# LIV effects on the quantum stochastic motion in an acoustic FRW-geometry

M. A. Anacleto<EMAIL_ADDRESS>Departamento de Física, Universidade
Federal de Campina Grande Caixa Postal 10071, 58429-900 Campina Grande,
Paraíba, Brazil  C. H. G. Bessa<EMAIL_ADDRESS>Departamento de Física,
Universidade Federal de Campina Grande Caixa Postal 10071, 58429-900 Campina
Grande, Paraíba, Brazil  F. A. Brito<EMAIL_ADDRESS>Departamento de
Física, Universidade Federal de Campina Grande Caixa Postal 10071, 58429-900
Campina Grande, Paraíba, Brazil  Departamento de Física, Universidade Federal
da Paraíba, Caixa Postal 5008, 58051-970 João Pessoa, Paraíba, Brazil  A. E.
Mateus<EMAIL_ADDRESS>Departamento de Física, Universidade Federal
de Campina Grande Caixa Postal 10071, 58429-900 Campina Grande, Paraíba,
Brazil  E. Passos<EMAIL_ADDRESS>Departamento de Física, Universidade
Federal de Campina Grande Caixa Postal 10071, 58429-900 Campina Grande,
Paraíba, Brazil  J. R. L. Santos<EMAIL_ADDRESS>Departamento de
Física, Universidade Federal de Campina Grande Caixa Postal 10071, 58429-900
Campina Grande, Paraíba, Brazil

###### Abstract

It is well known in the literature that vacuum fluctuations can induce a
random motion of particles which is sometimes called quantum Brownian motion
or quantum stochastic motion. In this paper, we consider Lorentz Invariance
Violation (LIV) in an acoustic spatially flat Friedman-Robertson-Walker (FRW)
geometry. In particular, we are looking for the LIV effects in the stochastic
motion of scalar and massive test particles. This motion is induced by a
massless quantized scalar field on this geometry, which in turn is derived
from an Abelian Higgs model with LIV. Deviations in the velocity dispersion of
the particles proportional to the LIV parameter are found.

##  I Introduction

It is well accepted today that in a complete theory of quantum gravity a
fundamental concept like the Lorentz invariance (LI), which arises naturally
from Lorentz transformations in special relativity, should be violated in some
regime. In general, it is quite believed that this violation, also known as
the Lorentz invariance violation (LIV), should appear in the energy range
close to the Planck scale. It was Kostelecky and collaborators ks89a  who
proposed one of the first effective theories where the LIV was permitted. They
developed a model that extends the standard model into a new one where the LIV
could appear. In this extended model, all sectors are corrected by small
coefficients or parameters which can be adjusted according to the requirements
of the theory under consideration. The search for phenomenological models
where the LIV could be detectable was developed by many authors ks89a ; ks89b
; ks91 ; kl01 ; ck98 ; bk04 ; k03 ; bbm05 ; abp10 ; abp11 ; Anacleto:2019rfn .
Although no violation has appeared so far, such studies have proved to be very
useful once they enable us to stipulate the values of the LIV parameters. In
this context, we mention works related to quantum gravity effects at low
energy scales. Among many possibilities we call attention to studies related
directly to string theory ks89a ; ks89b ; ks91 , loop quantum gravity atu00 ,
Horava-Lifshitz theory h09 , noncommutativity chklo01 ; bbq08 ; n09 ; bl16 ,
and also models related to modified gravity lsma20  and special relativity
cg06 ; msca14 ; ms02 . All of these approaches are attempts to search for LIV
effects that could be in principle testable with future technologies.

In this framework, the main purpose of this paper is to show a different model
where LIV effects could appear. The model consists of an acoustic spatially
flat Friedmann-Robertson-Walker (FRW) geometry derived from an Abelian Higgs
model with LIV in the scalar sector. In this scenario, we study the motion of
a scalar test particle under the influence of a massless quantized and vacuum
fluctuating scalar field. It is well known in the literature that vacuum
fluctuations can induce a random motion of particles which is sometimes called
quantum Brownian motion111To avoid confusion with other models which use the
same nomenclature in but are related to non-equilibrium quantum thermodynamic
models ehm09 ; bcgaa18 ; dc19 , we decide to call this effect here as quantum
stochastic motion., or quantum stochastic motion. So, the type of motion we
will deal with in the present paper is characterized by the calculation of the
velocity dispersion of the particles ($\langle\Delta v^{2}\rangle$) where a
Langevin type equation is considered and the particles dispersion velocity is
directly associated with the Hadamard function of the quantum fields. Thus, in
a LI perspective, this stochastic motion in Minkowski spacetime is exhibited
only when some physical boundary or non-trivial topology is present gs99 ;
jr92 ; YuFord2004 ; YuChen2004 ; Seriu2009 ; Seriu2008 ; Lorenci2019 ; clrr19
; pf11 ; lrs16 ; clrrs18 ; cy04 ; zy05 ; br20 ; mm19 ; DaShinLee2005 ;
DaShinLee2008 ; DaShinLee2014 , but for a time-dependent expanding background,
as a FRW type, it could appear without any boundary intervention and with
trivial topology Bessa2017 ; Bessa2009 ; abbfp20 . Particularly, for the
scalar particles, which are the kind of particles we are interested in, this
stochastic motion shows up only when they are submitted to an external
classical non-fluctuating force ($f_{ext}$). In this scenario, these particles
are usually named as ‘bound particles’ once this $f_{ext}$ can prevent them to
feel locally the expansion. In a different and more realistic scenario, this
external force could vanish and the particles could follow their geodesics. In
this case, the particles are named ‘free particles’ and their stochastic
motion is null once their velocity dispersion is zero ($\langle\Delta
v^{2}\rangle=0$), i.e., the free particles present a dispersion identical to
the one found in the Minkowski case without boundaries and with trivial
topology. In this context, the main result of this paper is to show that when
we take into account the LIV model, the quantum stochastic motion appears for
both types of particles discussed above, i.e. $\langle\Delta
v^{2}\rangle_{LIV}\neq 0$, for bound and free particles. A similar result was
found in Ref. abbfp20  for a (3+1)-dimensional non-commutative Abelian Higgs
model.

So this article is organized as follows: in Sect. II we review some basic
results found in Ref. Bessa2017  where the study of the stochastic motion for
scalar particles was developed in an expanding Bose-Einstein condensate (BEC)
that mimics the FRW geometry in the LI context. Then in Sect. III we show,
following the steps described in Ref. abp10 , how to find a similar acoustic
FRW geometry from an Abelian Higgs model with LIV terms. This new acoustic
geometry will be used in Sect. IV to reanalyze the problem discussed in Sect.
II but now in a LIV perspective. In this case, small corrections in the
particle’s velocity dispersion proportional to the LIV parameters are found
when $f_{ext}\neq 0$ and $f_{ext}=0$. In Sect. V we discuss the main results
and give the conclusions. In this paper we use units where $\hbar=c=1$.

##  II Quantum stochastic motion in an acoustic FRW geometry: Short review

In this section, we review the main results of Ref. Bessa2017  where the
stochastic motion of a scalar point particle that was under the influence of a
quantized vacuum fluctuating and massless scalar field in a FRW geometry was
considered. This study was carried out in a LI framework using a Bose-Einstein
condensate (BEC). In this context, a well-known result is the possibility to
mimic some aspects of a FRW universe using superfluids with remarkable
experimental results ekjsc18 . Thus, in Refs. Bessa2017 ; Visser:1997ux ;
Barcelo2011 ; bffr08 ; jwvg07 ; pfl10 , a linearization in the basic equation
of motion of an inviscid and irrotational BEC was applied. This linearization
was done in the fluid constituents such as the field $\phi=\phi_{0}+\phi_{1}$,
the phase $S=S_{0}+S_{1}$, fluid density $\rho=\rho_{0}+\rho_{1}$, and with a
vanished velocity flow $\vec{v}=\hbar\nabla\phi_{0}/m=0$. Following this
procedure, we can find an equation of motion similar to a Klein-Gordon
equation in curved spacetime

$\Box\phi_{1}=0,$ (1)

where
$\Box=\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu})$
with an effective spatially flat FRW metric:

$ds_{eff}^{2}=-c_{0}^{2}dt^{2}+a_{eff}^{2}(t)(dx^{2}+dy^{2}+dz^{2}),$ (2)

where the dimensionless effective scale factor is
$a_{eff}(t)=(c_{0}/c_{s}(t))^{2}$, with $c_{s}(t)=4\pi\hbar\rho_{0}L(t)/m$
been the time dependent sound speed, $c_{0}=c_{s}(t=t_{0})$, the sound speed
in certain initial time, and $L(t)$ a time dependent scattering length. Note
that, in order to have an analog of expanding universe, $a_{eff}$ must
increase with time when $c_{s}(t)$ decreases. As usual, a conformal
transformation of the type, $dt=ad\eta$, is possible to be made222For
briefness, we write from now on $ds_{eff}=ds$ and $a_{eff}=a$.. Thus we obtain
a conformal FRW analog geometry represented by:

$ds^{2}=a^{2}(\eta)\left(-c_{0}^{2}d\eta^{2}+dx^{2}+dy^{2}+dz^{2}\right),$ (3)

where $\eta$ is the conformal time.

In this model, the atoms that compound the fluid are treated as scalar point
particles that interact with a quantized massless scalar field, which
represents the acoustic perturbations driven by the phonons in this effective
time-dependent geometry. Thus, as was pointed out in Ref. Bessa2017 , these
particles fell the fluid expansion according to the effective metric (2) or
(3). In this context, the quantized field could induce metric fluctuations.
However, the contributions that come from these fluctuations are usually
secondary to the motion of the point particles considered in these types of
models abbfp20

The equation of motion for a single particle in certain $i$-direction is given
by

$m\frac{Du^{i}}{d\tau}={f^{\prime}}^{i},$ (4)

where $i\in\\{x,y,z\\}$, $D/d\tau$ is a covariant derivative, $u^{i}$ is the
coordinate $3$-velocity, and $m$ is the particle mass. From equation above,
the left hand side is

$m\dfrac{du^{i}}{dt}+2m\dfrac{\dot{a}}{a}u^{i}={f^{\prime}}^{i},$ (5)

where the proper time $\tau\approx t$, for non-relativistic motion, and Eq.
(2) was used to calculate the Cristofell symbols
$\Gamma^{i}_{\alpha\beta}=\dot{a}/a$. In the right hand side of Eq. (5), the
force term ${f^{\prime}}^{i}$ can be split into two parts, as follows:
${f^{\prime}}^{i}=f^{i}+f_{ext}^{i}$. Here, $f^{i}$ is a term which is
directly related with a scalar field and it can be derived from the Klein-
Gordon equation as ppv11

$f^{i}=qg^{i\nu}\nabla_{\nu}\phi_{1},$ (6)

where is valid for a flat or conformally flat spacetime Bessa2017 ; ppv11 ,
the covariant derivative is the partial derivative
($\nabla_{\mu}\phi=\partial_{\mu}\phi$), and $q$ is a scalar charge which
gives the strength of the interaction between particles and the field. The
other term, $f_{ext}^{i}$, is a classical external non-fluctuating force. In
fact, $f_{ext}^{i}$ can be considered equal to zero, without loss of
generality. In this case, the particles are called ‘free particles’, in such
way that they follow their own geodesic. However, in some situations $f_{ext}$
can be different from zero, or more conveniently $f_{ext}\approx\dot{a}/a$, in
such a way that the particles do not feel locally the expansion. In this case
they are named ‘bound particles’. This nomenclature was adopted in Refs.
Bessa2009 ; Bessa2017 ; abbfp20  and we will follow it in the present paper.

###  II.1 Bound particles

Let us start reviewing the bound particles case. For this purpose consider
that the scalar particles are under the influence of an external force
$f_{ext}=2m\frac{\dot{a}}{a}u^{i}$, in such way that Eq. (5) reduces to

$\frac{du^{i}}{dt}=\frac{f^{i}}{m}.$ (7)

After integrating the equation above and considering a zero initial velocity,
$u^{i}(t_{0})=0$, we can evaluate the dispersion for the coordinate velocity
of the scalar particle. Thereafter, we evaluate the dispersion for the
physical $3$-velocity, namely, proper velocity ($v^{i}$), in this expanding
geometry. The relation between $u^{i}$ and $v^{i}$ is well known and it is
associated with the coordinate length ($r$) and proper length ($l$), that is,
$l=a_{f}r$, where the sub-indexes indicates that the scale factor was
evaluated in a final time ($t_{f}$), i.e., when the expansion ceases. Thus,
$v^{i}=a_{f}u^{i}$ and taking into account Eq. (6), we obtain

$\langle(\Delta
v^{i})^{2}\rangle=\dfrac{q^{2}a_{f}^{2}}{m^{2}}\left[\partial_{i_{1}}\partial_{i_{2}}\int_{\eta_{i}}^{\eta_{f}}\int_{\eta_{i}}^{\eta_{f}}d\eta_{1}d\eta_{2}a^{-2}(\eta_{1})a^{-2}(\eta_{2})\langle\phi_{1}(\eta_{1},r_{1})\phi_{1}(\eta_{2},r_{2})\rangle_{M}\right]_{r_{1}\rightarrow
r_{2}}.$ (8)

This is the proper velocity dispersion333 The velocity dispersion was defined
as usual: $\langle(\Delta v^{i})^{2}\rangle=\langle
v^{i}(r_{1},t_{1})v^{i}(r_{2},t_{2})\rangle-\langle
v^{i}(r_{1},t_{1})\rangle\langle v^{i}(r_{2},t_{2})\rangle$ where a change
between coordinate time $t$ and conformal time $\eta$ was made and the
coincidence limit in the spatial coordinates ($r_{1}\rightarrow r_{2}$) was
taken, after the spatial partial derivatives were applied. Note that the
scalar field $\phi$ obeys the relation
$\langle\phi(t_{1},r_{1})\phi(t_{2},r_{2})\rangle\neq
0,\langle\phi(t,r)\rangle=0$. The sub-indexes $M$, indicates that the
correlation function for the massless scalar field was evaluated in a
$4$-dimensional Minkowski spacetime and by the conformal metric (3) it is
related to the $FRW$ correlation function by a conformal factor $\Omega$
Birrell1984 , that comes from the relation
$g_{\mu\nu}=\Omega^{2}\eta_{\mu\nu}$. From Eq. (3), $\Omega={a(\eta)}$, thus,
from Ref. Birrell1984 , we obatin,

$\langle\phi_{1}(\eta_{1},r_{1})\phi_{1}(\eta_{2},r_{2})\rangle_{FRW}=\Omega^{-1}(\eta_{1},r_{1})\Omega^{-1}(\eta_{2},r_{2})\langle\phi(\eta_{1},r_{1})\phi(\eta_{2},r_{2})\rangle_{M}=a^{-1}(\eta_{1})a^{-1}(\eta_{2})\langle\phi_{1}(\eta_{1},r_{1})\phi_{1}(\eta_{2},r_{2})\rangle_{M}.$
(9)

Where $\langle\phi_{1}(\eta_{1},r_{1})\phi_{1}(\eta_{2},r_{2})\rangle_{M}$ is
the Hadamard function in Minkowski spacetime, which is given by:

$\langle\phi_{1}(\eta_{1},r_{1})\phi_{1}(\eta_{2},r_{2})\rangle_{M}=\frac{1}{4\pi^{2}}\frac{1}{[-c_{0}^{2}(\eta_{1}-\eta_{2})^{2}+r^{2}]},$
(10)

with $r^{2}=\Delta x^{2}+\Delta y^{2}+\Delta z^{2}$.

To integrate Eq. (8) we must choose an appropriate effective scale factor
$a(\eta)$. In many papers Bessa2017 ; abbfp20 ; jwvg07 ; pfl10  the authors
consider an asymptotically bound dimensionless scale factor to model the
effective $FRW$ geometry444Recently, a model that reproduces in the laboratory
some features of an inflationary universe was performed ekjsc18  with a scale
factor similar to Eq. (11) , whose form is

$a^{2}(\eta)=a_{0}^{2}+a_{1}^{2}\tanh\left(\frac{\eta}{\eta_{0}}\right),$ (11)

where $a^{2}(\eta)$ is flat in the asymptotic regions
$(\eta\rightarrow\pm\infty)$ and $a_{0}$, $a_{1}$ are dimensionless constants,
with the former producing a displacement of $a(\eta)$ avoiding the divergence
when $\eta=0$ and the latter modifying the space in the limits
$\eta\rightarrow\pm\infty$. The $\eta_{0}$ is a constant parameter with
dimension of time that smooths the transition between the asymptotic limits.
Thus, we can evaluate Eq. (8) using this scale factor. Proceeding this way, we
take Eqs. (10) and (11) in Eq. (8), and to integrate it we followed the same
steps present in Appendix B of Ref. Bessa2009 . Then, the proper velocity
dispersion is

$\langle(\Delta
v^{i})^{2}\rangle=\dfrac{2q^{2}B}{\pi^{4}m^{2}c_{0}^{4}\eta_{0}^{2}}\left(\zeta(3)-\dfrac{\pi^{4}}{90}\right),$
(12)

where

$B=\frac{c_{0}}{4c_{sf}}\left(1-\frac{c_{sf}}{c_{0}}\right)^{2},$ (13)

is a dimensionless constant defined in terms of the initial ($c_{0}$) and
final ($c_{sf}$) sound speed, besides $\zeta(n)$ is the zeta function. Note
that, when $c_{sf}=c_{0}$, $B=0$ a null dispersion is found. This is the
expected result of a Minkowski spacetime with trivial topology YuFord2004 ;
YuChen2004 ; Bessa2009 .

In the Sect. IV we will re-analyse this problem and will find extra terms
(corrections) to the velocity dispersion coming from an expanding LIV model
similar to Eq. (3).

###  II.2 Free particles

The second case studied in Ref. Bessa2017  was the quantum stochastic motion
of the free particles. As previously presented, this case is characterized by
the fact that $f_{ext}=0$. So, Eq. (5) can be written as

$\frac{1}{a^{2}}\frac{d}{dt}(a^{2}u^{i})=\frac{f^{i}}{m},$ (14)

where $f^{i}$ is given by Eq. (6).

To integrate this equation and to find the proper velocity dispersion
($(\Delta v^{i})^{2}$) we follow the same procedure discussed in the bound
particles section. Thus, the velocity dispersion is

$\langle(\Delta
v^{i})^{2}\rangle=\dfrac{q^{2}}{m^{2}a_{f}^{2}}\partial_{i1}\partial_{i2}\left[\int_{0}^{\eta_{f}}\int_{0}^{\eta_{f}}d\eta_{1}d\eta_{2}\langle\phi_{1}(\eta_{1},r_{1})\phi_{1}(\eta_{2},r_{2})\rangle_{M}\right]_{r_{1}\rightarrow
r_{2}}.$ (15)

Note that the scale factor $a$ does not appears in the integrand above. Using
Eq. (10) and taking the coincidence limit after the integration, we find a
divergent result. To avoid this divergence, a renormalization is required. It
consists in subtract the Minkowski contribution in the Hadamard function.
Thus, the integrand will be null yielding to the trivial result

$\langle(\Delta v^{i})^{2}\rangle=0.$ (16)

In other words, there is no stochastic motion for free scalar particles when
the LI is preserved.

In the Sect. IV we will take into account the LIV model in the expanding
geometry. We will find a non-null velocity dispersion. This indicates that the
model proposed in this paper could be a source to look for LIV effects.

##  III Analog model to the FRW geometry in a LIV perspective

In the previous section, we have reviewed the formalism related to the quantum
stochastic motion of scalar particles. This motion is induced by the quantum
vacuum fluctuations of a massless scalar field in an expanding time-dependent
background. It was discussed that this background can mimic some aspects of a
FRW geometry when a linearization in the parameters of the basic equation of a
BEC is applied. Since, as far as we know, the Lorentz invariance is preserved
in a BEC, our goal in this section is to show that a similar time-dependent
expanding background can be accomplished when we take into account a model
where LIV is permitted. For this purpose, we consider an extension in the
Abelian Higgs model with a term that violates the Lorentz invariance in the
scalar sector. We will follow the procedure present in Section 2 of Ref. abp10
. Thus, consider a Lagrangian of the type,

$\mathcal{L}=-\dfrac{1}{4}F_{\mu\nu}F^{\mu\nu}+|D_{\mu}\phi|^{2}+M^{2}|\phi|^{2}-b|\phi|^{4}+k^{\mu\nu}D_{\mu}\phi^{*}D_{\nu}\phi\,,$
(17)

with

$k_{\mu\nu}=\begin{pmatrix}\beta&\alpha&\alpha&\alpha\\\
\alpha&\beta&\alpha&\alpha\\\ \alpha&\alpha&\beta&\alpha\\\
\alpha&\alpha&\alpha&\beta\\\ \end{pmatrix}.$ (18)

The tensor $k_{\mu\nu}$ introduces the Lorentz symmetry breaking terms,
represented by the real parameters $\alpha$ and $\beta$, which were coupled to
the scalar field. In this Lagrangian, gravity is not present and the term
$F^{\mu\nu}$ is the Maxwell tensor, $D_{\mu}=\partial_{\mu}-ieA_{\mu}$ with
$e$ and $b$ being interaction terms, and $A_{\mu}$ is the $4$-potential.

In order to simplify the model, we will only consider the diagonal terms in
$k_{\mu\nu}$ matrix, i.e. $\alpha=0,\beta\neq 0$. A decomposition of the type:
$\phi(x^{\prime},t^{\prime})=\sqrt{\rho(x^{\prime},t^{\prime})}exp(iS(x^{\prime},t^{\prime}))$,
which is known as Madelung representation, can be done with the prime
representing the coordinates in the flat spacetime Lagrangian. Notice that the
fluid density is now defined as $\rho=|\phi|^{2}$. This representation gives
us a fluid description and after substituting $\phi$ in Eq. (17) we obtain the
corresponding hydrodynamic equations of motion:

$-\partial_{t^{\prime}}\left[\tilde{\beta}_{+}\rho(\dot{S}-eA_{t^{\prime}})\right]+\partial_{i^{\prime}}\left[\tilde{\beta}_{-}\rho(\partial^{i^{\prime}}S-eA^{i^{\prime}})\right]=0$
(19)

and

$\dfrac{(\tilde{\beta}_{+}\partial_{t^{\prime}}^{2}-\tilde{\beta}_{-}\partial_{i^{\prime}}^{2})\sqrt{\rho}}{\sqrt{\rho}}+\tilde{\beta}_{+}(\dot{S}-eA_{t^{\prime}})^{2}-\tilde{\beta}_{-}(\partial_{i^{\prime}}S-eA_{i^{\prime}})^{2}+M^{2}-2b\rho=0,$
(20)

where $\tilde{\beta}_{\pm}\equiv 1\pm\beta$ and $\dot{S}=\frac{\partial
S}{\partial t^{\prime}}$.

Let us now consider a perturbation around the density given by
$\rho=\rho_{0}+\rho_{1}$. We also can change the phase for $S=S_{0}+S_{1}$,
and consequently we have $\phi=\phi_{0}+\phi_{1}$, where
$\rho_{1}\ll\rho_{0}$, $S_{1}\ll S_{0}$, and $\phi_{1}\ll\phi_{0}$. So for
simplicity, $M=0$, equations above become

$\displaystyle\partial_{t^{\prime}}$
$\displaystyle\left[\left(-\dfrac{b\rho_{0}}{2\tilde{\beta}_{-}}+\dfrac{\mathcal{D}_{2}\rho_{0}}{2\tilde{\beta}_{-}}-\dfrac{\tilde{\beta}_{+}}{\tilde{\beta}_{-}}\omega_{0}^{2}\right)\dot{S}_{1}-\omega_{0}\vec{v}_{0}\cdot\nabla
S_{1}\right]$ (21) $\displaystyle+$
$\displaystyle\nabla\cdot\left[-\omega_{0}\vec{v}_{0}\dot{S}_{1}+\left(\dfrac{b\rho_{0}}{2\tilde{\beta}_{+}}-\dfrac{\mathcal{D}_{2}\rho_{0}}{2\tilde{\beta}_{+}}\right)\nabla
S_{1}-\dfrac{\tilde{\beta}_{-}}{\tilde{\beta}_{+}}\vec{v}_{0}\cdot\nabla
S_{1}\vec{v}_{0}\right]=0.$

These fluctuations of the fluid are similar to the BEC case studied in Sect.
II and in other analogue gravity models Barcelo2011 . Furthermore, we obtain a
phonon description when these fluctuations (or equivalently $\phi_{1}$) are
quantized, as we shall see shortly. Note that we have defined
$\mathcal{D}_{2}=\tilde{\beta}_{+}D_{t2}+\tilde{\beta}_{-}D_{i2}$,
$\omega_{0}=-\dot{S}_{0}+eA_{t}$ and $\vec{v_{0}}=\nabla S_{0}+e\vec{A}$ is
the local velocity field. The term $\mathcal{D}_{2}$ is very small and
following Ref. abp10  it can be dropped out.

Following the usual procedure, we can see that Eq. (21) mimics a Klein-Gordon
equation in curved spacetime with the metric

$d\bar{s}^{2}=\frac{b\rho_{0}(\tilde{\beta}_{-})^{1/2}}{2c_{s}}\left[-\frac{c_{s}^{2}}{\tilde{\beta}_{+}}d{t^{\prime}}^{2}+\frac{\tilde{\beta}_{+}}{\tilde{\beta}_{-}}\left(d{x^{\prime}}^{2}+d{y^{\prime}}^{2}+d{z^{\prime}}^{2}\right)\right].$
(22)

Here we have assummed a non-relativistic limit where the time dependent sound
speed in the fluid is $c_{s}^{2}=\frac{b\rho_{0}}{2w_{0}^{2}}\ll 1$, and the
background flow is null ($\vec{v}=0$).

Absorbing the constants above into the coordinates
$t^{\prime},x^{\prime},y^{\prime},z^{\prime}$ and in $d\bar{s}$ we find the
usual flat metric evaluated in a Lorentz invariant (LI) context

$d\bar{s}^{2}=-c_{s}^{2}d{t^{\prime}}^{2}+d{x^{\prime}}^{2}+d{y^{\prime}}^{2}+d{z^{\prime}}^{2}.$
(23)

This is clearly different from the metric obtained in Ref. abp10 , in which
the flux velocity does not vanish, and an acoustic Kerr-like black hole metric
with the LIV corrections is found. Despite of the LI metric above, we will see
that the $\beta$ factor present in Eq. (17) influences the equation of motion
of the particles anyway. Finally, to write this metric in FRW form (3), we
simply follow the discussion presented in Sect. II. In this case we have a
Klein-Gordon equation in curved spacetime with

$\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}\tilde{g}^{\mu\nu}\partial_{\nu}\phi_{1})=0,$
(24)

where the effective metric tensor is now
$\tilde{g}^{\mu\nu}=g^{\mu\nu}+k^{\mu\nu}$.

In the next section we will use this metric to study the quantum stochastic
motion of the scalar particle described in Sect. II.

##  IV Quantum stochastic motion in a FRW acoustic geometry: LIV effects

In this section, we consider Eq. (24), and follow the same methodology
described in Sect. II to find a new $f^{i}$ term. Now, it will take into
account the LIV corrections in the following way:

$f^{i}=q({g}^{i\mu}+k^{i\mu})\partial_{\mu}\phi_{1}.$ (25)

Note that when $k^{\mu\nu}=0$, Eq. (6) is recovered, which is valid for a flat
or conformally flat spacetime where LI is preserved. Since we use metric (2),
the left hand side of Eq. (5) does not change. So we have now the following
equation of motion

$m\dfrac{du^{i}}{dt}+2m\dfrac{\dot{a}}{a}u^{i}=q[a^{-2}(t)+\beta]\partial_{i}\phi_{1}+f^{i}_{ext},$
(26)

where $\dot{a}=da/dt$. This is the equation of motion of the scalar particle
taking into account the LIV effects where $\beta$ comes from Eq. (24).

In this context, in the next two sub-sections, we will consider the acoustic
metric evaluated in the previous section to study the stochastic motion of
point-like massive scalar particles in this expanding background that mimics
some features of a spatially flat FRW geometry. Thus, to evaluate the
dispersion velocity of the particles with mass $m$ of the compounds (treated
as scalar point particles) of the fluid with density $\rho$, we will consider
again the bound and free particles defined in Sect. II.

###  IV.1 Bound Particles

Let us start this section with the bound particles case, as discussed in Sect.
II, a single particle is under the effect of a classical non-fluctuating force
given by: $f^{i}_{ext}=2mu^{i}\dot{a}/a$. Thus the equation of motion for the
particles reduces to

$m\dfrac{du^{i}}{dt}=q[a^{-2}(t)+\beta]\partial_{i}\phi_{1},$ (27)

where only the spatial derivatives are considered once the metric $g^{\mu\nu}$
and the $k^{\mu\nu}$ matrix are diagonal. Thus, the corresponding velocity-
velocity correlation function using the LIV effective metric is

$\displaystyle\langle
u^{i}(t_{1},r_{1})u^{i}(t_{2},r_{2})\rangle_{LIV}=\frac{q^{2}}{m^{2}}\int\int
dt_{1}dt_{2}\left\\{\left[a^{-2}(t_{1})+\beta\right]\left[a^{-2}(t_{2})+\beta\right]\right\\}$
(28)
$\displaystyle\times\partial_{i_{1}}\partial_{i_{2}}\langle\phi_{1}(t_{1},r_{1})\phi_{1}(t_{2},r_{2})\rangle_{FRW}$

Following the procedure presented in Sect. II, we first make a conformal
transformation involving the coordinate time $t$ and the conformal time $\eta$
($dt=ad\eta$) and make use of Eqs. (9) and (3) to obtain the two-point
function in the FRW spacetime in terms of the Hadamard function in Minkowski
spacetime

$\langle\phi_{1}(\eta_{1},r_{1})\phi_{1}(\eta_{2},r_{2})\rangle_{FRW}=a^{-1}(\eta_{1})a^{-1}(\eta_{2})\langle\phi_{1}(\eta_{1},r_{1})\phi_{1}(\eta_{2},r_{2})\rangle_{M},$
(29)

Using Eq. (29) in Eq. (28) together with the change, $dt=ad\eta$, we obtain,

$\displaystyle\langle\langle
u^{i}(t_{1},r_{1})u^{i}(t_{2},r_{2})\rangle\rangle_{LIV}$ $\displaystyle=$
$\displaystyle\dfrac{q^{2}}{m^{2}}\int d\eta_{2}a^{-2}(\eta_{2})\int
d\eta_{1}a^{-2}(\eta_{1})\partial_{i_{1}}\partial_{i_{2}}\langle\phi_{1}(\eta_{1},r_{1})\phi_{1}(\eta_{2},r_{2})\rangle_{M}+$
(30) $\displaystyle+$ $\displaystyle\dfrac{q^{2}\beta}{m^{2}}\int
d\eta_{2}\int
d\eta_{1}a^{-2}(\eta_{1})\partial_{i_{1}}\partial_{i_{2}}\langle\phi_{1}(\eta_{1},r_{1})\phi_{1}(\eta_{2},r_{2})\rangle_{M}+$
$\displaystyle+$ $\displaystyle\dfrac{q^{2}\beta}{m^{2}}\int d\eta_{1}\int
d\eta_{2}a^{-2}(\eta_{2})\partial_{i_{1}}\partial_{i_{2}}\langle\phi_{1}(\eta_{1},r_{1})\phi_{1}(\eta_{2},r_{2})\rangle_{M}+$
$\displaystyle+$ $\displaystyle\dfrac{q^{2}\beta^{2}}{m^{2}}\int d\eta_{1}\int
d\eta_{2}\partial_{i_{1}}\partial_{i_{2}}\langle\phi_{1}(\eta_{1},r_{1})\phi_{1}(\eta_{2},r_{2})\rangle_{M}.$

Note that the scale factor does not appears in the last term of Eq. (30).
Thus, the integration diverges in the coincidence limit. This implies that a
renormalization is needed. So the Minkowski contribution must be subtracted
from Hadamard function presented in the last integral, giving a null
contribution. Thus, the remained finite integrals are

$\displaystyle\langle v^{i}(t_{1},r_{1})v^{i}(t_{2},r_{2})\rangle_{LIV}$
$\displaystyle=$
$\displaystyle\dfrac{q^{2}a_{f}^{2}}{m^{2}}\int_{-\infty}^{\infty}d\eta_{2}a^{-2}(\eta_{2})\int_{-\infty}^{\infty}d\eta_{1}a^{-2}(\eta_{1})\partial_{1i}\partial_{2i}\langle\phi_{1}(\eta_{1},r_{1})\phi_{1}(\eta_{2},r_{2})\rangle_{M}+$
(31) $\displaystyle+$
$\displaystyle\dfrac{q^{2}a_{f}^{2}\beta}{m^{2}}\int_{0}^{\eta_{f}}d\eta_{2}\int_{-\infty}^{\infty}d\eta_{1}a^{-2}(\eta_{1})\partial_{1i}\partial_{2i}\langle\phi_{1}(\eta_{1},r_{1})\phi_{1}(\eta_{2},r_{2})\rangle_{M}+$
$\displaystyle+$
$\displaystyle\dfrac{q^{2}a_{f}^{2}\beta}{m^{2}}\int_{0}^{\eta_{f}}d\eta_{1}\int_{-\infty}^{\infty}d\eta_{2}a^{-2}(\eta_{2})\partial_{1i}\partial_{2i}\langle\phi_{1}(\eta_{1},r_{1})\phi_{1}(\eta_{2},r_{2})\rangle_{M},$

where we considered the proper velocity ($v^{i}=a_{f}u^{i}$).

Particularizing $i$ to the $x$ direction, the results in the $y$ and $z$
direction are the same, and taken the partial derivatives with respect to
$x_{1}$ and $x_{2}$, we obtain,

$\displaystyle\langle v^{i}(t_{1},r_{1})v^{i}(t_{2},r_{2})\rangle_{LIV}$
$\displaystyle=$
$\displaystyle\dfrac{q^{2}a_{f}^{2}}{2\pi^{2}m^{2}}\int_{-\infty}^{\infty}d\eta_{2}a^{-2}(\eta_{2})\int_{-\infty}^{\infty}d\eta_{1}a^{-2}(\eta_{1})\left\\{f_{2}(\eta,r)+4\Delta
x^{2}f_{3}(\eta,r)\right\\}+$ (32) $\displaystyle+$
$\displaystyle\dfrac{q^{2}a_{f}^{2}\beta}{m^{2}}\int_{0}^{\eta_{f}}d\eta_{2}\int_{-\infty}^{\infty}d\eta_{1}a^{-2}(\eta_{1})\left\\{f_{2}(\eta,r)+4\Delta
x^{2}f_{3}(\eta,r)\right\\}+$ $\displaystyle+$
$\displaystyle\dfrac{q^{2}a_{f}^{2}\beta}{m^{2}}\int_{0}^{\eta_{f}}d\eta_{1}\int_{-\infty}^{\infty}d\eta_{2}a^{-2}(\eta_{2})\left\\{f_{2}(\eta,r)+4\Delta
x^{2}f_{3}(\eta,r)\right\\},$

where $f_{n}$ was defined as,

$f_{n}(\eta,r)=\frac{1}{[c_{0}^{2}(\eta_{1}-\eta_{2})^{2}-r^{2}]^{n}}.$ (33)

Now we use the scale factor given by Eq. (11) and to evaluate the integral
above we follow again a contour integration presented in appendix B of Ref.
Bessa2009 . The result of such integral is:

$\displaystyle\langle(\Delta
v^{x})^{2}\rangle_{LIV}=\frac{2q^{2}B}{\pi^{4}m^{2}c_{0}^{4}\eta_{0}^{2}}\left(\zeta(3)-\frac{\pi^{4}}{90}\right)+\frac{2q^{2}\beta}{3c_{0}^{4}\pi
m^{2}\eta_{0}^{2}}\left(\frac{c_{0}}{c_{sf}}B\right)^{1/2}F_{k}(\eta),$ (34)

where the coincidence limit was taken in the expression above and

$F_{k}(\eta)=\dfrac{1}{2\pi^{3}}\left[\psi^{(2)}\left(\frac{ih}{\pi}+\frac{1}{2}\right)-\psi^{(2)}\left(\frac{2ih+2i\eta+\pi}{2\pi}\right)\right].$
(35)

with $h=\ln(\sqrt{c_{0}/c_{sf}})$ and $\psi^{(n)}$ being the $n$th-polygamma
function. We can re-arrange the terms in Eq. (34) to obtain,

$\displaystyle\langle(\Delta v^{x})^{2}\rangle_{LIV}$ $\displaystyle=$
$\displaystyle\dfrac{2q^{2}B}{\pi
m^{2}c_{0}^{4}\eta_{0}^{2}}\left\\{\dfrac{1}{\pi^{3}}\left(\zeta(3)-\dfrac{\pi^{4}}{90}\right)+\dfrac{\beta
B^{-\frac{1}{2}}}{3}\left(\dfrac{c_{0}}{c_{sf}}\right)^{\frac{1}{2}}Re\left[F_{k}(\eta)\right]\right\\},$
(36)

or, in terms of Eq. (12),

$\displaystyle\langle(\Delta v^{x})^{2}\rangle_{LIV}$ $\displaystyle=$
$\displaystyle\langle(\Delta v^{x})^{2}\rangle+\frac{2q^{2}\beta B^{1/2}}{3\pi
m^{2}c_{0}^{4}\eta_{0}^{2}}\left(\frac{c_{0}}{c_{sf}}\right)^{1/2}Re\left[F_{k}(\eta)\right]$
(37)

where the first term in the right-hand side is Eq. (12) and the second term is
the up to the first order corrections in $\beta$ that comes from the LIV model
represented by Lagrangian (17). So the relevant correction contribution comes
strictly from the theory represented by the Lagrangian (17), making a boost in
the velocity dispersion.

Now let us plot Eq. (36). Such a procedure can be done by defining a new
function, $G(\eta)$, where

$\displaystyle G(\eta)=\frac{\pi
m^{2}c_{0}^{4}\eta_{0}^{2}}{2q^{2}B}\langle(\Delta v^{x})^{2}\rangle_{LIV}$
$\displaystyle=$
$\displaystyle\frac{1}{\pi^{3}}\left(\zeta(3)-\frac{\pi^{4}}{90}\right)$
$\displaystyle+$
$\displaystyle\beta\left[\frac{3}{\pi^{3}}\left(\zeta(3)-\frac{\pi^{4}}{90}\right)+\frac{B^{-1/2}}{3}\left(\frac{c_{0}}{c_{sf}}\right)^{1/2}F_{k}(\eta)\right].$

In Fig. 1 we plot the function $G(\eta)$ defined in Eq. (IV.1), which is
directly related to $\langle(\Delta v^{x})^{2}\rangle_{LIV}$. Here we choose
$\beta=3.6\times 10^{-8}$ as presented in Ref. mtn08 . Moreover, we have the
constraint that $c_{sf}<c_{0}$, in order to the expansion occur. Given some
numerical values to the sound speed we are able to evaluate the dimensionless
quantities $B$ and $h$, both present in Eq. (IV.1). Thus, we have for the
dashed curve $c_{0}/c_{sf}=1.1$, $B=0.00227$ and $h=0.0467$, whereas for the
solid curve $c_{0}/c_{sf}=1.2$, $B=0.00833$ and $h=0.0911$.

Figure 1: Dispersion for bounded particles for different values for the sound
speed and $\beta=3.6\times 10^{-8}$. The dispersion decreases when LIV is
considered.

Note that, comparing the result found in Sect. II, where LI is preserved, the
LIV decreases the value of the velocity dispersion when the particles are
bounded for some external force. Decreasing, then, the uncertainty about this
quantity. For large values of $\eta$ we see that $G(\eta)$ goes to a constant.
In fact, as our time variable is found in Eq. (35), taking the limit where
$\eta\rightarrow\infty$, we have a constant value given by the polygamma
function.

###  IV.2 Free particles

In this section we will study the free particles case ($f^{i}_{ext}=0$). The
equation of motion (26) for a single particle can be expressed in the
following way

$\frac{1}{a^{2}}\dfrac{d}{dt}(a^{2}u^{i})=\frac{q}{m}[a^{-2}(t)+\beta]\partial_{i}\phi_{1},$
(39)

when $\beta=0$, we recover an equation valid for a flat or conformally flat
spacetime with LI preserved.

The corresponding velocity-velocity correlation function is now

$\displaystyle\langle
u^{i}(t_{1},r_{1})u^{i}(t_{2},r_{2})\rangle_{LIV}=\frac{q^{2}}{m^{2}a_{f}^{4}}\int\int
dt_{1}dt_{2}a^{2}(t_{1})a^{2}(t_{2})\left\\{\left[a^{-2}(t_{1})+\beta\right]\left[a^{-2}(t_{2})+\beta\right]\right\\}$
(40)
$\displaystyle\times\partial_{i_{1}}\partial_{i_{2}}\langle\phi_{1}(t_{1},r_{1})\phi_{1}(t_{2},r_{2})\rangle_{FRW}$

Taking the following transformation in time, $dt=ad\eta$, we obtain the
expression,

$\displaystyle\langle u^{i}(t_{1},r_{1})u^{i}(t_{2},r_{2})\rangle_{LIV}$
$\displaystyle=$
$\displaystyle\frac{q^{2}}{m^{2}a_{f}^{4}}\left[\int_{0}^{\eta_{f}}d\eta_{2}\int_{0}^{\eta_{f}}d\eta_{1}\partial_{1i}\partial_{2i}\langle\phi_{1}(\eta_{1},r_{1})\phi_{1}(\eta_{2},r_{2})\rangle_{M}\right.$
(41) $\displaystyle+$
$\displaystyle\left.\beta\int_{0}^{\eta_{f}}d\eta_{2}\int_{-\infty}^{\infty}d\eta_{1}a^{2}(\eta_{1})\partial_{1i}\partial_{2i}\langle\phi_{1}(\eta_{1},r_{1})\phi_{1}(\eta_{2},r_{2})\rangle_{M}\right.$
$\displaystyle+$
$\displaystyle\left.\beta\int_{0}^{\eta_{f}}d\eta_{1}\int_{-\infty}^{\infty}d\eta_{2}a^{2}(\eta_{2})\partial_{1i}\partial_{2i}\langle\phi_{1}(\eta_{1},r_{1})\phi_{1}(\eta_{2},r_{2})\rangle_{M}\right.$
$\displaystyle+$
$\displaystyle\left.\beta^{2}\int_{-\infty}^{\infty}d\eta_{2}a^{2}(\eta_{2})\int_{-\infty}^{\infty}d\eta_{1}a^{2}(\eta_{1})\partial_{1i}\partial_{2i}\langle\phi_{1}(\eta_{1},r_{1})\phi_{1}(\eta_{2},r_{2})\rangle_{M}\right].$

Note that, the first term in the right hand side has no scale factor in the
integrand. So a renormalization procedure is needed in this term. Thus, the
first integral does not contributes and we have,

$\displaystyle\langle u^{i}(t_{1},r_{1})u^{i}(t_{2},r_{2})\rangle_{LIV}$
$\displaystyle=$
$\displaystyle\frac{q^{2}}{m^{2}a_{f}^{4}}\left[\beta\int_{0}^{\eta_{f}}d\eta_{2}\int_{-\infty}^{\infty}d\eta_{1}a^{2}(\eta_{1})\partial_{1i}\partial_{2i}\langle\phi_{1}(\eta_{1},r_{1})\phi_{1}(\eta_{2},r_{2})\rangle_{M}\right.$
(42) $\displaystyle+$
$\displaystyle\left.\beta\int_{0}^{\eta_{f}}d\eta_{1}\int_{-\infty}^{\infty}d\eta_{2}a^{2}(\eta_{2})\partial_{1i}\partial_{2i}\langle\phi_{1}(\eta_{1},r_{1})\phi_{1}(\eta_{2},r_{2})\rangle_{M}\right.$
$\displaystyle+$
$\displaystyle\left.\beta^{2}\int_{-\infty}^{\infty}d\eta_{2}a^{2}(\eta_{2})\int_{-\infty}^{\infty}d\eta_{1}a^{2}(\eta_{1})\partial_{1i}\partial_{2i}\langle\phi_{1}(\eta_{1},r_{1})\phi_{1}(\eta_{2},r_{2})\rangle_{M}\right].$

Now putting Eq. (9) into Eq. (42) choosing $i=x$, the result for $y$ and $z$
is the same, and writing in terms of the proper velocity ($v^{i}$), where
$v^{i}=a_{f}u^{i}$, we obtain,

$\displaystyle\langle v^{i}(t_{1},r_{1})v^{i}(t_{2},r_{2})\rangle_{LIV}$
$\displaystyle=$
$\displaystyle\dfrac{q^{2}\beta}{2\pi^{2}m^{2}a_{f}^{2}}\int_{-\infty}^{\infty}d\eta_{2}a^{2}(\eta_{2})\int_{-\infty}^{\infty}d\eta_{1}a^{2}(\eta_{1})\left\\{f_{2}(\eta,r)+4\Delta
x^{2}f_{3}(\eta,r)\right\\}+$ (43) $\displaystyle+$
$\displaystyle\dfrac{q^{2}\beta}{2\pi^{2}m^{2}a_{f}^{2}}\int_{0}^{\eta_{f}}d\eta_{2}\int_{-\infty}^{\infty}d\eta_{1}a^{2}(\eta_{1})\left\\{f_{2}(\eta,r)+4\Delta
x^{2}f_{3}(\eta,r)\right\\}+$ $\displaystyle+$
$\displaystyle\dfrac{q^{2}\beta^{2}}{2\pi^{2}m^{2}a_{f}^{2}}\int_{0}^{\eta_{f}}d\eta_{1}\int_{-\infty}^{\infty}d\eta_{2}a^{2}(\eta_{2})\left\\{f_{2}(\eta,r)+4\Delta
x^{2}f_{3}(\eta,r)\right\\}.$

with $f_{n}$ given by Eq. (33).

Making use of Eq. (11) and taking the coincidence limit we are able to make
first an integration by parts and then apply the same contour integration of
the previous sections to obtain,

$\langle(\Delta v^{x})^{2}\rangle_{LIV}=\dfrac{2q^{2}\beta C}{3\pi
m^{2}c_{0}^{4}\eta_{0}^{2}}Re\left[F_{k}(\eta)\right]+\dfrac{2q^{2}\beta^{2}B}{\pi^{4}m^{2}c_{0}^{4}\eta_{0}^{2}}\zeta(3),$
(44)

with

$F_{k}(\eta)=\frac{14\zeta(3)+\psi^{(2)}\left(\frac{i\eta}{\pi}+\frac{1}{2}\right)}{2\pi^{3}}$
(45)

and $C=\frac{1}{2}\left(1-\frac{c_{sf}}{c_{0}}\right).$ Again, the relevant
contribution is the first-order correction in $\beta$ that comes from
Lagrangian (17). This contribution produces a boost in the velocity dispersion
that would vanish if $\beta=0$.

Note that a non-null dispersion was obtained which is different from the
result found in Sect. II. This is perhaps the most important result of the
present manuscript, since this model shows a fundamental effect coming from
the LIV theory.

Now let us plot Eq. (44). Proceeding this way, define a new function,
$G^{\prime}(\eta)$, where

$\displaystyle G^{\prime}(\eta)=\frac{3\pi
m^{2}c_{0}^{4}\eta_{0}^{2}}{2q^{2}}\langle(\Delta
v^{x})^{2}\rangle_{LIV}=C\beta F_{k}(\eta)$ (46)

Figure 2: Dispersion for free particles for different values for the sound
speed and $\beta=3.6\times 10^{-8}$. The dispersion increases when LIV is
considered.

The solid and dashed curves was plotted with $C=0.04545$ and $C=0.08333$,
respectively, with the speed ratio $c_{0}/c_{sf}=1.1$ and $1.2$. It was used
$\beta=3.6\times 10^{-8}$ in both cases. Note that this result is consistent
with the free case described in Sect. II only when $\eta=0$, since a null
dispersion is found. However, as the time increases, the dispersion is no
longer zero, which disagrees with Sect. II. This result is a direct
consequence of the LIV. Thus, in the LIV scenario particles that follows their
geodesics (free particles) can posses a stochastic motion due to quantum
fluctuations. So LIV holds a fundamental influence in the non-trivial quantum
effect described in this paper.

##  V Summary and discussion

In the present paper, we have considered the stochastic motion of scalar test
particles coupled to a massless fluctuating scalar field in an acoustic
spatially flat FRW geometry. In this context, in Sect. II, we reviewed the
main results present in Ref. Bessa2017 , where to model this geometry, it was
considered a fluid (BEC) where the LI was preserved. In this context, we
studied stochastic motion in two distinct situations: in the first one, we
have considered an external classical and non-fluctuating force acting on the
particles in this geometry. This force prevented them from feel locally the
expansion. We named them as bound particles. In the second situation, no
external force was acting on the particles, in this case, they are free and
follow their geodesics, and therefore they were named free particles. In
particular, for the case studied in this section, the stochastic motion exists
only for bound particles since for the free particles the dispersion vanishes
($\langle\Delta v^{2}\rangle=0$) Bessa2017 ; Bessa2009 .

In Sect. III, following the same steps of Ref. abp10 , we showed that it is
also possible to find an acoustic spatially flat FRW geometry where the LI is
broken, i.e., in a LIV context. Then, in Sect. IV, we studied the same problem
described in Sect. II using the metric found in Sect. III. Thus, comparing the
results found in Sect. II with the ones found in Sect. IV, we observed that
for bound particles the LIV corrections decrease their velocity dispersion and
also the uncertainty on the motion. We can see this fact from Eq. (37) and
Fig. 1. The reader can note that the curve for the dispersion decreases in the
limit of long times. Another point worth mentioning is the dispersion relation
between the sound speed in the beginning ($c_{0}$) and the end of the
expansion ($c_{sf}$). A change in the ratio ($c_{0}/c_{sf}$) causes a shift
that might increase or decrease the dispersion. Nonetheless, the most
important result in this paper is for the free particles in the LIV framework.
This case is more realistic than the bound particles, and a null dispersion
was found when the LI is taken into account. However, as can be seen, in Eq.
(44) and in Fig. 2, when LIV is considered a non-null dispersion is found. We
interpret this as a direct consequence of the LIV, and the dependence with the
initial and final sound speeds is also illustrated in Fig. 2.

Another important point related to our model appears in the $\beta$
corrections presented in Eqs. (37) and (44). This term comes from Lagrangian
(17), and it appears directly in the particle’s equation of motion.
Nevertheless, the acoustic metric plays a fundamental role in our results, not
only because it is derived directly from Eq. (17) but also because in
dispersion calculations the scale factor that appears in the integrands allows
us to obtain finite and non-zero results.

As expected from a theory with LIV, the corrections found in the velocity
dispersion of the particles are very small, and consequently, their detection
is not an easy task. However, our model shows that the LIV plays a fundamental
role related to the robustness of the motion of the particles, as in the case
of the free particles, whereas the non-zero result obtained disagrees with the
one previously found in the literature, where the Lorentz invariance is
preserved. Since experimental tests for quantum gravity effects are hard to
find, the present model denotes a mean where such effects could be explored.
Besides, the quantum stochastic motion is a subtle and non-trivial quantum
scenario effect that is, per se, a topic of interest.

###### Acknowledgements.

We would like to thank CNPq, CAPES and CNPq/PRONEX/FAPESQ-PB (Grants no.
165/2018 and 015/2019), for partial financial support. MAA, FAB, EP and JRLS
acknowledge support from CNPq (Grants no. 306962/2018-7 and 433980/2018-4,
312104/2018-9, 304852/2017-1, 420479/2018-0). The authors would like to thank
J. P. Spinelly and F. G. Costa for helpful comments.

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# On the convergence of the continuous version of the Moudafi’s viscosity
approximation method

Ramzi May  Mathematics Department, College of Science, King Faisal University,
P.O. 380, Ahsaa 31982, Kingdom of Saudi Arabia<EMAIL_ADDRESS>
(Date: September 27, 2023)

###### Abstract.

We study the asymptotic behavior of the trajectories of the continuous
dynamical system (CDS) associated to the the discrete viscosity approximation
method for fixed point problem of nonexpansive mapping (DDS) which was
introduced by Moudafi in 2000 [A. Moudafi, Viscosity approximation methods for
fixed points problems, J. Math. Anal. Appl. 241 (2000), 46-55]. We establish
that the trajectories $x(t)$ of the system (CDS) and the sequences $(x_{n})$
generated by the the discrete process (DDS) have a very similar asymptotic
behaviors.

###### Key words and phrases:

Hilbert space; Nonexpansive mappings; Viscosity approximation method;
Asymptotic behavior

##  1\. Introduction

Let $\mathcal{H}$ be a real Hilbert space with inner product
$\langle.,.\rangle$ and associated norm $\left\|.\right\|$. Throughout this
paper, $C$ is a nonempty closed and convex subset of $\mathcal{H}$,
$f:C\rightarrow C$ is a strictly contraction mapping with coefficient
$\alpha\in[0,1[$ i.e.,

$\left\|f(x)-f(y)\right\|\leq\alpha\left\|x-y\right\|\forall x,y\in C,$

and $T:C\rightarrow C$ is a nonexpansive mapping i.e.,

$\left\|T(x)-T(y)\right\|\leq\left\|x-y\right\|\forall x,y\in C.$

We assume moreover that the set $Fix(T)=\\{x\in C:T(x)=x\\}$ of fixed points
of $T$ is nonempty. We recall that Kirk [9] and Browder [2] independently
proved in 1965 that if in addition the set $C$ is bounded in $\mathcal{H}$
then $T$ has at least one fixed point.

A wide variety of problems in convex optimization and variational analysis can
be reduced to problems of fixed points of nonexpansives mapping (see for
instance [4],[4],[6],[16] and references therein). Hence, It is therefore of
interest to construct numerical efficient algorithms, continuous or discrete,
that approximate such fixed points. Let us first notice that the classical and
natural iterative process $x_{n+1}=T(x_{n})$ may not converge to a fixed point
of $T$. To see this, it suffices to consider the simplest example when $T=-I$,
where $I$ stands for the identity operator of $\mathcal{H}$. However, for any
initial data $x_{0}\in C$, the solution $x(.)$ of the continuous dynamical

(1.1) $\left\\{\begin{array}[c]{l}x^{\prime}(t)+x(t)=T(x(t)),t\geq 0\\\
x(0)=x_{0},\end{array}\right.$

associated to the discrete process $x_{n+1}=T(x_{n})$ converges weakly in
$\mathcal{H}$ as $t\rightarrow+\infty$ to a fixed point of $T$ (see [3]). We
notice that the explicit Euler discretization with variable step size
$h_{n}=\theta_{n}$ of the dynamical system (1.1) leads to the classical
Krasnoselskii-Mann algorithm (see [8] and [10]):

(1.2) $x_{n+1}=\theta_{n}x_{n}+(1-\theta_{n})T(x_{n}).$

It is Well-known (see for instance [7]) that, for any initial data $x_{1}\in
C$, the sequence $\\{x_{n}\\}$ generated by (1.2) converges weakly in
$\mathcal{H}$ to a fixed point of $T$ provided that the sequence
$\\{\theta_{n}\\}$ belongs to the interval $[0,1]$ and satisfies the condition

  1. (C0)

$\sum_{n=0}^{\infty}(1-\theta_{n})\theta_{n}=\infty$.

In a the pioneer paper [11], Halpern introduced an algorithm that converges
strongly to a particular and well-defined fixed point of a nonexpansive
mapping. Precisely, he considered the particular case when $C$ is the closed
unit ball of $\mathcal{H}$ and established that, for every $x_{1}\in C$, the
sequence defined recursively by the process $x_{n+1}=(1-\theta_{n})T(x_{n})$
converges strongly to the element of the set $Fix(T)$ with minimum norm if
$\theta_{n}=\frac{1}{n^{\theta}}$ with $0<\theta<1$. Later in 1977, Lions [13]
generalized and improved Halpern’s convergence result. In fact, he proved that
for every anchor point $u\in C$ and any initializing data $x_{1}\in C$, the
sequence $\\{x_{n}\\}$ generated by the process

$x_{n+1}=\theta_{n}u+(1-\theta_{n})T(x_{n})$

converges strongly to the closest element $u^{\star}$ of the set $Fix(T)$ to
$u$ provided the sequence $\\{\theta_{n}\\}$ belongs to $(0,1]$ and satisfies
the conditions:

  1. (C1)

$\theta_{n}\rightarrow 0$ as $n\rightarrow\infty,$

  2. (C2)

$\sum_{n=1}^{\infty}\theta_{n}=\infty,$

  3. (C3)

$\frac{\left|\theta_{n+1}-\theta_{n}\right|}{\theta_{n}^{2}}\rightarrow 0$ as
$n\rightarrow\infty,$

In 2000, Moudafi [14] introduced the called viscosity approximation method.
Precisely, he considered the iterative process

(DDS) $x_{n+1}=\theta_{n}f(x_{n})+(1-\theta_{n})T(x_{n}),~{}n\geq 1,$

and proved that if $(\theta_{n})_{n}$ satisfies the conditions (C1), (C2) and

  1. (C4)

$\frac{\left|\theta_{n+1}-\theta_{n}\right|}{\theta_{n}\theta_{n+1}}\rightarrow
0$ as $n\rightarrow\infty,$

then, for any $x_{1}$ in $C$, the sequence $\\{x_{n}\\}$ generated by (DDS)
converges strongly in $\mathcal{H}$ to the unique solution $q^{\ast}$ of the
variational problem

(VP) $\left\\{\begin{array}[c]{l}q^{\ast}\in Fix(T)\\\ \langle
f(q^{\ast})-q^{\ast},z-q^{\ast}\rangle\leq 0,\forall z\in
Fix(T).\end{array}\right.$

Later in 2004, Xu [17] improved Moudafi’s convergence result by replacing the
condition (C4) by the weaker one:

  1. (C5)

$\frac{\left|\theta_{n+1}-\theta_{n}\right|}{\theta_{n}}\rightarrow 0$ as
$n\rightarrow\infty$ or
$\sum_{n=1}^{\infty}\left|\theta_{n+1}-\theta_{n}\right|<\infty.$

In our present work, we consider the continuous dynamical system

(CDS)
$\left\\{\begin{array}[c]{l}x^{\prime}(t)+x(t)=\theta(t)f(x(t))+(1-\theta(t))T(x(t)),t\geq
0\\\ x(0)=x_{0},\end{array}\right.$

associated to the discrete algorithm (DDS) where
$\theta:[0,\infty)\rightarrow(0,1]$ is a regular function. We prove that, if
the function $\theta(.)$ satisfies the continuous version of the discreet
conditions (C1), (C2) and (C5), then for any initial data $x_{0}\in C$, the
system (CDS) has a unique global solution $x\in C^{1}([0,\infty),\mathcal{H})$
which converges strongly in $\mathcal{H}$ as $t\rightarrow+\infty$ to the
unique solution $q^{\ast}$ of the problem (VP). Moreover, in the particular
case where $\theta(t)=\frac{K}{(1+t)^{\nu}}$ with $K>0$ and $\nu\in(0,1],$ we
establish an estimate on the rate of convergence of
$\left\|T(x(t))-x(t)\right\|$ as $t\rightarrow+\infty.$ Such result can be
considered as the continuous version of a recently result, established by
Lieder [12] , on the rate of convergence of the sequence
$\\{x_{n}-T(x_{n})\\}$ for the process (DDS) in the case when the function $f$
is constant and the sequence $\\{\theta_{n}\\}$ is given by
$\theta_{n}=\frac{1}{n+2}.$

The rest of the paper is organized as follows. In the next section, we recall
some classical notions and results from functional analysis and convex
analysis that are useful in the sequel of the paper. In the third section, we
study the strong convergence of the trajectory of the system (CDS) under the
continuous version of the discrete conditions (C1), (C2) and (C5). In the last
section, we investigate the stability of the system (CDS) under relatively
small perturbations and we establish an estimation on the rate of the
convergence of the residual term $x(t)-T(x(t))$ in the particular case when
$\theta(t)=\frac{K}{(1+t)^{\nu}}$ with $K>0$ and $0<\nu\leq 1$.

##  2\. Preliminaries

In this section, we recall some classical definitions and results from convex
and functional analysis and derive some simple lemmas that will be needed in
proving the main results of this paper.

We first recall the definition and the main properties of the metric
projection onto a nonempty, closed and convex subset of the Hilbert space
$\mathcal{H}$.

######  Lemma 2.1 ([15, Proposition 1.37]).

Let $K$ be a nonempty, closed and convex subset of $\mathcal{H}$. Then the
following assertions hold:

  1. (1)

For every $x\in\mathcal{H},$ there exists a unique $P_{K}(x)\in K$ such that

$\left\|x-P_{K}(x)\right\|\leq\left\|x-y\right\|\ ~{}\forall y\in K.$

The operator $P_{K}:\mathcal{H}\rightarrow K$ is called the metric projection
onto $K$.

  2. (2)

For every $x\in\mathcal{H},$ $P_{K}(x)$ is the unique element of $K$
satisfying

(2.1) $\langle P_{K}(x)-x,P_{K}(x)-y\rangle\leq 0,\text{ for every }y\in K.$

  3. (3)

The operator $P_{K}:\mathcal{H}\rightarrow K$ is nonexpansive i.e.,

(2.2) $\left\|P_{K}(x)-P_{K}(y)\right\|\leq\left\|x-y\right\|,\text{ for all
}x,y\in\mathcal{H}.$

The second result is a classical property of the set of fixed points of a
nonexpansive mapping

######  Lemma 2.2 ([1, Proposition 4.13]).

Let $K$ be a closed convex and nonempty subset of $\mathcal{H}$. If
$A:K\rightarrow K$ is a nonexpansive mapping then $F_{ix}(A)=\\{x\in
K:A(x)=x\\}$ is a closed and convex subset of $\mathcal{H}$.

The next result is a particular case of the general demi-closedness property
for nonexpansive mappings

######  Lemma 2.3 ([1, Corollary 4.18]).

Let $K$ be a closed convex and nonempty subset of $\mathcal{H}$,
$A:K\rightarrow K$ a nonexpansive mapping, and $\\{x_{n}\\}$ a sequence in
$K$. If $\\{x_{n}\\}$ converges weakly in $\mathcal{H}$ to some element
$\bar{x}$ and $\\{x_{n}-A(x_{n})\\}$ converges strongly to $0$ in
$\mathcal{H}$, then $x\in F_{ix}(A)$.

We will now prove a variant of Gronwall’s inequality that will be used
frequently in the sequel.

######  Lemma 2.4.

Let $u,v,w:[0,\infty)\rightarrow[0,\infty)$ be three continuous functions. If
the function $u$ is absolutely continuous and satisfies, for almost every
$t\geq 0,$ the differential inequality

$u^{\prime}(t)+2v(t)u(t)\leq 2w(t)\sqrt{u(t)}.$

Then, for very $t\geq 0,$

(2.3) $\sqrt{u(t)}\leq
e^{-V(t)}\sqrt{u(0)}+e^{-V(t)}\int_{0}^{t}e^{V(s)}w(s)ds,$

where $V(t)=\int_{0}^{t}v(\tau)d\tau.$

###### Proof.

Let $\varepsilon>0.$ It is clear that the function $u_{\varepsilon}$ defined
on $[0,\infty)$ by $u_{\varepsilon}(t)=\sqrt{u(t)+\varepsilon}$ is absolutely
continuous and satisfies the estimations

$\displaystyle u_{\varepsilon}^{\prime}(t)$
$\displaystyle=\frac{u^{\prime}(t)}{2\sqrt{u(t)+\varepsilon}}$
$\displaystyle\leq-v(t)\frac{u(t)}{\sqrt{u(t)+\varepsilon}}+w(t)\frac{\sqrt{u(t)}}{\sqrt{u(t)+\varepsilon}}$
$\displaystyle\leq-v(t)u_{\varepsilon}(t)+v(t)\frac{\varepsilon}{\sqrt{u(t)+\varepsilon}}+w(t)$
$\displaystyle\leq-v(t)u_{\varepsilon}(t)+w(t)+v(t)\sqrt{\varepsilon}.$

Therefore, for almost every $t\geq 0,$

$\left(e^{V(t)}u_{\varepsilon}(t)\right)^{\prime}\leq
e^{V(t)}w(t)+\sqrt{\varepsilon}\left(e^{V(t)}\right)^{\prime}.$

Integrating the later differential inequality, we obtain

$\sqrt{u(t)+\varepsilon}\leq
e^{-V(t)}\sqrt{u(0)+\varepsilon}+e^{-V(t)}\int_{0}^{t}e^{V(s)}w(s)ds+\sqrt{\varepsilon}(1-e^{-V(t)}),~{}\forall
t\geq 0.$

Hence, by letting $\varepsilon\rightarrow 0,$ we get the desired inequality
(2.3). ∎

We close this section by proving the following simple result that will be
useful in the proof of the existence of the solutions $x(.)$ of the
differential system (CDS).

######  Lemma 2.5.

Let $K$ be a nonempty closed and convex subset of the Hilbert space
$\mathcal{H}$. Let $a<b$ be two reals numbers, $w:[a,b]\rightarrow[0,\infty)$
a continuous function such that $\int_{a}^{b}w(t)dt>0$, and
$v:[a,b]\rightarrow\mathcal{H}$ a continuous functions such that $v(t)\in K$
for every $t\in[a,b].$ Then

$I:=\frac{1}{\int_{a}^{b}w(t)dt}\int_{a}^{b}v(s)w(s)ds\in K.$

###### Proof.

For every $N\in\mathbb{N},$ set $I_{N}=\sum_{k=0}^{N-1}v(x_{k,N})\alpha_{k,N}$
with $x_{k,N}=a+k\frac{b-a}{N}$ and
$\alpha_{k,N}=\frac{1}{\int_{a}^{b}w(t)dt}\int_{x_{K,N}}^{x_{k+1,N}}w(s)ds.$
From the convexity of the set $K,$ it follows that $I_{N}\in K$ for every
$N\in\mathbb{N}$. On the other hand, the uniform continuity of the function
$v$ implies directly that the sequence $\\{I_{N}\\}$ converges strongly in
$\mathcal{H}$ to $I.$ Hence, by using the fact that $K$ is a closed subset of
$\mathcal{H}$, we conclude that $I\in K.$ ∎

##  3\. Strong convergence of the trajectories of the dynamical system(CDS)

In this section, we study the asymptotic behavior of the solution to the
dynamical system

(CDS)
$\left\\{\begin{array}[c]{l}x^{\prime}(t)+x(t)=\theta(t)f(x(t))+(1-\theta(t))T(x(t)),t\geq
0\\\ x(0)=x_{0},\end{array}\right.$

where $\theta:[0,\infty)\rightarrow(0,1]$ is an absolutely continuous function
and $x_{0}\in C$ is a given initial data.

Before stating our main result, let us first precise the notion of a
trajectory of the system (CDS).

######  Definition 3.1.

A trajectory of the system (CDS) is a continuously differentiable function
$x:[0,\infty[\rightarrow\mathcal{H}$ that satisfies the following properties:

  1. (1)

$x(t)\in C$ for every $t\geq 0,$

  2. (2)

$x(0)=x_{0},$

  3. (3)

$x^{\prime}(t)+x(t)=\theta(t)f(x(t))+(1-\theta(t))T(x(t))$ for every $t\geq
0.$

We now state and prove the main result of the paper.

######  Theorem 3.1.

The system (CDS) has a unique trajectory $x(.).$ Moreover, if in addition the
function $\theta(.)$ satisfies the following conditions

  1. (C’1)

$\theta(t)\rightarrow 0$ as $t\rightarrow\infty,$

  2. (C’2)

$\int_{0}^{+\infty}\theta(t)dt=\infty,$

  3. (C’5)

$\int_{0}^{+\infty}\left|\theta^{\prime}(t)\right|dt<\infty$ or
$\frac{\theta^{\prime}(t)}{\theta(t)}\rightarrow 0$ as $t\rightarrow\infty,$

then $x(t)$ converges strongly in $\mathcal{H}$ as $t\rightarrow\infty$ to
$q^{\ast}$ the unique solution of the variational problem

(VP) $\left\\{\begin{array}[c]{l}q^{\ast}\in Fix(T)\\\ \langle
f(q^{\ast})-q^{\ast},z-q^{\ast}\rangle\leq 0,\forall z\in
Fix(T).\end{array}\right.$

###### Proof.

Let us first recall, for the convenience of the readers, the classical proof
of the existence and uniqueness of the solution $q^{\ast}$ of the problem
(VP). First, from Lemma 2.2, $Fix(T)$ is a closed and convex nonempty subset
of $\mathcal{H}$, then the projection operator $P_{Fix(T)}$ is well-defined.
Moreover, from the second assertion of Lemma 2.1, the problem (VP) is
equivalent to $q^{\ast}$ is a fixed point of the mapping $P_{Fix(T)}\circ
f:Fix(T)\rightarrow Fix(T).$ Now, since $P_{Fix(T)}$ is nonexpansive, the
mapping $P_{Fix(T)}\circ f$ is a strict contraction with coefficient $\alpha$
and therefore, according to the classical theorem of Banach, it has a unique
fixed point. This proves the existence and the uniqueness of the solution
$q^{\ast}$ of (VP).

We divide the second part of the proof into many steps.

The first step: We prove here the existence and the uniqueness of the
trajectory $x(t)$ of the system (CDS). To do this, we consider the Cauchy
problem

(3.1) $\left\\{\begin{array}[c]{l}x^{\prime}(t)+x(t)=g(t,x(t)),t\geq 0\\\
x(0)=x_{0},\end{array}\right.$

where $g:[0,\infty)\times\mathcal{H}\rightarrow\mathcal{H}$ is the mapping
defined by

$g(t,x)=\theta(t)f(P_{C}(x))+(1-\theta(t))T(P_{C}(x)).$

It is clear that the function $g$ is continuous and satisfies , for every
$t\geq 0$ and $x_{1},x_{2}\in\mathcal{H},$ the estimations

(3.2) $\displaystyle\left\|g(t,x_{1})-g(t,x_{2})\right\|$
$\displaystyle\leq(1-\gamma\theta(t))\left\|x_{1}-x_{2}\right\|$
$\displaystyle\leq\left\|x_{1}-x_{2}\right\|,$

with

$\gamma=1-\alpha.$

Hence, according to the classical theorem of Cauchy-Lipschitz, system (3.1)
has a unique global solution $x\in C^{1}([0,\infty),\mathcal{H}).$ Let $t>0$
be a fixed real. From (3.1),

$\displaystyle x(t)$
$\displaystyle=e^{-t}x_{0}+e^{-t}\int_{0}^{t}e^{s}g(s,x(s))ds$
$\displaystyle=e^{-t}x_{0}+(1-e^{-t})\int_{0}^{t}g(s,x(s))w_{t}(s)d(s)$

where $w_{t}(s)=1_{[0,t]}(s)\frac{e^{s}}{1-e^{-t}}ds$. Since, for every
$s\in[0,t],g(s,x(s))\in C$, Lemma 2.5 ensures that
$\int_{0}^{t}g(s,x(s))w_{t}(s)d(s)\in C$, which implies, thanks again to the
convexity of $C,$ that $x(t)\in C.$ This proves that $x(.)$ is a trajectory of
the system (CDS) in the sense of the definition 3.1. The uniqueness of the the
trajectory of (CDS) follows from the uniqueness of the solution of the Cauchy
problem (3.1) and the trivial fact that every trajectory of (CDS) is also a
solution to (3.1).

The second step: In this step, we will prove that the trajectory $x(.)$ is
bounded i.e., $x\in L^{\infty}([0,\infty),\mathcal{H}).$ To this end, we
consider the function $u$ defined on $[0,\infty)$ by
$u(t)=\left\|x(t)-q^{\ast}\right\|^{2}.$ Using the fact that $x(.)$ is a
solution of (3.1) and the estimation (3.2), we easily obtain the following
estimations

$\displaystyle u(t)$ $\displaystyle=2\langle
x^{\prime}(t),x(t)-q^{\ast}\rangle$
$\displaystyle=2\langle-x(t)+g(t,x(t)),x(t)-q^{\ast}\rangle$
$\displaystyle=-2u(t)+2\langle
g(t,q^{\ast})-q^{\ast},x(t)-q^{\ast}\rangle+2\langle
g(t,x(t))-g(t,q^{\ast}),x(t)-q^{\ast}\rangle$
$\displaystyle=-2u(t)+2\theta(t)\langle
f(q^{\ast})-q^{\ast},x(t)-q^{\ast}\rangle+2\langle
g(t,x(t))-g(t,q^{\ast}),x(t)-q^{\ast}\rangle$ (3.3)
$\displaystyle\leq-2u(t)+2\theta(t)\langle
f(q^{\ast})-q^{\ast},x(t)-q^{\ast}\rangle+2(1-\gamma\theta(t))\left\|x(t)-q^{\ast}\right\|^{2}$
$\displaystyle\leq-2\gamma\theta(t)u(t)+2\theta(t)\left\|f(q^{\ast})-q^{\ast}\right\|\left\|x(t)-q^{\ast}\right\|$
$\displaystyle=-2\gamma\theta(t)u(t)+2\theta(t)\left\|f(q^{\ast})-q^{\ast}\right\|\sqrt{u(t)}.$

Hence, by applying Lemma 2.4, we get

$\sqrt{u(t)}\leq
e^{-\gamma\Theta(t)}\sqrt{u(0)}+\frac{\left\|f(q^{\ast})-q^{\ast}\right\|}{\gamma}(1-e^{-\gamma\Theta(t)}),$

where

(3.4) $\Theta(t)=\int_{0}^{t}\theta(s)ds.$

We thus conclude that

$\sup_{t\geq
0}\left\|x(t)-q^{\ast}\right\|\leq\max\left(\left\|x_{0}-q^{\ast}\right\|,\frac{\left\|f(q^{\ast})-q^{\ast}\right\|}{\gamma}\right).$

The third step: We prove here that $x^{\prime}(t)$ converges strongly in
$\mathcal{H}$ to $0$ as $t\rightarrow\infty.$ The main idea of the proof is
inspired by the proof of [5, Lemma 8]. Let $\delta>0.$ We define the function
$\omega$ on $[0,\infty)$ by $\omega(t)=\left\|x(t+\delta)-x(t)\right\|^{2}.$
Clearly,

$\displaystyle\omega^{\prime}(t)$ $\displaystyle=2\langle
x^{\prime}(t+\delta)-x^{\prime}(t),x(t+\delta)-x(t)\rangle$
$\displaystyle=-2\omega(t)+2\langle
g(t+\delta,x(t+\delta))-g(t,x(t)),x(t+\delta)-x(t)\rangle$
$\displaystyle=-2\omega(t)+2\langle
g(t,x(t+\delta))-g(t,x(t)),x(t+\delta)-x(t)\rangle$ $\displaystyle+2\langle
g(t+\delta,x(t+\delta))-g(t,x(t+\delta)),x(t+\delta)-x(t)\rangle$
$\displaystyle\leq-2\gamma\theta(t)\omega(t)+2\left\|g(t+\delta,x(t+\delta))-g(t,x(t+\delta))\right\|\sqrt{\omega(t)}$
(3.5)
$\displaystyle\leq-2\gamma\theta(t)\omega(t)+2M\left|\theta(t+\delta)-\theta(t)\right|\sqrt{\omega(t)},$

where

(3.6) $M:=\sup_{s\geq
0}\left(\left\|f(x(s))\right\|+\left\|T(x(s)\right\|\right).$

We notice here that $M$ is finite since $x(.)\in
L^{\infty}([0,\infty),\mathcal{H})$ and $f$ and $T$ are Lipschitiz continuous
functions. Applying now Lemma 2.4 to the inequality (3.5), we deduce that for
every $t\geq 0,$

$\left\|x(t+\delta)-x(t)\right\|\leq
e^{-\gamma\Theta(t)}\left\|x(\delta)-x(0)\right\|+Me^{-\gamma\Theta(t)}\int_{0}^{t}e^{\gamma\Theta(s)}\left|\theta(s+\delta)-\theta(s)\right|ds$

where $\Theta$ is the function defined by (3.4).

Dividing the last inequality by $\delta$ and letting $\delta\rightarrow 0,$ we
obtain

(3.7) $\left\|x^{\prime}(t)\right\|\leq
e^{-\gamma\Theta(t)}\left\|x^{\prime}(0)\right\|+Me^{-\gamma\Theta(t)}\int_{0}^{t}e^{\gamma\Theta(s)}\left|\theta^{\prime}(s)\right|ds\forall
t\geq 0.$

From the condition (C’2), $\Theta(t)\rightarrow\infty$ as
$t\rightarrow\infty,$ hence in order to prove that
$\left\|x^{\prime}(t)\right\|\rightarrow 0$ as $t\rightarrow\infty$ it
suffices to prove that
$r(t):=e^{-\gamma\Theta(t)}\int_{0}^{t}e^{\gamma\Theta(s)}\left|\theta^{\prime}(s)\right|ds\rightarrow
0$ as $t\rightarrow\infty.$ Here we make use of the condition (C’5). We
therefore consider the two following cases:

The case when $\int_{0}^{\infty}\left|\theta^{\prime}(s)\right|ds<\infty$.

Let $A>0.$ Since the function $\Theta$ is increasing then for every $t\geq A$

$r(t)\leq
e^{-\gamma\Theta(t)}\int_{0}^{A}e^{\gamma\Theta(s)}\left|\theta^{\prime}(s)\right|ds+\int_{A}^{t}\left|\theta^{\prime}(s)\right|ds.$

This inequality clearly implies

$\overline{\lim_{t\rightarrow\infty}}~{}r(t)\leq\int_{A}^{\infty}\left|\theta^{\prime}(s)\right|ds.$

Therefore, by letting $A\rightarrow\infty,$ we get the desired result
$\lim_{t\rightarrow\infty}r(t)=0.$

The case when
$\lim_{t\rightarrow\infty}\frac{\left|\theta^{\prime}(t)\right|}{\theta(t)}=0$.

Let $A>0.$ for every $t\geq A,$

$\displaystyle r(t)$ $\displaystyle\leq
e^{-\gamma\Theta(t)}\int_{0}^{A}e^{\gamma\Theta(s)}\left|\theta^{\prime}(s)\right|ds+e^{-\gamma\Theta(t)}\int_{A}^{t}e^{\gamma\Theta(s)}\theta(s)ds~{}\sup_{s\geq
A}\frac{\left|\theta^{\prime}(s)\right|}{\theta(s)}$ $\displaystyle\leq
e^{-\gamma\Theta(t)}\int_{0}^{A}e^{\gamma\Theta(s)}\left|\theta^{\prime}(s)\right|ds+\frac{1}{\gamma}(1-e^{-\gamma(\Theta(t)-\Theta(A)})\sup_{s\geq
A}\frac{\left|\theta^{\prime}(s)\right|}{\theta(s)}$ (3.8) $\displaystyle\leq
e^{-\gamma\Theta(t)}\int_{0}^{A}e^{\gamma\Theta(s)}\left|\theta^{\prime}(s)\right|ds+\frac{1}{\gamma}\sup_{s\geq
A}\frac{\left|\theta^{\prime}(s)\right|}{\theta(s)}.$

Hence by letting $t\rightarrow\infty$ we get

$\lim_{t\rightarrow\infty}r(t)\leq\frac{1}{\gamma}\sup_{s\geq
A}\frac{\left|\theta^{\prime}(s)\right|}{\theta(s)}.$

Thus we can conclude by letting $A$ go to $\infty$.

The fourth step: We will show that

$r^{\ast}:=\overline{\lim_{t\rightarrow\infty}}\langle
f(q^{\ast})-q^{\ast},x(t)-q^{\ast}\rangle\leq 0.$

Since $x(.)\in L^{\infty}([0,\infty),\mathcal{H}),$ there exist
$x_{\infty}\in\mathcal{H}$ and a sequence of positive real numbers
$\\{t_{n}\\}$ which tend to $\infty$ such that $(\\{x(t_{n})\\}$ converges
weakly in $\mathcal{H}$ to $x_{\infty}$ and

(3.9) $\displaystyle r^{\ast}$ $\displaystyle=\lim_{n\rightarrow\infty}\langle
f(q^{\ast})-q^{\ast},x(t_{n})-q^{\ast}\rangle$ (3.10) $\displaystyle=\langle
f(q^{\ast})-q^{\ast},x(\infty)-q^{\ast}\rangle.$

On the other hand, since $x(.)$ is a trajectory of (CDS),

$\displaystyle\left\|x(t)-T(x(t))\right\|$
$\displaystyle\leq\theta(t)\left(\left\|f(x(t))\right\|+\left\|T(x(t))\right\|\right)+\left\|x^{\prime}(t)\right\|$
(3.11) $\displaystyle\leq 2M~{}\theta(t)+\left\|x^{\prime}(t)\right\|,$

for every $t\geq 0$, where the constant $M$ is given by (3.6). Hence the
condition (C’${}_{1})$ combined with the fact that
$\left\|x^{\prime}(t)\right\|\rightarrow 0$ as $t\rightarrow\infty$ implies
that the sequence $\\{x(t_{n})-T(x(t_{n})\\}$ converges strongly in
$\mathcal{H}$ to $0.$ Therefore, by invoking Lemma 2.3, we deduce that
$x_{\infty}\in Fix(T)$ which in turn implies that

$r^{\ast}=\langle f(q^{\ast})-q^{\ast},x_{\infty}-q^{\ast}\rangle\leq 0.$

The fifth step: Finally, we establish that $x(t)\rightarrow q^{\ast}$ strongly
in $\mathcal{H}$ as $t\rightarrow\infty.$

Define $u(t)=\left\|x(t)-q^{\ast}\right\|^{2}.$ From (3.3), we have for every
$t\geq 0,$

$u^{\prime}(t)+2\gamma\theta(t)u(t)\leq 2\theta(t)w(t),$

where

$w(t):=\max\\{\langle f(q^{\ast})-q^{\ast},x(t)-q^{\ast}\rangle,0\\}.$

By integrating the last differential inequality, we get

$u(t)\leq
e^{-2\gamma\Theta(t)}u(0)+2e^{-2\gamma\Theta(t)}\int_{0}^{t}e^{2\gamma\Theta(s)}\theta(s)w(s)ds,t\geq
0.$

From the previous step, $w(t)\rightarrow 0$ as $t\rightarrow\infty.$ Hence, by
following the procedure yielding to the estimation (3.8), we easily get
$e^{-2\gamma\Theta(t)}\int_{0}^{t}e^{2\gamma\Theta(s)}\theta(s)w(s)ds$
$\rightarrow 0$ as $t\rightarrow\infty.$ We therefore conclude that
$u(t)\rightarrow 0$ as $t\rightarrow\infty.$ This ends the proof. ∎

##  4\. Stability and rate of convergence of the trajectory of the dynamical
system (CDS)

In the first part of this section, we prove that the dynamical system (CDS) is
stable under the effect of a relatively small perturbation. Precisely, we
prove the following result.

######  Theorem 4.1.

Let $h:[0,\infty)\rightarrow\mathcal{H}$ be a continuous function such that
$h\in L^{1}([0,\infty),\mathcal{H})$ or
$\frac{\left\|h(t)\right\|}{\theta(t)}\rightarrow 0$ as $t\rightarrow\infty.$
Then for every initial data $x_{0}\in C,$ the perturbed dynamical system

(PCDS)
$\left\\{\begin{array}[c]{l}x^{\prime}(t)+x(t)=P_{C}(\theta(t)f(x(t))+(1-\theta(t))T(x(t))+h(t)),t\geq
0\\\ x(0)=x_{0}\end{array}\right.$

has a unique solution $y\in C^{1}([0,\infty),H)$ such that $y(t)\in C$ for
every $t\geq 0.$ Moreover, $y(t)$ converges strongly in $\mathcal{H}$ as
$t\rightarrow\infty$ to $q^{\ast}$ the unique solution to the variational
problem (VP).

###### Proof.

By proceeding exactly as in the second step of the proof of Theorem 3.1, we
can establish that (PCDS) has a unique unique solution $y\in
C^{1}([0,\infty),\mathcal{H})$ which verifies $y(t)\in C$ for every $t\geq 0.$
Let now $x(.)$ be the unique trajectory of (CDS). We consider the function $v$
defined on $[0,\infty)$ by $v(t)=\left\|x(t)-y(t)\right\|^{2}.$ For every
$t\geq 0,$

$\displaystyle v^{\prime}(t)$ $\displaystyle=2\langle
x^{\prime}(t)-y^{\prime}(t),x(t)-y(t)\rangle$ (4.1)
$\displaystyle=-2v(t)+2\langle\xi(t),x(t)-y(t)\rangle$

with

$\xi(t)=P_{C}(\theta(t)f(y(t))+(1-\theta(t))T(y(t))+h(t))-(\theta(t)f(x(t))+(1-\theta(t))T(x(t))).$

Using the facts that $\delta(t):=\theta(t)f(x(t))+(1-\theta(t))T(x(t))\in C$
(which implies that $P_{C}(\delta(t))=\delta(t)$) and $P_{C}$ is nonexpansive,
we easily get

$\left\|\xi(t)\right\|\leq(1-\gamma\theta(t))\left\|x(t)-y(t)\right\|+\left\|h(t)\right\|.$

Hence, by combining this last estimate with (4.1) and using the Cauchy-Schwarz
inequality, we obtain

$v^{\prime}(t)\leq
2\gamma\theta(t)v(t)+2\left\|h(t)\right\|\sqrt{v(t)},~{}\forall t\geq 0.$

Therefore, by applying Lemma 2.4, we deduce that, for every $t\geq 0,$

$v(t)\leq
e^{-\gamma\Theta(t)}v(0)+e^{-\gamma\Theta(t)}\int_{0}^{t}e^{\gamma\Theta(s)}\left\|h(s)\right\|ds,$

which implies, as in the proof of Theorem 3.1, that $v(t)\rightarrow 0$ as
$t\rightarrow\infty.$ Recalling finally that, from Theorem 3.1,
$x(t)\rightarrow q^{\ast}$ strongly in $\mathcal{H}$ as $t\rightarrow\infty$,
we conclude that $y(t)$ converges strongly in $\mathcal{H}$ as well to the
same limit $q^{\ast}$ as $t\rightarrow\infty.$ ∎

We will now establish an estimation on the rate of convergence of
$x(t)-T(x(t))$ to $0$ in the particular case when
$\theta(t)=\frac{K}{(1+t)^{\nu}}$ where $K>0$ and $0<\nu\leq 1.$

######  Theorem 4.2.

Assume that $\theta(t)=\frac{K}{(1+t)^{\nu}}$ with $0<\nu<1$ and $K>0$ or
$\nu=1$ and $K>\frac{1}{1-\alpha}.$ Let $x(.)$ be the trajectory of the
dynamical system (CDS). Then there exists a constant $C_{\nu}>0$ such that for
every $t\geq 0$

(4.2) $\left\|x(t)-T(x(t))\right\|\leq\frac{C_{\nu}}{(1+t)^{\nu}}.$

###### Proof.

In the proof of Theorem 3.1 (see (3.7) and (3.11)), we have established that
there exists a constant $M>0$ such that for every $t\geq 0,$

(4.3) $\left\|x(t)-T(x(t))\right\|\leq
M~{}\theta(t)+\left\|x^{\prime}(t)\right\|$

and

(4.4) $\left\|x^{\prime}(t)\right\|\leq
e^{-\gamma\Theta(t)}\left\|x^{\prime}(0)\right\|+Me^{-\gamma\Theta(t)}\int_{0}^{t}e^{\gamma\Theta(s)}\left|\theta^{\prime}(s)\right|ds,$

where

$\gamma=1-\alpha,$

and

$\Theta(t)=\int_{0}^{t}\theta(s)ds.$

Let us now estimate the vanishing rate of the key term

$r(t):=e^{-\gamma\Theta(t)}\int_{0}^{t}e^{\gamma\Theta(s)}\left|\theta^{\prime}(s)\right|ds.$

To do this, we distinguish the two cases.

The first case: $\theta(t)=\frac{K}{1+t}$ with $K>\frac{1}{\gamma}.$

For every $t\geq 0,$

(4.5) $e^{-\gamma\Theta(t)}=\frac{1}{(1+t)^{K\gamma}},$ (4.6)
$r(t)=\frac{K}{(1+t)^{K\gamma}}\int_{0}^{t}\frac{ds}{(1+s)^{2-K\gamma}}\leq\frac{K}{K\gamma-1}\frac{1}{1+t}$

The second case: $\theta(t)=\frac{K}{(1+t)^{\nu}}$ with $0<\nu<1$ and $K>0.$

In this case, for every $t\leq 0,$

(4.7) $e^{-\gamma\Theta(t)}=e^{-\kappa(1+t)^{1-\nu}},$ $r(t)=K\nu
e^{-\kappa(1+t)^{1-\nu}}\int_{0}^{t}\frac{e^{\kappa(1+s)^{1-\nu}}ds}{(1+s)^{1+\nu}},$

with $\kappa=\frac{\gamma K}{\nu}.$ Now a simple application of the
L’Hospital’s rule gives

(4.8)
$\lim_{t\rightarrow\infty}\frac{\int_{0}^{t}\frac{e^{\kappa(1+s)^{1-\nu}}}{(1+s)^{1+\nu}}ds}{\frac{1}{\kappa(1-\nu)}\frac{e^{\kappa(1+t)^{1-\nu}}}{1+t}}=1.$

This implies that there exits a constant $M^{\prime}>0$ independent of $t$
such that

(4.9) $r(t)\leq\frac{M^{\prime}}{1+t},~{}\forall t\geq 0.$

Finally, by combining the estimations (4.3)-(4.9) we obtain the required
result (4.2). ∎

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