^{i}_{t})^{2k}]=$
$\displaystyle-2k\mathbb{E}[(X^{i}_{t})^{2k-1}V^{\prime}(X^{i}_{t})]-\frac{k}{N}\sum_{j=1}^{N}\mathbb{E}[(X^{i}_{t})^{2k-1}W^{\prime}(X^{i}_{t}-X^{j}_{t})]$
$\displaystyle+k(2k-1)\mathbb{E}[(X^{i}_{t})^{2k-2}],$

where

$\displaystyle{}-\mathbb{E}[(X^{i}_{t})^{2k-1}V^{\prime}(X^{i}_{t})]$
$\displaystyle=-\mathbb{E}[(X^{i}_{t})^{2k-1}(V^{\prime}(X^{i}_{t})-V^{\prime}(0))]-\mathbb{E}[(X^{i}_{t})^{2k-1}V^{\prime}(0)]$
$\displaystyle\leq-
C_{V}\mathbb{E}[|X^{i}_{t}|^{2k}]+|V^{\prime}(0)|\mathbb{E}[|X^{i}_{t}|^{2k-1}],$
(52)

follows from the strong convexity of $V$, i.e. $V^{\prime\prime}\geq C_{V}>0$,
whereas $\|W^{\prime}\|_{\infty}<\infty$ implies

$-\mathbb{E}[(X^{i}_{t})^{2k-1}W^{\prime}(X^{i}_{t}-X^{j}_{t})]\leq\|W^{\prime}\|_{\infty}\mathbb{E}[|X^{i}_{t}|^{2k-1}].$

We get

$\displaystyle\frac{d}{dt}\mathbb{E}[|X^{i}_{t}|^{2k}]\leq$
$\displaystyle-2kC_{V}\mathbb{E}[|X^{i}_{t}|^{2k}]+2k|V^{\prime}(0)|\mathbb{E}[|X^{i}_{t}|^{2k-1}]+k\|W^{\prime}\|_{\infty}\mathbb{E}[|X^{i}_{t}|^{2k-1}]$
$\displaystyle+k(2k-1)\mathbb{E}[|X^{i}_{t}|^{2k-2}],$ (53)

where

$\displaystyle\mathbb{E}[|X^{i}_{t}|^{2k-l}]$
$\displaystyle=\mathbb{E}[|X^{i}_{t}|^{2k-l}(\mathds{1}(|X^{i}_{t}|\leq
C)+\mathds{1}(|X^{i}_{t}|>C))]$ $\displaystyle\leq
C^{2k-l}+C^{-l}\mathbb{E}[|X^{i}_{t}|^{2k}],\quad l=1,2,$

for all $C>0$. Denote $m_{t}(2k):=\mathbb{E}[|X^{i}_{t}|^{2k}]$. Then it holds
that

$\displaystyle\frac{d}{dt}m_{t}(2k)$
$\displaystyle\leq-2kC_{V}m_{t}(2k)+k(2|V^{\prime}(0)|+\|W^{\prime}\|_{\infty})(C^{2k-1}+C^{-1}m_{t}(2k))$
$\displaystyle\quad+k(2k-1)(C^{2k-2}+C^{-2}m_{t}(2k))$
$\displaystyle=-Ak(m_{t}(2k)-B),$

where

$\displaystyle A$
$\displaystyle:=2C_{V}-(2|V^{\prime}(0)|+\|W^{\prime}\|_{\infty})C^{-1}-(2k-1)C^{-2},$
$\displaystyle B$
$\displaystyle:=A^{-1}((2|V^{\prime}(0)|+\|W^{\prime}\|_{\infty})C^{2k-1}+(2k-1)C^{2k-2})$

do not depend on $t$. For some fixed $\varepsilon\in(0,1)$, set

$C=C(k):=\left(\frac{(2k-1)}{2C_{V}\varepsilon}\right)^{\frac{1}{2}},\quad
k\in\mathbb{N},$

so that

$\displaystyle A=A(k)\sim 2C_{V}(1-\varepsilon)>0,\qquad
B=B(k)\sim\frac{2C_{V}\varepsilon C(k)^{2k}}{A(k)},$

where $\sim$ denotes asymptotic equality as $k\to\infty$. The Grönwall
inequality

$\frac{d}{dt}m_{t}(2k)=\frac{d}{dt}(m_{t}(2k)-B(k))\leq-A(k)k(m_{t}(2k)-B(k))$

gives

$\displaystyle m_{t}(2k)\leq B(k)+(m_{0}(2k)-B(k))e^{-A(k)kt},$

which in turn implies that, for all large enough $k$, uniformly in $t$,

$m_{t}(2k)\leq\max(m_{0}(2k),B(k)).$

Based on (51) and the subsequent analysis, we conclude that $\sup_{t\geq
0}\mathbb{E}[\exp(\pm cX_{t}^{i})]$ is bounded as long as

$\sup_{t\geq
0}\sum_{k=1}^{\infty}\frac{c^{2k}}{(2k)!}m_{t}(2k)\lesssim\sum_{k=1}^{\infty}\frac{c^{2k}}{(2k)!}(m_{0}(2k)+B(k))$

is also bounded. We observe that $\sum_{k=1}^{\infty}c^{2k}m_{0}(2k)/(2k)!$
converges because of (51) and our assumption $\mathbb{E}[\exp(\pm
cX_{0}^{i})]<\infty$. To test the convergence of the series
$\sum_{k=1}^{\infty}c^{2k}B(k)/(2k)!$ we use the ratio test also known as
d’Alembert’s criterion:

$\displaystyle\frac{c^{2k+2}B(k+1)/(2k+2)!}{c^{2k}B(k)/(2k)!}$
$\displaystyle\sim\frac{c^{2}C(k+1)^{2k+2}}{C(k)^{2k}(2k+1)(2k+2)}$
$\displaystyle=\frac{c^{2}(2k+1)^{k+1}}{(2C_{V}\varepsilon)(2k-1)^{k}(2k+1)(2k+2)}\frac{}{}\to
0$

as $k\to\infty$. The result for the exponential moments of $X_{t}^{i}$
follows.

One can follow the preceding proof to get $\sup_{t\geq 0}\mathbb{E}[\exp(\pm
c\overline{X}_{t})]<\infty$ from $\mathbb{E}[\exp(\pm
c\overline{X}_{0})]<\infty$, where recall $(\overline{X}_{t})_{t\geq 0}$ is a
solution of

$\overline{X}_{t}=\overline{X}_{0}-\int_{0}^{t}V^{\prime}(\overline{X}_{s})+\frac{1}{2}W^{\prime}\ast\overline{\Pi}_{s}(\overline{X}_{s})ds+B_{t}$

with $\overline{\Pi}_{t}:=\mathcal{L}(\overline{X}_{t})$. Indeed, Itô’s lemma
for $(\overline{X}_{t})^{2k}$ gives

$\displaystyle\frac{d}{dt}\mathbb{E}[(\overline{X}_{t})^{2k}]=$
$\displaystyle-2k\mathbb{E}[(\overline{X}_{t})^{2k-1}(V^{\prime}(\overline{X}_{t})+\frac{1}{2}W^{\prime}\ast\overline{\Pi}_{t}(\overline{X}_{t})]$
$\displaystyle+k(2k-1)\mathbb{E}[(\overline{X}_{t})^{2(k-1)}],$

where, in the same manner as in (6.3), we have

$-\mathbb{E}[(\overline{X}_{t})^{2k-1}V^{\prime}(\overline{X}_{t})]\leq-
C_{V}\mathbb{E}[|\overline{X}_{i}|^{2k}]+|V^{\prime}(0)|\mathbb{E}[|\overline{X}^{i}_{t}|^{2k-1}],$

and since $\|W^{\prime}\|_{\infty}<\infty$, we get

$\displaystyle-\mathbb{E}[(\overline{X}_{t})^{2k-1}W^{\prime}\ast\overline{\Pi}_{t}(\overline{X}_{t})]\leq\|W^{\prime}\|_{\infty}\mathbb{E}[|\overline{X}_{t}|^{2k-1}].$

Thus, we obtain the inequality (53) for $\mathbb{E}[|\overline{X}_{t}|^{2k}]$
instead of $\mathbb{E}[|X^{i}_{t}|^{2k}]$. The rest of the proof remains the
same.

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