# Electromagnetic multipole moments of the $P_{c}^{+}(4380)$ pentaquark in
light-cone QCD

U. Özdem<EMAIL_ADDRESS>Department of Physics, Dogus University,
Acibadem-Kadikoy, 34722 Istanbul, Turkey  K. Azizi<EMAIL_ADDRESS>Department of Physics, Dogus University, Acibadem-Kadikoy, 34722 Istanbul,
Turkey  School of Physics, Institute for Research in Fundamental Sciences
(IPM), P. O. Box 19395-5531, Tehran, Iran

###### Abstract

We calculate the electromagnetic multipole moments of the $P_{c}^{+}(4380)$
pentaquark by modeling it as the diquark-diquark-antiquark and
$\bar{D}^{*}\Sigma_{c}$ molecular state with quantum numbers
$J^{P}=\frac{3}{2}^{-}$. In particular, the magnetic dipole, electric
quadrupole and magnetic octupole moments of this particle are extracted in the
framework of light-cone QCD sum rule. The values of the electromagnetic
multipole moments obtained via two pictures differ substantially from each
other, which can be used to pin down the underlying structure of
$P_{c}^{+}(4380)$. The comparison of any future experimental data on the
electromagnetic multipole moments of the $P_{c}^{+}(4380)$ pentaquark with the
results of the present work can shed light on the nature and inner quark
organization of this state.

Pentaquarks, Electromagnetic form factors, Multipole moments, Molecule,
Diquark-diquark-antiquark

##  I Introduction

Since the discovery of the X(3872), many charmonium/bottomonium-like XYZ
states have been reported in the experiment. Some of these hadrons were
suggested to have internal structures more complex than the simple $\bar{q}q$
configuration for mesons or qqq/$\bar{q}\bar{q}\bar{q}$ configuration for
baryon/antibaryons in the conventional picture of the naive quark model, and
they are good candidates of exotic hadrons. In the newly observed family of
XYZ, there are some decay channels that break the isospin symmetry and affect
the identification of the traditional charmonium/bottomonium states
negatively. The investigation of the properties of these states is one of the
most attractive and active branches of hadron physics. For some reviews on the
theoretical and experimental progress on the properties of these new states
see Refs. Nielsen  _et al._ (2010); Swanson (2006); Voloshin (2008); Klempt
and Zaitsev (2007); Godfrey and Olsen (2008); Faccini  _et al._ (2012);
Esposito  _et al._ (2015); Chen  _et al._ (2016a); Ali  _et al._ (2017);
Esposito  _et al._ (2016); Olsen  _et al._ (2017); Lebed  _et al._ (2017). In
2015, the LHCb Collaboration discovered two candidates of the hidden-charm
pentaquark states, $P_{c}^{+}(4380)$ and $P_{c}^{+}(4450)$, in the invariant
mass spectrum of $J/\psi\,p$ in the $\Lambda_{b}^{0}\rightarrow
J/\psi\,K^{-}\,p$ decay Aaij  _et al._ (2015). According to the LHCb
measurements the $P_{c}^{+}(4380)$ has a mass of $4380\pm 8\pm 29$ MeV and a
width of $205\pm 18\pm 86$ MeV, while the $P_{c}^{+}(4450)$ has a mass of
$4449.8\pm 1.7\pm 2.5$ MeV and a width of $39\pm 5\pm 19$ MeV. The preferred
spin-parity assignments of the $P_{c}(4380)$ and $P_{c}(4450)$ are
$J^{P}=3/2^{-}$ and $5/2^{+}$, respectively. The minimal quark content of the
pentaquarks is $c\bar{c}uud$ because these states decay into $J/\psi\,p$, and
hence they are good candidates of exotic hidden-charm pentaquarks. After the
discovery of LHCb Collaboration there have been intensive theoretical studies
to explain the properties of these states. The spectroscopic parameters and
decays of the $P_{c}^{+}(4380)$ and $P_{c}^{+}(4450)$ pentaquarks have been
studied with different models and approaches Yang and Ping (2017); Burns
(2015); Lü and Dong (2016); Wang  _et al._ (2016a); Shimizu and Harada (2017);
Shen  _et al._ (2016); Azizi  _et al._ (2017); Roca  _et al._ (2015); Chen
_et al._ (2015a); Huang  _et al._ (2016); Meißner and Oller (2015); Xiao and
Meißner (2015); He (2016); Chen  _et al._ (2015b); Wang  _et al._ (2016b);
Chen  _et al._ (2016b); Yamaguchi and Santopinto (2017); He (2017); Zhu and
Qiao (2016); Lebed (2015); Anisovich  _et al._ (2015); Maiani  _et al._
(2015); Ghosh  _et al._ (2015); Wang (2016); Scoccola  _et al._ (2015); Liu
and Zahed (2017); Guo  _et al._ (2015); Liu  _et al._ (2016); Guo  _et al._
(2016); Bayar  _et al._ (2016); Lin  _et al._ (2017); Eides  _et al._ (2018);
Azizi  _et al._ (2018); Park  _et al._ (2017); Ortega  _et al._ (2017);
Shimizu  _et al._ (2016); Chen  _et al._ (2016c). Different theoretical models
give consistent mass results with the experimental observations. Hence, more
spectroscopic and decay parameters are needed to be calculated and compared
with the experimental data. In Azizi  _et al._ (2018) it is shown that the
molecular picture of $\bar{D}^{*}\Sigma_{c}$ for $P_{c}^{+}(4380)$ gives
consistent results for both the mass and width with the experimental data.

As we mentioned above, chasing the announcement of the observation of
pentaquarks there have been extensive amount of studies on their features.
However to acquire a deep understanding on their inner structure, which are
still not precise yet, we are in need of more experimental and theoretical
studies which may shed light on their features. In order to understand the
internal structure of the hadrons in the nonperturbative regime of QCD, the
essential challenges are the specification of the dynamical and statical
properties of hadrons such as their electromagnetic multipole moments,
coupling constants, masses and so on, both theoretically and experimentally.
Many theoretical models precisely predict the mass and decay width of the
multiquark states, but the internal structure of these states is still
uncertain. In other words, the mass and decay width alone can not distinguish
the internal structure of the multiquark states. Remember that the
electromagnetic multipole moments are equally significant dynamical
observables of the multiquark states. The electromagnetic multipole moments
are directly related with the charge and current distributions in the hadrons
and these parameters are directly connected to the spatial distributions of
quarks and gluons inside the hadrons. Their magnitude and sign provide
important information on structure, size and shape of hadrons. There are many
studies in the literature committed to the study the electromagnetic multipole
moments of the standard hadrons, but unfortunately relatively little are known
about the electromagnetic multipole moments of the exotic hadrons. There are a
few studies in the literature where the magnetic dipole moment of the
pentaquarks are studied Kim and Praszalowicz (2004); Huang  _et al._ (2004);
Liu  _et al._ (2004); Wang  _et al._ (2005, 2006, 2016a); Bijker  _et al._
(2004); Li  _et al._ (2004).

In this study, the magnetic dipole, electric quadrupole and magnetic octupole
moments of the pentaquark state $P_{c}^{+}(4380)$ (hereafter we will denote
this state as $P_{c}$) is extracted by using the diquark-diquark-antiquark and
$\bar{D}^{*}\Sigma_{c}$ molecular interpolating currents in the framework of
the light cone QCD sum rule (LCSR). The LCSR has already been successfully
applied to extract properties of hadrons for decades such as, form factors,
coupling constants and the electromagnetic multipole moments. In this
approach, the properties of the hadrons are expressed in terms of the light-
cone distribution amplitudes (DAs) and the vacuum condensates [for details,
see for instance Chernyak and Zhitnitsky (1990); Braun and Filyanov (1989);
Balitsky  _et al._ (1989)]. Since the electromagnetic multipole moments are
expressed in terms of the features of the DAs and the QCD vacuum, any
uncertainty in these parameters reflects the uncertainty of the estimations of
the electromagnetic multipole moments.

The rest of the paper is organized as follows: In section II, the calculation
of the sum rules in LCSR will be presented. In the last section, we
numerically analyze the sum rules obtained for the electromagnetic multipole
moments and discuss the obtained results. The explicit expressions of the
electromagnetic form factors defining the magnetic dipole, electric quadrupole
and magnetic octupole moments are moved to the Appendix A.

##  II The electromagnetic multipole moments of $P_{c}$ pentaquark in LCSR

In this section we derive the LCSR for the magnetic dipole, electric
quadrupole and magnetic octupole moments of the $P_{c}$ pentaquark. For this
purpose, we consider a correlation function in the presence of the external
electromagnetic field ($\gamma$),

$\Pi_{\mu\nu}(q)=i\int d^{4}xe^{ip\cdot x}\langle
0|\mathcal{T}\\{J_{\mu}(x)\bar{J}_{\nu}(0)\\}|0\rangle_{\gamma},$ (1)

where $J_{\mu}$ is the interpolating current of $P_{c}$ pentaquark. In the
diquark-diquark-antiquark and molecular pictures, it is given as Wang (2016);
Chen  _et al._ (2015b)

$\displaystyle J_{\mu}^{Di}(x)$ $\displaystyle=$
$\displaystyle\varepsilon^{abc}\varepsilon^{ade}\varepsilon^{bfg}\big{[}u^{T}_{d}(x)C\gamma_{5}d_{e}(x)\,u^{T}_{f}(x)C\gamma_{\mu}c_{g}(x)\,C\bar{c}^{T}_{c}(x)\big{]},$
$\displaystyle J_{\mu}^{Mol}(x)$ $\displaystyle=$
$\displaystyle\big{[}\bar{c}_{d}(x)\gamma_{\mu}d_{d}(x)\big{]}\big{[}\epsilon_{abc}(u_{a}^{T}(x)C\gamma_{\alpha}u_{b}(x))\gamma^{\alpha}\gamma_{5}c_{c}(x)\big{]},$
(2)

where $C$ is the charge conjugation matrix; and $a$, $b$… are color indices.

The correlation function, given in Eq. (1), can be obtained in terms of
hadronic parameters, known as hadronic representation. Furthermore it can be
obtained in terms of the quark-gluon parameters and distribution amplitudes
(DAs) of the photon in the deep Euclidean region, known as QCD representation.

The hadronic side of the correlation function can be obtained by inserting
complete sets of the hadronic pentaquarks, between the interpolating currents
in Eq. (1), with the same quantum numbers as the ${P_{c}}$ interpolating
currents, i.e.,

$\displaystyle\Pi^{Had}_{\mu\nu}(p,q)$ $\displaystyle=$
$\displaystyle\frac{\langle 0\mid
J_{\mu}\mid{P_{c}}(p)\rangle}{[p^{2}-m_{{P_{c}}}^{2}]}\langle{P_{c}}(p)\mid{P_{c}}(p+q)\rangle_{\gamma}\frac{\langle{P_{c}}(p+q)\mid\bar{J}_{\nu}\mid
0\rangle}{[(p+q)^{2}-m_{{P_{c}}}^{2}]},$ (3)

where q is the momentum of the photon. The matrix element of the interpolating
current between the vacuum and the $P_{c}$ pentaquark is defined as

$\langle 0\mid
J_{\mu}(0)\mid{P_{c}}(p,s)\rangle=\lambda_{{P_{c}}}u_{\mu}(p,s),$ (4)

where $\lambda_{{P_{c}}}$ is the residue and $u_{\mu}(p,s)$ is the Rarita-
Schwinger spinor. Summation over spins of ${P_{c}}$ pentaquark is applied as:

$\displaystyle\sum_{s}u_{\mu}(p,s)\bar{u}_{\nu}(p,s)=-\Big{(}p\\!\\!\\!/+m_{{P_{c}}}\Big{)}\Big{[}g_{\mu\nu}-\frac{1}{3}\gamma_{\mu}\gamma_{\nu}-\frac{2\,p_{\mu}p_{\nu}}{3\,m^{2}_{{P_{c}}}}+\frac{p_{\mu}\gamma_{\nu}-p_{\nu}\gamma_{\mu}}{3\,m_{{P_{c}}}}\Big{]}.$
(5)

The transition matrix element
$\langle{P_{c}}(p)\mid{P_{c}}(p+q)\rangle_{\gamma}$ entering Eq. (3) can be
parameterized in terms of four Lorentz invariant form factors as follows Weber
and Arenhovel (1978); Nozawa and Leinweber (1990); Pascalutsa  _et al._
(2007); Ramalho  _et al._ (2009); Azizi (2009); Aliev  _et al._ (2009):

$\displaystyle\langle{P_{c}}(p)\mid{P_{c}}(p+q)\rangle_{\gamma}$
$\displaystyle=$
$\displaystyle-e\bar{u}_{\mu}(p)\left\\{F_{1}(q^{2})g_{\mu\nu}\varepsilon\\!\\!\\!/-\frac{1}{2m_{{P_{c}}}}\left[F_{2}(q^{2})g_{\mu\nu}+F_{4}(q^{2})\frac{q_{\mu}q_{\nu}}{(2m_{{P_{c}}})^{2}}\right]\varepsilon\\!\\!\\!/q\\!\\!\\!/\right.$
$\displaystyle+$
$\displaystyle\left.F_{3}(q^{2})\frac{1}{(2m_{{P_{c}}})^{2}}q_{\mu}q_{\nu}\varepsilon\\!\\!\\!/\right\\}u_{\nu}(p+q),$

where $\varepsilon$ is the polarization vector of the photon.

In principle, using the above equations, we can obtain the final expression of
the hadronic side of the correlation function, but we come across with two
difficulties: all Lorentz structures are not independent and the correlation
function can also receive contributions from spin-1/2 particles, which should
be eliminated. Actually, the matrix element of the current $J_{\mu}$ between
vacuum and spin-1/2 pentaquarks is nonzero and is specified as

$\langle 0\mid J_{\mu}(0)\mid
B(p,s=1/2)\rangle=(Ap_{\mu}+B\gamma_{\mu})u(p,s=1/2).$ (7)

As is seen the unwanted spin-1/2 contributions are proportional to
$\gamma_{\mu}$ and $p_{\mu}$. By multiplying both sides with $\gamma^{\mu}$
and using the condition $\gamma^{\mu}J_{\mu}=0$ one can determine the constant
A in terms of B. To remove the spin-1/2 pollutions and obtain only independent
structures in the correlation function, we apply the ordering for Dirac
matrices as
$\gamma_{\mu}p\\!\\!\\!/\varepsilon\\!\\!\\!/q\\!\\!\\!/\gamma_{\nu}$ and
eliminate terms with $\gamma_{\mu}$ at the beginning, $\gamma_{\nu}$ at the
end and those proportional to $p_{\mu}$ and $p_{\nu}$ Belyaev and Ioffe
(1983). As a result, using Eqs. (3)-(II) for hadronic side we obtain,

$\displaystyle\Pi^{Had}_{\mu\nu}(p,q)$ $\displaystyle=$
$\displaystyle-\frac{\lambda_{{}_{{P_{c}}}}^{2}}{[(p+q)^{2}-m_{{}_{{P_{c}}}}^{2}][p^{2}-m_{{}_{{P_{c}}}}^{2}]}\Bigg{[}-g_{\mu\nu}p\\!\\!\\!/\varepsilon\\!\\!\\!/q\\!\\!\\!/\,F_{1}(q^{2})+m_{{P_{c}}}g_{\mu\nu}\varepsilon\\!\\!\\!/q\\!\\!\\!/\,F_{2}(q^{2})+\frac{F_{3}(q^{2})}{4m_{{P_{c}}}}q_{\mu}q_{\nu}\varepsilon\\!\\!\\!/q\\!\\!\\!/\,$
(8) $\displaystyle+$
$\displaystyle\frac{F_{4}(q^{2})}{4m_{{P_{c}}}^{3}}(\varepsilon.p)q_{\mu}q_{\nu}p\\!\\!\\!/q\\!\\!\\!/\,+\mbox{other
independent structures}\Bigg{]}.$

The magnetic dipole, $G_{M}(q^{2})$, electric quadrupole, $G_{Q}(q^{2})$, and
magnetic octupole, $G_{O}(q^{2})$, form factors are defined in terms of the
form factors $F_{i}(q^{2})$ in the following way Weber and Arenhovel (1978);
Nozawa and Leinweber (1990); Pascalutsa  _et al._ (2007); Ramalho  _et al._
(2009); Azizi (2009); Aliev  _et al._ (2009):

$\displaystyle G_{M}(q^{2})$ $\displaystyle=$
$\displaystyle\left[F_{1}(q^{2})+F_{2}(q^{2})\right](1+\frac{4}{5}\tau)-\frac{2}{5}\left[F_{3}(q^{2})+F_{4}(q^{2})\right]\tau\left(1+\tau\right),$
$\displaystyle G_{Q}(q^{2})$ $\displaystyle=$
$\displaystyle\left[F_{1}(q^{2})-\tau
F_{2}(q^{2})\right]-\frac{1}{2}\left[F_{3}(q^{2})-\tau
F_{4}(q^{2})\right]\left(1+\tau\right),$ $\displaystyle G_{O}(q^{2})$
$\displaystyle=$
$\displaystyle\left[F_{1}(q^{2})+F_{2}(q^{2})\right]-\frac{1}{2}\left[F_{3}(q^{2})+F_{4}(q^{2})\right]\left(1+\tau\right),$
(9)

where $\tau=-\frac{q^{2}}{4m^{2}_{{P_{c}}}}$. At $q^{2}=0$, the multipole form
factors are obtained in terms of the functions $F_{i}(0)$ as:

$\displaystyle G_{M}(0)$ $\displaystyle=$ $\displaystyle F_{1}(0)+F_{2}(0),$
$\displaystyle G_{Q}(0)$ $\displaystyle=$ $\displaystyle
F_{1}(0)-\frac{1}{2}F_{3}(0),$ $\displaystyle G_{O}(0)$ $\displaystyle=$
$\displaystyle F_{1}(0)+F_{2}(0)-\frac{1}{2}[F_{3}(0)+F_{4}(0)].$ (10)

The magnetic dipole ($\mu_{{P_{c}}}$), electric quadrupole ($Q_{{P_{c}}}$) and
magnetic octupole ($O_{{P_{c}}}$) moments are defined in the following way:

$\displaystyle\mu_{{P_{c}}}$ $\displaystyle=$
$\displaystyle\frac{e}{2m_{{P_{c}}}}G_{M}(0),$ $\displaystyle Q_{{P_{c}}}$
$\displaystyle=$ $\displaystyle\frac{e}{m_{{P_{c}}}^{2}}G_{Q}(0),$
$\displaystyle O_{{P_{c}}}$ $\displaystyle=$
$\displaystyle\frac{e}{2m_{{P_{c}}}^{3}}G_{O}(0).$ (11)

The next step is to calculate the correlation function in Eq. (1) in terms of
quark-gluon parameters as well as the photon DAs in the deep Euclidean region.
For this purpose, the interpolating currents are inserted into the correlation
function and after the contracting out the quark pairs using Wick theorem the
following results are obtained:

$\displaystyle\Pi^{QCD}_{\mu\nu}(p)$ $\displaystyle=$ $\displaystyle
i\,\varepsilon^{abc}\varepsilon^{a^{\prime}b^{\prime}c^{\prime}}\varepsilon^{ade}\varepsilon^{a^{\prime}d^{\prime}e^{\prime}}\varepsilon^{bfg}\varepsilon^{b^{\prime}f^{\prime}g^{\prime}}\int
d^{4}xe^{ip\cdot x}\langle
0|\Bigg{\\{}Tr\Big{[}\gamma_{5}S_{d}^{ee^{\prime}}(x)\gamma_{5}\tilde{S}_{u}^{dd^{\prime}}(x)\Big{]}Tr\Big{[}\gamma_{\mu}S_{c}^{gg^{\prime}}(x)\gamma_{\nu}\tilde{S}_{u}^{ff^{\prime}}(x)\Big{]}\tilde{S}_{c}^{c^{\prime}c}(-x)$
(12) $\displaystyle-
Tr\Big{[}\gamma_{5}S_{d}^{ee^{\prime}}(x)\gamma_{5}\tilde{S}_{u}^{fd^{\prime}}(x)\gamma_{\mu}S_{c}^{gg^{\prime}}(x)\gamma_{\nu}\tilde{S}_{u}^{df^{\prime}}(x)\Big{]}\tilde{S}_{c}^{c^{\prime}c}(-x)\Bigg{\\}}|0\rangle_{\gamma},$

in the diquark-diquark-antidiquark picture, and

$\displaystyle\Pi_{\mu\nu}^{\mathrm{QCD}}(p)$ $\displaystyle=$
$\displaystyle-i\,\epsilon^{abc}\epsilon^{a^{\prime}b^{\prime}c^{\prime}}\,\int
d^{4}xe^{ip\cdot x}\langle
0|\Bigg{\\{}\mathrm{Tr}\Big{[}\gamma_{\mu}S_{d}^{dd^{\prime}}(x)\gamma_{\nu}S_{c}^{d^{\prime}d}(-x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{\beta}\widetilde{S}_{u}^{aa^{\prime}}(x)\gamma_{\alpha}S_{u}^{bb^{\prime}}(x)]\Big{(}\gamma^{\alpha}\gamma_{5}S_{c}^{cc^{\prime}}(x)\gamma_{5}\gamma^{\beta}\Big{)}$
(13)
$\displaystyle-\mathrm{Tr}\Big{[}\gamma_{\mu}S_{d}^{dd^{\prime}}(x)\gamma_{\nu}S_{c}^{d^{\prime}d}(-x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{\beta}\widetilde{S}_{u}^{ba^{\prime}}(x)\gamma_{\alpha}S_{u}^{ab^{\prime}}(x)\Big{]}\Big{(}\gamma^{\alpha}\gamma_{5}S_{c}^{cc^{\prime}}(x)\gamma_{5}\gamma^{\beta}\Big{)}\Bigg{\\}}|0\rangle_{\gamma},$

in the molecular picture, where

$\widetilde{S}_{c(q)}^{ij}(x)=CS_{c(q)}^{ij\mathrm{T}}(x)C,$

with $S_{q(c)}(x)$ being the quark propagator. The light (q) and heavy (c)
propagators are given as Balitsky and Braun (1989)

$\displaystyle
S_{q}(x)=i\frac{{x\\!\\!\\!/}}{2\pi^{2}x^{4}}-\frac{\bar{q}q}{12}-\frac{\bar{q}\sigma.Gq}{192}x^{2}-\frac{ig_{s}}{32\pi^{2}x^{2}}G^{\mu\nu}(x)\Big{[}\hbox
to0.0pt{/\hss}{x}\sigma_{\mu\nu}+\sigma_{\mu\nu}\hbox
to0.0pt{/\hss}{x}\Big{]},$ (14)

and

$\displaystyle S_{c}(x)$
$\displaystyle=\frac{m_{c}^{2}}{4\pi^{2}}\Bigg{[}\frac{K_{1}\Big{(}m_{c}\sqrt{-x^{2}}\Big{)}}{\sqrt{-x^{2}}}+i\frac{{x\\!\\!\\!/}~{}K_{2}\Big{(}m_{c}\sqrt{-x^{2}}\Big{)}}{(\sqrt{-x^{2}})^{2}}\Bigg{]}-\frac{g_{s}m_{c}}{16\pi^{2}}\int_{0}^{1}dv\,G^{\mu\nu}(vx)\Bigg{[}(\sigma_{\mu\nu}{x\\!\\!\\!/}+{x\\!\\!\\!/}\sigma_{\mu\nu})\frac{K_{1}\Big{(}m_{c}\sqrt{-x^{2}}\Big{)}}{\sqrt{-x^{2}}}$
$\displaystyle+2\sigma_{\mu\nu}K_{0}\Big{(}m_{c}\sqrt{-x^{2}}\Big{)}\Bigg{]},$
(15)

where $K_{i}$ are modified the second kind Bessel functions and $G^{\mu\nu}$
is the gluon field strength tensor. Note that with the above form of the light
quark propagator and considering Eqs. (12) and (13), which represent the quark
propagators between vacuum and the photon states, we include all the possible
contributions.

The correlation function includes different types of contributions. In the
first part, the photon interacts with one of the light or heavy quarks,
perturbatively. In this case, the propagator of the quark that interacts with
the photon, perturbatively is replaced by

$\displaystyle S^{free}(x)\rightarrow\int d^{4}y\,S^{free}(x-y)\,\hbox
to0.0pt{/\hss}{\\!A}(y)\,S^{free}(y)\,,$ (16)

with $S^{free}(x)$ representing the first term of the light or heavy quark
propagator, and the remaining four propagators in Eqs. (12) and (13) are
replaced with the full quark propagators including the free (perturbative)
part as well as the interacting parts (with gluon or QCD vacuum) as
nonperturbative contributions. The full perturbative contribution is obtained
by applying the above replacement for the perturbatively interacting quark
propagator with the photon and replacing the remaining propagators by their
free parts.

In the second type, one of the light quark propagators in Eqs. (12) and (13),
describing the photon emission at large distances, is replaced by

$\displaystyle
S_{\alpha\beta}^{ab}(x)\rightarrow-\frac{1}{4}\big{[}\bar{q}^{a}(x)\Gamma_{i}q^{b}(x)\big{]}\big{(}\Gamma_{i}\big{)}_{\alpha\beta},$
(17)

and the remaining propagators are replaced with the full quark propagators.
Here, $\Gamma_{i}$ are the full set of Dirac matrices. Once Eq. (17) is
plugged into Eqs. (12) and (13) , there appear matrix elements such as
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}q(0)\right|0\rangle$ and
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}G_{\alpha\beta}q(0)\right|0\rangle$,
representing the nonperturbative contributions. These matrix elements can be
expressed in terms of photon wave functions with definite twists, whose
expressions are given in Ref. Ball  _et al._ (2003). The QCD side of the
correlation function can be obtained in terms of quark-gluon properties using
Eqs. (12)-(17) and after applying the Fourier transformation to transfer the
calculations to the momentum space.

The two representations, the QCD and hadronic sides, of the correlation
function, in two different kinematical regions are then matched using
dispersion relation. Then we carry out the double Borel transforms with
respect to the variables $p^{2}$ and $(p+q)^{2}$ on both sides of the
correlation function in order to suppress the contributions of the higher
states and continuum, and use the quark-hadron duality assumption. By matching
the coefficients of the structures
$g_{\mu\nu}p\\!\\!\\!/\varepsilon\\!\\!\\!/q\\!\\!\\!/$,
$g_{\mu\nu}\varepsilon\\!\\!\\!/q\\!\\!\\!/$,
$q_{\mu}q_{\nu}\varepsilon\\!\\!\\!/q\\!\\!\\!/$ and
$(\varepsilon.p)q_{\mu}q_{\nu}p\\!\\!\\!/q\\!\\!\\!/$, respectively for the
$F_{1}$, $F_{2}$, $F_{3}$ and $F_{4}$ we find LCSR for these four invariant
form factors. The explicit expressions of the sum rules for these form factors
are given in the Appendix A. For the sake of simplicity only the results
obtained from the diquark-diquark-antiquark picture are given. The results of
the molecular picture have more or less has similar forms.

##  III Numerical analysis and conclusion

Present section is devoted to the numerical analysis for the magnetic dipole,
electric quadrupole and magnetic octupole moments of the $P_{c}$ pentaquark.
We use $m_{u}=m_{d}=0$, $m_{c}=(1.28\pm 0.03)\,GeV$ Patrignani  _et al._
(2016), $m_{P_{c}}=4.38\pm 0.37~{}GeV$ Patrignani  _et al._ (2016),
$f_{3\gamma}=-0.0039~{}GeV^{2}$ Ball  _et al._ (2003),
$\langle\bar{u}u\rangle=\langle\bar{d}d\rangle=(-0.24\pm 0.01)^{3}\,GeV^{3}$
Ioffe (2006), and $\langle g_{s}^{2}G^{2}\rangle=0.88~{}GeV^{4}$ Nielsen  _et
al._ (2010). To obtain a numerical prediction for the electromagnetic
multipole moments, we also need to specify the values of the residue of the
$P_{c}$ pentaquark. The residue is obtained from the mass sum rule as
$\lambda_{P_{c}}=(1.55\pm 0.28)\times 10^{-3}~{}GeV^{6}$ Wang (2016) for the
diquark-diquark-antiquark picture and $\lambda_{P_{c}}=(0.98\pm 0.05)\times
10^{-3}~{}GeV^{6}$ Azizi  _et al._ (2017) for molecular picture. The
parameters used in the photon distribution amplitudes are given in Ball  _et
al._ (2003).

The predictions for the magnetic dipole, electric quadrupole and magnetic
octupole moments depend on two auxiliary parameters; the Borel mass parameter
$M^{2}$ and continuum threshold $s_{0}$. According to the standard
prescriptions in the method used the predictions should weakly depend on these
helping parameters. The continuum threshold represents the scale at which, the
excited states and continuum start to contribute to the correlation function.
To specify the working interval of the continuum threshold, we impose the
conditions of pole dominance and operator product expansion (OPE) convergence.
Our numerical computations lead to the interval [22-24] $GeV^{2}$ for this
parameter. To specify the working region of the Borel parameter one needs to
take into account two criteria: convergence of the series of OPE and effective
suppression of the higher states and continuum . The above requirements
restrict the working region of the Borel parameter to $5~{}GeV^{2}\leq
M^{2}\leq 7~{}GeV^{2}$. In Fig. 1, we plot the dependencies of the magnetic
dipole, electric quadrupole and magnetic octupole moments on $M^{2}$ at
several fixed values of the continuum threshold $s_{0}$. As can be seen from
this figure, the corresponding electromagnetic multipole moments show overall
weak dependence on the variations of the Borel mass parameter in its working
regions. However, the dependence of the results on the continuum threshold is
considerable.

In this part we would like to discuss the the amount of the perturbative and
different nonperturbative contributions to the whole results. Our numerical
calculations show that almost $85\%$ of the total contribution belongs to the
perturbative part and the remaining $15\%$ corresponds to the nonperturbative
contributions: almost $17\%$ of the total nonperturbative contributions comes
from the terms containing quark condensates $\langle\bar{q}q\rangle$, $5\%$
belongs to those containing gluon condensates $\langle g_{s}^{2}G^{2}\rangle$,
$77\%$ belongs to the terms including the DAs parameters and remaining $1\%$
corresponds to the higher dimensional operators, where because of their
negligible contributions we will not present these terms in the Appendix.

Figure 1:  The dependence of the magnetic dipole, electric quadrupole and
magnetic octupole moments for $P_{c}$ pentaquark; on the Borel parameter
squared $M^{2}$ at different fixed values of the continuum threshold.

Our final results for the magnetic dipole, electric quadrupole and magnetic
octupole moments are given in Table I. The errors in the given results arise
due to the variations in the calculations of the working regions of $M^{2}$
and $s_{0}$ as well as the uncertainties in the values of the input parameters
and the photon DAs. We shall remark that the main source of uncertainties is
due to the variations of the results with respect to $s_{0}$. As previously
mentioned, the continuum threshold is not totally arbitrary but it depends on
the energy of the first excited state. We don’t have enough information on the
mass of the first excited state in the channel under consideration. Hence we
choose its working interval such that the above mentioned criteria of the sum
rules be satisfied. Our analyses show that in the selected region for $s_{0}$,
the dependence of the results on this parameter is very weak compared to the
regions out of its working window. We also would like to note that in Table I
and Fig. 1, the absolute values are given since it is not possible to specify
the sign of the residue from the mass sum rules. Hence, it is not possible to
predict the signs of the magnetic dipole, electric quadrupole and magnetic
octupole moments.

Picture |  $|\mu_{P_{c}}|[\mu_{N}]$ |  $|Q_{P_{c}}|[fm^{2}]$ |  $|O_{P_{c}}|[fm^{3}]$
---|---|---|---
Diquark |  $1.30\pm 0.50$ |  $0.017\pm 0.05$ |  $0.0002\pm 0.00006$
Molecule |  $3.35\pm 1.35$ |  $0.23\pm 0.06$ |  $0.017\pm 0.004$
Table 1: Results of the magnetic dipole, electric quadrupole and magnetic
octupole moments of $P_{c}$ pentaquark.

In conclusion, we have calculated the electromagnetic multipole moments of the
$P_{c}^{+}(4380)$ pentaquark by modeling it as the diquark-diquark-antiquark
and molecular state of $\bar{D}^{*}\Sigma_{c}$ with quantum numbers
$J^{P}=\frac{3}{2}^{-}$. The magnetic dipole, electric quadrupole and magnetic
octupole moments of this particle have been extracted in the framework of
light-cone QCD sum rule. The values of the electromagnetic multipole moments
obtained via two pictures show large differences from each other, which can be
used to pin down the underlying structure of $P_{c}^{+}(4380)$. In other
words, as many models give compatible results on the mass and width with the
experimental data preventing us assigning exact inner structure for
pentaquarks, the experimental measurement of the electromagnetic multipole
moments of the $P_{c}^{+}(4380)$ pentaquark indeed can help us precisely
distinguish its inner structure. The electromagnetic multipole moments of
$P_{c}^{+}(4380)$ can be extracted through the process
$\gamma^{(*)}p\rightarrow P_{c}^{+}(4380)\rightarrow
P_{c}^{+}(4380)\,\gamma\rightarrow J/\psi\,p\,\gamma$ like those of
$\Delta^{+}$ baryon.

##  IV Acknowledgements

This work has been supported by the Scientific and Technological Research
Council of Turkey (TÜBİTAK) under the Grant No. 115F183.

## Appendix A: Explicit forms of the sum rules for $F_{i}^{Di}$

In this appendix, we present the explicit expressions for the sum rules
$F_{i}^{Di}$:

$\displaystyle F_{1}^{Di}$
$\displaystyle=-\frac{e^{m^{2}_{P_{c}}/M^{2}}}{\lambda_{P_{c}}^{2}}\Bigg{\\{}\frac{m_{c}\,\langle\bar{q}q\rangle}{3774873\,\pi^{5}}\Bigg{[}-40\,\pi^{2}\,f_{3\gamma}(e_{d}+10e_{u})\,I_{2}[\mathcal{V}]\,I[0,4,4,0]+3(e_{d}+14e_{u})\,I_{4}[\tilde{S}]\,I[0,5,4,0]$
$\displaystyle+192\,e_{c}\,\Big{(}I[0,5,2,2]-2\,I[0,5,2,3]+I[0,5,2,4]-2\,I[0,5,3,2]+2\,I[0,5,3,3]+I[0,5,4,2]\Big{)}\Bigg{]}$
$\displaystyle-\frac{\langle
g_{s}^{2}G^{2}\rangle}{108716359680\,\pi^{7}}\Bigg{[}20\,\pi^{2}\,f_{3\gamma}\Big{(}4\,(32e_{d}+41e_{u})\,I[0,4,4,0]+3\,(9e_{d}+20e_{u})\,I[0,4,5,0]\Big{)}I_{2}[\mathcal{V}]+(72e_{d}+720e_{u}$
$\displaystyle-702e_{c})\,I[0,5,2,1]-(324e_{d}+2184e_{u}-2673e_{c})\,I[0,5,2,2]+(492e_{d}+2456e_{u}-3511e_{c})\,I[0,5,2,4]$
$\displaystyle-(300e_{d}+1240e_{u}-1811e_{c})\,I[0,5,2,4]+(60e_{d}+248e_{u}-271e_{c})\,I[0,5,2,5]+(216e_{d}+2160e_{u}-2106)$
$\displaystyle\times
I[0,5,3,1]+(684e_{d}+4728e_{d}-5693e_{c})\,I[0,5,3,2]-(624e_{d}+3424e_{u}-4483e_{c})\,I[0,5,3,3]+(156e_{d}$
$\displaystyle+856e_{u}-893e_{c})\,I[0,5,3,4]+(216e_{d}+2160e_{u}-2106e_{c})\,I[0,5,4,1]-(396e_{d}+2904e_{u}-3375e_{c})$
$\displaystyle\times
I[0,5,4,2]+(132e_{d}+968e_{u}-973e_{c})\,I[0,5,4,3]-(72e_{d}+720e_{u}-702e_{c})\,I[0,5,5,1]+(36e_{d}+360e_{u}$
$\displaystyle-351e_{c})\,I[0,5,5,2]\Bigg{]}$
$\displaystyle-\frac{f_{3\gamma}}{3019898880\,\pi^{5}}(13e_{d}+58e_{u})\,I_{2}[\mathcal{V}]\,I[0,6,5,0]$
$\displaystyle+\frac{e_{c}}{880803840\,\pi^{7}}\Bigg{[}4\,I[0,7,2,3]-13\,I[0,7,2,4]+15\,I[0,7,2,5]-7\,I[0,7,2,6]+I[0,7,2,7]-12\,I[0,7,3,3]$
$\displaystyle+27\,I[0,7,3,4]-18\,I[0,7,3,5]+3\,I[0,7,3,6]+12\,I[0,7,4,3]+3\,I[0,7,4,5]-4\,I[0,7,5,3]+I[0,7,5,4]\Bigg{]}\Bigg{\\}},$
(18) $\displaystyle F_{2}^{Di}$
$\displaystyle=\frac{m_{P_{c}}\,e^{m^{2}_{P_{c}}/M^{2}}}{\lambda_{P_{c}}^{2}}\Bigg{\\{}-\frac{m_{c}\,\langle\bar{q}q\rangle}{125829120\,\pi^{5}}\Bigg{[}-120\,(e_{d}+10e_{u})\,f_{3\gamma}\,\pi^{2}\,I_{2}[\mathcal{V}]\,I[0,4,4,0]+(3e_{d}+14e_{u})I_{4}[\mathcal{S}]+2e_{d}\,I_{4}[\mathcal{T}_{1}]$
$\displaystyle\times
I[0,5,4,0]+64\,e_{c}\,\Big{(}I[0,5,2,2]-2\,I[0,5,2,3]+I[0,5,2,4]-2\,I[0,5,3,2]+2\,I[0,5,3,3]+I[0,5,4,2]\Big{)}\Bigg{]}$
$\displaystyle+\frac{\langle
g_{s}^{2}G^{2}\rangle}{108716359680\,\pi^{7}}\Bigg{[}20\,f_{3\gamma}\pi^{2}\,\Big{\\{}-4\Big{(}(52e_{d}-19e_{u})I_{2}[\mathcal{A}]-9(32e_{d}+41e_{u})I_{2}[\mathcal{V}]+4(3e_{d}+2e_{u})\,I_{6}[\psi^{\nu}]\Big{)}\,I[0,4,4,0]$
$\displaystyle+3\Big{(}8(-34e_{d}+e_{u})I_{2}[\mathcal{A}]+9(9e_{d}+220e_{u})\,I_{2}[\mathcal{V}]\Big{)}\,I[0,4,5,0]\Big{\\}}-\Big{\\{}(-72e_{d}+702e_{c}-720e_{u})\,I[0,5,2,1]+(324e_{d}$
$\displaystyle-2673e_{c}+2184e_{u})\,I[0,5,2,2]+(-492e_{d}+3511e_{c}-2456e_{u})\,I[0,5,2,3]+(300e_{d}-1811e_{c}+1240e_{u})\,I[0,5,2,4]$
$\displaystyle+(-60e_{d}+271e_{c}-248e_{u})\,I[0,5,2,5]+(216e_{d}-2106e_{c}+2160e_{u})\,I[0,5,3,1]+(-684e_{d}+5697e_{c}-4728eu)$
$\displaystyle\times
I[0,5,3,2]+(624e_{d}-4484e_{c}+3424e_{u})\,I[0,5,3,3]+(-156e_{d}+893e_{c}-856e_{u})\,I[0,5,3,4]$
$\displaystyle+$
$\displaystyle(-216e_{d}+2106e_{c}-2160e_{u})\,I[0,5,4,1]+(396e_{d}-3375e_{c}+2904e_{u})\,I[0,5,4,2]+(-132e_{d}+973e_{c}-968e_{u})$
$\displaystyle\times
I[0,5,4,3]+(72e_{d}-702e_{c}+720e_{u})\,I[0,5,5,1]+(-36e_{d}+351e_{c}-360e_{u})\,I[0,5,5,2]\Big{\\}}\Bigg{]}$
$\displaystyle+\frac{f_{3\gamma}}{3019898880\,\pi^{5}}(13e_{d}+58e_{u})I_{2}[\mathcal{V}]\,I[0,6,5,0]$
$\displaystyle-\frac{e_{c}}{880803840\,\pi^{7}}\Bigg{[}4\,I[0,7,2,3]-13\,I[0,7,2,4]+15\,I[0,7,2,5]-7\,I[0,7,2,6]+I[0,7,2,7]-12\,I[0,7,3,3]$
$\displaystyle+27\,I[0,7,3,4]-18\,I[0,7,3,5]+3\,I[0,7,3,6]+12\,I[0,7,4,3]-15\,I[0,7,4,4]+3\,I[0,7,4,5]-4\,I[0,7,5,3]$
$\displaystyle+I[0,7,5,4]\Bigg{]}\Bigg{\\}},$ (19) $\displaystyle F_{3}^{Di}$
$\displaystyle=\frac{4\,m_{P_{c}}\,e^{m^{2}_{P_{c}}/M^{2}}}{\lambda_{P_{c}}^{2}}\Bigg{\\{}\frac{m_{c}\,\langle
g_{s}^{2}G^{2}\rangle}{5435817984\,\pi^{7}}\Bigg{[}2\,f_{3\gamma}\,\pi^{2}\,I_{2}[\mathcal{V}]\,\big{(}4\,(5e_{d}-16e_{u})\,I[0,3,3,0]+3\,(3e_{d}+5e_{u})\,I[0,3,4,0]\big{)}+3\,f_{3\gamma}\,\pi^{2}$
$\displaystyle\times
I_{4}[\mathcal{V}]\,\big{(}14\,(e_{d}+4e_{u})\,I[0,3,3,0]-(9e_{d}+194e_{u})\,I[0,3,4,0]\big{)}(-36e_{d}+351e_{c}-360e_{u})\,I[0,4,1,2]+(108e_{d}$
$\displaystyle-977e_{c}+1080e_{u})\,I[0,4,1,3]+(-108e_{d}+901e_{c}-1080e_{u})\,I[0,4,1,4]+(36e_{d}-275e_{c}+360e_{u})\,I[0,4,1,5]$
$\displaystyle+(108e_{d}-1053e_{c}+1080e_{u})\,I[0,4,2,2]+(-216e_{d}+1954e_{c}-2160e_{u})\,I[0,4,2,3]+(108e_{d}-901e_{c}+1080e_{u})$
$\displaystyle\times
I[0,4,2,4]+(-108e_{d}+1053e_{c}-1080e_{u})\,I[0,4,3,2]+(108e_{d}-977e_{c}+1080e_{u})\,I[0,4,3,3]$
$\displaystyle+(36e_{d}-351e_{c}+360e_{u})\,I[0,4,4,2]+(-24e_{d}-240e_{u})\,I[1,3,1,3]+(72e_{d}+720e_{u})\,I[1,3,1,4]+(-72e_{d}$
$\displaystyle-720e_{u})\,I[1,3,1,5]+(24e_{d}+240e_{u})I[1,3,1,6]+(72e_{d}+720e_{u})\,I[1,3,2,3]+(-144e_{d}-1440e_{u})\,I[1,3,2,4]$
$\displaystyle+(72e_{d}+720e_{u})\,I[1,3,2,5]+(-72e_{d}-720e_{u})\,I[1,3,3,3]+(72e_{d}+720e_{u})\,I[1,3,3,4]+(24e_{d}+240e_{u})$
$\displaystyle\times I[1,3,4,3]\Bigg{]}$
$\displaystyle-\frac{m_{c}\,f_{3\gamma}}{251658240\pi^{5}}\Bigg{[}(e_{d}+4e_{u})\,I_{2}[\mathcal{V}]+6(3e_{d}+14e_{u})\,I_{4}[\mathcal{V}]\Bigg{]}\,I[0,5,4,0]$
$\displaystyle+\frac{m_{c}\,e_{c}}{31457280\pi^{7}}\Bigg{[}I[0,6,1,4]-3\,I[0,6,1,5]+3\,I[0,6,1,6]-I[0,6,1,7]-3\,\Big{(}I[0,6,2,4]-2\,I[0,6,2,5]$
$\displaystyle+I[0,6,2,6]-I[0,6,3,4]+I[0,6,3,5]\big{)}-I[0,6,4,4]\Bigg{]}\Bigg{\\}},$
(20)

and

$\displaystyle F_{4}^{Di}$
$\displaystyle=\frac{4\,m^{3}_{P_{c}}\,e^{m^{2}_{P_{c}}/M^{2}}}{\,\lambda_{P_{c}}^{2}}\Bigg{\\{}\frac{m_{c}\,\langle
g_{s}^{2}G^{2}\rangle}{452984832\pi^{7}}\Bigg{[}3\,f_{3\gamma}\,\pi^{2}\,I_{4}[\mathcal{V}]\Big{(}(3e_{d}-8e_{u})\,I[0,2,3,0]+13e_{u}\,I[0,2,4,0]\Big{)}+4\,(e_{d}+10e_{u})$
$\displaystyle\times\Big{(}I[0,3,1,3]-3\,I[0,3,1,4]+3\,I[0,3,1,5]-\,I[0,3,1,6]-3\big{(}I[0,3,2,3]-2\,I[0,3,2,4]+I[0,3,2,5]-I[0,3,3,3]$
$\displaystyle+I[0,3,3,4]\big{)}-I[0,3,4,3]\Big{)}\Bigg{]}\Bigg{\\}},$ (21)

where, $m_{c}$ is the mass of the c quark, $e_{q}$ is the electric charge of
the corresponding quark, $\langle\bar{q}q\rangle$ and $\langle
g_{s}^{2}G^{2}\rangle$ are quark and gluon condensates, respectively.

The functions $I[n,m,l,k]$, $I_{1}[\mathcal{A}]$, $I_{2}[\mathcal{A}]$,
$I_{3}[\mathcal{A}]$, $I_{4}[\mathcal{A}]$, $I_{5}[\mathcal{A}]$, and
$I_{6}[\mathcal{A}]$ are defined as:

$\displaystyle I[n,m,l,k]$
$\displaystyle=\int_{4m_{c}^{2}}^{s_{0}}ds\int_{0}^{1}dt\int_{0}^{1}dw~{}e^{-s/M^{2}}~{}s^{n}\,(s-4\,m_{c}^{2})^{m}\,t^{l}\,w^{k},$
$\displaystyle I_{1}[\mathcal{A}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{A}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta^{\prime}(\alpha_{q}+\bar{v}\alpha_{g}-u_{0}),$
$\displaystyle I_{2}[\mathcal{A}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{A}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta^{\prime}(\alpha_{\bar{q}}+v\alpha_{g}-u_{0}),$
$\displaystyle I_{3}[\mathcal{A}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{A}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta(\alpha_{q}+\bar{v}\alpha_{g}-u_{0}),$
$\displaystyle I_{4}[\mathcal{A}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{A}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta(\alpha_{\bar{q}}+v\alpha_{g}-u_{0}),$
$\displaystyle I_{5}[\mathcal{A}]$
$\displaystyle=\int_{0}^{1}du~{}A(u)\delta^{\prime}(u-u_{0}),$ $\displaystyle
I_{6}[\mathcal{A}]$ $\displaystyle=\int_{0}^{1}du~{}A(u).$

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# On averaging and mixing for stochastic PDEs

Guan Huang  Yau Mathematical Sciences Center, Tsinghua University, Beijing,
China<EMAIL_ADDRESS>and  Sergei Kuksin  Institut de
Mathémathiques de Jussieu–Paris Rive Gauche, CNRS, Université Paris Diderot,
UMR 7586, Sorbonne Paris Cité, F-75013, Paris, France & School of Mathematics,
Shandong University, Jinan, Shandong, China<EMAIL_ADDRESS>
###### Abstract.

We examine the convergence in the Krylov–Bogolyubov averaging for nonlinear
stochastic perturbations of linear PDEs with pure imaginary spectrum and show
that if the involved effective equation is mixing, then the convergence is
uniform in time.

##  1\. Introduction

The Krylov–Bogolyubov averaging for stochastic PDEs which we are concerned
with in this work, means the following. Starting with a linear PDE on a torus
(or on a bounded domain) with pure imaginary spectrum we consider its
$\varepsilon$-small nonlinear stochastic perturbation. Then the above
mentioned averaging describes the behaviour of the distributions of actions of
solutions for the perturbed equation on time-intervals of order
$\varepsilon^{-1}$. Here the actions of solutions are made by the halves of
squared norms of their Fourier coefficients with respect to the basis, made by
eigenfunctions of the original linear system. Description of the limit is made
via an auxiliary effective equation which is another nonlinear stochastic
equation whose nonlinearity is made from resonant terms of the nonlinear part
of the perturbation. The effective equation may be mixing, and then as time
goes to infinity its solutions converge in distributions to a statistical
equilibrium, given by a measure in a function space. An observation which we
make in this work is that in the mixing case the convergence in distribution
of actions of solutions for the perturbed equation to those of solution for
the effective equation, described by the Krylov–Bogolyubov averaging, is
uniform in time.

The Krylov–Bogolyubov averaging under discussion applies to various classes of
stochastic PDEs, depending on the type of the original unperturbed linear
system. In Sections 2-3 we discuss in details the averaging for stochastic
complex Ginzburg–Landau (CGL) equations, regarded as perturbations of linear
Schrödinger equations, and in Section 4 briefly repeat the argument for
stochastic nonlinear wave equations.

Notation. For a Banach space $B$ and $R>0$ we denote $\bar{B}_{R}(B)=\\{b\in
B:|b|_{B}\leq R\\};$ for a metric space $M$, $\mathcal{P}(M)$ stands for the
space of probability Borel measures on $M$. By $\rightharpoonup$ we denote the
weak convergence of measures and by $\mathcal{D}(\xi)$ we denote the
distribution of a random variable $\xi$. For a function $f$ and a measure
$\mu$ we denote $\langle f,\mu\rangle=\int\\!f\,d\mu.$

##  2\. CGL: the setting and result

We consider a stochastic CGL equation on a torus
$T^{D}:=\mathbb{R}/(L_{1}\mathbb{Z})\times\mathbb{R}/(L_{2}\mathbb{Z})\times\cdots\times\mathbb{R}/(L_{D}\mathbb{Z})$,
$L_{1},\dots,L_{D}>0$,

$u_{t}+i(-\Delta+V(x))u=\varepsilon\mu\Delta u+\varepsilon\mathscr{P}(\nabla
u,u)+\sqrt{\varepsilon}\eta(t,x),\;\;\;u=u(t,x),\;x\in T^{D},$ (2.1)

where $\mu\in\\{0,1\\}$, $\mathscr{P}:\mathbb{C}^{{D}+1}\to\mathbb{C}$ is a
$C^{\infty}$-smooth function, $\varepsilon\in(0,1]$ is a small parameter, the
random force $\eta(t,x)$ is white in time and regular in $x$, and the
potential $V(x)$ is a real smooth function. If $\mu=0$, the nonlinearity
$\mathscr{P}(\nabla u,u)$ should not depend on $\nabla u$. For simplicity we
assume that $\mu=1$ (the case $\mu=0$ can be treated similarly). Again only to
simplify presentation we also assume that $V(x)>0$ for all $x$.

For any $s\in\mathbb{R}$ we denote by $H^{s}$ the Sobolev space of complex
functions on $T^{D}$, provided with the norm $\|\cdot\|_{s}$,

$\|u\|_{s}^{2}=\langle(-\Delta)^{s}u,u\rangle+\langle
u,u\rangle,\;\text{if}\;s\geqslant 0,$

where $\langle\cdot,\cdot\rangle$ is the real scalar product in
$L^{2}(T^{D};\mathbb{C})$,

$\langle u,v\rangle=\Re\int_{T^{d}}u\bar{v}\,{\rm{d}}x,\quad u,v\in
L^{2}(T^{D};\mathbb{C}).$

Let $\\{\mathbf{e}_{l}(x),l\in\mathbb{N}\\}$ be the usual trigonometric basis
of the space $L^{2}(T^{D})$, parametrized by natural numbers. Then
$-\Delta\mathbf{e}_{l}=\kappa_{l}\mathbf{e}_{l}$, $\kappa_{l}\geq 0$, and we
assume that $0=\kappa_{1}<\kappa_{2}\leq\kappa_{3},\dots$. We take the force
term $\eta(t,x)$ in (2.1) to be of the form

$\eta(t,x)=\frac{\partial}{\partial t}\xi(t,x),\quad\xi(t,x):=\sum_{l\geqslant
1}b_{l}\beta_{l}(t)\mathbf{e}_{l}(x).$ (2.2)

Here $\beta_{l}(t)=\beta_{l}^{R}(t)+i\beta_{l}^{I}(t)$, where
$\beta_{l}^{R}(t)$, $\beta_{l}^{I}(t)$, $l\geqslant 1$, are independent real-
valued standard Brownian motions, defined on a complete probability space
$(\Omega,\mathcal{F},\mathbf{P})$ with a filtration
$\\{\mathcal{F}_{t};t\geqslant 0\\}$.111So $\\{\beta_{l}(t)\\}$ are standard
independent complex Brownian motions.  As a function of $x$, $\xi(t,x)$ is
assumed to be smooth in the sense that the real numbers $b_{l},l\geq 1$,
decays to zero faster than any negative degree of $l$.

Introducing the slow time $\tau=\varepsilon t$, we rewrite eq. (2.1) as

$\dot{u}+\varepsilon^{-1}(-\Delta+V(x))u=\Delta u+\mathscr{P}(\nabla
u,u)+\dot{\xi}(\tau,x),\quad u=u(\tau,x),\;x\in T^{D},$ (2.3)

where the upper dot stands for ${\partial}/{\partial\tau}$, and $\xi(\tau,x)$
is as in (2.2) with $t:=\tau$ and with another set of standard independent
complex Brownian motions $\beta_{l}$. Here and below we write stochastic PDEs
with additive noise as nonlinear PDEs with forcing terms of the form (2.2).

######  Definition 2.1.

If $L$ and $E$ are Banach spaces with norms $|\cdot|_{L}$ and $|\cdot|_{E}$,
then $\text{Lip}_{m}(L,E)$, $m\geq 0$, is the collection of maps $F:L\to E$
such that for any $R\geqslant 1$,

$\sup_{R>0}\Big{(}(1+|R|)^{-m}\big{(}\text{Lip}(F|_{\bar{B}_{R}(L)})+\sup_{v\in\bar{B}_{R}(L)}|F(v)|_{E}\big{)}\Big{)}<\infty,$

where $\text{Lip}(f)$ is the Lipschitz constant of a mapping $f$.

We make the following assumption concerning the well-posedness of eq. (2.3).
There and everywhere below in our paper

$\begin{split}\text{\it either always the indices $s$ of involved Sobolev
spaces $H^{s}$ are integer,}\\\ \text{\it or always they are any real
numbers.}\end{split}$

######  Assumption 2.2.

There exist numbers $0<s_{1}<s_{2}<+\infty$ and $\bar{m}\in\mathbb{N}$ such
that for each $s\in(s_{1},s_{2})$,

  1. (1)

the mapping $H^{s}\to H^{s-1}$: $u\mapsto\mathscr{P}(\nabla u,u)$ belongs to
$\text{Lip}_{\bar{m}}(H^{s},H^{s-1})$;

  2. (2)

for any $\varepsilon\in(0,1]$ and $u_{0}\in H^{s}$ equation (2.3) has a unique
strong solution $u^{\omega}(\tau;u_{0})$, equal $u_{0}$ at $\tau=0$, defined
for $\tau\geqslant 0$, and

$\mathbf{E}\sup_{\theta\leqslant\tau\leqslant\theta+1}\|u(\tau;u_{0})\|_{s}^{2\bar{m}}\leqslant
C_{s}(\|u_{0}\|_{s}),\quad\forall\theta\geqslant 0,$

where $C_{s}(\cdot)$ is a continuous and non-decreasing function.

  3. (3)

If we work in the category of Sobolev spaces with integer indices, then the
integer segment $(s_{1},s_{2})\cap\mathbb{Z}$ contains at least two points.

Under the above assumptions the family of solutions
$\\{u^{\omega}(\tau;u_{0}),u_{0}\in H^{s}\\}$ defines in the spaces $H^{s}$,
$s\in(s_{1},s_{2})$, Markov processes.

Assumption 2.2 is satisfied for many nonlinearities $\mathscr{P}$. In
particular it holds if

$\mathscr{P}(\nabla
u,u)=-u+{\mathfrak{z}}f_{p}(|u|^{2})u,\quad{\mathfrak{z}}\in\mathbb{C},\
|{\mathfrak{z}}|=1,\Im{\mathfrak{z}}\leq 0,\ \Re{\mathfrak{z}}\leq 0,$ (2.4)

where $f_{p}(r)$ is a non-decreasing smooth function on $\mathbb{R}$, equal
$r^{p}$ for $r\geq 1$. The degree $p\geq 0$ is any if $D=1,2$, and $p<2/(D-2)$
if $D\geq 3$. See [4, Section 5].

We denote by $A_{V}$ the Schrödinger operator

$A_{V}u:=-\Delta u+V(x)u.$

Let $\\{\lambda_{l}\\}_{l\geqslant 1}$ be its eigenvalues, ordered in such a
way that

$0<\lambda_{1}\leqslant\lambda_{2}\leqslant\lambda_{3}\leqslant\cdots\,$

(we recall that $V>0$) and let $\\{\varphi_{l},\;l\geqslant 1\\}\subset
L^{2}({T^{D}})$ be an orthonormal basis, formed by the corresponding
eigenfunctions. We denote $\Lambda=(\lambda_{1},\lambda_{2},\dots)$ and call
$\Lambda$ the frequency vector of eq. (2.3). For a complex-valued function
$u\in H^{s}$ we denote by

$\Psi(u):=v=(v_{1},v_{2},\dots),\quad v_{k}\in\mathbb{C},$

the vector of its Fourier coefficients with respect to the basis
$\\{\varphi_{l}\\}_{l\geqslant 1}$: $u=\sum_{l\geqslant 1}v_{l}\varphi_{l}$.
Note that $\Psi$ is a real operator: it maps real functions $u(\cdot)$ to real
vectors $v$. In the space of complex sequences $v=(v_{1},v_{2},\dots)$ we
introduce the norms

$\left|v\right|^{2}_{s}=\sum_{k=1}^{\infty}\left(|\lambda_{k}|^{s}+1\right)|v_{k}|^{2},\quad
s\in\mathbb{R}\,,$

and denote $h^{s}=\\{v:|v|_{s}<\infty\\}$. Then $\Psi$ defines an isomorphism
between the spaces $H^{s}$ and $h^{s}$, for any $s$.

Now we write Eq. (2.3) in the $v$-variables:

$\dot{v}_{k}+\varepsilon^{-1}i\lambda_{k}v_{k}=-\lambda_{k}v_{k}+P_{k}(v)+\sum_{l=1}^{\infty}\Psi_{kl}b_{l}\dot{\beta}_{l},\quad
k\in\mathbb{N}.$ (2.5)

Here the $k$-th equation is obtained as the $L^{2}$-scalar product with
$\varphi_{k}$ of eq. (2.3), where $u=\Psi^{-1}v$. So

$P(v)=(P_{k}(v),\;k\in\mathbb{N})=\Psi\Big{(}V(x)u+\mathscr{P}(\nabla
u,u)\Big{)},\quad u=\Psi^{-1}v,$

and $\Psi_{kl}=\langle\varphi_{k},\mathbf{e}_{l}\rangle$ (thus
$\big{(}\Psi_{kl}\big{)}$ is the matrix of the operator $\Psi$ with respect to
the trigonometric basis in $L^{2}(T^{D})$ and the natural basis in $h^{0}$).

Our task is to study the dynamics of eq. (2.5) when $\varepsilon\ll 1$. An
efficient way to deal with this problem is through the interaction
representation, which means transition from variables $\\{v_{k}(\tau)\\}$ to
variables $\\{a_{k}(\tau)\\}$, where

$a_{k}(\tau)=e^{i\varepsilon^{-1}\lambda_{k}\tau}v_{k}(\tau),\quad k\geq 1.$

In the $a$-variables equations (2.5) read

$\dot{a}_{k}(\tau)=-\lambda_{k}a_{k}+e^{i\varepsilon^{-1}\lambda_{k}\tau}P_{k}(\Phi_{-\varepsilon^{-1}\Lambda\tau}a)+e^{i\varepsilon^{-1}\lambda_{k}\tau}\sum_{l=1}^{\infty}\Psi_{kl}b_{l}\dot{\beta}_{l},\quad
k\in\mathbb{N}\,,\;\;\tau\geq 0,$ (2.6)

where for a vector
$\theta=(\theta_{k},\;k\in\mathbb{N})\in\mathbb{R}^{\infty}$, $\Phi_{\theta}$
stands for the rotation in $h^{s}$, defined by

$\Phi_{\theta}v=v^{\prime},\quad
v^{\prime}_{k}=e^{i\theta_{k}}v_{k}\quad\forall\,k\,.$ (2.7)

Clearly operators $\Phi_{\theta}$ define isometries of all spaces $h^{s}$. By
$a^{\varepsilon}(\tau;v_{0})=\big{(}a_{k}^{\varepsilon}(\tau;v_{0}),k\geq 1)$
we denote a solution $u(\tau;u_{0})$, written in the $a$-variables. It solves
system (2.6) with the initial data

$a(0)=v_{0}:=\Psi(u_{0}),$

and in view of Assumption 2.2.(2),

$\mathbf{E}\sup_{\theta\leqslant\tau\leqslant\theta+1}|a^{\varepsilon}(\tau;v_{0})|_{s}^{2\bar{m}}\leqslant
C^{\prime}_{s}(|v_{0}|_{s}),\quad\forall\theta\geqslant 0.$ (2.8)

In order to describe the dynamics of eq. (2.6) with $\varepsilon\ll 1$ we
introduce an effective equation:

$\dot{a}_{k}=-\lambda_{k}a_{k}+R_{k}(a)+\sum_{l=1}^{\infty}B_{kl}\dot{\beta}_{l},\quad
k\in\mathbb{N},\;\;\;\tau\geq 0.$ (2.9)

Here $\\{B_{kl},k,l\geqslant 1\\}$ is the principal square root of infinite
matrix $\\{A_{kl},\;k,l\geqslant 1\\}$, 222 The matrix $(A_{kl})$ defines a
non-negative compact self-adjoint operator in the space $l^{2}$. So its
principal square root (which defines another non-negative compact self-adjoint
operator) exists.

$A_{kl}=\begin{cases}\sum_{j\geqslant 1}b_{j}^{2}\Psi_{kj}\Psi_{lj},&\text{if
}\lambda_{k}=\lambda_{l},\\\ 0,&\text{else},\end{cases}$

and

$R(a):=(R_{k}(a),\;k\in\mathbb{N})=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\Phi_{\Lambda
t}P(\Phi_{-\Lambda t}a)dt$ (2.10)

(see [4, Lemma 4] concerning this limit). Under (1) of Assumption 2.2, for
$s\in(s_{1},s_{2})$, the mapping $h^{s}\to h^{s-1}:v\mapsto R(v)$ belongs to
$\text{Lip}_{\bar{m}}(h^{s},h^{s-1})$, see [4]. Therefore for any $a_{0}\in
H^{s}$ with $s\in(s_{1},s_{2})$, a strong solution $a^{0}(\tau;a_{0})$ of eq.
(2.9), equal $a_{0}$ at $\tau=0$, is unique and exists at least locally. In
[4, Theorem 2 and Proposition 1] we proved the following result:

######  Theorem 2.3.

If Assumption 2.2 holds, then for any $s_{1}<s_{*}<\bar{s}<s_{2}$ and any
$v_{0}\in h^{\bar{s}}$ we have:

i) eq. (2.9) has a unique strong solution $a^{0}(\tau;v_{0})$, $\tau\geq 0$,
equal to $v_{0}$ at $\tau=0$. It belongs to $C([0,\infty),h^{\bar{s}})$ a.s.,
and for any $\theta\geq 0$,

$\mathbf{E}\sup_{\tau\in[\theta,\theta+1]}|a^{0}(\tau;v_{0})|_{\bar{s}}^{2\bar{m}}\leqslant
C^{\prime}_{\bar{s}}(|v_{0}|_{\bar{s}}).$ (2.11)

ii) For any $T>0$,

$\mathcal{D}\big{(}a^{\varepsilon}(\tau;v_{0})\\!\mid_{[0,T]}\\!\big{)}\rightharpoonup\mathcal{D}\big{(}a^{0}(\tau;v_{0})\\!\mid_{[0,T]}\\!\big{)}\quad\text{in}\;\;\mathcal{P}\big{(}C([0,T]);h^{s_{*}})\big{)}\;\;\text{as}\;\;\varepsilon\to
0.$ (2.12)

######  Remark 2.4.

1) In [4] it was proved only the second assertion of the theorem, implying
that $a^{0}$ is a solution of eq. (2.9) in $h^{s_{*}}$. Then (2.8) and Fatou’s
lemma imply (2.11). Indeed, for $M\in\mathbb{N}$ let
$\Pi_{M}(a_{1},a_{2},\dots)=(a_{1},\dots,a_{M},0,\dots)$. Then by (2.8) and
convergence (2.12), for each $\theta\geq 0$ and any $M,N\in\mathbb{N}$

$\mathbf{E}\sup_{\tau\in[\theta,\theta+1]}N\wedge|\Pi_{M}a^{0}(\tau;v_{0})|_{\bar{s}}^{2\bar{m}}\leqslant
C^{\prime}_{\bar{s}}(|v_{0}|_{\bar{s}}).$

Sending first $N\to\infty$ and then $M\to\infty$ and using Fatou’s lemma we
recover (2.11). Since $a^{0}$ is a solution in $h^{s_{*}}$, then dues to
(2.11) it is a solution in $h^{\bar{s}}$. Uniqueness of a solution is obvious.

2) It follows immediately that (2.12) also holds for a solution
$a^{\varepsilon}(\tau;v_{0})$ if the initial data $v_{0}$ is a r.v. in
$h^{\bar{s}}$, independent from the random field $\xi$. Moreover, a simple
analysis of the proof in [4] implies that if $|v_{0}^{\omega}|_{\bar{s}}\leq
M$ a.s., then the rate of convergence (2.12) depends only on $M$ (and, of
course, on $s_{*}<\bar{s}$).

3) Theorem’s assertion with the same proof remains true if in the r.h.s. of
eq. (2.3) we replace the viscosity $\Delta u$ by the hyperviscosity
$-(-\Delta+1)^{r}u$, $r\in\mathbb{N}$ (provided that Assumption 2.2 holds for
the equation). The equations with hyperviscosity are important for some
applications, see below Example 2.11.

4) In [4] the bounds
$\mathbf{E}\sup_{\theta\leqslant\tau\leqslant\theta+1}\|u(\tau;u_{0})\|_{s}^{m}$
are assumed for all $m$-th moments of solutions, $m\in\mathbb{N}$. It was done
only for simplicity, and under the additional restriction in Assumption
2.2.(1), which specifies the growth of the the nonlinearity $\mathscr{P}$,
only the bounds for the moments with $m\leq 2\bar{m}$ are needed for the
proof. This difference is rather insignificant since using the standard
techniques of exponential supermartingales it is usually easy to obtain bounds
for all $m$-th moments after the second moments are estimated.

Again, solutions $a^{0}(\tau;v_{0})$ of (2.9) define in the space $h^{s_{*}}$
a Markov process.

Our goal in this work is to prove that if effective equation (2.9) is mixing,
then convergence (2.12) is such that
$\mathcal{D}(a^{\varepsilon}(\tau;v_{0}))\rightharpoonup\mathcal{D}(a^{0}(\tau;v_{0}))$
uniformly in $\tau\geq 0$, with respect to the dual-Lipschitz distance. We
recall

######  Definition 2.5.

Let $M$ be a complete and separable metric space. For any two measures
$\mu_{1}$, $\mu_{2}\in\mathcal{P}(M)$ the dual-Lipschitz distance between them
is

$\|\mu_{1}-\mu_{2}\|_{L,M}^{*}:=\sup_{f\in C(M),\,|f|_{L,M}\leqslant
1}\big{|}\langle f,\mu_{1}\rangle-\langle f,\mu_{2}\rangle\big{|}\leqslant 2,$

where $|f|_{L,M}=\text{Lip}f+\|f\|_{C(M)}$.

For future usage we note that for $s\geq s^{\prime}$ space
$\mathcal{P}(h^{s})$ is naturally embedded in $\mathcal{P}(h^{s^{\prime}})$,
and it easily follows from the definition that for
$\mu_{1},\mu_{2}\in\mathcal{P}(h^{s})$ the distance
$\|\mu_{1}-\mu_{2}\|_{L,h^{s^{\prime}}}^{*}$ is a non-decreasing function of
$s^{\prime}\leq s$.

To proceed we have to assume that the effective equation is mixing, and the
rate of mixing is uniform for initial data from bounded sets:

######  Assumption 2.6.

For some $s_{*}\in(s_{1},s_{2})$ effective equation (2.9) is mixing in the
space $h^{s_{*}}$ with a stationary measure
$\mu^{0}\in\mathcal{P}(h^{s_{*}})$, and for each $M>0$ and
$v\in\bar{B}_{M}(h^{s_{*}})$, we have

$\|\mathcal{D}(a^{0}(\tau;v))-\mu^{0}\|_{L,h^{s_{*}}}^{*}\leqslant\mathfrak{g}_{M}(\tau),$
(2.13)

where $\mathfrak{g}$ is a continuous function of $(M,\tau)$ which goes to zero
when $\tau\to\infty$.

Relation (2.13) is a mild specification of the mixing in eq. (2.9), and a
proof of the latter in fact usually establishes the former.

Our main result is the following:

######  Theorem 2.7.

Under Assumptions 2.2 and 2.6, for any $\bar{s}\in(s_{*},s_{2})$ and any
$v_{0}\in h^{\bar{s}}$

$\lim_{\varepsilon\to 0}\sup_{\tau\geqslant
0}\|\mathcal{D}(a^{\varepsilon}(\tau;v_{0}))-\mathcal{D}(a^{0}(\tau;v_{0}))\|_{L,h^{s_{*}}}^{*}=0,$

where $a^{\varepsilon}(\tau;v_{0})$ and $a^{0}(\tau;v_{0})$ solve respectively
equations (2.6) and (2.9) with initial conditions
$a^{\varepsilon}(0;v_{0})=a^{0}(0;v_{0})=v_{0}$. Moreover, for any $M>0$ the
above convergence is uniform for $v_{0}\in\bar{B}_{M}(h^{\bar{s}})$.

For $v=(v_{1},v_{2},\dots)\in h^{s}$ we introduce the vector of action
variables $I(v)=(I_{1}(v),I_{2}(v),\dots)$, where
$I_{k}(v)=\frac{1}{2}|v_{k}|^{2}$, $k=1,2,\dots$. Then $I(v)\in
h_{I}^{s}\cap\mathbf{R}_{+}^{\infty}$, where $h^{s}_{I}$ is the weighted
$l^{1}$-space with the norm
$|I|_{I,s}=\sum_{k=1}^{\infty}(|\lambda_{k}|^{s}+1)|I_{k}|.$ Since the
interaction representation does not change the actions, then for the action
variables of solutions for the original equations (2.5) we have

######  Corollary 2.8.

Under the assumptions of Theorem 2.7 actions of a solution
$v^{\varepsilon}(\tau;v_{0})$ for eq. (2.5) with $v_{0}\in h^{\bar{s}}$,
$\bar{s}>s_{*}$ satisfy

$\lim_{\varepsilon\to 0}\,\sup_{\tau\geqslant
0}\|\mathcal{D}\big{(}I(v^{\varepsilon}(\tau;v_{0}))\big{)}-\mathcal{D}\big{(}(I(a^{0}(\tau;v_{0}))\big{)}\|_{L,h_{I}^{s_{*}}}^{*}=0.$

###  2.1. Examples

For a finite-dimensional stochastic equation

$\dot{a}_{k}(\tau)=e^{i\varepsilon^{-1}\lambda_{k}\tau}P_{k}(\Phi_{-\varepsilon^{-1}\Lambda\tau}a)+e^{i\varepsilon^{-1}\lambda_{k}\tau}\sum_{l=1}^{N}\mathcal{M}_{kl}\dot{\beta}_{l}(\tau),\quad
k=1,\dots,N,$ (2.14)

where $a(\tau)=(a_{1},\dots,a_{N})(\tau)\in\mathbb{C}^{N}$ and
$\Lambda=(\lambda_{1},\dots,\lambda_{N})$, satisfying a natural analogy of
Assumptions 2.2, 2.6, a natural version of Theorem 2.7 holds. If the matrix
$(\mathcal{M}_{kl})$ is non-degenerate and Assumption 2.2 is met, then a
convenient sufficient condition for Assumption 2.6 follows from the Khasminski
criterion for mixing (details will be given in paper on the finite-dimensional
stochastic averaging under preparation).333 In [2, Theorem 2.9] the uniform in
time convergence as above is proved for a class of systems (2.14). The proof
in [2] is based on the observation that the corresponding $v$-equations are
mixing with a rate of mixing, independent from $\varepsilon$.  For infinite-
dimensional stochastic systems an instrumental criterion of mixing is not
known yet. We believe that as in finite dimensions, Assumption 2.6 holds for
effective equations for various equations (2.3), satisfying Assumption 2.2, at
least if the random forces (2.2) is such that all $b_{l}$’s are non-zero.
Examples below are given to support this belief.

######  Example 2.9.

Consider eq. (2.3), where $\mathscr{P}(\nabla u,u)=\mathscr{P}(u)$ has the
form (2.4) with $\mathfrak{z}=-1$. Since the function $f_{p}$ is monotone,
then mapping $\mathscr{P}$ also is monotone in the sense that

$\langle\mathscr{P}(u)-\mathscr{P}(v),u-v\rangle\geq 0\quad\forall u,v\in
L^{\infty}(T^{D};\mathbb{C}).$

From definition (2.10) we see that operator $R$ is monotone as well. Taking
any two vectors $v^{1},v^{2}\in h^{s}$ and denoting
$w(\tau)=a^{0}(\tau;v^{1})-a^{0}(\tau;v^{2})$ we immediately derive from the
equation for $w$ that
$(d/dt)|w^{\omega}(\tau)|_{0}^{2}\leq-\tfrac{1}{2}\lambda_{1}|w^{\omega}(\tau)|_{0}^{2}.$
So

$|w^{\omega}(\tau)|_{0}\leq
e^{-\lambda_{1}\tau}|v^{1}-v^{2}|_{0}\quad\forall\,\tau\geq 0,\
\forall\omega.$

This very strong a-priori estimate implies Assumption 2.6 after some non-
complicated work.

######  Example 2.10.

Now let $\mathscr{P}(\nabla u,u)=\mathscr{P}(u)$ has the form (2.4), where
$\Re\mathfrak{z}<0$ and $\Im\mathfrak{z}<0$. Then relation (2.13) with
${\mathfrak{g}}_{M}(\tau)=C(M)e^{-\kappa\tau},$ $\kappa>0$, and with
sufficiently large integer $s_{*}$ follows from the abstract theorems in [10,
Theorem 2.1] and [5, Theorem 3.1.7] since solutions $a^{0}$ inherit estimates
for solutions $a^{\varepsilon}$ via convergence (2.12). We leave to the reader
details of this derivation (which are not quite trivial).

######  Example 2.11.

Consider eq. (2.3) with $\mathscr{P}(\nabla u,u)=-i\rho|u|^{2}u$, with $V=0$
and with a hyperviscosity instead of viscosity:

$\dot{u}-i\varepsilon^{-1}\Delta
u=-(-\Delta+1)^{r}u-i\rho|u|^{2}u+\dot{\xi}(\tau,x),\quad
x\in\mathbb{T}^{D}_{L}=\mathbb{R}^{D}/(L\mathbb{Z}^{D}).$ (2.15)

Here $r\in\mathbb{N}$, $\rho>0$ is a scaling factor and $\xi$ has the form
$\eqref{eta}$, where all $b_{l}$’s are non-zero. Now $A_{V}=-\Delta$, so its
eigenfunctions are exponents $e^{i2\pi s\cdot x}$, $s\in
L^{-1}\mathbb{Z}^{D}$. The action variables $I_{s}(\tau)$ of solutions
$u(\tau,x)$ are naturally parametrised by $s\in L^{-1}\mathbb{Z}^{D}$. The
theory of wave turbulence (WT), among other things, examine the behaviour of
the expectations of the actions $\mathbf{E}I_{s}(\tau)$ under the limit of WT:

$\varepsilon\to 0,\quad L\to\infty,$

when $\rho$ is properly scaled with $\varepsilon$ and $L$. See [9] and the
introduction in [3]. For any dimension $D$ solutions of eq. (2.15) with $r$
sufficiently large in terms of $D$ satisfy Assumption 2.2 by the same
straightforward proof as for eq. (2.3) with $\mathscr{P}$ as in (2.4), and the
equation also meets Assumption 2.6 with sufficiently large integer $s_{*}$ in
view of the abstract theorems, mentioned in Example 2.10. So Corollary 2.8
applies to eq. (2.15) with $\rho$ and $L$ fixed, under the limit
$\varepsilon\to 0$ and imply that the expectations of the actions of solutions
for (2.15) converge to those of solutions for the corresponding effective
equation, uniformly in time. The limit $\varepsilon\to 0$ in eq. (2.15) and in
similar equations is known in nonlinear physics as the limit of discrete
turbulence. See [9], [6] and [3, Sections 1.2, 12.1].

##  3\. Proof of Theorem 2.7

Below we always assume Assumptions 2.2 and 2.6. We fix $\bar{s}>s_{*}$ and
$v_{0}\in h^{\bar{s}}$ as in Theorem 2.7. We abbreviate
$\|\cdot\|_{L,h^{s_{*}}}^{*}$ to $\|\cdot\|_{L}^{*}$ and
$a^{\epsilon}(\tau;v_{0})$ to $a^{\epsilon}(\tau)$. For any $T^{\prime}\geq 0$
we denote by $a_{T^{\prime}}^{0}(\tau)$ a weak solution of the effective
equation (2.9) such that

$\mathcal{D}(a^{0}_{T^{\prime}}(0))=\mathcal{D}(a^{\varepsilon}(T^{\prime})).$

Note that solution $a^{0}_{T^{\prime}}(\tau)$ depends on $\varepsilon$ and
that $a^{0}_{0}(\tau)=a^{0}(\tau;v_{0})$.

The following lemma follows from Theorem 2.3, Assumption 2.2.(2) and Remark
2.4.2).

######  Lemma 3.1.

For any $\delta>0$ and $T>0$ there exists
$\varepsilon_{1}=\varepsilon_{1}(\delta,T)>0$ such that if
$\varepsilon\leq\varepsilon_{1}$, then

$\sup_{\tau\in[0,T]}\|\mathcal{D}(a^{\varepsilon}(T^{\prime}+\tau))-\mathcal{D}(a_{T^{\prime}}^{0}(\tau))\|_{L,h^{s_{*}}}^{*}\leqslant\delta/2,\quad\forall\,T^{\prime}\geqslant
0.$ (3.1)

Since by Assumption 2.6, $\mathcal{D}(a^{0}(\tau;0))\to\mu^{0}$ in
$\mathcal{P}(h^{s_{*}})$ as $\tau\to\infty$, then from estimate (2.11) and
Fatou’s lemma we derive that

$\langle|v|_{\bar{s}}^{2\bar{m}},\mu^{0}\rangle\leqslant
C^{\prime}_{\bar{s}}(0):=C_{0}.$ (3.2)

We need the following statement for the mixing in eq. (2.9).

######  Lemma 3.2.

For any solution $a^{0}(\tau;\mu)\in h^{s*}$, $\tau\geqslant 0$, of effective
equation (2.9) such that $\mathcal{D}(a^{0}(0;\mu))=:\mu$, where the measure
$\mu$ satisfies

$\langle|a|_{s_{*}}^{2\bar{m}},\mu(da)\rangle=\mathbf{E}|a^{0}(0;\mu)|_{s_{*}}^{2\bar{m}}\leqslant
M\quad\text{ for some $M>0$, }$

we have

$\|\mathcal{D}\big{(}a^{0}(\tau;\mu)\big{)}-\mu^{0}\|_{L}^{*}\leq
g_{M}(\tau,d)\quad\text{if}\;\;\|\mu-\mu^{0}\|_{L}^{*}\leq d\leq 2.$ (3.3)

Here the function $g:\mathbb{R}_{+}^{3}\to\mathbb{R}_{+}$, $(M,\tau,d)\to
g_{M}(\tau,d)$ is continuous, vanishes with $d$, converges to zero when
$\tau\to\infty$ and is such that for each fixed $M\geqslant 0$ the function
$(\tau,d)\to g_{M}(\tau,d)$ is uniformly continuous in $d$ for
$(\tau,d)\in[0,\infty)\times[0,2]$.

A proof of the lemma is rather straightforward but lengthy. It is given below
in Subsection 3.1.

We denote

$M_{*}:=C^{\prime}_{\bar{s}}(|v_{0}|_{\bar{s}}),$

where $C^{\prime}_{\bar{s}}(|v_{0}|_{\bar{s}})$ is as in (2.8) and (2.11).
Constants in the estimates below depend on $M_{*}$, but this dependence
usually is not indicated.

######  Lemma 3.3.

Take any $\delta>0$ and choose a $T^{*}=T^{*}(\delta)>0$, satisfying

$g_{M_{*}}(T,2)\leq\delta/4\qquad\forall\,T\geq T^{*}.$

Then there exists $\varepsilon_{2}=\varepsilon_{2}(\delta)>0$ such that if
$\varepsilon\leq\varepsilon_{2}$ and
$\|\mathcal{D}(a^{\varepsilon}(T^{\prime}))-\mu^{0}\|_{L}^{*}\leqslant\delta$
for some $T^{\prime}\geq 0$, then

$\|\mathcal{D}(a^{\varepsilon}(T^{\prime}+T^{*}))-\mu^{0}\|_{L}^{*}\leqslant{\delta},$
(3.4)

and

$\sup_{\tau\in[T^{\prime},T^{\prime}+T^{*}]}\|\mathcal{D}(a^{\varepsilon}(\tau))-\mu^{0}\|_{L}^{*}\leqslant\tfrac{\delta}{2}+\sup_{\tau\geq
0}g_{M_{*}}(\tau,\delta).$ (3.5)

######  Proof.

Let us choose
$\varepsilon_{2}(\delta)=\varepsilon_{1}(\frac{\delta}{2},T^{*}(\delta))$,
where $\varepsilon_{1}(\cdot)$ is as in Lemma 3.1. Then we get from (3.1),
(2.11), (3.3) with $M=M_{*}$ and the definition of $T^{*}$ that for
$\varepsilon\leq\varepsilon_{2}$,

$\begin{split}\|\mathcal{D}(a^{\varepsilon}(T^{\prime}+T^{*}))-\mu^{0}\|_{L}^{*}\leqslant\|\mathcal{D}(a^{\varepsilon}(T^{\prime}+T^{*}))&-\mathcal{D}(a^{0}_{T^{\prime}}(T^{*}))\|_{L}^{*}\\\
&+\|\mathcal{D}(a^{0}_{T^{\prime}}(T^{*}))-\mu^{0}\|_{L}^{*}\leqslant{\delta}.\end{split}$

This proves (3.4). Next, in view of (3.1) and (3.3),

$\begin{split}&\sup_{\theta\in[0,T^{*}]}\|\mathcal{D}(a^{\varepsilon}(T^{\prime}+\theta))-\mu^{0}\|_{L}^{*}\\\
&\leqslant\sup_{\theta\in[0,T^{*}]}\|\mathcal{D}(a^{\varepsilon}(T^{\prime}+\theta))-\mathcal{D}(a_{T^{\prime}}^{0}(\theta))\|_{L}^{*}+\sup_{\theta\in[0,T^{*}]}\|\mathcal{D}(a_{T^{\prime}}^{0}(\theta))-\mu^{0}\|_{L}^{*}\\\
&\leqslant\frac{\delta}{2}+\max_{\theta\in[0,T^{*}]}g_{M_{*}}(\theta,\delta).\end{split}$

This implies (3.5). ∎

We are now ready to prove Theorem 2.7.

######  Proof.

[of Theorem 2.7.] Let us fix arbitrary $\delta>0$ and take some
$0<\delta_{1}\leq\delta/4$. Below in the proof the functions
$\varepsilon_{1}$, $\varepsilon_{2}$ and $T^{*}$ are as in Lemmas 3.1 and 3.3.

i) By the definition of $T^{*}=T^{*}(\delta_{1})$, (2.11) and (3.3),

$\|\mathcal{D}\big{(}a^{0}_{T^{\prime}}(\tau))-\mu^{0}\|_{L}^{*}\leq\delta_{1}/4\quad\forall\,\tau\geq
T^{*},$ (3.6)

for any $T^{\prime}\geq 0$.

ii) By Lemma 3.1, if
$\varepsilon\leq\varepsilon_{1}=\varepsilon_{1}(\frac{\delta_{1}}{2},T^{*})>0$,
then

$\sup_{0\leq\tau\leq
T^{*}}\|\mathcal{D}\big{(}a^{\varepsilon}(\tau))-\mathcal{D}\big{(}a^{0}(\tau;v_{0})\big{)}\|_{L}^{*}\leq\tfrac{\delta_{1}}{2}.$
(3.7)

In particular, in view of (3.6) with $T^{\prime}=0$,

$\|\mathcal{D}\big{(}a^{\varepsilon}(T^{*})\big{)}-\mu^{0}\|_{L}^{*}<\delta_{1}.$
(3.8)

iii) By (3.8) and (3.4) with $\delta=\delta_{1}$ and with $T^{\prime}=nT^{*}$,
$n=1,2,\dots$ we get inductively that

$\|\mathcal{D}\big{(}a^{\varepsilon}(nT^{*})\big{)}-\mu^{0}\|_{L}^{*}\leq\delta_{1}\quad\forall\,n\in\mathbb{N},$
(3.9)

if $\varepsilon\leq\varepsilon_{2}=\varepsilon_{2}(\delta_{1})$.

iv) Now by (3.9) and (3.5) with $\delta=\delta_{1}$, for any $n\in\mathbb{N}$
and $0\leq\theta\leq T^{*}$,

$\|\mathcal{D}\big{(}a^{\varepsilon}(nT^{*}+\theta))-\mu^{0}\|_{L}^{*}\leq\delta_{1}/2+\sup_{\theta\geq
0}g_{M_{*}}(\theta,\delta_{1}),$ (3.10)

if $\varepsilon\leq\varepsilon_{2}(\delta_{1})$.

v) Finally, if
$\varepsilon\leq\varepsilon_{\\#}(\delta_{1})=\min\big{\\{}\varepsilon_{1},\varepsilon_{2}\big{\\}}$,
then by (3.7) if $\tau\leq T^{*}$ and by (3.6)+(3.10) if $\tau\geq T^{*}$ we
have that

$\|\mathcal{D}\big{(}a^{\varepsilon}(\tau))-\mathcal{D}\big{(}a^{0}(\tau;v_{0})\big{)}\|_{L}^{*}\leq\delta_{1}+\sup_{\theta\geq
0}g_{M_{*}}(\theta,\delta_{1})\qquad\forall\,\tau\geq 0.$

By Lemma 3.2, for $M_{*}$ fixed the function $g_{M_{*}}(\theta,d)$ is
uniformly continuous in $d$ and vanishes at $d=0$. So there exists
$\delta^{*}=\delta^{*}(\delta)$, which we may assume to be $\leq\delta/4$,
such that if $\delta_{1}=\delta^{*}$, then
$g_{M_{*}}(\theta,\delta_{1})\leq\delta/2$ for every $\theta\geqslant 0$. Then
by the estimate above,

$\|\mathcal{D}\big{(}a^{\varepsilon}(\tau))-\mathcal{D}\big{(}a^{0}(\tau;v_{0})\big{)}\|_{L}^{*}\leq\delta\quad\text{if}\quad\varepsilon\leq\varepsilon_{*}(\delta):=\varepsilon_{\\#}(\delta^{*}\big{(}\delta)\big{)}>0,$

for every positive $\delta$. This proves the theorem’s assertion. ∎

###  3.1. Proof of Lemma 3.2

In this proof we write solutions of effective equation (2.9) as $a(\tau)$
(rather than $a^{0}(\tau)$).

i) At this step, for any non-random $v^{1},v^{2}\in\bar{B}_{M}(h^{s_{*}})$ we
denote $a_{j}^{\omega}(\tau)=a^{\omega}(\tau,v^{j})$, $j=1,2$, and examine the
distance $\|\mathcal{D}(a_{1}(\tau))-\mathcal{D}(a_{2}(\tau))\|_{L}^{*}$ as a
function of $\tau$ and $|v^{1}-v^{2}|_{s_{*}}$. We assume that
$|v^{1}-v^{2}|_{s_{*}}\leq\bar{d}$ for some $\bar{d}\geq 0$ and set
$w^{\omega}(\tau)=a_{1}^{\omega}(\tau)-a_{2}^{\omega}(\tau)$. We have

$\dot{w}^{\omega}=-A_{V}w^{\omega}+R(a_{1}^{\omega})-R(a_{2}^{\omega}).$

So by Duhamel’s principle,

$w^{\omega}(\tau)=e^{-A_{V}\tau}w(0)+\int_{0}^{\tau}e^{-A_{V}(\tau-t)}(R(a^{\omega}_{1}(t))-R(a^{\omega}_{2}(t)))dt.$

Here $|w(0)|_{s_{*}}\leq\bar{d}$, and by (1) of Assumption 2.2

$|R(a_{1}^{\omega}(\tau))-R(a^{\omega}_{2}(\tau))|_{s_{*}-1}\leq
C|w^{\omega}(\tau)|_{s_{*}}\,X^{\omega}(\tau),\quad
X^{\omega}(\tau)=1+|a^{\omega}_{1}(\tau)|_{s_{*}}^{\bar{m}}+|a^{\omega}_{2}(\tau)|_{s_{*}}^{\bar{m}}.$

For $\theta>0$ the norm of the operator $e^{-A_{V}\theta}:H^{s_{*}-1}\to
H^{s_{*}}$ is bounded by $\chi(\theta)$, where

$\chi(\theta)=\begin{cases}C\theta^{-1/2},&0<\theta\leq 1,\\\
Ce^{-c\theta},&\theta>1,\end{cases}$

with some $C,c>0$. So if $\max_{0\leq t\leq\tau}X^{\omega}(t)\leq K,$ then

$|w^{\omega}(\tau)|_{s_{*}}\leq\bar{d}e^{-\lambda_{1}\tau}+KC\int_{0}^{\tau}\chi(\tau-l)|w^{\omega}(l)|_{s_{*}}dl.$

Applying Gronwall’s lemma we derive from here that

$|w^{\omega}(\tau)|_{s_{*}}\leq\bar{d}(1+CKe^{c_{1}K}),\quad\forall\,\tau\geq
0.$ (3.11)

for some positive constants $C,c_{1}$.

Denote $Y(T)=\sup_{0\leq t\leq T}|X^{\omega}(t)|$. By (2.11),

$\mathbf{E}Y(T)\leqslant 2(1+C^{\prime}_{s_{*}}(M))(T+1).$

For $K>0$ let $\Omega_{K}(T)$ be the event $\\{Y(T)\geq K\\}$. Then
$\mathbf{P}(\Omega_{K}(T))\leq 2(1+C^{\prime}_{s_{*}}(M))(T+1)K^{-1},$ and
$|a_{1}(\tau)-a_{2}(\tau)|_{s_{*}}=|w^{\omega}(\tau)|_{s_{*}}$ satisfies
(3.11) for $\omega\notin\Omega_{K}(T)$. From here we see that if $f\in
C_{b}(h^{s_{*}})$ is such that $|f|_{C_{b}(h^{s_{*}})}\leq 1$ and Lip$\,f\leq
1$, then

$\begin{split}\mathbf{E}\big{(}f(a_{1}(\tau))&-f(a_{2}(\tau))\big{)}\leq
2\mathbf{P}(\Omega_{K}(\tau))+\bar{d}(1+CKe^{c_{1}K})\\\ &\leqslant
4(1+C^{\prime}_{s_{*}}(M))(\tau+1)K^{-1}+\bar{d}(1+CKe^{c_{1}K})\quad\forall\,K>0.\end{split}$
(3.12)

Let us denote by $g^{1}_{M}(\tau,\bar{d})$ the function in the r.h.s. above
with $K=\ln\ln\big{(}\bar{d}^{-1}\vee 3\big{)}$. This is a continuous function
of $(M,\tau,\bar{d})\in\mathbb{R}_{+}^{3}$, vanishing when $\bar{d}=0$. Due to
(2.13) and (3.12),

$\begin{split}\|\mathcal{D}(a(\tau;v^{1}))-&\mathcal{D}(a(\tau;v^{2}))\|_{L}^{*}=\|\mathcal{D}(a_{1}(\tau))-\mathcal{D}(a_{2}(\tau))\|_{L}^{*}\\\
&\leq(2{\mathfrak{g}}_{M}(\tau))\wedge g^{1}_{M}(\tau,\bar{d})\wedge
2=:g_{M}^{2}(\tau,\bar{d}),\end{split}$ (3.13)

if $v_{1},v_{2}\in\bar{B}_{M}(h^{s_{*}})$ and
$|v^{1}-v^{2}|_{s_{*}}\leq\bar{d}$, for any $M,\bar{d}>0$. The function
$g^{2}$ is continuous in the variables $(M,\tau,\bar{d})$, vanishes with
$\bar{d}$ and goes to zero when $\tau\to\infty$ since $\mathfrak{g}_{M}(\tau)$
does.

ii) At this step we consider a solution $a^{0}(\tau;\mu)=:a(\tau;\mu)$ of
(2.9) as in the lemma and examine the l.h.s. of (3.3) as a function of $\tau$.
For any $K>0$ consider the conditional probabilities
$\mu_{K}=\mu(\cdot\mid\bar{B}_{K}(h^{s_{*}}))$ and
$\bar{\mu}_{K}=\mu(\cdot\mid h^{s_{*}}\setminus\bar{B}_{K}(h^{s_{*}}))$. Then
$\mu=A_{K}\mu_{K}+\bar{A}_{K}\bar{\mu}_{K},$ where
$A_{K}=\mu(\bar{B}_{K}(h^{s_{*}}))$ and
$\bar{A}_{K}=\mathbf{P}\\{|a(0)|_{s_{*}}>K\\}\leqslant M/K^{2\bar{m}}$ as
$\mathbf{E}|a(0)|_{s_{*}}^{2\bar{m}}\leq M$. So

$\mathcal{D}(a(\tau,\mu))=A_{K}\mathcal{D}(a(\tau;\mu_{K}))+\bar{A}_{K}\mathcal{D}(a(\tau;\bar{\mu}_{K})).$
(3.14)

In view of Assumption 2.6,

$\begin{split}\|\mathcal{D}(a(\tau;\mu_{K}))-\mu^{0}\|_{L}^{*}&=\|\int\big{[}\mathcal{D}(a(\tau;v))\big{]}\mu_{K}(dv)-\mu^{0}\|_{L}^{*}\\\
&\leq\int\|\mathcal{D}(a(\tau;v))-\mu^{0}\|_{L}^{*}\mu_{K}(dv)\leq\mathfrak{g}_{K}(\tau).\end{split}$

Therefore due to (3.14),

$\begin{split}&\|\mathcal{D}(a(\tau,\mu))-\mu^{0}\|_{L}^{*}\leqslant
A_{K}\|\mathcal{D}(a(\tau,\mu_{K}))-\mu^{0}\|_{L}^{*}+\bar{A}_{K}\|\mathcal{D}(a(\tau,\bar{\mu}_{K}))-\mu^{0}\|_{L}^{*}\\\
&\leq\|\mathcal{D}(a(\tau,\mu_{K}))-\mu^{0}\|_{L}^{*}+2\bar{A}_{K}\leq\mathfrak{g}_{K}(\tau)+2\frac{M}{K^{2\bar{m}}}\quad\text{for
$K>0$ and $\tau\geq 0$. }\end{split}$

Let $K_{1}(\tau)>0$ be a continuous non-decreasing function such that
$K_{1}(\tau)\to\infty$ and $\mathfrak{g}_{K_{1}(\tau)}(\tau)\to 0$ as
$\tau\to\infty$ (it exists since $\mathfrak{g}_{K}(\tau)$ is a continuous
function of $(K,\tau)$, going to 0 as $\tau\to\infty$ for each fixed $K$).
Then choosing in the estimate above $K=K_{1}$ we get

$\|\mathcal{D}(a(\tau;\mu))-\mu^{0}\|_{L}^{*}\leqslant\mathfrak{g}_{K_{1}(\tau)}(\tau)+\frac{2M}{K_{1}(\tau)^{2\bar{m}}}=:{\hat{\mathfrak{g}}}_{M}(\tau).$
(3.15)

Obviously ${\hat{\mathfrak{g}}}_{M}(\tau)\geq 0$ is a continuous function on
$\mathbb{R}_{+}^{2}$ , converging to $0$ as $\tau\to\infty$.

iii) Now we examine the l.h.s. of (3.3) as a function of $\tau$ and $d$.
Recall that the Kantorovich distance between measures $\nu_{1},\nu_{2}$ on
$h^{s_{*}}$ is

$\|\nu_{1}-\nu_{2}\|_{K}=\sup_{\text{Lip}\,f\leq 1}\langle
f,\nu_{1}\rangle-\langle f,\nu_{2}\rangle\leq\infty.$

Obviously $\|\nu_{1}-\nu_{2}\|_{L}^{*}\leq\|\nu_{1}-\nu_{2}\|_{K}$. Since the
$2\bar{m}$-th moments of $\mu$ and $\mu^{0}$ are bounded by $M\vee C_{0}$ by
(3.2) and the assumption on $\mu$ and since $\|\mu-\mu^{0}\|_{L}^{*}\leq d$,
then

$\|\mu-\mu^{0}\|_{K}\leq\tilde{C}(M\vee
C_{0})^{\gamma_{1}}d^{\gamma_{2}}:=\tilde{d},\quad\gamma_{1}=\tfrac{1}{2\bar{m}},\;\gamma_{2}=\tfrac{2\bar{m}-1}{2\bar{m}};$
(3.16)

see [1, Section 11.4] and [11, Chapter 7]. By the Kantorovich–Rubinstein
theorem (see [11, 1]) there exist r.v.’s $\xi$ and $\xi_{0}$, defined on a new
probability space
$(\Omega^{\prime},\mathcal{F}^{\prime},\mathbf{P}^{\prime})$, such that
$\mathcal{D}(\xi)=\mu$, $\mathcal{D}(\xi_{0})=\mu^{0}$ and

$\mathbf{E}\,|\xi_{1}-\xi_{0}|_{s_{*}}=\|\mu-\mu^{0}\|_{K}.$ (3.17)

Then using (3.13) and denoting by $a_{st}(\tau)$ a stationary solution of
(2.9), $\mathcal{D}(a_{st}(\tau))\equiv\mu^{0}$, we have:

$\begin{split}\|\mathcal{D}(a(\tau))-\mu^{0}\|_{L}^{*}=\|\mathcal{D}a(\tau;\mu^{0})-\mathcal{D}(a_{st}(\tau))\|_{L}^{*}\leq\mathbf{E}^{\omega^{\prime}}\|\mathcal{D}(a(\tau;\xi^{\omega^{\prime}}))-\mathcal{D}(a(\tau;\xi_{0}^{\omega^{\prime}}))\|_{L}^{*}\\\
\leq\mathbf{E}^{\omega^{\prime}}g^{2}_{\bar{M}}(\tau,|\xi^{\omega^{\prime}}-\xi_{0}^{\omega^{\prime}}|_{s_{*}}),\qquad{\bar{M}}={\bar{M}}^{\omega^{\prime}}=|\xi^{\omega^{\prime}}|_{s_{*}}\vee|\xi_{0}^{\omega^{\prime}}|_{s_{*}}.\end{split}$

As $\mathbf{E}^{\omega^{\prime}}{\bar{M}}^{2\bar{m}}\leq 2(M\vee C_{0})$, then
for any $K>0$,

$\mathbf{P}^{\omega^{\prime}}(Q^{\prime}_{K})\leq 2K^{-2\bar{m}}(M\vee
C_{0}),\qquad Q^{\prime}_{K}=\\{\bar{M}\geq K\\}\subset\Omega^{\prime}.$

Since $g^{2}\leq 2$ and for $\omega^{\prime}\notin Q^{\prime}_{K}$ we have
$|\xi^{\omega^{\prime}}|_{s_{*}},|\xi_{0}^{\omega^{\prime}}|_{s_{*}}\leq K$,
then

$\|\mathcal{D}(a(\tau))-\mu^{0}\|_{L}^{*}\leq 4K^{-2\bar{m}}(M\vee
C_{0})+\mathbf{E}^{\omega^{\prime}}g_{K}^{2}(\tau,|\xi^{\omega^{\prime}}-\xi_{0}^{\omega^{\prime}}|_{s_{*}})\quad\forall\,K>0.$

For an $r>0$ let us denote
$\Omega^{\prime}_{r}=\\{|\xi^{\omega^{\prime}}-\xi_{0}^{\omega^{\prime}}|_{s_{*}}\geq
r\\}$. Then by (3.17) and (3.16),
$\mathbf{P}^{\omega^{\prime}}\Omega^{\prime}_{r}\leq\tilde{d}r^{-1}$. So

$\|\mathcal{D}(a(\tau))-\mu^{0}\|_{L}^{*}\leq 4K^{-2\bar{m}}(M\vee
C_{0})+2\tilde{d}r^{-1}+g^{2}_{K}(\tau,r),\qquad\text{ for any $K,r>0$. }$
(3.18)

Let $g_{0}(l)$ be a positive continuous function on $\mathbb{R}_{+}$ such that
$g_{0}(l)\to\infty$ as $l\to+\infty$ in such a way that
$|C^{\prime}_{s_{*}}\big{(}g_{0}(l)\big{)}(\ln\ln l)^{-1/2}|\leqslant
2C^{\prime}_{s_{*}}(0)$ for $l\geqslant 3$. With $r=\tilde{d}^{1/2}$ and
$K=g_{0}(r^{-1})$, we denote the r.h.s of (3.18) as $g^{3}_{M}(\tau,r)$ (so we
substitute in (3.18) $\tilde{d}=r^{2}$ and $K=g_{0}(r^{-1})$). By (3.18) and
the definition of $g^{2}$ (see (3.13)), we have

$\begin{split}&\|\mathcal{D}(a(\tau))-\mu^{0}\|_{L}^{*}\leq
g_{M}^{3}(\tau,r)\leqslant 4(g_{0}(r^{-1}))^{-2\bar{m}}(M\vee C_{0})+2r\\\
&+4(\tau+1)\big{(}1+C^{\prime}_{s_{*}}\big{(}g_{0}(r^{-1})\big{)}\big{)}\big{(}\ln\ln(r^{-1}\vee
3)\big{)}^{-1}+rC\ln\ln(r^{-1}\vee 3)\exp\big{(}c_{1}\ln\ln(r^{-1}\vee
3)\big{)}.\end{split}$

As $r\to 0$ the second and fourth terms converge to zero. By the choice of
$g_{0}$, the first term clearly converges to zero with $r$, so does the third
term, which is $\leqslant
8(1+\tau)(1+C^{\prime}_{s_{*}}(0))(\ln\ln(r^{-1}))^{-1/2}$ for
$r\leqslant\frac{1}{3}$. Hence $g_{M}^{3}(\tau,r)$ defines a continuous
function on $\mathbb{R}_{+}^{3}$, vanishing with $r$. Using (3.16) let us
write $r=\tilde{d}^{1/2}$ as $r=R_{M}(d)$, where $R$ is a continuous function
$\mathbb{R}_{+}^{2}\to\mathbb{R}_{+}$, non-decreasing in $d$ and vanishing
with $d$. Setting $g^{4}_{M}(\tau,d)=g^{3}_{M}(\tau,R_{M}(d\wedge 2))$ and
using that $\|\mu-\mu^{0}\|_{L}^{*}\leq 2$, we get from the above that

$\|\mathcal{D}(a(\tau))-\mu^{0}\|_{L}^{*}\leq
g^{4}_{M}(\tau,\|\mu-\mu^{0}\|_{L}^{*}).$

Finally, evoking (3.15) we arrive at (3.3) with $g_{M}=g_{M}^{5}$, where

$g^{5}_{M}(\tau,d)=g^{4}_{M}(\tau,d)\wedge\hat{\mathfrak{g}}_{M}(\tau)\wedge
2,\quad 0\leq d\leq 2.$

The function $g^{5}$ is continuous, vanishes with $d$ and converges to zero as
$\tau\to\infty$. For any fixed $M>0$ this convergence is uniform in $d$ due to
the term $\hat{\mathfrak{g}}_{M}(\tau)$. So for a fixed $M>0$ the function
$(\tau,d)\mapsto g^{5}_{M}(\tau,d)$ extends to a continuous function on the
compact set $[0,\infty]\times[0,2]$ (where it vanishes when $\tau=\infty$).
Thus $g^{5}_{M}$ is uniformly continuous in $d$, and the lemma is proved.

##  4\. NLW: the setting and result

In this section we briefly discuss stochastic nonlinear wave (NLW) equations.
Following [7, 8] we consider the following equations on a smooth bounded
domain $\mathfrak{D}\subset\mathbb{R}^{3}$:

$\partial_{t}^{2}u+\gamma\partial_{t}u-\Delta u=-\gamma f(u)+\gamma
h(x)+\sqrt{\gamma}\,\eta(t,x),\quad x\in\mathfrak{D},$ (4.1)

supplemented with the Dirichlet boundary condition on $\partial\mathfrak{D}$.
Here $\gamma\in(0,1]$ is a small parameter, $h(x)$ is a function in
$H^{1}_{0}(\mathfrak{D};\mathbb{R})$ and the nonlinearity $f$ is
$C^{2}$-smooth. The random force $\eta(t,x)$ is a white noise of time of the
form

$\eta(t,x)=\frac{\partial}{\partial
t}\sum_{j=1}^{\infty}b_{j}\beta^{R}_{j}(t)\mathfrak{e}_{j}(x).$

Here $\\{\beta^{R}_{j}(t),j\geqslant 1\\}$ is a sequence of independent
standard real Brownian motions, $\\{\mathfrak{e}_{j}(x),j\geqslant 1\\}$ is an
orthonormal basis in $L^{2}(\mathfrak{D};\mathbb{R})$ composed of
eigenfunctions of the Laplacian operator; that is
$-\Delta\mathfrak{e}_{j}=\lambda_{j}\mathfrak{e}_{j}$ with
$0<\lambda_{1}\leqslant\lambda_{2}\leqslant\dots$. The set
$\\{b_{j},j\geqslant 1\\}$ is a sequence of positive real numbers, satisfying

$\mathcal{B}:=\sum_{j\geq 1}\lambda_{j}b_{j}^{2}<+\infty.$ (4.2)

The nonlinear term $f$ meets the following growth condition,

$|f^{\prime\prime}(u)|\leqslant C(|u|^{\rho-1}+1),\;\;\;u\in\mathbb{R},$ (4.3)

where $C$ and $\rho<2$ are positive constants, as well as the dissipativity
conditions

$F(u)\geqslant-\kappa u^{2}-C,\;\;\;u\in\mathbb{R},$ (4.4)
$f(u)u-F(u)\geqslant-\kappa u^{2}-C,\;u\in\mathbb{R},$ (4.5)

where $F$ is the primitive of $f$ and $\kappa$ is a positive constant.

######  Lemma 4.1.

([7, Lemma 4.5]). Under the condition (4.3) we have

$\|f(u)-f(v)\|_{0}^{2}\leqslant
C(\|u\|^{2\rho}_{1-s_{\rho}}+\|v\|^{2\rho}_{1-s_{\rho}}+1)\|u-v\|_{1-s_{\rho}}^{2},\quad
s_{\rho}=\tfrac{2-\rho}{2(\rho+1)}.$

Let us denote $\partial_{t}u=Lv$, where $L=(-\Delta)^{1/2}$, and set
$\xi=u+iv$. Then in terms of $\xi$ equation (4.1) reads

$\partial_{t}\xi=iL\xi-\gamma i\Im\xi-\gamma iL^{-1}f(\Re\xi)-\gamma
iL^{-1}\eta(t,x).$

Introducing the slow time $\tau=\gamma t$, we have

$\partial_{\tau}\xi=\gamma^{-1}iL\xi-i\Im\xi-
iL^{-1}f(\Re\xi)-i\tilde{\eta}(\tau,x),$ (4.6)

where
$\tilde{\eta}(\tau,x)=\sum_{j=1}^{\infty}\tilde{b}_{j}\dot{\beta}^{R}_{j}(\tau)\mathfrak{e}_{j}(x)$
with $\tilde{b}_{j}=b_{j}\lambda_{j}^{-1/2}$, $j\geqslant 1$.

The NLW equation (4.1) (without the $\gamma$-factor in the r.h.s) was studied
in [7, 8], where the global well-posedness and estimates for the norms of
solutions are established in the Sobolev space $H^{s}\times
H^{s-1}=\\{(u,\dot{u})\\}$, $s\in[1,2-\frac{\rho}{2}]$. A simple analysis of
the proof of [7, Proposition 3.4] or [8, Proposition 5.4] implies the
following statement on the global well-posedness of eq. (4.6):

######  Theorem 4.2.

Assume conditions (4.3), (4.4) and (4.5). Then for any
$s\in[1,2-\frac{\rho}{2})$ and $\xi_{0}\in H^{s}(\mathfrak{D};\mathbb{C})$,
equation (4.6) has a unique strong solution $\xi^{\gamma}(\tau;\xi_{0})$,
equal to $\xi_{0}$ at $\tau=0$ and defined for $\tau\geqslant 0$. Uniformly in
$\gamma\in(0,1]$ this solution satisfies

$\mathbf{E}\sup_{\tau\in[\theta,\theta+1]}\|\xi^{\gamma}(\tau;\xi_{0})\|_{s}^{2m}\leqslant
C_{s}(m,\|\xi_{0}\|_{s},\mathcal{B}),\;\;\;\forall\theta\geqslant
0,\;m\in\mathbb{N},$ (4.7)

where $C_{s}$ is a continuous function, non-decreasing in all its arguments.

Now let us write eq. (4.6) in terms of Fourier coefficients with respect to
the basis $\\{\mathfrak{e}_{j}(x)\\}$: $\xi=\sum_{j\geqslant
1}v_{j}\mathfrak{e}_{j}$. As in Section 2, we use $h^{s}$ to denote the space
of sequences of complex Fourier coefficients and denote by $\Psi$ the map
$H^{s}(\mathfrak{D};\mathbb{C})\to h^{s}$, $\xi\mapsto v:=(v_{j},j\geqslant
1)$. Then we have

$\dot{v}_{k}=\gamma^{-1}i\lambda_{k}^{1/2}v_{k}-i\Im
v_{k}+i\hat{P}_{k}(v)+i\tilde{b}_{k}\beta_{k}^{R}(\tau),\quad k\in\mathbb{N},$
(4.8)

where $\hat{P}(v)=(\hat{P}_{k}(v),k\in\mathbb{N})=\Psi(L^{-1}f(\Re\xi))$ with
$\xi=\Psi^{-1}v$. Passing to the interaction representation

$a_{k}(\tau)=e^{-i\gamma^{-1}\lambda_{k}^{1/2}\tau}v_{k}(\tau),\quad
k\geqslant 1,$

we obtain the following system of equations for the $a$-variables

$\dot{a}_{k}(\tau)=ie^{-i\gamma^{-1}\lambda_{k}^{1/2}\tau}\Big{(}-\Im\big{(}e^{i\gamma^{-1}\lambda_{k}^{1/2}\tau}a_{k}\big{)}+\hat{P}_{k}(\Phi_{\gamma^{-1}\hat{\Lambda}\tau}a)+\tilde{b}_{k}\dot{\beta}_{k}^{R}\Big{)},\;\;\;k\in\mathbb{N},$
(4.9)

where $\hat{\Lambda}=(\lambda_{k}^{1/2},k\geqslant 1)$ and operator $\Phi$ is
defined in (2.7). Let us calculate an effective equation for (4.9), following
[4].

i) To calculate the effective diffusion we decomplexify components
$a_{k}=a_{k}^{R}+ia_{k}^{I}$ of the $a$-vector as
$(a_{k}^{R}+a_{k}^{I})\in\mathbb{R}^{2}$ and write the dispersion matrix of
the equation as a block-diagonal real matrix with the blocks
$\tilde{b}_{k}\begin{pmatrix}\cos\varphi_{k}&-\sin\varphi_{k}\\\
0&0\end{pmatrix},$ where
$\varphi_{k}=\varphi^{k}(\tau)=\gamma^{-1}\lambda_{k}^{1/2}\tau$ for
$k=1,2,\dots.$ So the diffusion matrix of eq. (4.9) – let us call it $A(\tau)$
– is formed by blocks
$\tilde{b}_{k}^{2}\begin{pmatrix}\cos^{2}\varphi_{k}&-\cos\varphi_{k}\sin\varphi_{k}\\\
-\cos\varphi_{k}\sin\varphi_{k}&\sin^{2}\varphi_{k}\end{pmatrix}$. Consider
the limit $\mathcal{A}(\tau)=\lim_{\gamma\to
0}\frac{1}{\tau}\int_{0}^{\tau}A(l)\,dl.$ This is a $\tau$–independent block-
diagonal real matrix with the blocks
$\tfrac{1}{2}\tilde{b}_{k}^{2}\,\text{id}$, and this is the diffusion matrix
of the effective equation. Corresponding dispersion matrix has blocks
$\tfrac{1}{\sqrt{2}}\tilde{b}_{k}\,\text{id}$. Coming back to the complex
coordinates we see that the noise in the effective equation is
$\tfrac{1}{\sqrt{2}}\sum\tilde{b}_{k}\,\dot{\tilde{\beta}}_{k}(\tau),$ where
$\\{\tilde{\beta}_{k}(\tau)\\}$ are standard independent complex Brownian
motions.

ii) The drift in the effective equation is a sum of two terms. The second one
is

$\tilde{R}({a})=(\tilde{R}_{k}({a}),k\geqslant
1):=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\Phi_{-\hat{\Lambda}t}\hat{P}(\Phi_{\hat{\Lambda}t}{a})dt$

(cf. (2.10)). The $k$-th component of the first term is

$\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}e^{-i\lambda_{k}^{1/2}t}\big{(}-i\Im(e^{i\lambda_{k}^{1/2}t}{a}_{k})\big{)}dt=-i\tfrac{1}{2}(\Im{a}_{k}-i\Re{a}_{k})=-\tfrac{1}{2}{a}_{k}.$

iii) So the effective equation for (4.9) reads

$\dot{a}_{k}(\tau)=-\tfrac{1}{2}{a}_{k}+i\tilde{R}_{k}({a})+\tfrac{1}{\sqrt{2}}\tilde{b}_{k}d\tilde{\beta}_{k},\qquad
k\in\mathbb{N}.$ (4.10)

By Lemma 4.1 and [4, Lemma 4], we have

$|\tilde{R}(a_{1})-\tilde{R}(a_{2})|_{1}\leqslant
C\big{(}|a_{1}|_{1-s_{\rho}}^{\rho}\wedge|a_{2}|_{1-s_{\rho}}^{\rho}+1\big{)}|a_{1}-a_{2}|_{1-s_{\rho}},\;\;\;a_{1},a_{2}\in
h^{1}.$ (4.11)

In particular, the mapping ${a}\mapsto\tilde{R}({a})$ belongs to
$\text{Lip}_{\rho+1}(h^{s},h^{s})$, $s\in[s_{\rho},1]$. Therefore effective
equation (4.10) is at least locally well-posed in $h^{s}$, $s\in[s_{\rho},1]$.
Let us denote $a^{0}(\tau;a_{0})$ its solution with an initial condition
$a^{0}(0;a_{0})=a_{0}$. Similarly let $a^{\gamma}(\tau;v_{0})$ be a solution
of eq. (4.9), equal $a_{0}$ at $\tau=0$. Then using (4.7) and arguing as in
[4] we get

######  Theorem 4.3.

Under the assumptions of Theorem 4.2 we have:

i) for any ${a}_{0}\in h^{1}$, eq. (4.10) has a unique strong solution
$a^{0}(\tau;{a}_{0})$, $\tau\geqslant 0$. It belongs to $C([0,\infty),h^{1})$
a.s., and for any $\theta\geqslant 0$ satisfies

$\mathbf{\sup}_{\tau\in[\theta,\theta+1]}|a^{0}(\tau;{a}_{0})|_{1}^{2m}\leqslant
C(m,|{a}|_{1},\mathcal{B}),\;\;\;\forall m\in\mathbb{N},$ (4.12)

where $C$ is a continuous function, increasing in all its arguments.

ii) For any $s>1$, ${a}_{0}\in h^{s}$ and $T>0$ we have

$\mathcal{D}\big{(}a^{\gamma}(\tau;{a}_{0})\\!\mid_{[0,T]}\\!\big{)}\rightharpoonup\mathcal{D}\big{(}a^{0}(\tau;{a}_{0})\\!\mid_{[0,T]}\\!\big{)}\quad\text{in}\;\;\mathcal{P}\big{(}C([0,T]);h^{1})\big{)}\;\;\text{as}\;\;\gamma\to
0.$ (4.13)

Moreover, for any $M>0$ the above convergence is uniform for
${a}_{0}\in\bar{B}_{M}(h^{s})$.

Repeating the argument in Section 3 we get the following analogy to Theorem
2.7:

######  Theorem 4.4.

Assume in addition to the assumption in Theorem 4.3 that the effective
equation (4.10) is mixing in the space $h^{1}$ and that the requirement as in
Assumption 2.6 is met. Then for any $s>1$ and $v_{0}\in h^{s}$ we have the
following convergence

$\lim_{\gamma\to 0}\sup_{\tau\geqslant
0}\|\mathcal{D}(a^{\gamma}(\tau;v_{0}))-\mathcal{D}(a^{0}(\tau;v_{0}))\|_{L,h^{1}}^{*}=0.$

Moreover, for any $M>0$ the above convergence is uniform for
$v_{0}\in\bar{B}_{M}(h^{s})$.

For the original equation (4.1) we introduce the vector of action variables
for its solution $\mathfrak{u}=(u,u_{t})$ as
$I(\mathfrak{u})=(I_{1}(\mathfrak{u}),\dots)$, where
$I_{k}(\mathfrak{u})=\frac{1}{2}|\langle
u,\mathfrak{e}_{k}\rangle|^{2}+\frac{1}{2}\lambda_{k}^{-1}|\langle
u_{t},\mathfrak{e}_{k}\rangle|^{2}$, $k=1,\dots$. Then we have the following
corollary in analogy to Corollary 2.8:

######  Corollary 4.5.

Under the assumption of Theorem 4.4, for any $s>1$ and
$\mathfrak{u}_{0}=[u_{1},u_{2}]\in H^{s}\times H^{s-1}$, the action-vector of
a solution $\mathfrak{u}^{\gamma}(t;\mathfrak{u}_{0})$ for equation (4.1),
equal $\mathfrak{u}_{0}$ at $t=0$, satisfies

$\lim_{\gamma\to 0}\,\sup_{\tau\geqslant
0}\|\mathcal{D}\big{(}I(\mathfrak{u}^{\gamma}(\tau\gamma^{-1};\mathfrak{u}_{0}))\big{)}-\mathcal{D}\big{(}(I(a^{0}(\tau;v_{0}))\big{)}\|_{L,h_{I}^{1}}^{*}=0,$

where $v_{0}=\Psi(u_{1}+iL^{-1}u_{2})$.

The result of Theorem 4.4 is conditional since it requires that eq. (4.9) is
mixing. It is not our goal in this short section to check the mixing property.
But we mention that since the effective equation (4.10) is similar to the NLW
equation (4.1), written in the form (4.6), and since by (4.13) solutions for
(4.10) inherit the estimates on solutions for (4.1), obtained in [7, 8], then
most likely the proof of the exponential mixing in eq. (4.1), given in [7]
(and based on an abstract theorem from [5]), applies to establish the
exponential mixing for eq. (4.10) and thus verify for it the analogy of
Assumption 2.6, required in Theorem 4.4.

## Acknowledgment

We are thankful to Armen Shirikyan for discussion. GH was supported by
National Natural Science Foundation of China (Significant project
No.11790273).

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