# Precise high moment asymptotics for parabolic Anderson model with log-
correlated Gaussian field

Yangyang Lyua,111E-mail address: lvyy16@mails.jlu.edu.cn(Yangyang Lyu).
_$a$ School of Mathematics, Jilin University, Changchun 130012, China_
Correspondence to: School of Mathematics, Jilin University, Changchun 130012,
China.

Abstract In this paper, we consider the continuous parabolic Anderson model
(PAM) driven by a time-independent log-correlated Gaussian field (LGF). We
obtain an asymptotic result of

$\mathbb{E}\exp\Bigg{\\{}\frac{1}{2}\sum\limits_{j,k=1}^{N}\int_{0}^{t}\int_{0}^{t}\gamma(B_{j}(s)-B_{k}(r))drds\Bigg{\\}}\qquad(N\rightarrow\infty)$

which is composed of the independent Brownian motions $\\{B_{j}(s)\\}$ and the
function $\gamma$ approximating to a logarithmic potential at $0$, such as the
covariances of massive free field and Bessel field. Based on the asymptotic
result, we get the precise high moment asymptotics for Feynman-Kac formula of
the PAM with LGF.

Keywords Precise high moment asymptotics; Large deviation; Log-correlated
Gaussian field; Massive free field; Bessel field

##  1 Introduction

In this paper, we consider the continuous parabolic Anderson model (PAM)

$\displaystyle\qquad\qquad\quad\left\\{\begin{array}[]{l}\frac{\partial
u}{\partial t}(t,x)=\frac{1}{2}\triangle
u(t,x)+u(t,x)V(x),\qquad(t,x)\in\mathrm{R}_{+}\times\mathrm{R}^{d},\\\
u(0,x)=u_{0}(x),\qquad x\in\mathrm{R}^{d},\end{array}\right.$ (3)

with the log-correlated Gaussian field $V(x)$ on $\mathrm{R}^{d}$, which is
given by a centered Gaussian family $\\{\langle
V,\varphi\rangle;\varphi\in\mathcal{S}(\mathrm{R}^{d})\\}$ with covariance

$\displaystyle\qquad\qquad\qquad\mathbb{E}[\langle V,\phi\rangle\langle
V,\psi\rangle]=\int_{\mathrm{R}^{d}\times\mathrm{R}^{d}}\phi(x)\psi(y)k(x,y)dxdy\qquad\forall\phi,\psi\in\mathcal{S}(\mathrm{R}^{d}).$
(4)

Here, $\mathcal{S}(\mathrm{R}^{d})$ is Schwartz space, $k(x,y)$ is a positive
definite kernel on $\mathrm{R}^{d}\times\mathrm{R}^{d}$ and

$\displaystyle k(x,y)=\log_{+}\frac{T}{|x-y|}+g(x,y),$ (5)

where $\log_{+}x=(\log x)\vee 0$, $\log$ is the natural logarithm, the
constant $T>0$ is called correlation length and $g(x,y)$ is a bounded function
on $\mathrm{R}^{d}\times\mathrm{R}^{d}$. For the details of LGF, see e.g. [19,
20, 21, 22]. Throughout the paper, we assume that the initial value $u_{0}(x)$
satisfies

$\displaystyle
0<\inf\limits_{x\in\mathrm{R}^{d}}u_{0}(x)\leq\sup\limits_{x\in\mathrm{R}^{d}}u_{0}(x)<+\infty.$
(6)

Because Stratonovich integral is closer to “physical integral” than Skorokhod
integral, we interpret the $uV$ in (3) in the Stratonovich’s sense. For the
details of Stratonovich and Skorokhod integral, see e.g. [11].

There are many achievements for the continuous PAM with other random fields,
such as Poisson potential, fractional noise, etc. (see e.g.[5, 6, 7, 12, 13]).
In view of the fruitful developments of LGF in quantum field theory, Wolfgang
König proposed the study of the PAM with LGF in his book [14]. As far as we
know, there are few results for the high moment asymptotics of solutions to
the model.

The precise high moment asymptotic problem for the solution $u(t,x)$ is to
find a suitable rate $\sigma(N)$ about positive integer $N$ such that the
quantity $\frac{1}{\sigma(N)}\log\mathbb{E}u^{N}(t,x)$ converges to a nonzero
constant as $N\rightarrow\infty$. To some extent, it is meaningful to study
the problem in physics. One hand, in [9, 10], the precise high moment
asymptotics is a key step in the process of computing spatial asymptotics. As
explained in [8], spatial asymptotics is one of the problems of “physical
intermittency” which is applied to describing the absence of diffusion of
waves in a disordered medium in physics. On the other hand, based on the
precise high moment asymptotic result, the precise asymptotics of tail
probability of $\log u(t,x)$ can be obtained by nonnegative large deviation
technique in [15]. Here, the $\log u(t,x)$ represents the free energy of
directed Brownian polymer in a continuous random environment. The high moment
asymptotic problem has attracted a lot of attention. There have been many
results for precise high moment asymptotics when the covariances of driven
Gaussian fields are homogeneous in time and space, or locally integrable in
time and bounded in space, see [1, 9, 10, 13, 15, 16] as references.

By the following (10) and the same method as [11], it’s trivial to prove that
if the Feynman-Kac formula of equation (3) exists then it is a mild solution
in the Stratonovich’s sense. However, according to [11, 12], the uniqueness of
the mild and weak solutions of PAM can’t be proved in the Stratonovich’s sense
now. Thus, this paper is only concerned with the precise high moment
asymptotics of the Feynman-Kac formula of equation (3). We remark that, in the
Young’s sense, it can be proved that the Feynman-Kac formula is the unique
mild solution for PAM with the Gaussian fields in [11, 25]. The similar result
might be also extended to the PAM with LGF, which will be our future work.

According to [12], the Feynman-Kac formula for the solution to equation (3) is

$\displaystyle
u(t,x)=\mathbb{E}^{B}\left[\exp\left\\{\int_{0}^{t}V(B^{x}(s))ds\right\\}u_{0}(B^{x}(t))\right],$
(7)

where $B^{x}$ is a $d$-dimensional Brownian motion independent of $V$ and
starting at $x\in\mathrm{R}^{d}$ and $\mathbb{E}^{B}$ is the expectation with
respect to the standard Brownian motion $B$. For all $\varepsilon>0$, let
$p_{\varepsilon}(x):=(2\pi\varepsilon)^{-\frac{d}{2}}\exp\\{-\frac{1}{2\varepsilon}|x|^{2}\\}$
and $V_{\varepsilon}(x):=\langle V(\cdot),p_{\varepsilon}(x-\cdot)\rangle$,
then for all $t>0$, the time integral in (7) is defined as the $L^{2}$-limit

$\displaystyle\int_{0}^{t}V(B(s))ds:=\lim\limits_{\varepsilon\downarrow
0}\int_{0}^{t}V_{\varepsilon}(B(s))ds-\mbox{
$L^{2}(\Omega,\mathcal{F},\mathbb{P})$}.$ (8)

It’s easy to get that (8) is well defined by (5). Conditioning on the Brownian
motion, the $\int_{0}^{t}V(B(s))ds$ is a centered Gaussian process with the
conditional covariance $\int_{0}^{t}\int_{0}^{t}k(B(s),B(r))dsdr$.
Furthermore, by reference to [12], the Feynman-Kac moment representation of
(7) is

$\displaystyle\mathbb{E}u^{N}(t,x)=\mathbb{E}\Bigg{[}\exp\Bigg{\\{}\frac{1}{2}\sum\limits_{j,k=1}^{N}\int_{0}^{t}\int_{0}^{t}k(B_{j}^{x}(s),B_{k}^{x}(r))drds\Bigg{\\}}\prod\limits_{j=1}^{N}u_{0}(B_{j}^{x}(t))\Bigg{]},$
(9)

where $\\{B_{j}^{x};j=1,\cdots,N\\}$ is a family of $d$-dimensional
independent B.M. starting at $x$. By reference to [12], the existence of (7)
and (9) is provided by the exponential integrability of
$\int_{0}^{t}\int_{0}^{t}k(B(s),B(r))dsdr$. In fact, by (5), there exists some
$\alpha\in(0,1)$ and $C>0$ such that

$\displaystyle|k(x,y)|\leq|x-y|^{-\alpha}+C,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
(10)
$\displaystyle\mbox{then\qquad\qquad}\mathbb{E}\exp\left\\{\int_{0}^{t}\int_{0}^{t}k(B(s),B(r))dsdr\right\\}\leq
e^{Ct^{2}}\mathbb{E}\exp\left\\{\int_{0}^{t}\int_{0}^{t}|B(s)-B(r)|^{-\alpha}dsdr\right\\}.\qquad\qquad$
(11)

Hence, the exponential integrability of
$\int_{0}^{t}\int_{0}^{t}k(B(s),B(r))dsdr$ is transformed into that of the
right-hand side in (11). To explain the exponential integrability of
$\int_{0}^{t}\int_{0}^{t}|B(s)-B(r)|^{-\alpha}dsdr$ and prove the following
Proposition 2.1, we need to introduce a general framework. Let $\gamma$ be the
Fourier transform of the tempered measure $\mu$ in
$\mathcal{S}^{\prime}(\mathrm{R}^{d})$, i.e.

$\displaystyle\qquad\qquad\qquad\langle\gamma,\phi\rangle=\int_{\mathrm{R}^{d}}\mathcal{F}(\phi)(\xi)\mu(d\xi),\qquad\qquad\forall\phi\in\mathcal{S}(\mathrm{R}^{d}).$
(12)

Here, $\mathcal{F}(\phi)(\xi):=\int_{\mathrm{R}^{d}}e^{i\xi\cdot x}\phi(x)dx$
is the Fourier transform of $\phi$ on $\mathrm{R}^{d}$. For all
$\varepsilon>0$, let
$\mu_{\varepsilon}(d\xi):=\exp\left\\{-\frac{\varepsilon}{2}|\xi|^{2}\right\\}\mu(d\xi)$
and $\gamma_{\varepsilon}(x):=\int_{\mathrm{R}^{d}}e^{i\xi\cdot
x}\mu_{\varepsilon}(d\xi)$, then we define

$\displaystyle\int_{0}^{t}\int_{0}^{t}\gamma(B(s)-B(r))dsdr:=\lim\limits_{\varepsilon\downarrow
0}\int_{0}^{t}\int_{0}^{t}\gamma_{\varepsilon}(B(s)-B(r))dsdr-\mbox{
$L^{1}(\Omega,\mathcal{F},\mathbb{P})$}.$ (13)

By (13) and Proposition 4.4 in [25], it can be obtained that the Dalang’s
condition $\int_{\mathrm{R}^{d}}\frac{1}{1+|\xi|^{2}}\mu(d\xi)<+\infty$ is
equivalent to

$\displaystyle\mathbb{E}\exp\left\\{\int_{0}^{t}\int_{0}^{t}\gamma(B(s)-B(r))dsdr\right\\}<+\infty.$
(14)

Becasue the Riesz potential $|\cdot|^{-\alpha}$ in (11) satisfies the Dalang’s
condition, $\int_{0}^{t}\int_{0}^{t}k(B(s),B(r))dsdr$ is exponentially
integrable. Hence, (7) and (9) are well defined. Based on it, we obtain the
following main result about the precise high moment asymptotics for the
Feynman-Kac formula (7) of the PAM with LGF.

######  Theorem 1.1.

For the Feynman-Kac formula (7), it has

$\displaystyle\lim\limits_{N\rightarrow\infty}\frac{1}{N\lambda_{N}}\log\mathbb{E}u^{N}(t,x)=\frac{t^{2}}{2}.$
(15)

Here, $\lambda_{N}$ is a function about positive integer $N$, which satisfies
$\lambda_{N}>e$ and $\frac{\log\lambda_{N}}{\lambda_{N}}=\frac{2}{N}$ when $N$
is enough large.

######  Remark 1.1.

There is usually no explicit solution for the equation
$\frac{\log\lambda_{N}}{\lambda_{N}}=\frac{2}{N}$. When $N$ is enough large,
the equation has two implicit solutions, and the $\lambda_{N}$ is unique if we
assume $\lambda_{N}>e$. Moreover, the $\lambda_{N}$ is a regularly varying
function about $N$ and satisfies $\frac{1}{\lambda_{N}}=o(\frac{1}{N})$ as
$N\rightarrow\infty$.

The paper is organized as follows. In Section 2, we prove Proposition 2.1
which is applied to proving Proposition 3.1. In Section 3, we prove
Proposition 3.1 which is an asymptotic result for the function approximating
to a logarithmic potential at $0$. Based on Proposition 3.1, Section 4 is the
proof of Theorem 1.1.

##  2 A general limit in the time-independent case

To prove Proposition 3.1, we need the following Proposition 2.1 which is a
general result for positive definite function in the time-independent case. In
view of the time-space case, i.e.
$\sum_{j,k=1}^{N}\int_{0}^{t}\int_{0}^{t}\gamma_{0}(s-r)\gamma(B_{j}(s)-B_{k}(r))dsdr$,
we call
$\sum_{j,k=1}^{N}\int_{0}^{t}\int_{0}^{t}\gamma(B_{j}(s)-B_{k}(r))dsdr$ and
$\sum_{1\leq j\neq k\leq N}\int_{0}^{t}\gamma(B_{j}(s)-B_{k}(s))ds$ time-
independent case ($\gamma_{0}\equiv 1$) and time-white case
($\gamma_{0}=\delta_{0}$, i.e. Dirac delta function at $0$), respectively.
Before Proposition 2.1, the time-white case has been considered in [1] . When
the $\mu$ is finite in Proposition 2.1, our method is partly inspired by [1]
and we use the semigroup inequality in [2]. When the $\mu$ is unbounded, we
can’t transform the time-independent case into the time-white case because the
$\sum_{1\leq j\neq k\leq N}\int_{0}^{t}\gamma(B_{j}(s)-B_{k}(s))ds$ may be not
exponentially integrable.

######  Proposition 2.1.

Assume that the $\gamma$ in (12) satisfies the Dalang’s condition, then for
any sequence $\\{t_{N}\\}_{N\in\mathbb{N}}$ satisfying
$t_{N}\rightarrow+\infty$ as $N\rightarrow\infty$, it holds that

$\displaystyle\lim\limits_{N\rightarrow\infty}\frac{1}{Nt_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{1}{2Nt_{N}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}$
(16)
$\displaystyle=\frac{1}{2}\sup\limits_{g\in\mathcal{F}_{d}}\Bigg{\\{}\int_{\mathrm{R}^{d}}\Bigg{|}\int_{\mathrm{R}^{d}}e^{i\xi\cdot
x}g^{2}(x)dx\Bigg{|}^{2}\mu(d\xi)-\int_{\mathrm{R}^{d}}|\nabla
g(x)|^{2}dx\Bigg{\\}}:=\mathcal{E}(\gamma)$ (17)

and the variation $\mathcal{E}(\gamma)<+\infty$, where we define the set of
functions $\mathcal{F}_{d}:=\left\\{g\in
W^{1,2}(\mathrm{R}^{d});\|g\|_{2}=1\right\\}$.

To prove Proposition 2.1, we need the following two Lemmas.

######  Lemma 2.1.

For any $t_{1},t_{2}>0$ and positive integer $m$, it holds that

$\displaystyle~{}\mathbb{E}\exp\Bigg{\\{}\frac{1}{t_{1}+t_{2}}\sum_{j,k=1}^{m}\int_{0}^{t_{1}+t_{2}}\int_{0}^{t_{1}+t_{2}}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle~{}\leq\mathbb{E}\exp\Bigg{\\{}\frac{1}{t_{1}}\sum_{j,k=1}^{m}\int_{0}^{t_{1}}\int_{0}^{t_{1}}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}\cdot\mathbb{E}\exp\Bigg{\\{}\frac{1}{t_{2}}\sum_{j,k=1}^{m}\int_{0}^{t_{2}}\int_{0}^{t_{2}}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}.$
(18)

proof: By the approximation in (13) and Jensen’s inequality, the following
inequality is almost surely established.

$\displaystyle\frac{1}{t_{1}+t_{2}}\int_{\mathrm{R}^{d}}\Bigg{|}\sum_{j=1}^{m}\int_{0}^{t_{1}+t_{2}}e^{i\xi\cdot
B_{j}(s)}ds\Bigg{|}^{2}\mu(d\xi)$
$\displaystyle\leq\frac{1}{t_{1}}\int_{\mathrm{R}^{d}}\Bigg{|}\sum_{j=1}^{m}\int_{0}^{t_{1}}e^{i\xi\cdot
B_{j}(s)}ds\Bigg{|}^{2}\mu(d\xi)+\frac{1}{t_{2}}\int_{\mathrm{R}^{d}}\Bigg{|}\sum_{j=1}^{m}\int_{t_{1}}^{t_{1}+t_{2}}e^{i\xi\cdot
B_{j}(s)}ds\Bigg{|}^{2}\mu(d\xi).$ (19)

Moreover, by (19) and strong Markov property, we have

$\displaystyle\mathbb{E}\exp\Bigg{\\{}\frac{1}{t_{1}+t_{2}}\sum_{j,k=1}^{m}\int_{0}^{t_{1}+t_{2}}\int_{0}^{t_{1}+t_{2}}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}$
$\displaystyle\leq\mathbb{E}\Bigg{[}\exp\Bigg{\\{}\frac{1}{t_{1}}\sum_{j,k=1}^{m}\int_{0}^{t_{1}}\int_{0}^{t_{1}}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}$
$\displaystyle\cdot\mathbb{E}\Bigg{[}\exp\Bigg{\\{}\frac{1}{t_{2}}\sum_{j,k=1}^{m}\int_{t_{1}}^{t_{1}+t_{2}}\int_{t_{1}}^{t_{1}+t_{2}}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}\Bigg{|}\mathfrak{F}_{t_{1}}\Bigg{]}\Bigg{]}$
$\displaystyle\leq\mathbb{E}\exp\Bigg{\\{}\frac{1}{t_{1}}\sum_{j,k=1}^{m}\int_{0}^{t_{1}}\int_{0}^{t_{1}}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}$
$\displaystyle\cdot\Bigg{(}\sup\limits_{\tilde{b}}\mathbb{E}^{\tilde{b}}\exp\Bigg{\\{}\frac{1}{t_{2}}\sum_{j,k=1}^{m}\int_{0}^{t_{2}}\int_{0}^{t_{2}}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}\Bigg{)},$
(20)

where $\tilde{b}=(b_{1},\cdots,b_{m})\in(R^{d})^{m}$ and
$\mathfrak{F}_{t_{1}}$ is the $\sigma$-algebra up to $t_{1}$. We claim that
for any $t>0$, it holds that

$\displaystyle\sup\limits_{\tilde{b}}\mathbb{E}^{\tilde{b}}\exp\Bigg{\\{}\frac{1}{t}\sum_{j,k=1}^{m}\int_{0}^{t}\int_{0}^{t}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}\leq\mathbb{E}\exp\Bigg{\\{}\frac{1}{t}\sum_{j,k=1}^{m}\int_{0}^{t}\int_{0}^{t}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}.$
(21)

Indeed, by Taylor expansion and Bochner’s representation, for all integer
$n\geq 1$ and $\tilde{b}\in(R^{d})^{m}$, we have

$\displaystyle\mathbb{E}^{\tilde{b}}\Bigg{[}\frac{1}{t}\sum_{j,k=1}^{m}\int_{0}^{t}\int_{0}^{t}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{]}^{n}$
$\displaystyle=\frac{1}{t^{n}}\int_{\mathrm{R}^{dn}}\int_{[0,t]^{2n}}\sum_{j_{1},\cdots,j_{n}=1\atop
k_{1},\cdots,k_{n}=1}^{m}\mathbb{E}\prod\limits_{l=1}^{n}\mathrm{e}^{i\xi_{l}\cdot(b_{k_{l}}-b_{j_{l}})}\cdot\mathrm{e}^{i\xi_{l}\cdot(B_{k_{l}}(s_{l})-B_{j_{l}}(r_{l}))}\prod\limits_{l=1}^{n}ds_{l}dr_{l}\mu(d\xi_{l})$
$\displaystyle\leq\frac{1}{t^{n}}\int_{\mathrm{R}^{dn}}\int_{[0,t]^{2n}}\sum_{j_{1},\cdots,j_{n}=1\atop
k_{1},\cdots,k_{n}=1}^{m}\mathbb{E}\mathrm{e}^{i\sum\limits_{l=1}^{n}\xi_{l}\cdot(B_{k_{l}}(s_{l})-B_{j_{l}}(r_{l}))}\prod\limits_{l=1}^{n}ds_{l}dr_{l}\mu(d\xi_{l})$
$\displaystyle=\mathbb{E}\Bigg{[}\frac{1}{t}\sum_{j,k=1}^{m}\int_{0}^{t}\int_{0}^{t}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{]}^{n}.$
(22)

Then, by (20) and (21), the proof is completed.

The following Lemma is a direct result of Lemma 2.1.

######  Lemma 2.2.

For any positive integer $m$, $n$ and $t>0$, it holds that

$\displaystyle\mathbb{E}\exp\Bigg{\\{}\frac{1}{n}\sum_{j,k=1}^{m}\int_{0}^{nt}\int_{0}^{nt}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}\leq\Bigg{(}\mathbb{E}\exp\Bigg{\\{}\sum_{j,k=1}^{m}\int_{0}^{t}\int_{0}^{t}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}\Bigg{)}^{n}.$
(23)

The proof of Proposition 2.1: Step 1. We prove (17) in the case that $\mu$ is
finite. Firstly, we prove the lower bound of (17). When $\mu$ is finite, by
Bochner’s representation, we have

$\displaystyle\mathbb{E}\exp\Bigg{\\{}\frac{1}{2Nt_{N}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}$
$\displaystyle=\mathbb{E}\exp\Bigg{\\{}\frac{Nt_{N}}{2}\int_{\mathrm{R}^{d}}\Bigg{|}\frac{1}{Nt_{N}}\sum\limits_{k=1}^{N}\int_{0}^{t_{N}}\mathrm{e}^{i\xi\cdot
B_{k}(s)}ds\Bigg{|}^{2}\mu(d\xi)\Bigg{\\}}.$ (24)

Let $\mathcal{H}$ be a subspace of complex-value Hilbert space
$L^{2}(\mathrm{R}^{d},\mu)$ satisfying $f(-x)=\overline{f(x)}$ and $\langle
f,h\rangle_{\mathcal{H}}:=\int_{\mathrm{R}^{d}}f(\xi)\overline{h(\xi)}\mu(d\xi)$
for all $f,h\in\mathcal{H}$. Then, by (24), $\|h\|_{\mathcal{H}}^{2}\geq
2\langle f,h\rangle_{\mathcal{H}}-\|f\|_{\mathcal{H}}^{2}$ for all
$f,h\in\mathcal{H}$ and Theorem 4.1.6 in [2], we have

$\displaystyle\liminf\limits_{N\rightarrow\infty}\frac{1}{Nt_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{1}{2Nt_{N}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}$
$\displaystyle\geq\sup\limits_{f\in\mathcal{H}}\left\\{\liminf\limits_{N\rightarrow\infty}\frac{1}{t_{N}}\log\mathbb{E}\exp\left\\{\int_{0}^{t_{N}}f^{*}(B(s))ds\right\\}-\frac{1}{2}\|f\|_{\mathcal{H}}^{2}\right\\},$
(25)
$\displaystyle=\sup\limits_{f\in\mathcal{H}}\sup\limits_{g\in\mathcal{F}_{d}}\left\\{\int_{\mathrm{R}^{d}}f^{*}(x)g^{2}(x)dx-\frac{1}{2}\int_{\mathrm{R}^{d}}|\nabla
g(x)|^{2}dx-\frac{1}{2}\|f\|_{\mathcal{H}}^{2}\right\\}$
$\displaystyle=\sup\limits_{g\in\mathcal{F}_{d}}\sup\limits_{f\in\mathcal{H}}\left\\{\langle
f,\mathcal{F}(g^{2})\rangle_{\mathcal{H}}-\frac{1}{2}\|f\|_{\mathcal{H}}^{2}-\frac{1}{2}\int_{\mathrm{R}^{d}}|\nabla
g(x)|^{2}dx\right\\}=\mathcal{E}(\gamma),$ (26)

where $f^{*}(x)=\int_{\mathrm{R}^{d}}e^{i\xi\cdot x}f(-\xi)\mu(d\xi)$ and the
last equality is due to
$\|h\|_{\mathcal{H}}^{2}=\sup\limits_{f\in\mathcal{H}}\\{2\langle
f,h\rangle_{\mathcal{H}}-\|f\|_{\mathcal{H}}^{2}\\}$.

Secondly, we prove the upper bound of (17) when $\mu$ is finite. We first
assume that $t_{N}/t$ is always an integer for all positive integer $N$ and
fixed $t>0$. By Lemma 2.2 and Bochner’s representation, it holds that

$\displaystyle\frac{1}{Nt_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{1}{2Nt_{N}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}$
$\displaystyle\leq\frac{1}{Nt}\log\mathbb{E}\exp\Bigg{\\{}\frac{Nt}{2}\int_{\mathrm{R}^{d}}\left|\frac{1}{N}\sum\limits_{k=1}^{N}\frac{1}{t}\int_{0}^{t}\mathrm{e}^{i\xi\cdot
B_{k}(s)}ds\right|^{2}\mu(d\xi)\Bigg{\\}}.$ (27)

For all positive integer $1\leq k\leq N$, define
$X_{k}:=\frac{1}{t}\int_{0}^{t}\mathrm{e}^{i\xi\cdot B_{k}(s)}ds$ and
$X\stackrel{{\scriptstyle d}}{{=}}X_{k}$. Let $\left\\{X,X_{k}\right\\}_{1\leq
k\leq N}$ be a sequence of i.i.d $\mathcal{H}$-valued random variables.
Apparently, the $X$ satisfies the exponential moment condition:
$\mathbb{E}\exp\left\\{\theta\left\|X\right\|_{\mathcal{H}}\right\\}<\infty$
for all $\theta>0.$ Let Cramér function and its Legendre transform be

$\Lambda(y):=\log\mathbb{E}\exp\left\\{\left\langle
y,X\right\rangle_{\mathcal{H}}\right\\}\quad\forall
y\in\mathcal{H}\quad\mbox{and}\quad\Lambda^{*}(x):=\sup\limits_{y\in\mathcal{H}}\left\\{\langle
y,x\rangle_{\mathcal{H}}-\Lambda(y)\right\\}\quad\forall x\in\mathcal{H},$

respectively. By Donsker-Varadhan’s large deviation principle (LDP) in a
separable Banach space (Theorem 5.3,[3]), the distributions
$\Big{\\{}\mathbb{P}\Big{(}\frac{1}{N}\sum\limits_{k=1}^{N}X_{k}\in\cdot\Big{)}\Big{\\}}_{N\in\mathbb{N}}$
satisfy LDP in $\mathcal{H}$ with the rate function $\Lambda^{*}(x)$. By
$\|X_{k}\|_{\mathcal{H}}^{2}\leq\mu(\mathrm{R}^{d})$ for all $1\leq k\leq N$
and Varadhan integral Lemma (theorem 4.3.1,[4]), we have

$\displaystyle\lim\limits_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{E}\exp\Bigg{\\{}\frac{Nt}{2}\int_{\mathrm{R}^{d}}\Bigg{|}\frac{1}{N}\sum\limits_{k=1}^{N}\frac{1}{t}\int_{0}^{t}\mathrm{e}^{i\xi\cdot
B_{k}(s)}ds\Bigg{|}^{2}\mu(d\xi)\Bigg{\\}}=\sup\limits_{h\in\mathcal{H}}\left\\{\frac{t}{2}\|h\|_{\mathcal{H}}^{2}-\Lambda^{*}(h)\right\\}.$
(28)

Here, we claim that for all $t>0$, it holds that

$\displaystyle\sup\limits_{h\in\mathcal{H}}\left\\{\frac{t}{2}\|h\|_{\mathcal{H}}^{2}-\Lambda^{*}(h)\right\\}=\sup\limits_{\|h\|_{\mathcal{H}}^{2}\leq\mu(\mathrm{R}^{d})}\left\\{\frac{t}{2}\|h\|_{\mathcal{H}}^{2}-\Lambda^{*}(h)\right\\}.$
(29)

In fact, the $\frac{1}{N}\sum\limits_{k=1}^{N}X_{k}$ satisfies lower bound of
LDP in $\mathcal{H}$. For the open set
$[\|\cdot\|_{\mathcal{H}}^{2}>\mu(\mathrm{R}^{d})]$ of $\mathcal{H}$, we have

$\displaystyle\liminf\limits_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{P}\Bigg{(}\Bigg{\|}\frac{1}{N}\sum\limits_{k=1}^{N}X_{k}\Bigg{\|}_{\mathcal{H}}^{2}>\mu(\mathrm{R}^{d})\Bigg{)}\geq-\inf\limits_{\|h\|_{\mathcal{H}}^{2}>\mu(\mathrm{R}^{d})}\Lambda^{*}(h).$
(30)

By (30) and the fact
$\Big{\|}\frac{1}{N}\sum\limits_{k=1}^{N}X_{k}\Big{\|}_{\mathcal{H}}^{2}\leq\mu(\mathrm{R}^{d})$,
we get $\Lambda^{*}(h)=+\infty$ on the set
$\\{h\in\mathcal{H};\|h\|_{\mathcal{H}}^{2}>\mu(\mathrm{R}^{d})\\}$. Hence,
the quantity in (29) reaches the maximum on the set
$\\{h\in\mathcal{H};\|h\|_{\mathcal{H}}^{2}\leq\mu(\mathrm{R}^{d})\\}$. We
notice that, for all $h\in\mathcal{H}$ and
$\|h\|_{\mathcal{H}}^{2}\leq\mu(\mathrm{R}^{d})$, the linearization of $h$ can
be rewritten as

$\displaystyle\|h\|_{\mathcal{H}}^{2}=\sup\limits_{\|f\|_{\mathcal{H}}^{2}\leq\mu(\mathrm{R}^{d})}\\{2\langle
f,h\rangle_{\mathcal{H}}-\|f\|_{\mathcal{H}}^{2}\\},$ (31)

which is due to that the quantity in (31) attains the maximum when $f=h$. By
(29), (31) and Legendre transform, we get

$\displaystyle\sup\limits_{\|h\|_{\mathcal{H}}^{2}\leq\mu(\mathrm{R}^{d})}\left\\{\frac{t}{2}\|h\|_{\mathcal{H}}^{2}-\Lambda^{*}(h)\right\\}$
$\displaystyle=$
$\displaystyle\sup\limits_{\|h\|_{\mathcal{H}}^{2}\leq\mu(\mathrm{R}^{d})}\Bigg{\\{}\sup\limits_{\|f\|_{\mathcal{H}}^{2}\leq\mu(\mathrm{R}^{d})}\left\\{-\frac{t}{2}\|f\|_{\mathcal{H}}^{2}+t\langle
f,h\rangle_{\mathcal{H}}\right\\}-\Lambda^{*}(h)\Bigg{\\}}$ (32)
$\displaystyle\leq$
$\displaystyle\sup\limits_{\|f\|_{\mathcal{H}}^{2}\leq\mu(\mathrm{R}^{d})}\left\\{-\frac{t}{2}\|f\|_{\mathcal{H}}^{2}+\Lambda(tf)\right\\}.$

Let $I(x):=\langle f,\mathrm{e}^{i\xi\cdot x}\rangle_{\mathcal{H}}$ for all
$f\in\mathcal{H}$, then $I(x)$ is a bounded and continuous function about $x$
on $\mathrm{R}^{d}$. By (27), (28) and (32), we only need to prove

$\displaystyle\limsup\limits_{t\rightarrow+\infty}\sup\limits_{\|f\|_{\mathcal{H}}^{2}\leq\mu(\mathrm{R}^{d})}\frac{1}{t}\log\mathbb{E}\exp\left\\{-\frac{t}{2}\|f\|_{\mathcal{H}}^{2}+\int_{0}^{t}I(B(s))ds\right\\}$
(33)
$\displaystyle\leq\frac{1}{2}\sup\limits_{g\in\mathcal{F}_{d}}\left\\{\int_{\mathrm{R}^{d}\times\mathrm{R}^{d}}\gamma(x-y)g^{2}(x)g^{2}(y)dxdy-\int_{\mathrm{R}^{d}}|\nabla
g(x)|^{2}dx\right\\}.$

Indeed, let
$C_{1}:=\sup\limits_{x\in\mathrm{R}^{d}\atop\|f\|_{\mathcal{H}}^{2}\leq\mu(\mathrm{R}^{d})}|I(x)|\leq\mu(\mathrm{R}^{d})$,
then there exists positive constants $C_{2},C_{3},C_{4}<+\infty$ such that for
all $t>1$, it has

$\displaystyle\mathbb{E}\exp\left\\{\int_{0}^{t}I(B(s))ds\right\\}\leq
e^{C_{1}}\mathbb{E}\exp\left\\{\int_{1}^{t}I(B(s))ds\right\\}\leq
e^{C_{1}}\Bigg{(}\mathbb{E}\Bigg{[}1_{|B(1)|\leq
t^{2}}\exp\left\\{\int_{1}^{t}I(B(s))ds\right\\}1_{|B(t)|\leq t^{2}}\Bigg{]}$
$\displaystyle+\mathbb{E}\Bigg{[}\exp\left\\{\int_{1}^{t}I(B(s))ds\right\\}1_{|B(1)|>t^{2}}\Bigg{]}+\mathbb{E}\Bigg{[}\exp\left\\{\int_{1}^{t}I(B(s))ds\right\\}1_{|B(t)|>t^{2}}\Bigg{]}\Bigg{)}$
$\displaystyle\leq
C_{2}e^{C_{1}}t^{2d}\exp\Bigg{\\{}(t-1)\sup\limits_{g\in\mathcal{F}_{d}}\Big{\\{}\int_{\mathrm{R}^{d}}I(x)g^{2}(x)dx-\frac{1}{2}\int_{\mathrm{R}^{d}}|\nabla
g(x)|^{2}dx\Big{\\}}\Bigg{\\}}+C_{3}e^{C_{1}t-\frac{t^{4}}{2}}+C_{4}e^{C_{1}t-\frac{t^{3}}{2}},$
(34)

which has been proved in page 101-102 of [2]. Now we use the semigroup
inequality (34), then

$\displaystyle\sup\limits_{\|f\|_{\mathcal{H}}^{2}\leq\mu(\mathrm{R}^{d})}\mathbb{E}\exp\left\\{-\frac{t}{2}\|f\|_{\mathcal{H}}^{2}+\int_{0}^{t}I(B(s))ds\right\\}$
$\displaystyle\leq
C_{2}e^{C_{1}}t^{2d}\exp\Bigg{\\{}(t-1)\sup\limits_{g\in\mathcal{F}_{d}}\Big{\\{}\sup\limits_{f\in\mathcal{H}}\Big{\\{}-\frac{1}{2}\|f\|_{\mathcal{H}}^{2}+\Big{\langle}f(\xi),\int_{\mathrm{R}^{d}}\mathrm{e}^{i\xi\cdot
x}g^{2}(x)dx\Big{\rangle}_{\mathcal{H}}\Big{\\}}-\frac{1}{2}\int_{\mathrm{R}^{d}}|\nabla
g(x)|^{2}dx\Big{\\}}\Bigg{\\}}$
$\displaystyle+C_{3}e^{C_{1}t-\frac{t^{4}}{2}}+C_{4}e^{C_{1}t-\frac{t^{3}}{2}}$
$\displaystyle=C_{2}e^{C_{1}}t^{2d}\exp\left\\{\frac{(t-1)}{2}\sup\limits_{g\in\mathcal{F}_{d}}\left\\{\int_{\mathrm{R}^{d}\times\mathrm{R}^{d}}\gamma(x-y)g^{2}(x)g^{2}(y)dxdy-\int_{\mathrm{R}^{d}}|\nabla
g(x)|^{2}dx\right\\}\right\\}+C_{3}e^{C_{1}t-\frac{t^{4}}{2}}$
$\displaystyle+C_{4}e^{C_{1}t-\frac{t^{3}}{2}}.$ (35)

Then, by (35) and the inequality $\log(a+b+c)\leq\log 3+\max\\{\log a,\log
b,\log c\\}$ for all $a,b,c>0$, we have

$\displaystyle\limsup\limits_{t\rightarrow+\infty}\sup\limits_{\|f\|_{\mathcal{H}}^{2}\leq\mu(\mathrm{R}^{d})}\frac{1}{t}\log\mathbb{E}\exp\left\\{-\frac{t}{2}\|f\|_{\mathcal{H}}^{2}+\int_{0}^{t}I(B(s))ds\right\\}$
$\displaystyle\leq\limsup\limits_{t\rightarrow+\infty}\max\Bigg{\\{}\frac{1}{2}\sup\limits_{g\in\mathcal{F}_{d}}\left\\{\int_{\mathrm{R}^{d}\times\mathrm{R}^{d}}\gamma(x-y)g^{2}(x)g^{2}(y)dxdy-\int_{\mathrm{R}^{d}}|\nabla
g(x)|^{2}dx\right\\}+\frac{\log(C_{2}e^{C_{1}}t^{2d})}{t},$ $\displaystyle
C_{1}-\frac{t^{3}}{2}+\frac{\log C_{3}}{t},C_{1}-\frac{t^{2}}{2}+\frac{\log
C_{4}}{t}\Bigg{\\}}=\mathcal{E}(\gamma).$ (36)

We prove the result for the general $t_{N}/t$. For any parameters $p,q>0$ and
$p+q=1$, by Jensen’s inequality, triangle inequality, (27) and (36), we have

$\displaystyle\limsup\limits_{N\rightarrow\infty}\frac{1}{Nt_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{1}{2Nt_{N}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}$
$\displaystyle\leq\limsup\limits_{p\rightarrow
1}\limsup\limits_{t\rightarrow+\infty}\limsup\limits_{N\rightarrow\infty}\frac{1}{N\lfloor\frac{t_{N}}{t}\rfloor
t}\log\mathbb{E}\exp\Bigg{\\{}\frac{1}{2pN\lfloor\frac{t_{N}}{t}\rfloor
t}\int_{\mathrm{R}^{d}}\left|\sum\limits_{k=1}^{N}\int_{0}^{\lfloor\frac{t_{N}}{t}\rfloor
t}\mathrm{e}^{i\xi\cdot
B_{k}(s)}ds\right|^{2}\mu(d\xi)+\frac{Nt\mu(\mathrm{R}^{d})}{2qt_{N}}\Bigg{\\}}$
$\displaystyle=\lim\limits_{p\rightarrow
1}\frac{1}{2}\sup\limits_{g\in\mathcal{F}_{d}}\left\\{\int_{\mathrm{R}^{d}\times\mathrm{R}^{d}}\frac{1}{p}\gamma(x-y)g^{2}(x)g^{2}(y)dxdy-\int_{\mathrm{R}^{d}}|\nabla
g(x)|^{2}dx\right\\}=\mathcal{E}(\gamma).$ (37)

Step 2. We prove that the variation $\mathcal{E}(\gamma)$ is finite when
$\mu(d\xi)$ is infinite. For all $t>0$, let

$\displaystyle
a(t):=\log\mathbb{E}\exp\left\\{\frac{1}{2t}\int_{0}^{t}\int_{0}^{t}\gamma(B(s)-B(r))dsdr\right\\},$

which is well defined because of the exponential integrability (14). By Lemma
2.1, we can obtain the subadditivity of $a(t)$, the definition of which is
referred in [2]. By the subadditivity and Lemma 1.3.4 in [2], the following
limit exists

$\displaystyle\Lambda:=\lim\limits_{t\rightarrow+\infty}\frac{1}{t}\log\mathbb{E}\exp\left\\{\frac{1}{2t}\int_{0}^{t}\int_{0}^{t}\gamma(B(s)-B(r))dsdr\right\\}<+\infty.$
(38)

We will use the same linearized method as the proof of lower bound in Step 1
and the $\mu_{\varepsilon}$, $\gamma_{\varepsilon}$ in (13). For all
$\varepsilon>0$, let the Hilbert space $\mathcal{H}_{\varepsilon}$ with
$\mu_{\varepsilon}$, similarly to $\mathcal{H}$ in Step 1, and
$f^{*}_{\varepsilon}(x):=\int_{\mathrm{R}^{d}}e^{i\xi\cdot
x}f(-\xi)\mu_{\varepsilon}(d\xi)$, we have

$\displaystyle\liminf\limits_{t\rightarrow+\infty}\frac{1}{t}\log\mathbb{E}\exp\left\\{\frac{1}{2t}\int_{0}^{t}\int_{0}^{t}\gamma_{\varepsilon}(B(s)-B(r))dsdr\right\\}$
$\displaystyle\geq\sup\limits_{f\in\mathcal{H}_{\varepsilon}}\left\\{\liminf\limits_{t\rightarrow+\infty}\frac{1}{t}\log\mathbb{E}\exp\left\\{\int_{0}^{t}f^{*}_{\varepsilon}(B(s))ds\right\\}-\frac{1}{2}\|f\|_{\mathcal{H}_{\varepsilon}}^{2}\right\\}$
$\displaystyle\geq\frac{1}{2}\sup\limits_{g\in\mathcal{F}_{d}}\left\\{\int_{\mathrm{R}^{d}\times\mathrm{R}^{d}}\gamma_{\varepsilon}(x-y)g^{2}(x)g^{2}(y)dxdy-\int_{\mathrm{R}^{d}}|\nabla
g(x)|^{2}dx\right\\}.$ (39)

For all $\varepsilon>0$, by Taylor expansion and Bochner’s representation, we
can get the inequality

$\displaystyle\mathbb{E}\exp\left\\{\frac{1}{2t}\int_{0}^{t}\int_{0}^{t}\gamma_{\varepsilon}(B(s)-B(r))dsdr\right\\}\leq\mathbb{E}\exp\left\\{\frac{1}{2t}\int_{0}^{t}\int_{0}^{t}\gamma(B(s)-B(r))dsdr\right\\}.$
(40)

Then, by (38)-(40) and monotone convergence theorem, we have

$\displaystyle\mathcal{E}(\gamma)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sup\limits_{g\in\mathcal{F}_{d}}\left\\{\int_{\mathrm{R}^{d}\times\mathrm{R}^{d}}\gamma(x-y)g^{2}(x)g^{2}(y)dxdy-\int_{\mathrm{R}^{d}}|\nabla
g(x)|^{2}dx\right\\}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sup\limits_{\varepsilon>0}\sup\limits_{g\in\mathcal{F}_{d}}\left\\{\int_{\mathrm{R}^{d}\times\mathrm{R}^{d}}\gamma_{\varepsilon}(x-y)g^{2}(x)g^{2}(y)dxdy-\int_{\mathrm{R}^{d}}|\nabla
g(x)|^{2}dx\right\\}\leq\Lambda<+\infty.$

Step 3. We complete the proof of the limit (17) when $\mu$ is infinite. For
all $\varepsilon>0$, by Taylor expansion and Bochner’s representation, we can
get the inequality

$\displaystyle\mathbb{E}\exp\Bigg{\\{}\frac{1}{2Nt_{N}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}\gamma_{\varepsilon}(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}$
$\displaystyle\leq\mathbb{E}\exp\Bigg{\\{}\frac{1}{2Nt_{N}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}.$
(41)

Then, by (26) and (41), we get the lower bound as the followings

$\displaystyle\liminf\limits_{N\rightarrow\infty}\frac{1}{Nt_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{1}{2Nt_{N}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}$
$\displaystyle\geq\sup\limits_{\varepsilon>0}\sup\limits_{g\in\mathcal{F}_{d}}\left\\{\int_{\mathrm{R}^{d}\times\mathrm{R}^{d}}\gamma_{\varepsilon}(x-y)g^{2}(x)g^{2}(y)dxdy-\frac{1}{2}\int_{\mathrm{R}^{d}}|\nabla
g(x)|^{2}dx\right\\}=\mathcal{E}(\gamma).$

For the upper bound, we make the decomposition
$\gamma=\gamma-\gamma_{\varepsilon}+\gamma_{\varepsilon}:=\overline{\gamma}_{\varepsilon}+\gamma_{\varepsilon}$,
where
$\overline{\gamma}_{\varepsilon}=\mathcal{F}(\overline{\mu}_{\varepsilon})$ in
$\mathcal{S}^{\prime}(\mathrm{R}^{d})$ and
$\overline{\mu}_{\varepsilon}(d\xi):=(1-\exp\left\\{-\frac{1}{2}\varepsilon|\xi|^{2}\right\\})\mu(d\xi)$.
For any parameters $p,q>0$ and $\frac{1}{p}+\frac{1}{q}=1$, by Hölder
inequality, we have

$\displaystyle\mathbb{E}\exp\Bigg{\\{}\frac{1}{2Nt_{N}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}$
$\displaystyle\leq\Bigg{(}\mathbb{E}\exp\Bigg{\\{}\frac{p}{2Nt_{N}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}\gamma_{\varepsilon}(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}\Bigg{)}^{\frac{1}{p}}$
(42)
$\displaystyle\cdot\Bigg{(}\mathbb{E}\exp\Bigg{\\{}\frac{q}{2Nt_{N}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}\overline{\gamma}_{\varepsilon}(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}\Bigg{)}^{\frac{1}{q}}.$
(43)

For the error (43), for any $q>0$, it suffices to prove

$\displaystyle\limsup\limits_{\varepsilon\rightarrow
0}\limsup\limits_{N\rightarrow\infty}\frac{1}{Nt_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{q}{2Nt_{N}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}\overline{\gamma}_{\varepsilon}(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}\leq
0.$ (44)

Moreover, by Lemma 2.2 and the similar computations to (37), we only need to
prove

$\displaystyle\limsup\limits_{\varepsilon\rightarrow
0}\limsup\limits_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{E}\exp\Bigg{\\{}\frac{q}{2N}\sum\limits_{j,k=1}^{N}\int_{0}^{1}\int_{0}^{1}\overline{\gamma}_{\varepsilon}(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}\leq
0.$ (45)

In fact, by the approximation in (13), the similar computations to (19) and
Jensen’s inequality, we have

$\displaystyle\mathbb{E}\exp\Bigg{\\{}\frac{qN}{2}\int_{\mathrm{R}^{d}}\left|\frac{1}{N}\sum_{j=1}^{N}\int_{0}^{1}e^{i\xi\cdot
B_{j}(s)}ds\right|^{2}\overline{\mu}_{\varepsilon}(d\xi)\Bigg{\\}}\leq\Bigg{(}\mathbb{E}\exp\Bigg{\\{}\frac{q}{2}\int_{\mathrm{R}^{d}}\left|\int_{0}^{1}e^{i\xi\cdot
B(s)}ds\right|^{2}\overline{\mu}_{\varepsilon}(d\xi)\Bigg{\\}}\Bigg{)}^{N}.$
(46)

By (46), Taylor expansion and Fatou’s Lemma, we find that (45) comes from the
following fact

$\displaystyle\limsup\limits_{\varepsilon\rightarrow
0}\mathbb{E}\exp\left\\{\frac{q}{2}\int_{\mathrm{R}^{d}}\left|\int_{0}^{1}e^{i\xi\cdot
B(s)}ds\right|^{2}\overline{\mu}_{\varepsilon}(d\xi)\right\\}\leq 1.$

At last, by (37) and (42)-(44), we have

$\displaystyle\limsup\limits_{N\rightarrow\infty}\frac{1}{Nt_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{1}{2Nt_{N}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}$
$\displaystyle\leq\limsup\limits_{p\rightarrow
1}\limsup\limits_{\varepsilon\rightarrow
0}\limsup\limits_{N\rightarrow\infty}\frac{1}{pNt_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{p}{2Nt_{N}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}\gamma_{\varepsilon}(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}$
$\displaystyle\leq\lim\limits_{p\rightarrow
1}\frac{1}{2p}\sup\limits_{g\in\mathcal{F}_{d}}\left\\{p\int_{\mathrm{R}^{d}}\Bigg{|}\int_{\mathrm{R}^{d}}e^{i\xi\cdot
x}g^{2}(x)dx\Bigg{|}^{2}\mu(d\xi)-\int_{\mathrm{R}^{d}}|\nabla
g(x)|^{2}dx\right\\}=\mathcal{E}(\gamma).$

##  3 An asymptotic result for asymptotically logarithmic function

To prove Theorem 1.1, we may first consider the case of “asymptotically
logarithmic” function, i.e. the function satisfying the following assumption,
and prove a general result by Proposition 2.1.

  1. (H)

The $\gamma$ is point-wise defined in $\mathrm{R}^{d}\backslash\\{0\\}$ and
bounded outside every neighborhood of $0$, and there exists some constant
$C>0$ such that

$\displaystyle\qquad\qquad\gamma(x)\sim
C\log\frac{1}{|x|}\quad\qquad(x\rightarrow 0).$ (47)

######  Proposition 3.1.

Assume the condition (H), then for all $t>0$, it holds that

$\displaystyle\lim\limits_{N\rightarrow\infty}\frac{1}{N\sigma_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{1}{2}\sum\limits_{j,k=1}^{N}\int_{0}^{t}\int_{0}^{t}\gamma(B_{j}(s)-B_{k}(r))drds\Bigg{\\}}=\frac{t}{2}.$
(48)

Here, $\sigma_{N}$ is a function about positive integer $N$, which satisfies
$\sigma_{N}>e$ and $\sigma_{N}^{-1}\log\sigma_{N}=\frac{2}{NCt}$ when $N$ is
enough large.

######  Remark 3.1.

Proposition 3.1 also includes the precise high moment asymptotic results for
the PAM with massive free field and Bessel field. In fact, their covariances
satisfy the condition (H), because their covariances are Bessel potential
$G_{s}(x)$ on $\mathrm{R}^{d}$ which approximates to the logarithmic potential
$C_{d}\log\frac{1}{|x|}$ at $0$ with $C_{d}>0$ when $s=d$, by [18]. For the
detailed definition of the two fields, see e.g. [17, 21, 23].

The proof of Proposition 3.1: We will use the truncated power function
$\phi_{l}(|x|):=(1-|x|)_{+}^{l}$ with $l>0$, which is positive definite on
$\mathrm{R}^{d}$ when $l\geq\lfloor\frac{d}{2}\rfloor+1$ by [26]. For any two
parameters $\delta,M>0$, let
$\gamma_{\delta}(x):=\phi_{l}(|\frac{x}{\delta}|)$ and
$\gamma_{M}(x):=\phi_{l}(|\frac{x}{M}|)$. By the condition (H), for any
$\varepsilon>0$, there exists a $\delta>0$ such that for all $|x|<\delta$, it
has

$\displaystyle(1-\varepsilon)C\log\frac{1}{|x|}\leq\gamma(x)\leq(1+\varepsilon)C\log\frac{1}{|x|}.$
(49)

To simplify notation, we let $C=1$ in (49) and $(fg)(x):=f(x)g(x)$ for all
$f,g$ on $\mathrm{R}^{d}$.

First, we prove the upper bound. Let $t_{N}=\sigma_{N}t$. For any parameters
$p,q>0$ and $\frac{1}{p}+\frac{1}{q}=1$, by Brownian scaling,
$\gamma=\gamma\gamma_{\delta}+\gamma(1-\gamma_{\delta})$ and Hölder inequality
, we have

$\displaystyle\mathbb{E}\exp\Bigg{\\{}\frac{1}{2}\sum\limits_{j,k=1}^{N}\int_{0}^{t}\int_{0}^{t}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}$
$\displaystyle\leq\Bigg{[}\mathbb{E}\exp\Bigg{\\{}\frac{p}{2\sigma_{N}^{2}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}(\gamma\gamma_{\delta})(\sigma_{N}^{-\frac{1}{2}}(B_{j}(s)-B_{k}(r)))dsdr\Bigg{\\}}\Bigg{]}^{\frac{1}{p}}$
(50)
$\displaystyle\cdot\Bigg{[}\mathbb{E}\exp\Bigg{\\{}\frac{q}{2\sigma_{N}^{2}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}(\gamma(1-\gamma_{\delta}))(\sigma_{N}^{-\frac{1}{2}}(B_{j}(s)-B_{k}(r)))dsdr\Bigg{\\}}\Bigg{]}^{\frac{1}{q}}.$
(51)

For the main term (50), by (49) and $0\leq\gamma_{\delta}\leq 1$, we have

$\displaystyle\mathbb{E}\exp\Bigg{\\{}\frac{p}{2\sigma_{N}^{2}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}(\gamma\gamma_{\delta})(\sigma_{N}^{-\frac{1}{2}}(B_{j}(s)-B_{k}(r)))dsdr\Bigg{\\}}\qquad\qquad\qquad\qquad\quad~{}~{}~{}$
$\displaystyle\leq\mathbb{E}\exp\Bigg{\\{}\frac{(1+\varepsilon)p}{2\sigma_{N}^{2}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}\log_{+}\frac{1}{|(B_{j}(s)-B_{k}(r))|}dsdr+\frac{(1+\varepsilon)pt^{2}N^{2}\log\sigma_{N}^{\frac{1}{2}}}{2}\Bigg{\\}}.$
(52)

Let $\sigma_{N}>e$ and $\frac{tN\log\sigma_{N}^{\frac{1}{2}}}{\sigma_{N}}=1$
when $N$ is enough large, then $\sigma_{N}$ exists and satisfies
$\frac{1}{\sigma_{N}}=o(\frac{1}{N})$ as $N\rightarrow\infty$. Then, by (52),
we have

$\displaystyle\limsup\limits_{N\rightarrow\infty}\frac{1}{pNt_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{p}{2\sigma_{N}^{2}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}(\gamma\gamma_{\delta})(\sigma_{N}^{-\frac{1}{2}}(B_{j}(s)-B_{k}(r)))dsdr\Bigg{\\}}$
$\displaystyle\leq\limsup\limits_{N\rightarrow\infty}\frac{1}{pNt_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{(1+\varepsilon)p}{2\sigma_{N}^{2}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}\log_{+}\frac{1}{|(B_{j}(s)-B_{k}(r))|}dsdr\Bigg{\\}}+\frac{(1+\varepsilon)}{2}.$
(53)

We claim that for all $p>0$, it holds that

$\displaystyle\limsup\limits_{N\rightarrow\infty}\frac{1}{pNt_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{(1+\varepsilon)p}{2\sigma_{N}^{2}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}\log_{+}\frac{1}{|(B_{j}(s)-B_{k}(r))|}dsdr\Bigg{\\}}\leq
0.$ (54)

Indeed, to use Proposition 2.1, we only need to find a positive definite
function satisfying the Dalang’s condition instead of $\log_{+}\frac{1}{|x|}$
in (54), because the exact kernel $\log_{+}\frac{1}{|x|}$ is positive definite
if and only if $d\leq 3$ in [24]. We may notice that there is some
$\alpha\in(0,1)$ and $C>0$ such that

$\displaystyle\log_{+}\frac{1}{|x|}\leq|x|^{-\alpha}+C.$ (55)

Furthermore, for all $\varepsilon_{1}>0$, it holds that
$\frac{1}{\sigma_{N}^{2}}=\frac{2}{\log\sigma_{N}}\frac{1}{Nt_{N}}\leq\frac{\varepsilon_{1}}{Nt_{N}}$
when $N$ is enough large. By (55) and Proposition 2.1, we have

$\displaystyle\limsup\limits_{N\rightarrow\infty}\frac{1}{pNt_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{(1+\varepsilon)p}{2\sigma_{N}^{2}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}\log_{+}\frac{1}{|(B_{j}(s)-B_{k}(r))|}dsdr\Bigg{\\}}$
$\displaystyle\leq\limsup\limits_{\varepsilon_{1}\rightarrow
0}\limsup\limits_{N\rightarrow\infty}\frac{1}{pNt_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{(1+\varepsilon)\varepsilon_{1}p}{2Nt_{N}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}(\left|B_{j}(s)-B_{k}(r)\right|^{-\alpha}+C)dsdr\Bigg{\\}}$
$\displaystyle\leq\limsup\limits_{\varepsilon_{1}\rightarrow
0}\frac{1}{2p}\sup\limits_{g\in\mathcal{F}_{d}}\Bigg{\\{}(1+\varepsilon)\varepsilon_{1}p\int_{\mathrm{R}^{d}\times\mathrm{R}^{d}}(|x-y|^{-\alpha}+C)g^{2}(x)g^{2}(y)dxdy-\int_{\mathrm{R}^{d}}|\nabla
g(x)|^{2}dx\Bigg{\\}}=0.$

For the error term (51), we only need to show that for all $q>0$, it holds
that

$\displaystyle\limsup\limits_{N\rightarrow\infty}\frac{1}{qNt_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{q}{2\sigma_{N}^{2}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}(\left|\gamma\right|(1-\gamma_{\delta}))(\sigma_{N}^{-\frac{1}{2}}(B_{j}(s)-B_{k}(r)))dsdr\Bigg{\\}}\leq
0.$ (56)

Indeed, for any $\varepsilon_{2}>0$, there exists a $\delta_{1}\in(0,\delta)$
such that $1-\gamma_{\delta}(x)\leq\varepsilon_{2}$ when $|x|<\delta_{1}$.
Since $\gamma$ is bounded outside every neighborhood of $0$, we can define
$C:=\sup\limits_{|x|\geq\delta_{1}}|\gamma|<+\infty$. By (49), we have

$\displaystyle|\gamma|(1-\gamma_{\delta})\leq(1+\varepsilon)\varepsilon_{2}\log\frac{1}{|x|}1_{|x|\leq\delta_{1}}+C.$
(57)

By (57) and the similar computations to (52), we obtain

$\displaystyle\mathbb{E}\exp\Bigg{\\{}\frac{q}{2\sigma_{N}^{2}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}(\left|\gamma\right|(1-\gamma_{\delta}))(\sigma_{N}^{-\frac{1}{2}}(B_{j}(s)-B_{k}(r)))dsdr\Bigg{\\}}\qquad\qquad\qquad\qquad\qquad\qquad\qquad~{}~{}$
$\displaystyle\leq\mathbb{E}\exp\Bigg{\\{}\frac{q(1+\varepsilon)\varepsilon_{2}}{2\sigma_{N}^{2}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}\log_{+}\frac{1}{|B_{j}(s)-B_{k}(r)|}dsdr+\frac{1}{2}q(1+\varepsilon)\varepsilon_{2}t^{2}N^{2}\log\sigma_{N}^{\frac{1}{2}}+\frac{1}{2}Cqt^{2}N^{2}\Bigg{\\}}.$
(58)

Moreover, by the same computations as (52)-(54) and (58), we have

$\displaystyle\limsup\limits_{N\rightarrow\infty}\frac{1}{qNt_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{q}{2\sigma_{N}^{2}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}(\left|\gamma\right|(1-\gamma_{\delta}))(\sigma_{N}^{-\frac{1}{2}}(B_{j}(s)-B_{k}(r)))dsdr\Bigg{\\}}$
$\displaystyle\leq\limsup\limits_{\varepsilon_{2}\rightarrow
0}\frac{(1+\varepsilon)\varepsilon_{2}}{2}=0.$

Then, by (50), (51), (53), (54) and (56), we get the upper bound

$\displaystyle\limsup\limits_{N\rightarrow\infty}\frac{1}{Nt_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{1}{2}\sum\limits_{j,k=1}^{N}\int_{0}^{t}\int_{0}^{t}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}\leq\limsup\limits_{\varepsilon\rightarrow
0}\frac{(1+\varepsilon)}{2}=\frac{1}{2}.$

Second, we prove the lower bound. For any parameter $p>1$, by the reverse
Hölder inequality, we have

$\displaystyle\mathbb{E}\exp\Bigg{\\{}\frac{1}{2}\sum\limits_{j,k=1}^{N}\int_{0}^{t}\int_{0}^{t}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}$
$\displaystyle\geq\Bigg{[}\mathbb{E}\exp\Bigg{\\{}\frac{p}{2\sigma_{N}^{2}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}(\gamma\gamma_{\delta})(\sigma_{N}^{-\frac{1}{2}}(B_{j}(s)-B_{k}(r)))dsdr\Bigg{\\}}\Bigg{]}^{\frac{1}{p}}$
(59)
$\displaystyle\cdot\Bigg{[}\mathbb{E}\exp\Bigg{\\{}\frac{-1}{2(p-1)\sigma_{N}^{2}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}(\gamma(1-\gamma_{\delta}))(\sigma_{N}^{-\frac{1}{2}}(B_{j}(s)-B_{k}(r)))dsdr\Bigg{\\}}\Bigg{]}^{(1-p)}.$
(60)

For any $M>1$, there exists a $N_{0}>0$ such that for all $N>N_{0}$, it has
$M\leq\delta\sigma_{N}^{\frac{1}{2}}$. When $N>N_{0}$, by (49) and
$0\leq\gamma_{M}\leq 1$, we get the following evaluation for (59)

$\displaystyle\mathbb{E}\exp\Bigg{\\{}\frac{p}{2\sigma_{N}^{2}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}(\gamma\gamma_{\delta})(\sigma_{N}^{-\frac{1}{2}}(B_{j}(s)-B_{k}(r)))dsdr\Bigg{\\}}$
$\displaystyle\geq\mathbb{E}\exp\Bigg{\\{}\frac{(1-\varepsilon)p}{2\sigma_{N}^{2}}\log\sigma_{N}^{\frac{1}{2}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}\gamma_{M}(B_{j}(s)-B_{k}(r))dsdr$
$\displaystyle+\frac{(1-\varepsilon)p}{2\sigma_{N}^{2}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}\log\frac{1}{|(B_{j}(s)-B_{k}(r))|}\gamma_{M}(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}$
$\displaystyle\geq\mathbb{E}\exp\Bigg{\\{}\frac{(1-\varepsilon)p}{2\sigma_{N}^{2}}\log\sigma_{N}^{\frac{1}{2}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}\gamma_{M}(B_{j}(s)-B_{k}(r))dsdr-p\log
M\frac{t^{2}N^{2}}{2}\Bigg{\\}}.$ (61)

Moreover, by (61), Proposition 2.1 and
$\frac{\log\sigma_{N}^{\frac{1}{2}}}{\sigma_{N}^{2}}=\frac{1}{Nt_{N}}$ (i.e.
$\frac{t^{2}N}{t_{N}}=\frac{2}{\log\sigma_{N}}$) when $N$ is enough large, we
have

$\displaystyle\liminf\limits_{N\rightarrow\infty}\frac{1}{pNt_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{p}{2\sigma_{N}^{2}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}(\gamma\gamma_{\delta})(\sigma_{N}^{-\frac{1}{2}}(B_{j}(s)-B_{k}(r)))dsdr\Bigg{\\}}$
$\displaystyle\geq\liminf\limits_{N\rightarrow\infty}\frac{1}{pNt_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{(1-\varepsilon)p}{2Nt_{N}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}\gamma_{M}(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}$
$\displaystyle\geq\frac{1}{2p}\sup\limits_{M>1}\sup\limits_{g\in\mathcal{F}_{d}}\Bigg{\\{}(1-\varepsilon)p\int_{\mathrm{R}^{d}\times\mathrm{R}^{d}}\gamma_{M}(x-y)g^{2}(x)g^{2}(y)dxdy-\int_{\mathrm{R}^{d}}|\nabla
g(x)|^{2}dx\Bigg{\\}}=\frac{1-\varepsilon}{2}.$ (62)

Here, the last step is due to $\gamma_{M}\uparrow 1$ as $M\rightarrow+\infty$.
For the error term (60), we only need to show that for all $p>1$, it holds
that

$\displaystyle\liminf\limits_{N\rightarrow\infty}\frac{-(p-1)}{Nt_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{-1}{2(p-1)\sigma_{N}^{2}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}(\gamma(1-\gamma_{\delta}))(\sigma_{N}^{-\frac{1}{2}}(B_{j}(s)-B_{k}(r)))dsdr\Bigg{\\}}\geq
0.$ (63)

Indeed, for all $p>1$, by (56), we have

$\displaystyle\limsup\limits_{N\rightarrow\infty}\frac{(p-1)}{Nt_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{-1}{2(p-1)\sigma_{N}^{2}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}(\gamma(1-\gamma_{\delta}))(\sigma_{N}^{-\frac{1}{2}}(B_{j}(s)-B_{k}(r)))dsdr\Bigg{\\}}$
$\displaystyle\leq\limsup\limits_{N\rightarrow\infty}\frac{(p-1)}{Nt_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{1}{2(p-1)\sigma_{N}^{2}}\sum\limits_{j,k=1}^{N}\int_{0}^{t_{N}}\int_{0}^{t_{N}}(\left|\gamma\right|(1-\gamma_{\delta}))(\sigma_{N}^{-\frac{1}{2}}(B_{j}(s)-B_{k}(r)))dsdr\Bigg{\\}}=0.$

At last, by (59), (60), (62) and (63), we get the lower bound

$\displaystyle~{}~{}~{}~{}~{}~{}\liminf\limits_{N\rightarrow\infty}\frac{1}{Nt_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{1}{2}\sum\limits_{j,k=1}^{N}\int_{0}^{t}\int_{0}^{t}\gamma(B_{j}(s)-B_{k}(r))dsdr\Bigg{\\}}\geq\liminf\limits_{\varepsilon\rightarrow
0}\frac{(1-\varepsilon)}{2}=\frac{1}{2}.$

##  4 The proof of Theorem 1.1

Because the kernel $k(x,y)$ may be non-stationary, we need to find a function
satisfying the condition (H) instead of the $k(x,y)$ in the moment
representation (9). By (5), there exists a constant $C>0$ such that
$\log_{+}\frac{1}{|x-y|}-C\leq k(x,y)\leq\log_{+}\frac{1}{|x-y|}+C$ on
$\mathrm{R}^{d}\times\mathrm{R}^{d}$. Furthermore, by (6) and
$\frac{1}{N\lambda_{N}}=o(\frac{1}{N^{2}})$ as $N\rightarrow\infty$ in Remark
1.1, we notice that when $N\rightarrow\infty$, it has

$\displaystyle\frac{1}{N\lambda_{N}}\log\mathbb{E}u^{N}(t,x)\sim\frac{t}{N\sigma_{N}}\log\mathbb{E}\exp\Bigg{\\{}\frac{1}{2}\sum\limits_{j,k=1}^{N}\int_{0}^{t}\int_{0}^{t}\log_{+}\frac{1}{|B_{j}(s)-B_{k}(r)|}drds\Bigg{\\}}.$
(64)

In fact, to simplify the notation, we use the $\lambda_{N}$ instead of the
$\sigma_{N}$ in Proposition 3.1. By
$\frac{\log\lambda_{N}}{\lambda_{N}}=\frac{2}{N}$ and
$\frac{\log\sigma_{N}}{\sigma_{N}}=\frac{2}{Nt}$ and
$\lambda_{N},\sigma_{N}<N^{2}$ when $N$ is enough large, we have
$\lim\limits_{N\rightarrow\infty}\frac{\sigma_{N}}{\lambda_{N}}=\lim\limits_{N\rightarrow\infty}\frac{t\log(Nt/2\log\sigma_{N})}{\log(N/2\log\lambda_{N})}=t$.
By (64), Theorem 1.1 is a direct result of Proposition 3.1.

##  5 Acknowledgments

The author would like to thank Professor Xia Chen for his help during the
completion of this paper. The manuscript is partially supported by NSFC Grant
No. 11871244.

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Complementary collider and astrophysical probes of multi-component Dark Matter

J. Hernández-Sánchez[E-mail<EMAIL_ADDRESS>1,
V.  Keus[E-mail<EMAIL_ADDRESS>2,3,4,
S.  Moretti[E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>^4,5,6,

D.  Sokołowska[E-mail<EMAIL_ADDRESS>^7,8

^1 Instituto de Física and Facultad de Ciencias de la Electrónica,

Benemérita Universidad Autónoma de Puebla,
Apdo. Postal 542, C.P. 72570 Puebla, México,

^2  Dublin Institute for Advanced Studies, School of Theoretical Physics,

10 Burlington road, Dublin, D04 C932, Ireland

^3 Department of Physics and Helsinki Institute of Physics,

Gustaf Hallstromin katu 2, FIN-00014 University of Helsinki, Finland

^4 School of Physics and Astronomy, University of Southampton,

Southampton, SO17 1BJ, United Kingdom

^5 Particle Physics Department, Rutherford Appleton Laboratory,

Chilton, Didcot, Oxon OX11 0QX, United Kingdom

^6 Department of Physics and Astronomy, Uppsala University,

Box 516, SE-751 20 Uppsala, Sweden

^7 University of Warsaw, Faculty of Physics, Pasteura 5, 02-093 Warsaw, Poland.

^8 International Institute of Physics, Universidade Federal do Rio Grande do Norte,

Campus Universitario, Lagoa Nova, Natal-RN 59078-970, Brazil

We study a new physics scenario with two inert and one active scalar doublets, hence a 3-Higgs Doublet Model (3HDM). We impose a $Z_2 \times Z'_2$ symmetry onto such a 3HDM with one inert doublet odd under the $Z_2$ transformation and the other odd under the $Z'_2$ one. Such a construction leads to a two-component Dark Matter (DM) model. It has been shown that, when there is a sufficient mass difference between the two DM candidates, it is possible to probe the light DM candidate in the nuclear recoil energy in direct detection experiments and the heavy DM component in the photon flux in indirect detection experiments.
With the DM masses at the electroweak scale, we show that, independently of astrophysical probes, this model feature can be tested at the Large Hadron Collider via scalar cascade decays in $2\ell + \cancel{E}_T$ final states.
We study several observable distributions whose shapes hint at the presence of the two different DM candidates.

§ INTRODUCTION

The Standard Model (SM) of particle physics has been extensively tested and is in great agreement with experiment with its last particle – the Higgs boson $h$ – discovered in 2012 with a mass of $\approx$ 125 GeV by the ATLAS and CMS experiments at the CERN Large Hadron Collider (LHC) <cit.>.
Although the properties of the observed state are in agreement with those of the SM Higgs boson, it is entirely possible that it is one member of an extended scalar sector.

In fact, although in agreement with experiment, the SM is understood to be incomplete with
one of its shortcomings being the lack of a viable Dark Matter (DM) candidate in its particle content.
The standard cosmological $\Lambda$CDM Model <cit.> requires DM to be a particle stable on cosmological time scales, cold (i.e., non-relativistic at the onset of galaxy formation), non-baryonic, neutral and weakly interacting:  such a state does not exist in the SM.
Within Beyond the SM (BSM) frameworks, many such candidates exist, with the most well-studied being the Weakly Interacting Massive Particles (WIMPs) <cit.> with masses between a few GeV and a few TeV.
WIMPs are usually stable due to the conservation of a discrete symmetry, such as scalar DM candidates in non-minimal Higgs frameworks which are stabilised by the conserved discrete symmetry of the scalar potential, see, e.g.,  <cit.>.

One of the simplest BSM scenarios which provides a scalar DM candidate is the Inert Doublet Model (IDM) <cit.>, which contains one inert doublet plus one active (Higgs) doublet, hence also known as I(1+1)HDM.
In this model, which has been studied extensively in the literature (see, e.g., <cit.>), the additional $SU(2)_W$ scalar doublet has the same SM quantum numbers as the SM Higgs doublet.

One of the possible vacuum alignments in this model is $(v,0)$ wherein the active (or Higgs)   doublet acquires a non-zero Vacuum Expectation Value (VEV) while the inert (or dark) doublet does not develop a VEV and therefore does not take part in  electroweak symmetry breaking.
The Lagrangian and the vacuum are symmetric under a $Z_2$ group under which only the inert doublet is odd.
Due to the conservation of this symmetry, the inert doublet provides a stable DM candidate: the lightest neutral $Z_2$-odd particle.

In the I(1+1)HDM, for masses of DM smaller than $m_h/2$, the dark sector communicates with the SM mainly through the Higgs boson exchange which is a characteristic of all Higgs-portal models <cit.>.
Such a set-up leads to the DM-Higgs coupling, $g_{{\rm DM}h}$, dictating the DM annihilation rate $\langle\sigma v\rangle$, the DM-nucleon scattering cross-section $\sigma_{\rm DM-N}$ and the Higgs invisible decays.
Simultaneous fulfilment of current experimental constraints for these three types of processes is a challenging task, as shown, e.g., in <cit.>. For heavier DM particles, the direct annihilation into pairs of gauge bosons usually results in a relic density below the observed value, with the exception of very heavy DM particles.

A possible solution to this problem is breaking the simple relation between the annihilation rate and  direct detection cross-section by introducing coannihilation processes between DM and other dark particles which are close in mass.
Coannihilation processes (through constructive and/or destructive interference) can decrease or increase the effective annihilation cross-section, which in turn change the DM relic density value.

In the I(1+1)HDM, the DM candidate could potentially coannihilate with the neutral or charged $Z_2$-odd particles. However, in a vast region of the parameter space, this coannihilation is too efficient, which leads to a total relic density below the observed value <cit.>.

In models with an extended inert sector, more coannihilation processes are available and lead to a much richer phenomenology, see, e.g.,  models with
extra inert singlets <cit.>
or inert doublets <cit.> and, within the framework of 3-Higgs Doublet Models (3HDMs), the I(2+1)HDM of Refs. <cit.>.
Proposed by Weinberg in 1976 <cit.>, 3HDMs are very well motivated scenarios <cit.> due to their implications for flavour physics, CP-violation, baryogenesis and inflation <cit.>.

Here, we study an I(2+1)HDM framework symmetric under a $Z_2\times Z'_2$ group with one inert doublet odd under $Z_2$ and even under $Z'_2$, and the other inert doublet even under $Z_2$ and odd under $Z'_2$.
The lightest particle from each inert doublet is a viable DM candidate, yielding a two-component DM model.
In a previous study <cit.>, we showed that other dark particles from both doublets influence the thermal evolution and decoupling rate of DM particles and significantly impact the final relic abundance.
(A similar analysis was performed in the context of a supersymmetric model in <cit.>.)

When there is a sufficient mass difference between the two DM candidates, which are both typically at the Electro-Weak (EW) scale, we showed that the light DM component can be probed by the nuclear recoil energy in direct detection experiments while the heavy DM  component appears through its contribution to the photon flux in indirect detection experiments.

In this paper, quite independently of astrophysical probes, we study collider signatures of the I(2+1)HDM, namely, scalar cascade decays in $2 \ell + \cancel{E}_T$ final states at the LHC. Specifically, we analyse several observable distributions whose shapes hint at the presence of the two different DM candidates.

The remainder of the paper is organised as follows.
In Section <ref>, we present the model, its scalar mass spectrum and discuss theoretical and experimental constraints on its  parameter space.
In Section <ref>, we discuss the experimental probes of the model and construct some Benchmark Points (BPs).
In Section <ref>, we present our results, and in Section <ref>, we conclude.

§ POTENTIAL, MASS SPECTRUM AND CONSTRAINTS

§.§ The scalar potential

The most general $Z_2\times Z'_2$ symmetric 3HDM potential has the following form <cit.>:

V    =    V_0+V_Z_2 ×Z'_2,

V_0    =    - μ^2_1 (ϕ_1^†ϕ_1) -μ^2_2 (ϕ_2^†ϕ_2) - μ^2_3(ϕ_3^†ϕ_3) + λ_11 (ϕ_1^†ϕ_1)^2+ λ_22 (ϕ_2^†ϕ_2)^2  + λ_33 (ϕ_3^†ϕ_3)^2

+ λ_12  (ϕ_1^†ϕ_1)(ϕ_2^†ϕ_2)
+ λ_23  (ϕ_2^†ϕ_2)(ϕ_3^†ϕ_3) + λ_31 (ϕ_3^†ϕ_3)(ϕ_1^†ϕ_1)

+ λ'_12 (ϕ_1^†ϕ_2)(ϕ_2^†ϕ_1)
+ λ'_23 (ϕ_2^†ϕ_3)(ϕ_3^†ϕ_2) + λ'_31 (ϕ_3^†ϕ_1)(ϕ_1^†ϕ_3),

V_Z_2 ×Z'_2   =     λ_1 (ϕ_1^†ϕ_2)^2 + λ_2(ϕ_2^†ϕ_3)^2 + λ_3 (ϕ_3^†ϕ_1)^2 + h.c.,
where $V_0$ is invariant under any phase rotation while $V_{Z_2 \times Z'_2}$ ensures the symmetry under the $Z_2 \times Z'_2$ group generated by
g_Z_2 = diag(-1,1,1)  ,     g_Z'_2 = diag(1,-1,1)  .
Under this charge assignment, all SM fields, including the Higgs doublet $\phi_3$, are even under both $Z_2$ and $Z_2'$.  The additional doublets, $\phi_1$ and $\phi_2$ are odd under $Z_2$ and $Z_2'$, respectively.   In this paper, we assume that all parameters in the potential are real, therefore, we do not introduce any explicit CP-violation in the scalar sector. In particular, we do not consider the possible effects of dark CP-violation, a feature that was introduced for the first time in <cit.>
and further studied in
which could indeed arise in an extended dark sector. However, there is still the possibility of spontaneous breaking of the CP symmetry in the active sector for particular choices of parameters, as discussed in <cit.>. This choice of vacuum is not considered in this paper as it cannot lead to the  appearance of two DM candidates in the model.

The Yukawa interactions are set to “Type-I” interactions, i.e.,  only the third doublet, $\phi_3$, will couple to fermions:
ℒ_Y = Γ^u_mn q̅_m,L ϕ̃_3 u_n,R + Γ^d_mn q̅_m,L ϕ_3 d_n,R
+  Γ^e_mn l̅_m,L ϕ_3 e_n,R + Γ^ν_mn l̅_m,L ϕ̃_3 ν_n,R + h.c.
Following the $Z_2 \times Z_2'$ charge assignment, this choice of Yukawa interaction is the only one which will not lead to breaking of the imposed discrete symmetries. This also ensures that there are no Flavour Changing Neutral Currents (FCNCs), since fields from the doublets which do not develop VEVs, $\phi_1$ and $\phi_2$,
will not couple to fermions.

§.§ Mass spectrum and physical parameters

The  vacuum state, i.e., the one with two automatically stable inert particles,  has the alignment $(0,0,v)$ in which the composition of the doublets are
ϕ_1= $\begin{scriptsize}$ H^+_1 $\end{scriptsize}$H_1+iA_1/√(2) ,  ϕ_2= $\begin{scriptsize}$ H^+_2 $\end{scriptsize}$H_2+iA_2/√(2) ,   ϕ_3= $\begin{scriptsize}$ H^+_3 $\end{scriptsize}$v+h+iA^0_3/√(2)   ,

with the extremum condition for this state reading as
v^2=μ_3^2/λ_33 .

The third doublet, $\phi_3$, plays the role of the SM Higgs doublet, with the Higgs particle $h$ having, by construction, tree-level interactions with gauge bosons and fermions identical to those of the SM Higgs boson. Its mass is fixed through the tadpole conditions to be
m^2_h= 2μ_3^2 = 2 v^2 λ_33
and the ${A^0}_3$ and ${H}^\pm_3$ states are the would-be Goldstone bosons.

The two inert doublets, $\phi_1$ and $\phi_2$, provide two DM candidates. Each doublet consists of two neutral particles[As it is the case in multi-scalar models with unbroken $Z_2$ symmetries, the  inert scalars $H_i$ and $A_i$ have opposite CP parity, as evident from their gauge interactions, however, it is not possible to establish their definite CP properties, as they do not couple to fermions.], $H_i$ and $A_i$, and one charged particle $H^\pm_i$ with $i=1,2$. For the two generations of inert scalars the mass spectrum is as follows:
m^2_H_1= -μ^2_1  +1/2(λ_31+λ'_31 +2λ_3 )v^2 ≡-μ_1^2 + Λ_3 v^2,

m^2_A_1= -μ^2_1  +1/2(λ_31+λ'_31 -2λ_3)v^2 ≡-μ_1^2 + Λ̅_3 v^2,

m^2_H^±_1= -μ^2_1 +1/2λ_31v^2
m^2_H_2= -μ^2_2  +1/2(λ_23+λ'_23 +2λ_2)v^2≡-μ_2^2 + Λ_2 v^2,

m^2_A_2= -μ^2_2  +1/2(λ_23+λ'_23 -2λ_2)v^2≡-μ_2^2 + Λ̅_2 v^2,

m^2_H^±_2= -μ^2_2 +1/2λ_23 v^2.

Parameters of the potential can be rephrased in terms of  physical observables, such as masses and couplings. The tree-level SM couplings in the gauge and fermionic sectors follow exactly the SM definitions. The relevant parameters arising from the extended scalar sector are: (i) masses of inert particles and the Higgs-DM couplings, which represent parameters from the visible sector; (ii) self-interaction parameters, which describe interaction within the dark sector. The full list is:
v^2, m_h^2, m^2_H_1, m^2_H_2, m^2_A_1, m^2_A_2, m^2_H^±_1, m^2_H^±_2, Λ_2, Λ_3, Λ_1, λ_11, λ_22, λ'_12, λ_12 .
The self-couplings $\lambda_{11}, \lambda_{22}, \lambda'_{12}, \lambda_{12}$ correspond exactly to the terms in Eq.  (<ref>), while the relations between remaining parameters and our chosen physical basis are as follows:

λ_1 = 2Λ_1 - (λ_12+λ'_12).

In principle, any particle among $(H_i, A_i, H^\pm_i)$ can be the lightest. Here, we dismiss the possibility of $H^\pm_i$ being the lightest, as it would mean that DM candidate is a charged particle. Choosing between $H_1$ and $A_1$ (or $H_2$ and $A_2$) is related only to a change of the sign of the quartic parameter $\lambda_{3}$ ($\lambda_2$) and has no impact on the ensuing phenomenology. Therefore, we will choose  $m_{H_i} < m_{A_i}, m_{H^\pm_i}$, which leads to the following relations between the parameters:
λ_2<0,    λ_3<0,    λ_31'+ 2λ_3<0,    λ_23'+ 2λ_2<0.
Notice that, unlike many $Z_N$ symmetric models, the two lightest states from two doublets are automatically stable, regardless of their mass hierarchy, as they are stabilised by different $Z_2$ symmetries.

§.§ Theoretical and experimental constraints

The parameters of the potential $V$ are subject to a number of  theoretical and experimental constraints (described in detail  in
<cit.>). Below, we summarise the constraints imposed on the model to ensure that all proposed BPs  are in agreement with current theoretical and experimental knowledge.

Stability of the potential
For the potential to be bounded from below (i.e., having a stable vacuum) the following conditions are required
λ_ii>0,   i =1,2,3,

λ_x > - 2 √(λ_11 λ_22),   λ_y > - 2 √(λ_11 λ_33),   λ_z > - 2 √(λ_22 λ_33),

√(λ_33) λ_x + √(λ_11) λ_z+√(λ_22) λ_y ≥0

λ_33 λ_x^2 + λ_11 λ_z^2+λ_22 λ_y^2 -λ_11 λ_22 λ_33 - 2 λ_x λ_y λ_z <0,

λ_x = λ_12+min(0,λ_12'-2|λ_1|),

λ_y = λ_31+min(0,λ_31'-2|λ_3|),

λ_z = λ_23+min(0,λ_23'-2|λ_2|).
As noted in <cit.>,  these conditions are in fact sufficient but
not necessary, as it is possible to construct examples of this model in which the potential is bounded from below, but which violate conditions
(<ref>)–(<ref>).  We do not explore such a region of parameter space in this work.

Global minimum condition
For $(0,0,v)$ to be a local minimum, all mass-squared values have to be positive, and, for it to be a global minimum, i.e., the true vacuum, its energy, $\mathcal{V}_{\texttt{2Inert}}$, has to be lower than the energy of any other possible minima, $\mathcal{V}_{\texttt{X}}$, that exist at the same time (see discussion in <cit.>). Following the chosen mass order and resulting relations in Eq. (<ref>) we arrive at the following conditions:
Local minimum if:       {
v^2=μ_3^2/λ_33 > 0

Λ_2 > μ^2_2/v^2

Λ_3 > μ^2_1/v^2.  .

Global minimum if, in addition:       𝒱_= -μ_3^4/4λ_33 < 𝒱_.

Perturbative unitarity We require that the scalar $2 \to 2$ scattering matrix is unitary, i.e., the absolute values of all eigenvalues of such a matrix for Goldstones, Higgs and dark states with specific hypercharge and isospin should be smaller than $8 \pi$. Furthermore, all quartic scalar couplings should be perturbative, i.e., $\lambda_i \leq 4 \pi$.

EW Precision Observables (EWPOs) We demand a 2$\sigma$, i.e., 95% Confidence Level (CL) agreement with EWPOs which are  parametrised through the EW oblique parameters $S,T,U$. Assuming an SM Higgs boson mass of $m_h$ = 125 GeV, the central values of the oblique parameters are given by <cit.>:
Ŝ = 0.05 ±0.11 ,    T̂ = 0.09 ±0.13,     Û=0.01±0.11.

In the I(1+1)HDM these constraints impose a strict order on the masses of the inert particles, with two neutral dark scalars being lighter than the charged particle. Furthermore, mass splitting between heavier neutral scalar and charged scalar is limited to roughly 50 GeV. However, in the case of a $Z_2 \times Z_2'$ 3HDM,  these conclusions are no longer necessary. Cancellations between contributions to $S,T,U$ parameters from the two generations of dark particles may lead to a different mass orderings, where either of $A_i$ or $H^\pm_i$ is the heaviest, as well as to an increased mass splittings between these particles (for a detailed discussion, see <cit.>).

Collider searches for new physics The presence of additional scalars, especially if they are sufficiently light, can influence properties of SM particles, e.g., their decay channels and widths. We forbid decays of EW gauge bosons into new scalars by enforcing:

m_H_i+m_H^±_i ≥ m_W^±,   m_A_i+m_H^±_i ≥ m_W^±,
m_H_i+m_A_i ≥ m_Z,
2 m_H_i^± ≥ m_Z.
Furthermore, we adopt LEP 2 searches for supersymmetric particles re-interpreted for the I(1+1)HDM in order to exclude the region of masses where the following conditions are simultaneously satisfied <cit.> ($i=1,2$):

m_A_i ≤ 100 ,
m_H_i ≤ 80 ,
Δm= |m_A_i-m_H_i| ≥ 8 ,
since this would lead to a visible di-jet or di-lepton signal.

The model also must agree with null results for additional neutral scalar searches at the LHC.
As discused in <cit.>, current searches at the LHC for multi-lepton final states with missing transverse energy, $\cancel{E}_T$, are, in general, not sensitive enough to probe the parameter space of this model. This is mainly due to a relatively large cut on $\cancel{E}_T$ used in  experimental analyses, which results in a reduced sensitivity to probe the viable parameter space of the I(2+1)HDM scenario. Notice also that, as new charged particles are inert and hence do not couple to fermions, they are not subject to many constraints present in the 2HDM framework, e.g., flavour bounds on the charged scalar mass from $b\to s \gamma$, are not applicable here.

Charged scalar mass and lifetime

We take a model independent lower estimate on the masses of all charged states: $m_{H^\pm_i} > 70$ GeV ($i=1,2$) <cit.>. Furthermore, in this work we will not consider scenarios with possibly long-lived charged particles and, following <cit.>, we set the limit for a charged state lifetime to be $\tau\,\leq\,10^{-7}\,{\rm s}$ .

Higgs mass and signal strengths The combined ATLAS and CMS result for the Higgs mass is <cit.>:
m_h = 125.09±0.21  (stat.) ±0.11  (syst.)   .
The Higgs particle detected at the LHC is  in excellent agreement with the SM predictions.
By construction, the $h$ state in the  vacuum in Eq. (<ref>) is SM-like and its (tree-level) couplings to gluons, massive gauge bosons and fermions are identical to the SM values.

The Higgs total width can be modified through additional decays into light inert scalars, $S$, by contributing to the $h\to SS$ decay channel when $m_S \leq m_h/2$ as well as through modifications to decay channels already present in the SM, in particular, the $h \to \gamma \gamma$ decay. In this work we take the upper limit on the Higgs total decay width to be <cit.>:
Γ_tot ≤9.1   MeV.
The partial decay width $\Gamma(h\to \gamma\gamma)$ is modified with the respect to the SM through the presence of two charged inert scalars. In this work, we use the combined ATLAS and CMS limit for the signal strength <cit.>:
μ_γγ = 1.14^+0.19_-0.018,
ensuring a 2$\sigma$ agreement with the observation.

The latest constraints on the Higgs invisible decays from CMS and ATLAS are <cit.>:
BR(h → inv.) < 0.19   (CMS),    0.26   (ATLAS).
These constraints significantly limit the allowed values of Higgs-inert couplings for light inert states.

DM constraints
The total relic density is given by the sum of the contributions from both DM candidates $H_1$ and $H_2$,

Ω_Th^2 = Ω_H_1h^2 + Ω_H_2h^2   ,
and is constrained by Planck data <cit.> to be:
Ω_DMh^2 = 0.1200 ±0.0012.

The current strongest upper limit on the Spin-Independent (SI) scattering cross-section of DM particles off of nuclei,
$\sigma_{\rm DM-N}$, is provided by the XENON1T experiment and is relevant for all regions of DM mass <cit.>. Regarding indirect detection searches, for light DM particles annihilating into $bb$ or $\tau\tau$, the strongest constraints come from the Fermi-LAT satellite, ruling out the canonical cross-section
$\langle \sigma v\rangle \approx 3\times 10^{-26}~{\rm cm}^3/{\rm s}$ for $m_{\rm DM} \lesssim 100 \mbox{ GeV}$
For heavier DM candidates the PAMELA and Fermi-LAT experiments provide similar limits of $ \langle \sigma v\rangle \approx 10^{-25}~{\rm cm}^3/{\rm s}$ for  $m_{\rm DM}=200 \mbox{ GeV} $ in the $bb,\tau\tau$ or $WW$ channels <cit.>.

§ EXPERIMENTAL PROBES OF THE MODEL

A detailed DM analysis of the $Z_2 \times Z'_2$ symmetric I(2+1)HDM was presented in <cit.>, with an emphasis on astrophysical
probes of it. Here we focus instead on the complementarity between astrophysical and collider tests of the model.

In our present analysis, we use the  package <cit.> to calculate the relic density of the two DM candidates. During the analysis we follow the standard assumptions included in the code, namely: 1) particles within the dark sector are in thermal equilibrium;  2) they have the same kinetic temperature as that of the SM particle bath;  3) the number densities  of DM particles can differ from the equilibrium values  once their number density multiplied by their annihilation cross-section becomes too small to compete with the expansion rate of the Universe.

The lightest particle in each family is a viable DM candidate. As discussed previously, without loss of generality, we can take $H_1$ from the first family and $H_2$ from the second family to be the respective DM candidates. However, other dark particles from both families have a significant impact on the final relic abundances of these two stable particles, as they influence the thermal evolution and decoupling rate of DM particles.

In the discussion that follows, $x_{a}$ denotes any relevant dark particle for a particular process from a respective dark sector. There are two[Due to the imposed symmetry there are no processes that would be classified as semi-annihilation, i.e., processes of the form $x_a x_b \to \textrm{SM } x_d$. ] distinct classes of processes that can influence number densities of dark particles in each sector.
In the first class there are (co)annihilation processes of the type:
x_a x_a → SM SM.
In this class of processes, we have the standard DM annihilation, $H_i H_i \to  \textrm{ SM SM}$ with $i=1,2$, whose product depends mostly on the mass of the DM particles: we observe mostly Higgs-mediated annihilation into fermions for relatively light DM ($m_{\rm DM} \lesssim m_h/2$), while heavier DM particles annihilate predominantly into gauge boson pairs, either directly or through Higgs $s$-channel.

In our calculations, we also include annihilation into virtual gauge bosons, as these processes have a significant impact on DM annihilation rates for medium DM masses ($m_h/2 \lesssim m_{\rm DM} \lesssim m_{W^\pm}$).
Furthermore, if the mass difference between the DM candidate and other neutral or charged inert scalars from the same generation is small, then coannihilation channels such as $H_i A_i \to Z \to \textrm{SM SM}$ play an essential role.
This is, in fact, the dominant class of processes for relatively high DM masses ($m_{\rm DM} \gtrsim 500$ GeV), exactly as it happens in the I(1+1)HDM. We would like to stress that, due to the $Z_2 \times Z'_2$ imposed symmetry in this class of processes, the two DM sectors are separated.
There are no vertices that involve fields from two separate families, e.g., Higgs or gauge bosons couple only to a pair of inert particles from the same generation.

The second class of processes is DM conversion in which a pair of heavier scalars from one generation converts, either directly or through interaction with an SM particle, into a pair of dark particles from the other generation:
x_a x_a →x_b x_b   .
Note that the conversion between two generations of DM particles occurs even if all self-interaction couplings are switched off. In Fig. <ref>, diagrams (a) and (b) represent the same initial/final state where a pair of $H_2$ particles are converted to a pair of $H_1$ particles through either Higgs-mediated or direct conversion.
Even if the self-interaction parameter $\Lambda_1$ was set to zero, there would still be a non-zero contribution coming from diagram (a), as long as both particles couple to the Higgs boson ($\Lambda_{2,3}\neq 0$).
We expect cancellations or enhancements depending on the relative sign of $\Lambda_2 \Lambda_3$ and $\Lambda_1$.

Furthermore, depending on the masses and couplings, we also need to take into account annihilation of heavier dark particles from the second generation directly into stable particles from the first generation, e.g., $A_2 A_2 \to h \to H_1 H_1$. All these processes are automatically included in our numerical analysis.

(m) at ( 0, 0);
(n) at (1.5,0);
(a) at (-1.5,-1.5) \(H_2\);
(b) at ( 3,-1.5) \(H_1\);
(c) at (-1.5, 1.5) \(H_2\);
(d) at ( 3, 1.5) \(H_1\);
(e) at ( 1, -2) \((a)\);
(f) at ( 3.5, 0) \(\propto \Lambda_2 \cdot \Lambda_3\);
(a) – [scalar] (m),
(b) – [scalar] (n),
(c) – [scalar] (m),
(d) – [scalar] (n),
(n) – [scalar, edge label'=\(h\)] (m),

(m) at ( 0, 0);
(a) at (-1.5,-1.5) \(H_2\);
(b) at ( 1.5,-1.5) \(H_1\);
(c) at (-1.5, 1.5) \(H_2\);
(d) at ( 1.5, 1.5) \(H_1\);
(e) at ( 0, -2) \((b)\);
(f) at ( 2, 0) \(\propto \Lambda_1\);
(a) – [scalar] (m) – [scalar] (c),
(b) – [scalar] (m) – [scalar] (d),

(m) at ( 0, 0);
(n) at (1.5,0);
(a) at (-1.5,-1.5) \(A_2\);
(b) at ( 3,-1.5) \(H_1\);
(c) at (-1.5, 1.5) \(H_2\);
(d) at ( 3, 1.5) \(A_1\);
(e) at ( 1, -2) \((c)\);
(f) at ( 3, 0) \(\propto g^2\);
(a) – [scalar] (m),
(b) – [scalar] (n),
(c) – [scalar] (m),
(d) – [scalar] (n),
(n) – [boson, edge label'=\(Z\)] (m),           ;

Example of DM conversion diagrams: (a) Higgs-mediated conversion of $H_2 H_2 \to H_1 H_1$, present always as long as $\Lambda_{2,3} \neq 0$; (b) direct DM conversion depending on the self-interaction parameter $\Lambda_1$; (c) $Z$-mediated conversion due to coannihilation channels.

As discussed in Section <ref> in Eq. (<ref>) we can parametrise the model by using masses of scalar particles and their couplings as follows.

* The masses of  the inert particles, $m^2_{H_1}, m^2_{H_2}, m^2_{A_1}, m^2_{A_2}, m^2_{H^\pm_1}, m^2_{H^\pm_2}$ dictate the annihilation patterns of DM particles. Depending on the absolute values of masses, but also on mass splittings between particles, we can expect different dominant channels of annihilation, coannihilation and conversion. Furthermore, both absolute and relative values of these masses will result in different possible collider signatures.

* The couplings of the DM particles to the Higgs boson, $\Lambda_2$ and $\Lambda_3$, govern not only DM annihilation and conversion but also influence possible invisible decays of the Higgs particle as well as direct and indirect detection of DM.
In our numerical analysis we find that, in particular, the following vertices have significant impact on DM phenomenology:
g_hH_1H_1    =    2λ_3 +λ_31 + λ'_31 = 2 Λ_3  ,

g_hH_2H_2    =    2λ_2 +λ_23 + λ'_23 = 2 Λ_2  ,

g_hA_1A_1    =    -2λ_3 +λ_31 + λ'_31 = 2Λ_3 + 2(m^2_A_1 - m^2_H_1)/v^2   ,

g_hA_2A_2    =     -2λ_2 +λ_23 + λ'_23 = 2Λ_2 + 2(m^2_A_2 - m^2_H_2)/v^2  .

* The self-couplings of dark particles, $\lambda_1, \lambda'_{12}, \lambda_{12}$ and $\lambda_{11}, \lambda_{22}$, play two different roles. The first set governs interactions between two different families and will have an observable impact on DM relic abundance through DM conversion processes. In particular, the following couplings have a crucial impact on DM phenomenology:
g_H_1H_1H_2H_2    =     - 2λ_1 +λ_12 + λ'_12 =  4Λ_1 -(λ_12 + λ'_12)  ,

g_A_1A_1H_2H_2  = g_A_2A_2H_1H_1    =     -4λ_1 +λ_12 + λ'_12 =-4Λ_1 + 2(λ_12 + λ'_12)   .

In turn, $\lambda_{11}$ and $\lambda_{22}$ do not directly contribute to any observable process and do not influence the DM abundance. They also have no impact on any observable collider processes. However, they have a fundamental impact on the range of other parameters through vacuum stability conditions.

Our numerical analysis shows that, due to the existence of the conversion processes, the total DM relic density receives its dominant contribution from $H_1$  while the contribution from $H_2$ is of a few percent of the total DM relic density.

§.§ BPs

In our previous analysis <cit.>, we showed that, as a consequence of both DM candidate masses being in the EW region, astrophysical tools could be used to probe different DM components in a complementary way. The lighter DM candidate, $H_1$, from which the total DM relic density receives dominant contribution, could have a detectable effect in the spectrum of the nuclear recoil energy measured at direct direction experiments (e.g., XENONnT/LZ or DARWIN) <cit.>.
Complementary to that, the heavier DM candidate, $H_2$, whose contribution to the total DM relic density is sub-dominant, may be detectable in its effect in enhancing the  photon flux coming from the galactic centre measured at indirect detection experiments (e.g., FermiLAT) <cit.>.

In the present paper, we focus on yet another complementary and independent probe of the two-component DM nature of this model, namely its collider signatures.
We study  scalar cascade decays into the two different DM candidates, $H_1$ and $H_2$.
Since the latter have comparable masses, we show that their presence is detectable simultaneously. Furthermore,  their mass difference leads to different shapes in the distributions of several observables such as $\cancel{E}_T$ and visible transverse momenta, as we will show.

[gluon] (0,0) – node[black,above,xshift=-0.6cm,yshift=0.4cm] $g$ (1,-1);
[gluon] (0,-2) – node[black,above,yshift=-1.0cm,xshift=-0.6cm] $g$ (1,-1);
[dashed] (1,-1) – node[black,above,xshift=0.0cm,yshift=0.0cm] $h$ (2.5,-1);
[dashed] (2.5,-1) – node[black,above,xshift=-0.2cm,yshift=0cm] $A_1$ (3.5,0);
[photon] (3.5,0) – node[black,above,xshift=0.5cm,yshift=-0.2cm] $Z^*$ (4.5,-1);
[dashed] (3.5,0) – node[black,above,xshift=0.6cm,yshift=-0.2cm] $H_1$ (4.5,1);

[dashed] (2.5,-1) – node[black,above,yshift=-0.4cm,xshift=-0.3cm] $A_1$ (2.5,-2.4);
[dashed] (2.5,-2.4) – node[black,above,xshift=0.6cm,yshift=-0.2cm] $H_1$ (3.5,-1.4);
[photon] (2.5,-2.4) – node[black,above,xshift=0.5cm,yshift=-0.2cm] $Z^*$ (3.5,-3.4);
[xshift=-0cm] (1,-1) node[circle,fill,inner sep=3pt](A) – (1,-1);

[gluon] (0,0) – node[black,above,xshift=-0.6cm,yshift=0.4cm] $g$ (1,-1);
[gluon] (0,-2) – node[black,above,yshift=-1.0cm,xshift=-0.6cm] $g$ (1,-1);
[dashed] (1,-1) – node[black,above,xshift=0.0cm,yshift=0.0cm] $h$ (2.5,-1);
[dashed] (2.5,-1) – node[black,above,xshift=-0.2cm,yshift=0cm] $A_2$ (3.5,0);
[photon] (3.5,0) – node[black,above,xshift=0.5cm,yshift=-0.2cm] $Z^*$ (4.5,-1);
[dashed] (3.5,0) – node[black,above,xshift=0.6cm,yshift=-0.2cm] $H_2$ (4.5,1);

[dashed] (2.5,-1) – node[black,above,yshift=-0.4cm,xshift=-0.3cm] $A_2$ (2.5,-2.4);
[dashed] (2.5,-2.4) – node[black,above,xshift=0.6cm,yshift=-0.2cm] $H_2$ (3.5,-1.4);
[photon] (2.5,-2.4) – node[black,above,xshift=0.5cm,yshift=-0.2cm] $Z^*$ (3.5,-3.4);
[xshift=-0cm] (1,-1) node[circle,fill,inner sep=3pt](A) – (1,-1);

Diagrams leading to the $4\ell + \Et$ final state mediated by the $h$ boson.

Two candidates for  relevant processes here would be $gg \to h \to A_{1}A_{1} \to Z^* H_{1} \, Z^* H_{1}$ and $gg \to h \to A_{2}A_{2} \to Z^* H_{2} \, Z^* H_{2}$, whose final states appear as $4\ell + \cancel{E}_T$,  as shown in Fig. <ref>.
Since the mass splitting of the two DM candidates, $m_{H_2}-m_{H_1}$, is larger than the $\cancel{E}_T$ resolution at the LHC, one could potentially see the effect of the two DM components in different distributions.
These channels are particularly interesting since they are sensitive to the $hA_1A_1$ and $hA_2A_2$ couplings, which highlight an important characteristic of the model, i.e., the fact that, due to the sub-dominant relic density of $H_2$, the couplings of the heavier DM family to the SM-like Higgs boson could be exceptionally large ($g_{hA_2A_2} \sim 0.5$) while still in agreement with all collider and astrophysical bounds.
However, the cross-sections for these processes are very small ($\sim \, 10^{-7}$ fb for the first family and $\sim \, 10^{-4}$ fb for the second family) to have any visible effect in collider searches. As a result, we do not discuss this process any further.

[particle] (10,0) – node[black,above,xshift=-0.6cm,yshift=0.4cm] $q$ (11,-0.75);
[antiparticle] (10,-1.5) – node[black,above,yshift=-1.0cm,xshift=-0.6cm] $\bar{q}$ (11,-0.75);
[photon] (11,-0.75) – node[black,above,xshift=0.0cm,yshift=0.0cm] $Z^*$ (12,-0.75);
[dashed] (12,-0.75) – node[black,above,yshift=0.1cm,xshift=0.8cm] $H_1$ (13,-0);
[dashed] (12,-0.75) – node[black,above,yshift=-0.3cm,xshift=-0.3cm] $A_{1}$ (12,-1.75);
[dashed] (12,-1.75) – node[black,above,yshift=0.1cm,xshift=0.8cm] $H_1$ (13,-1);

[photon] (12,-1.75) – node[black,above,xshift=0.6cm,yshift=-0.9cm] $Z^{(*)}$ (13,-2.5);

[particle] (10,0) – node[black,above,xshift=-0.6cm,yshift=0.4cm] $q$ (11,-0.75);
[antiparticle] (10,-1.5) – node[black,above,yshift=-1.0cm,xshift=-0.6cm] $\bar{q}$ (11,-0.75);
[photon] (11,-0.75) – node[black,above,xshift=0.0cm,yshift=0.0cm] $Z^*$ (12,-0.75);
[dashed] (12,-0.75) – node[black,above,yshift=0.1cm,xshift=0.8cm] $H_2$ (13,-0);
[dashed] (12,-0.75) – node[black,above,yshift=-0.3cm,xshift=-0.3cm] $A_{2}$ (12,-1.75);
[dashed] (12,-1.75) – node[black,above,yshift=0.1cm,xshift=0.8cm] $H_2$ (13,-1);
[photon] (12,-1.75) – node[black,above,xshift=0.6cm,yshift=-0.9cm] $Z^{(*)}$ (13,-2.5);

Diagrams leading to the $2\ell + \Et$ final state mediated by the $Z$ boson.

The ideal processes for our analysis are the $q \bar{q} \to Z^* \to H_1 A_1 \to H_1 H_1 Z^*$ and $q \bar{q} \to Z^* \to H_2 A_2 \to H_2 H_2 Z^*$ ones,
whose final states appear as $2\ell + \cancel{E}_T$, as shown in Fig. <ref>.
To highlight these signatures, we have chosen four BPs as shown in Tab. <ref>. The cross-sections for the $pp\to 2H_1 + 2\ell$ and $pp\to 2H_2 + 2\ell$ processes for the four BPs are shown in Tab. <ref>.

The four BPs, for which we have chosen $m_{H_1} = 50$ GeV, $m_{H_2} = 1000$ GeV and $g_{hH_1H_1}=0.00002$, and we have set $\lambda_{11}=0.11$, $\lambda_{22}=0.12$, $\lambda_{12}=0.121$, $\lambda'_{12}=0.13$, $\Lambda_1 = \Lambda_3 = 0.00001$, the SM Higgs mass $m_{h}=125$ GeV and the VEV $v=246$ GeV, are in agreement with all astrophysical and collider constraints.

We now proceed to a Monte Carlo (MC) analysis of these BPs.

§ NUMERICAL RESULTS

The integrated and differential cross-sections have been calculated using  <cit.> by adopting  a generic LHC parameter card.
Furthermore, we have used  <cit.>  for constructing our selection, which  involve  basic cuts for leptons $\ell$ and jets
$j$[The latter being produced from QCD Initial State Radiation (ISR) and clustered with  the Cambridge/Aachen algorithm with a 0.4 cone size <cit.>.]: in pseudorapidity $|\eta (\ell) |,  |\eta (j) | < 3$,  transverse momentum $p_T (\ell ), p_T(j) > 10$ GeV as well as separation $\Delta R (j, \ell) >0.5$.  Finally,  event rates have been computed considering $100$ fb$^{-1}$ of  luminosity for the LHC machine.

The cross-sections for the two processes $q \bar{q} \to Z^* \to H_1 A_1 \to H_1 H_1 Z^*$ and $q \bar{q} \to Z^* \to H_2 A_2 \to H_2 H_2 Z^*$ given  in Tab. <ref> are instrumental to enable the extraction of distributions sensitive to the simultaneous presence of the two DM candidates at the forthcoming
LHC Run 3 as they are both visible and comparable to each other.
As mentioned before, in all four BPs presented herein,  within each family the cross-section depends on both the absolute value of the masses involved and their mass splittings $M_{A_i}-M_{H_i}$ ($i = 1, 2$), with the latter having been chosen (in agreement with all astrophysical bounds) to be larger than or comparable to the expected experimental resolutions in (missing) transverse energy/momentum and (transverse  or invariant) mass. Therefore, we shall be looking for differential spectra with a distinctive shape from which the existence of  two different underpinning component distributions could be easily inferred.
As a proof of concept, we show such observables for both the aforementioned two processes limitedly to one BP, e.g.,  L2n2. (Results are qualitatively
similar for the other BPs, so we do not discuss these here.)
For this choice, it is worth highlighting that the inert masses involved are as follows:  $m_{H_1}=50$ GeV with $m_{A_1}-m_{H_1} =9.3$ GeV for one  doublet and $m_{H_2}=100$ GeV with $m_{A_2}-m_{H_2} =30.06$ GeV for the other.

$p p \to 2H_1+ 2 \ell$      $p p \to 2H_2+ 2 \ell$

L2n2   $ \sigma = 1.058 $ [fb]     $ \sigma = 1.929 $ [fb]

L2p1   $ \sigma = 1.043 $ [fb]    $ \sigma = 1.912 $ [fb]

L2p2    $ \sigma = 0.9169 $ [fb]    $ \sigma = 1.230 $ [fb]

L2p3    $ \sigma = 0.9797 $ [fb]   $ \sigma = 2.448 $ [fb]

The cross-sections for the processes yielding $2\ell + \cancel{E}_T$ final states, whose leading contribution come from the
$q \bar{q} \to Z^* \to H_1 A_1 \to H_1 H_1 Z^*$ and $q \bar{q} \to Z^* \to H_2 A_2 \to H_2 H_2 Z^*$ channels shown in Fig. <ref>.

In Fig. <ref>, we show the $ \cancel{E}_T$ spectrum on the left and that of the transverse momentum of either lepton on the right, for the two channels
$p p \to \ell \bar{\ell} + H_1H_1$
and  $p p \to \ell \bar{\ell}+ H_2H_2$ separately. The two sets of distributions are strikingly different so that, even when summed, one could clearly deduce the
presence of two different DM candidates with different masses. In the $ \cancel{E}_T$ histogram, one can see two different peaks, a sharp one around 25 GeV for the $H_1$ case
and a smoother one around 35 GeV for the $H_2$ case, this tracking the fact that $m_{H_1}\ll m_{H_2}$ (even though the presence of two DM particles in the event in each case spoils somewhat the correlations). In the
$p_T^\ell $ spectra, it can be noticed the much sharper decrease of the $H_1$ distribution with respect to the $H_2$ one, this correlating to the fact that $m_{A_1}-m_{H_1} \ll m_{A_2}-m_{H_2} $.

Missing transverse energy (left) and transverse momentum of either  lepton (right) in the case of  $H_1$ and $H_2$ separately.

In Fig. <ref>, one can notice that leptons are similarly central (see the pseudorapidity plot of either of these on the left) while their separation
is noticeably different (see their cone size on the right) between the two DM candidates, the former spectrum showing that the overall events are not generally boosted but  the $Z^*$ itself can be so and differently between the $H_1$ and $H_2$ cases.

Pseudorapidity of either lepton (left) and separation between the leptons (right)  in the case of  $H_1$ and $H_2$ separately.

Fig. <ref> shows the invariant mass of the leptons, defined through $M_{\ell\ell}^2=(p_{\ell^+}+p_{\ell^-})^2$,  which is also correlated with the $A_i$ and $H_i$ mass splitting in each process, thus
explaining the low mass peak for the $H_1$ case and the high mass peak for the $H_2$ case.

The invariant mass of the final state leptons  in the case of  $H_1$ and $H_2$ separately.

Fig. <ref> shows the transverse mass of the final state leptons, defined through $M_T^2(\ell \ell) = (\sum_i  E_{Ti})^2 - (\sum_i p_{Ti})^2$ , which
is rather strongly correlated to  the $A_1$ and $A_2$ masses for the $H_1$ and $H_2$ cases, respectively.

Therefore, there exist several distributions which are significantly different in shape between the two DM candidates while also yielding sizeable event rates for
LHC Run 3 luminosities, so that one could not only establish the presence of $H_1$ and $H_2$ simultaneously but also attempt to extract the underlying inert mass spectra from fitting the ensuing data to the I(2+1)HDM predictions. All this is clearly subject to validation through a MC analysis in presence of both reducible and
irreducible backgrounds from  the SM.

The transverse mass of the final state leptons and DM candidates  in the case of  $H_1$ and $H_2$ separately.

§ CONCLUSIONS

In this paper, we have studied a I(2+1)HDM framework symmetric under a $Z_2\times Z'_2$ group with one inert doublet being odd under $Z_2$ and even under  $Z'_2$ and the other inert doublet being even under $Z_2$ and odd under $Z'_2$, while all SM particles transform trivially under the $Z_2\times Z'_2$ symmetry.
The lightest particle from each inert doublet is a viable DM candidate, resulting in a two-component DM model.
In a recent publication <cit.>, we showed that, when there is a sufficient mass difference between the two DM candidates, which are both typically at the EW scale, the light DM component can be probed by the nuclear recoil energy in direct detection experiments while the heavy DM  component appears through its contribution to the photon flux in indirect detection experiments.

Here, in addition to such astrophysical probes, we have shown that certain collider signatures can serve the same purpose; in each family, the decay of the heavier neutral inert state into the lighter one (i.e., the  DM candidate) plus an off-shell (secondary) $Z$ boson (decaying to di-lepton pairs) in association with an additional identical DM candidate emerging in parallel from primary production via an off-shell (primary) $Z$ boson.
The smoking-gun signature of this two-component DM scenario is thus $2\ell + \cancel{E}_T$, which can in fact be pursued already at the upcoming Run 3 of the LHC. Herein, one could
study a variety of differential distributions stemming  from this final state that would have a distinctive shape carrying the imprint of two underlying components, each
corresponding to a different DM candidate. Indeed, those having an energy dimension could even be used to extract the underpinning inert state masses involved.

While these conclusions have been obtained without a signal-to-background analysis, we encourage ATLAS and CMS experimentalists to look into this novel phenomenology, as it would manifest itself in a well-studied final state at  the LHC machine.

§.§.§ Acknowledgement

SM acknowledges support from the STFC Consolidated Grant ST/L000296/1 and is partially financed through the NExT Institute.
DS is supported by the National Science Center, Poland, through the HARMONIA project under contract UMO-2015/18/M/ST2/00518.
VK acknowledges financial support from the Academy of Finland projects “Particle cosmology and gravitational waves” No. 320123
and “Particle cosmology beyond the Standard Model” No. 310130.
J H-S acknowledges support from SNI-CONACYT, PRODEP-SEP and VIEP-BUAP.
The authors acknowledge the use of the IRIDIS High-Performance Computing Facility and associated support services at the University of Southampton in the completion of this work.