11institutetext: Departamento de Investigación en Física
Universidad de Sonora
Hermosillo, Sonora, MEXICO
11email<EMAIL_ADDRESS>22institutetext: EESA Num. 1,
Tornquist, Pcia. de Buenos Aires
ARGENTINA

Arreaga-Garcia, Topa

# Simulating radially outward winds within a turbulent gas clump

Guillermo Arreaga-García1 Silvio Oreste Topa2

###### Abstract

By using the particle-based code Gadget2, we follow the evolution of a gas
clump, in which a gravitational collapse is initially induced. The particles
representing the gas clump have initially a velocity according to a turbulent
spectrum built in a Fourier space of 643 grid elements. In a very early stage
of evolution of the clump, a set of gas particles representing the wind,
suddenly move outwards from the clump’s center. We consider only two kinds of
winds, namely: one with spherical symmetry and a second one being a bipolar
collimated jet. In order to assess the dynamical change in the clump due to
interaction with the winds, we show iso-velocity and iso-density plots for all
our simulations.

###### Keywords:

winds,turbulence,collapse,hydrodynamics,simulations

##  1 Introduction

Stars are born in large gas structures made of molecular hydrogen. These gas
structures are named clumps; see Ref. [1]. These clumps have typical sizes and
masses of a few pc (parsecs) and a few hundred or even thousands of
$M_{\odot}$ (one solar mass), respectively111A parsec (pc) is equivalent to
$3.08\,\times 10^{18}\,$cm and a solar mass $M_{\odot}$ is equivalent to
$1.99\,\times 10^{33}\,$g..

The physical process by which the molecular gas is transformed from a gas
structure into some stars is mainly gravitational collapse, whose main effects
on the gas clump are that the gas density is increased while its size is
reduced. At some point during this transformation process from gas to star,
the densest gas structures settled down in more or less dynamically stable gas
objects called protostars222Any star does radiate its own energy produced by
thermonuclear reactions in its interior, but a protostar does not. This is the
main difference between a star and a protostar; but they can share some
dynamical properties as they are stable structures of different stages of the
same formation process.. For instance, the number density of a typical large
clump structure ranges around $10^{3}$ molecules per cm-3 whilst that of a
typical protostar ranges around $10^{14}$ molecules per cm-3. To achieve a
better understanding of the huge change in scales, we mention that the number
density of a typical star is around $10^{24}$ molecules per cm-3. The results
of a set of numerical simulations aimed to study the gravitational collapse of
a spherically symmetric, rigidly rotating, isolated, interstellar gas core are
presented in Ref. [2].

The process of gravitational collapse is not the only process acting upon the
gas structures in the interstellar medium, as many other phenomena can have a
great influence on the evolution of the clump, among others: (i) the highly
ionized gas ejected by the explosion of supernovas; (ii) the bipolar
collimated winds ejected by a massive protostar; (iii) rapidly expanding HII
regions which hit the slower and less dense gas structures.

Recently, in Ref.[3], a set of SPH simulations were conducted to study star
formation triggered by an expanding HII region within a spherical gas
structure with uniform density that was not globally collapsing. The expanding
shock plays the role of a snowplow, which sweeps out the gas of the
surrounding medium. The density of the swept out gas increases as a
consequence of this agglomeration process, and a gravitational collapse may
then be locally initiated. Small gas over-densities can be formed in this way,
which may achieve the protostar stage. Because these two processes are
complementary in forming proto-stars, this star formation scenario was named
the collect and collapse model.

Furthermore, in order to study the effects of proto-stellar outflows on the
turbulence of a star forming region, in Refs.[4] and [5], magneto-
hydrodynamics simulations (MHD) were conducted with a mesh based code which
implements the Adaptive Mesh Refinement technique (AMR).

In this paper, we investigate the change in the dynamical configuration of a
typical turbulent clump when a wind of particles is outwardly ejected from the
central region of the clump. The most important difference between the present
paper and Ref. [3] is the turbulent nature of our clump. Turbulence makes a
big change in the spatial distribution as the clump becomes filamentary and
flocculent.

##  2 The physical system

In this section we briefly describe the physics of the clump and the winds,
which will be considered in the following sections.

###  2.1 The initial clump

We here consider a typical spherical clump with a radius $R_{0}=2\,$ pc and
mass $M_{0}=1219\,M_{\odot}$. Initially, it has a radially uniform density
distribution with an average density given by $\rho_{0}=2.4\times 10^{-21}\;$
g cm-3, which is equivalent to a number density $n_{0}\approx 600\,$ molecules
cm-3 for molecular hydrogen with molecular mass $\mu=4.0\times
10^{-24}\,$gr/mol. The size and mass of this clump are chosen here to be
typical in the statistical sense, in accordance with Ref. [1].

The free fall time $t_{ff}$ is defined as the time needed for an external
particle to reach the center of the clump when gravity is the only force
pulling the particle. In this idealized gravitational collapse, we have

$t_{ff}\approx\sqrt{\frac{3\,\pi}{32\,G\,\rho_{0}}}$ (1)

where $G$ is Newton’s universal gravitational constant. For our clump, we have
$t_{ff}=1.3\times 10^{6}\,$ years.

Following Ref. [6], the dynamical properties of the initial distribution of
the gas are usually characterized by $\alpha$, the ratio of the thermal energy
to the gravitational energy, and $\beta$, that of the rotational energy to the
gravitational energy. For a spherical clump, the approximate total
gravitational potential energy is
$<E_{grav}>\approx-\frac{3}{5}\;\frac{G\,M_{0}^{2}}{R_{0}}$. The average total
thermal energy $<E_{therm}>$ ( kinetic plus potential interaction terms of the
molecules) is $<E_{therm}>\approx\frac{3}{2}{\cal
N}\,k\,T=\frac{3}{2}M_{0}\,c_{0}^{2}$, where $k$ is the Boltzmann constant,
$T$ is the equilibrium temperature, ${\cal N}$ is the total number of
molecules in the gas and $c_{0}$ is the speed of sound, see Section 3.4 for a
more precise definition.

The kinetic energy $<E_{kin}>$ can be estimated by $M_{0}\,v_{av}^{2}/2$,
where $v_{av}$ is the average translational velocity of the clump. In order to
have both energies of the same order of magnitude,
$<E_{kin}>\approx<E_{grav}>$, the gas elements of the clump must attain
average velocities within the range

$v_{av}/c_{0}\approx 3-3.5$ (2)

or $v_{av}\approx 1.6\,$km/s, for a speed of sound given by

$\begin{array}[]{l}c_{0}=0.54\,{\rm km}/{\rm s}\;\equiv 54862.91\;{\rm
cm}/{\rm s}\end{array}$ (3)

so that the corresponding temperature associated with the clump is $T\approx
25\,$K.

It is possible to define the crossing time by means of

$t_{cr}\approx\frac{R_{0}}{c_{0}}=3.56\,\times 10^{6}\,{\rm yr}\,$ (4)

which sets a time scale for a sound wave to travel across the clump. To make
the crossing time comparable in magnitude to the free fall time of Eq. 1, the
front wave must have velocities around $v_{req}/c_{0}\approx 2.6$ or
$v_{req}\approx 1.45\,$ km/s, which are velocities a little bit slower than
the ones estimated above, see Eq. 2. Anyway, in this paper we will treat
propagation velocities of gas particles ranging around $2-3$ Mach.

###  2.2 The wind.

In this paper we consider two kinds of winds: the first kind has a fully
spherical symmetry and the second kind is a bipolar collimated jet.

The dynamical characteristics of the wind strongly depends on its type of
source. All stars eject winds of the first kind, which are driven by the
stellar radiation. For instance, in cool stars, like the ones observed in the
AGB (asymtotic giant branch) of the Galaxy, the winds cause a mass loss in the
range $10^{-8}-10^{-4}\,M_{\odot}/$yr whereas the terminal wind velocities are
around $10-45\,$km/s. In OB stars, the mass loss ranges over
$10^{-6}-10^{-4}\,M_{\odot}$/yr and the terminal wind velocities can go up to
thousands of km/sec.

Supernovas dump around $10^{4}$ joules of thermal and kinetic energy into the
interstellar medium. But there are many types of supernovas, so that the mass
losses and terminal velocities are very different, see Ref. [7]. For example,
for a supernova whose progenitor was a He star, the mass loss and terminal
velocities are within the ranges $10^{-7}-10^{-4}\,M_{\odot}$/yr and
$100-1000$ km/sec, respectively. When the progenitor was a RSG star, then
their values ranges over $10^{-5}-10^{-4}\,M_{\odot}$/yr and $10-40$ km/s,
respectively.

It seems that all protostars eject highly collimated jets of gas during their
formation process by gravitational collapse. The origin of these jets is still
unclear but it may be that the accretion disk and magnetic field around the
protostars play a crucial role in determining the velocities and the degree of
collimation of the jets. For the molecular winds associated with protostarts
of Class 0 and Class 1, the characteristic velocities are around 20 km/sec.
However, for optical jets of highly ionized gas, the typical jet velocities
are a few hundred km/sec. See Ref. [8] and the references therein.

##  3 The Computational Method

In this section we briefly describe the way we set up the physical system
outlined above in computational terms.

###  3.1 The initial configuration of particles.

We set $N=10$ million SPH particles for representing the gas clump. By means
of a rectangular mesh we make the partition of the simulation volume in small
elements each with a volume $\Delta x\,\Delta y\,\Delta z$; at the center of
each volume we place a particle (the $i$th, say), with a mass determined by
its location according to the density profile being considered, that is:
$m_{i}=\rho(x_{i},y_{i},z_{i})*\Delta x\,\Delta y\,\Delta z$ with $i=1,...,N$.
Next, we displace each particle from its location by a distance on the order
of $\Delta x/4.0$ in a random spatial direction.

As was stated earlier, in this paper we only consider a uniform density clump,
for which $\rho(x_{i},y_{i},z_{i})\equiv\rho_{0}$, for all the simulations
(see Section 2.1). Therefore, all the particles have the same mass
irrespective of whether a wind or clump particle.

###  3.2 The initial turbulent velocity of particles.

To generate the turbulent velocity spectrum for the clump particles, we follow
a procedure based on the papers [9] and [10]. We set a second mesh
$N_{x},N_{y},N_{z}$ with the size of each element given by $\delta
x=R_{0}/N_{x}$, $\delta y=R_{0}/N_{y}$ and $\delta z=R_{0}/N_{z}$. In Fourier
space, the partition is $\delta K_{x}=1.0/\left(N_{x}\times\delta x\right)$ ,
$\delta K_{y}=1.0/\left(N_{y}\times\delta y\right)$ and $\delta
K_{z}=1.0/\left(N_{z}\times\delta z\right)$. Each Fourier mode has the
components $K_{x}=i_{K_{x}}\delta K_{x}$, $K_{y}=i_{K_{y}}\,\delta K_{y}$ and
$K_{z}=i_{K_{z}}\,\delta K_{z}$, where the indices $i_{K_{x}}$, $i_{K_{y}}$ e
$i_{K_{z}}$ take values in $[-N_{x}/2,N_{x}/2]$, $[-N_{y}/2,N_{y}/2]$ and
$[-N_{z}/2,N_{z}/2]$, respectively. The wave number magnitude is
$K=\sqrt{K_{x}^{2}+K_{y}^{2}+K_{z}^{2}}$, and so $K_{min}=0$ and
$K_{max}=\frac{\sqrt{3}\,N_{x}}{2\,R_{0}}$. The Fourier wave can equally be
described by a wave length $\lambda=2\,\pi/K$, then we see that
$K\approx\frac{1}{R_{0}}$ and $\lambda\approx R_{0}$.

Following [10], the components of the particle velocity are

$\begin{array}[]{l}\vec{v}=\Sigma_{-K_{max}}^{K_{max}}\left|K^{\frac{-n-2}{2}}\right|\times\par\left\\{\par\begin{array}[]{l}\left[K_{z}\,C_{K_{y}}\sin\left(\vec{K}\cdot\vec{r}+\Phi_{K_{y}}\right)-K_{y}\,C_{K_{z}}\sin\left(\vec{K}\cdot\vec{r}+\Phi_{K_{z}}\right)\right]\mbox{for}\,v_{x}\\\
\left[-K_{x}\,C_{K_{z}}\sin\left(\vec{K}\cdot\vec{r}+\Phi_{K_{z}}\right)+K_{z}\,C_{K_{x}}\sin\left(\vec{K}\cdot\vec{r}+\Phi_{K_{x}}\right)\right]\mbox{for}\,v_{y}\\\
\left[-K_{x}\,C_{K_{y}}\sin\left(\vec{K}\cdot\vec{r}+\Phi_{K_{y}}\right)+K_{y}\,C_{K_{x}}\sin\left(\vec{K}\cdot\vec{r}+\Phi_{K_{x}}\right)\right]\mbox{for}\,v_{z}\\\
\end{array}\par\right.\end{array}$ (5)

where the spectral index $n$ was fixed at $n=-1$ and thus we have
$v^{2}\approx K^{-3}$. The vector $\vec{C}_{K}$ whose components are denoted
by $\left(C_{K_{x}},C_{K_{y}},C_{K_{z}}\right)$, take values obeying a
Rayleigh distribution. The wave phase vector, $\Phi_{K}$, given by
$\left(\Phi_{K_{x}},\Phi_{K_{y}},\Phi_{K_{z}}\right)$ takes random values on
the interval $[0,2\,\pi]$. The components of the vector $\vec{C}$ are
calculated by means of
$C=\sigma\times\sqrt{-2.0\times\log\left(1.0-u\right)}$, where $u$ is a random
number in $(0,1)$. $\sigma$ is a fixed parameter with value $\sigma=1.0$.

###  3.3 The set up of the particle wind.

Let us consider the equation of mass conservation for a set of particles
moving radially outwards, that is

$\dot{M}=4\,\pi\,r^{2}\,\rho(r)\times v(r)$ (6)

We fix the mass loss $\dot{M}$ as a parameter of the simulation and also fix
the wind density to have the uniform value $\rho_{0}$. We then determine the
wind velocities according to Eq. 6. As the velocity magnitude diverges for
particles around $r\approx 0$, we set a cut velocity value such that the
maximum velocity allowed in our simulations is $v_{max}$.

Of course, there are other possibilities, which will be considered elsewhere:
one is to fix the radial density $\rho(r)$ and/or the velocity profile $v(r)$
in order to obtain the mass loss $\dot{M}$ as a result. Besides, for modeling
an expanding $H_{II}$ region, the authors of Ref. [3] proposed another and
more complicated velocity function, but anyway it gives a constant expansion
velocity at the last stages of time evolution, so that the average velocity of
the shocked shell considered by [3] is $v_{cc}/c_{0}\approx 5.6$ or
$v_{cc}\approx 3.7\,$ km/s.

###  3.4 Initial energies

In a particle based code, we approximate the thermal energy of the clump by
calculating the sum over all the $N$ particles described in Section 3.1, that
is

$E_{therm}=\sum_{i=1}^{N}\,\frac{3}{2}\,\frac{P_{i}(\rho)\,m_{i}}{\rho_{i}}\,,$
(7)

where $P_{i}$ is the pressure associated with particle $i$ with density
$\rho_{i}$ by means of the equation of state given in Eq. 12. In a similar
way, the approximate potential energy is

$E_{pot}=\sum_{i=1}^{N}\,\frac{1}{2}\,m_{i}\,\Phi_{i}\,.$ (8)

where $\Phi_{i}$ is the gravitational potential of particle $i$. For the clump
considered in this paper, the values of the speed of sound $c_{0}$ (see Eq. 3
) and the level of turbulence are chosen so that the energy ratios have the
numerical values

$\begin{array}[]{l}\alpha\equiv\frac{E_{therm}}{\left|E_{grav}\right|}=0.3\vspace{0.25
cm}\\\ \beta\equiv\frac{E_{kin}}{\left|E_{grav}\right|}=1.0\end{array}$ (9)

###  3.5 Resolution and thermodynamical considerations

Following Refs. [11] and [12], in order to avoid artificial fragmentation, the
$SPH$ code must fulfil certain resolution criteria, imposed on the Jeans
wavelength $\lambda_{J}$, which is given by

$\lambda_{J}=\sqrt{\frac{\pi\,c^{2}}{G\,\rho}}\;,$ (10)

where $c$ is the instantaneous speed of sound and $\rho$ is the local density.
To obtain a more useful form for a particle based code, the Jeans wavelength
$\lambda_{J}$ is transformed into the Jeans mass given by

$M_{J}\equiv\frac{4}{3}\pi\;\rho\left(\frac{\lambda_{J}}{2}\right)^{3}=\frac{\pi^{\frac{5}{2}}}{6}\frac{c^{3}}{\sqrt{G^{3}\,\rho}}\;.$
(11)

In this paper, the values of the density and speed of sound are updated
according to the following equation of state

$p=c_{0}^{2}\,\rho\left[1+\left(\frac{\rho}{\rho_{crit}}\right)^{\gamma-1}\,\right],$
(12)

as proposed by [13], where $\gamma\,\equiv 5/3$ and for the critical density
we assume the value $\rho_{crit}=5.0\times 10^{-14}\,$ g cm-3.

For the turbulent clump under consideration, we have $m_{r}\approx
M_{J}/(2N_{neigh})\approx 7.47\times 10^{33}\,$ g, where we take
$N_{neigh}=40$.

In this paper, the mass of an SPH particle is $m_{p}=1.98\,\times 10^{29}\,$g,
so that $m_{p}/m_{r}=2.5\,\times 10^{-4}$ and therefore the Jeans resolution
requirement is satisfied very easily.

In a previous paper of collapse reported in Ref.[14], by means of a
convergence study, we demonstrated the correctness of a regular cartesian grid
to make collapse calculation, as is used here to make the partition of the
simulation domain in small volume elements, each of which has a SPH particle
located not necessarily in its center, see Sect.3.1.

##  4 The evolution code

We carry out the time evolution of the initial distribution of particles with
the fully parallel Gadget2 code, which is described in detail by Ref.[15].
Gadget2 is based on the $tree-PM$ method for computing the gravitational
forces and on the standard $SPH$ method for solving the Euler equations of
hydrodynamics. $Gadget2$ incorporates the following standard features: (i)
each particle $i$ has its own smoothing length $h_{i}$; (ii) the particles are
also allowed to have individual gravitational softening lengths
$\epsilon_{i}$, whose values are adjusted such that for every time step
$\epsilon_{i}\,h_{i}$ is of order unity. Gadget2 fixes the value of
$\epsilon_{i}$ for each time-step using the minimum value of the smoothing
length of all particles, that is, if $h_{min}=min(h_{i})$ for $i=1,2...N$,
then $\epsilon_{i}=h_{min}$.

The $Gadget2$ code has an implementation of a Monaghan-Balsara form for the
artificial viscosity, see Ref.[16] and Ref.[17]. The strength of the viscosity
is regulated by setting the parameter $\alpha_{\nu}=0.75$ and
$\beta_{\nu}=\frac{3}{2}\,\times\alpha_{\nu}$, see Equation (14) in Ref.[15].
We here fix the Courant factor to 0.1.

We now mention here that the public Gadget2 code used in this paper, presents
some potential technical problems when wind particles are simultaneously
evolved with clump particles; these problems are caused by the different mass
scales involved, as the discretized version of the Navier-Stokes
hydrodynamical equations are written in the so called density-entropy
formulation, see Ref.[18]: mainly that the particle time-step becomes
prohibitively small to achieve the overall evolution of the system.

We finally mention that the initial condition code was written in ANSI-C and
it makes use of a Fourier mesh of 643 grid elements in order to obtain the
turbulent velocity field of 10 million SPH particles. It takes 300 CPU hours
running on one INTEL-Xeon processor at 3.2 GHz. All the simulations presented
in this work were conducted on a high performance Linux cluster with 46 dual
cores located at the University of Sonora (ACARUS). The simulations were
calculated over 300 CPU hours on a fully parallelized scheme using 12 cores
per each run. The proprietary program pvwave-8.0 was used as the main the
visualization tool for showing the results of this paper.

##  5 Results

To present the results of our simulations, we consider a slice of particles
around the equatorial plane of the clump; with these particles (around 10,000)
we make plots containing two complementary panels: one to show colored regions
of iso-density and the other one to show the velocity field of the particles.
We also make $3D$ plots built with the 500,000 densest particles of each
simulation. Later, we present plots with the velocity distributions and radial
velocity profile, for which we use all the simulation particles.

In Table 1 we show the values we use to define the simulations and also we
give some results described below. In this Table and in all the subsequent
figures, we use the following labels: ”Tur” to indicate the gas clump; when
the winds appear within the clump, we use the label ”Tur+Wind” to refer to the
spherically symmetric case and the label ”Tur+Wind+Col” for the bipolar jet.

Table 1: The models. Label | $t_{start}/t_{ff}$ | $\dot{M}\;[M_{\odot}/{\rm yr}]$ | $\frac{R_{s}}{R_{0}}$ | $M_{g}\,[M_{\odot}]$ | $M_{\infty}\,[M_{\odot}]$ | $v_{max}/c_{s}$ | $v_{\infty}/c_{s}$
---|---|---|---|---|---|---|---
T | 0.0 | —– | —– | 1219 | 333 | 100 | 10
Tur+Wind | 0.05 | $1.0\,\times 10^{-4}$ | $0.1$ | 6.84 | 359 | 100 | 11
Tur+Wind+Col | 0.05 | $1.0\,\times 10^{-3}$ | $0.05$ | 4.51 | 422 | 100 | 72
|
---|---
|
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Figure 1:  Iso-density plots for the turbulent clump.

###  5.1 The evolution of the turbulent clump.

One of the characteristics of turbulence is the appearance of a filamentary
and flocculent structure across the clump, a structure which can already be
seen in the first two panels of Fig.1. Because of the initial conditions
chosen for the clump, there is a clear tendency to a global collapse towards
its central region, as can be seen in the last panels of Fig.1.

We emphasize now a very important fact occurring at the outer regions of the
turbulent clump. The turbulent clump is not in hydrodynamic equilibrium and
there is no external pressure acting upon the clump, then the outermost
particles have a non equilibrated thermal pressure. Therefore, the outer clump
particles expand outwards. So, we have to keep in mind this expansion effect
for the problem at hand, as we shall quantify the mass of the clump swept out
by the winds.

|  |
---|---|---
Figure 2:  (left) the velocity distribution of the particles; (middle) the
velocity radial profile and (right) the peak density time evolution.

###  5.2 The effects of winds in the evolution of the turbulent clump.

As one can see in the two first panels of Fig. 2, a very small fraction of gas
particles can attain velocities much higher than those velocities provided
from the turbulence alone, which are around $2$ Mach. Eighty per cent of the
simulation particles have velocities less than $v/cs<4$.

The winds are suddenly activated at the time $t/t_{ff}=0.05$, when the clump
has already acquired a fully flocculent aspect, which is a consequence of the
huge number of gas collisions produced randomly across the clump. This time
also marks the occurrence of the highest peak in the clump’s density curve,
shown in the third panel of Fig.2

Besides, we notice by looking at the third panel Fig. 2, that the global
collapse of the clump does not change even when the winds are introduced, as
the peak density curve of each run goes to higher values. However, the wind of
the first kind makes the collapse take place slower as its density peak curve
shows less steepness in the middle stages of its evolution.

|
---|---
|
|
Figure 3:  Iso-density and velocity plots for the model ”Tur+Wind”.

The iso-density plots for the simulation $Tur+Wind$ are shown in Fig. 3. By
the time $t/t_{ff}=0.36$, we see a void created in the central region of the
clump because both the wind particles and those particles which are swept out
move towards the outer parts of the clump. However, gravity and viscosity act
together in such a way that the particles quickly fill the void, as can be
seen in the last panel of Fig. 3.

When we consider the model $Tur+Wind+Col$, so that a collimated gas of
particles is ejected, we see that the effects on the clump are more
significant, but essentially the same phenomena as seen in the model
$Tur+Wind$ take place, as can be seen in Fig. 4.

In Fig.5 we present $3d$ plots of the densest particles for both models. In
Figs.6 and 7 we present $3d$ plots at two different times in which one can
distinguish the wind and the gas particles.

We emphasize that the fraction of gas swept out by the winds is really
significant: hundreds of solar masses move even far beyond the clump radius
$R_{0}$. In the fourth column of Table 1, we show that the initial mass
contained within the radius $R_{s}$( which defines the outer boundary of the
initial wind configuration) is around 7 and 5 $M_{\odot}$, respectively. It is
then surprising to notice that the total mass dragged outside $R_{0}$ by the
end of the simulation time is around 359 and 422 $M_{\odot}$, respectively, as
is shown in the fifth column of Table 1.

|
---|---
|
|
|
Figure 4:  Iso-density and velocity plots for the model ”Tur+Wind+Col”.

Lastly, in columns 6 and 7 of Table 1, we show the velocities attained by
those particles located far outside the initial clump. Furthermore, we mention
that these velocities are not terminal, but are only the velocities during the
time we follow these simulations.

##  6 Discussion

We mentioned in Sect. 3.4 that the clump under consideration here is initially
given a ratios of thermal ($\alpha$ ) and kinetic energies ($\beta$) to the
gravitational energy, respectively, such that the clump collapse is greatly
favored. In fact, as we see in Sect.5.1, the clump presents a strong collapse
towards the central region, in which no fragmentation is observed. This
behavior can be seen in other turbulent simulations, when the turbulent Jeans
mass ( in analogy to the thermal Jeans mass) is large, so that only one
turbulent Jeans mass is ”contained” in the total clump mass, see Ref.[21].

|
---|---
Figure 5:  $3D$ plot of the densest particles for models (left) $Tur+Win$ and
(right) $Tur+Win+Col$ at the corresponding snapshots shown in the second lines
of Figs. 3 and 4, respectively.

The winds are activated in a very early stage of the clump evolution, when
turbulence is still ongoing. Soon after, turbulence quickly decays, so that a
purely gravity driven collapse will take place in the clump. As we see in
Sect.5.2, the winds act firstly as a disruptive perturbation on the clump’s
near environment. Despite of this, the clump particles quickly reform this
evacuated central region. Now, in Ref.[22] we found that the wind models show
a strong tendency to form accretion centers in the central region of the
clump. It was noticed there that great differences appeared when we compared
the number and location of the accretion centers obtained for the turbulent
clump and for the clump in the presence of wind. Unfortunately, these results
are not included in this paper for a lack of space. Although we try to
indicate them by means of the first panel of Fig.5.

Thus, in view of these results, we consider that the expected fate of the
turbulent clump is really affected by the wind-clump interaction, as compared
with the purely gravity driven collapse, despite that it seems to be the
dominant physics in determining the time evolution of the clump. This behavior
appears to be the case in general, at least for observed gas structures around
0.1 pc in size, as is discussed in Ref.[21].

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Figure 6:  $3D$ plot of the turbulent clump (in blue) with winds (in yellow)
and formed accretion centers (in red) for models (top) $Tur+Win$ and (bottom)
$Tur+Win+Col$, corresponding to the plots shown in the second line of Figs.3
and 4, respectively.

It must be noted that in this paper, as a first approximation, the wind
emission is a unique event, as it is emitted just for only one time. It would
be interesting to make that the winds be created and emitted in each time
step, so that a continuous emission rate can be simulated. In this case, one
would expect that the collapse delay be larger and that the disruption wind
effects be more important too.

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---|---
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Figure 7:  $3D$ plot of the turbulent clump (in blue) with winds (in yellow)
for models (top) $Tur+Win$ and (bottom) $Tur+Win+Col$, corresponding to last
snapshot available.

##  7 Concluding Remarks

A star formation scenario based only on the collapse of turbulence gas
structures gives a very highly efficient transformation of the gas into
protostars. This points is in contradiction with observations. Besides, a
physical system showing simultaneously in-fall and outflow motions are
observed in cluster like NGC 1333 and NGC 2264, see Refs.[19] and [20],

Because of this, another scenario must be considered, or at least a
theoretical complement to the turbulent model is needed. As we have shown in
this paper, the winds must be considered as an additional ingredient to
complement the turbulent model with the hope that this new model can alleviate
some of the problems mentioned above as they make a delay in the runaway
collapse of the clump, see Fig. 2.

We have shown here that all the wind models show a strong tendency to form
accretion centers in the central region of the clump. It must be noted that
some differences appear when we compare the accretion centers obtained for the
wind models, as can be seen in Fig. 5, 6 and 7.

#### Acknowledgments.

We would like to thank ACARUS-UNISON for the use of their computing facilities
in the preparation of this paper.

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# Random periodic solutions of non-autonomous stochastic differential
equations

Zhao Dong1,2, Weili Zhang1,2,∗ and Zuohuan Zheng 3,1,2

1 Academy of Mathematics and Systems Science, Chinese Academy of Sciences,
Beijing, 100190, China

2 School of Mathematical Sciences, University of Chinese Academy of Sciences,
Beijing, 100049, China

3 College of Mathematics and Statistics, Hainan Normal University, Haikou,
Hainan 571158, China

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
††* Corresponding author

Abstract: In this paper, we study the existence of random periodic solutions
for nonlinear stochastic differential equations with additive white noise. We
extend the input-to-state characteristic operator of the system to the non-
autonomous stochastic differential equation via the pull-back of the
discretised stochastic differential equation. We then use the completeness of
the measurable function space which we construct skillfully and the Banach
fixed point theorem to prove the existence of a fixed point of the gain
operator. And we prove that the image for the input-to-state characteristic
operator at this fixed point is a random periodic solution for the forward
stochastic flow generated by the non-autonomous stochastic differential
equation. Finally, we present some examples.

MSC: 37H05; 28D10; 60G17

Keywords: Random periodic solution, stochastic flow, stochastic differential
equation, pull-back, metric dynamical system.

##  1 Introduction

Periodic solution has been a central concept of the qualitative theory of
ordinary differential equations and the deterministic dynamic system theory,
it originated from Poincaré’s pioneering work ([17]). The deterministic
autonomous dynamical system does not depend on the initial time, but depends
on the time interval. However, the remarkable feature of deterministic non-
autonomous dynamical system is that its evolution depends on both the current
time and the initial time. Therefore, this kind of systems require a two-
parameter semi-group to characterize, which makes it describe more natural
phenomena. Denote $\triangle:=\\{(t,s)\in\mathbb{R}^{2},s\leq t\\}$. For a
deterministic non-autonomous dynamical system $u(t,s):X\rightarrow X$ over
time $(t,s)\in\triangle$, where $X$ is a metric space. A periodic solution for
$u$ is a periodic function $y:\mathbb{R}\rightarrow X$ with periodic $T>0$
such that

$u(t,s)y(s)=y(t)\ \ \ \text{and}\ \ \ y(s+T)=y(s)\ \ \ \text{for}\ \text{all}\
(t,s)\in\triangle.$

As the random counterpart of periodic solution, H.Z. Zhao and Z-H. Zheng first
proposed the definition of random periodic solution for a $C^{1}$-cocycle
([19]). Later the concept of random periodic solutions for semi-flows
generated by non-autonomous stochastic differential equations and stochastic
partial differential equations were given in [8] and [6], respectively. Also,
they led to more progress on investigations of various issues in autonomous
stochastic differential equations and non-autonomous stochastic differential
equations. They include numerical analysis of random periodic solutions and
periodic measures of stochastic differential equations ([3], [4]);
anticipating random periodic solutions of stochastic differential equations
([5]); random periodic processes, periodic measures and ergodicity ([7]) etc.

Let $(\Omega,{\mathcal{F}},P)$ be a probability space, $X$ be a Polish space
and ${\mathcal{B}}(X)$ be its Borel $\sigma$-algebra. Denote by
$(\Omega,{\mathcal{F}},P,(\theta_{s})_{s\in\mathbb{R}})$ a metric dynamical
system and $\theta_{s}:\Omega\rightarrow\Omega$ is assumed to be measurably
invertible for all $s\in\mathbb{R}$.

######  Definition 1.1 ([19])

A random periodic solution of periodic $T>0$ for the random dynamical system
$\varphi:\mathbb{R}\times\Omega\times X\rightarrow X$ is an
${\mathcal{F}}$-measurable map $Y:\mathbb{R}\times\Omega\rightarrow X$ such
that for almost all $\omega\in\Omega$,

$\varphi(t,\theta_{s}\omega)Y(s,\omega)=Y(t+s,\omega),\ \
Y(s+T,\omega)=Y(s,\theta_{T}\omega),\ \ \rm{for}\ \rm{any}\ \
t,s\in\mathbb{R}.$ (1.1)

Consider a forward stochastic flow $\varphi:\triangle\times\Omega\times
X\rightarrow X$ of periodic $T$, which satisfies the following standard
condition

$\varphi(s,s)=\rm{Id},\ \
\varphi(t,s,\omega)=\varphi(t,r,\omega)\circ\varphi(r,s,\omega),$ (1.2)

and the periodic property

$\varphi(t+T,s+T,\omega)=\varphi(t,s,\theta_{T}\omega),$ (1.3)

for all $s\leq r\leq t$, $s,r,t\in\mathbb{R}$.

######  Definition 1.2 ([8])

A random periodic solution of periodic $T>0$ for the forward stochastic flow
$\varphi:\triangle\times\Omega\times X\rightarrow X$ is an
${\mathcal{F}}$-measurable map $Y:\mathbb{R}\times\Omega\rightarrow X$ such
that for almost all $\omega\in\Omega$,

$\varphi(t,s,\omega)Y(s,\omega)=Y(t,\omega),Y(s+T,\omega)=Y(s,\theta_{T}\omega),\
\ \rm{for}\ \rm{any}\ \ (t,s)\in\triangle.$ (1.4)

In this paper, we consider the following nonlinear stochastic system:

$dX_{t}=[AX_{t}+h(t,X_{t})]dt+\sigma(t)dW_{t},$ (1.5)

where $W_{t}(\omega)=(W_{t}^{1}(\omega),...,W_{t}^{m}(\omega))$ is a two-side
time Wiener process with values in $\mathbb{R}^{m}$ on the canonical Wiener
space $(\Omega,{\mathcal{F}},({\mathcal{F}}^{t})_{t\in\mathbb{R}},P)$, i.e.,
${\mathcal{F}}$ is the Borel $\sigma$-algebra of
$\Omega=C_{0}(\mathbb{R},\mathbb{R}^{m})=\\{\omega:\omega(t)$ continuous,
$\omega(0)=0,t\in\mathbb{R}\\}$; ${\mathcal{F}}_{s}^{t}$ is the least complete
$\sigma$-field for which all $W_{u}-W_{v}$, $s\leq v\leq u\leq t$ are
measurable and ${\mathcal{F}}^{t}={\mathcal{F}}_{-\infty}^{t}=\bigvee_{s\leq
t}{\mathcal{F}}_{s}^{t}$; $P$ is the Wiener measure. $A=(a_{ij})_{d\times d}$
is a $(d\times d)$-dimensional matrix.
$h:\mathbb{R}\times\mathbb{R}^{d}\rightarrow\mathbb{R}_{+}^{d}$,
$h(t+T,X_{t})=h(t,X_{t})$, for any $t\in\mathbb{R},x\in\mathbb{R}^{d}$, $T>0$
is a constant. $\sigma(t)=\sigma(t+T)$ is a $(d\times m)$-dimensional matrix.

In [9], J.F. Jiang and X. Lv studied small-gain results for nonlinear
stochastic system driven by additive white noise of the form with
$h(t,x)\equiv h(x),\sigma(t)\equiv\sigma$ in (1.5), and $h$ is order-
preserving or anti-order-preserving. Motivated by their work, we extend their
results to non-autonomous stochastic differential equation (1.5). In our case,
the solution of (1.5) is a forward stochastic flow which leads to the input-
to-state characteristic operator ${\mathcal{K}}(u)$ not only depending on
$\omega\in\Omega$ but also depending on $t\in\mathbb{R}$. Thus, in order to
get the fixed point of the gain operator, we construct a measurable function
space skillfully. And we prove that the image for the input-to-state
characteristic operator at this fixed point is a random periodic solution for
the non-autonomous stochastic differential equation (1.5).

The rest of this paper is organized as follows. In section 2, we review some
preliminary concepts and definitions, present the assumptions for the non-
autonomous stochastic differential equation (1.5), and define an operator of
the system via the pull-back of the discretised stochastic differential
equation. In section 3, we establish some auxiliary lemmas and proposition,
and present the definition of another operator and its properties. In section
4, the main result Theorem 4.2 is proved. In section 5, we present some
examples.

##  2 Preliminaries and Assumptions

Before starting our main results, we introduce some basic concepts and
notations.

######  Definition 2.1 ([1])

A family of mapping on the sample space $\Omega$,
$\theta_{t}:\Omega\rightarrow\Omega,t\in\mathbb{R}$, is called a measurable
dynamical system if the following conditions are satisfied
(i) Identity property: $\theta_{0}=\rm{Id}$;
(ii) Flow property: $\theta_{t+s}=\theta_{t}\circ\theta_{s}$;
(iii) Measurability: $(\omega,t)\rightarrow\theta_{t}\omega$ is measurable.
It is called a measure-preserving or metric dynamical system if, furthermore
(iv) Measure-preserving preserving property: $P(\theta_{t}(A))=P(A)$, for
every $A\in{\mathcal{F}}$ and $t\in\mathbb{R}$.
In this case, $P$ is called an invariant measure with respect to the dynamical
system $\theta_{t}$.

######  Definition 2.2 ([1])

A measurable random dynamical system (RDS) on the measurable space
$(X,{\mathcal{B}}(X))$ over a metric dynamical system
$(\Omega,{\mathcal{F}},P,(\theta_{t})_{t\in\mathbb{R}})$ with time
$\mathbb{R}$ is a mapping

$\Phi:{\mathbb{R}}\times\Omega\times X\rightarrow X,\
(t,\omega,x)\mapsto\Phi(t,\omega,x)$

with the following properties:
(i) Measurability: $\Phi$ is
$({\mathcal{B}}({\mathbb{R}})\otimes{\mathcal{F}}\otimes{\mathcal{B}}(X),{\mathcal{B}}(X))$-measurable.
(ii) Cocycle property: The mappings
$\Phi(t,\omega):=\Phi(t,\omega,\cdot):X\rightarrow X$ form a cocycle over
$\theta(\cdot)$, i.e. they satisfy

$\Phi(0,\omega)={id_{X}},\ \ \text{for}\ \text{all}\ \omega\in\Omega,$
$\Phi(t+s,\omega)=\Phi(t,\theta_{s}\omega)\circ\Phi(s,\omega),\ \ \text{for}\
\text{all}\ s,t\in\mathbb{R},\ \omega\in\Omega.$

Here $\circ$ means composition of mappings.

######  Definition 2.3 ([12])

Let $\varphi(t,s,x,\omega),s,t\in\mathbb{R},x\in\mathbb{R}^{d}$ be a
continuous $\mathbb{R}^{d}$-valued random field defined on the probability
space $(\Omega,{\mathcal{F}},P)$. Then for almost all $\omega$,
$\varphi(t,s,\omega)\equiv\varphi(t,s,\cdot,\omega)$ defines a continuous map
from $\mathbb{R}^{d}$ into itself for any $s,t$. It is called a stochastic
flow if there exists a null set $N$ of $\Omega$ such that for any $\omega\in
N^{c}$, the family of continuous maps
$\\{\varphi(t,s,\omega):s,t\in\mathbb{R}\\}$ defines a flow i.e. it satisfies
the following properties:
(i) $\varphi(u,s,\omega)=\varphi(u,t,\omega)\circ\varphi(t,s,\omega)$ holds
for all $s,t,u$, where $\circ$ denotes the composition of maps.
(ii) $\varphi(s,s,\omega)=$ identity map for all $s$.
Further if $\varphi(t,s,\omega)$ satisfies (iii), it is called a stochastic
flow of homeomorphisms.
(iii) the map $\varphi(t,s,\omega):\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ is
an onto homeomorphism for all $s,t\in\mathbb{R}$.
For the analysis of a stochastic flow, it is convenient to divide the flow
into forward flow $\varphi(t,s),s\leq t$ and the backward flow
$\varphi(t,s),s\geq t$ and discuss them separately.

Consider the corresponding linear ordinary differential equation:

$dX_{t}=AX_{t}dt.$ (2.1)

Let $\Phi_{j}(t)=(\Phi_{1j}(t),...,\Phi_{dj}(t))^{T}$ be the solution of
equation (2.1) with initial value $X(0)=e_{j},j=1,...,d$. Then the $d\times d$
matrix $\Phi(t)=(\Phi_{1}(t),...,\Phi_{d}(t))=(\Phi_{ij}(t))_{d\times d}$ is
the fundamental matrix of equation (2.1) and $\Phi(t+s)=\Phi(t)\circ\Phi(s)$
for all $t,s\geq 0$.

Now we propose the assumptions on $A,h,\sigma$ in (1.5) as follows.

$\boldsymbol{(A)}$ $A$ is cooperative, i.e., $a_{ij}\geq 0$ for all
$i,j\in\\{1,...,d\\}$ and $i\neq j$, and $A$ is stable, i.e., all real part of
its eigenvalues are negative:

$Re\ \mu\leq\lambda<0\ \ \ \ \ \rm{for}\ \rm{all}\ \rm{eigenvalues}\ \mu\
\rm{of}\ A.$ (2.2)

$\boldsymbol{(H_{1})}$ $h\in
C_{b}^{1}(\mathbb{R}\times\mathbb{R}^{d},\mathbb{R}_{+}^{d})$, i.e., the
function $h$ and its derivatives are both bounded, and $h$ is order-preserving
in $\mathbb{R}^{d}$, i.e., for any $t\in\mathbb{R}$

$h(t,x_{1})\leq_{\mathbb{R}_{+}^{d}}h(t,x_{2})\ \ \text{whenever}\
x_{1}\leq_{\mathbb{R}_{+}^{d}}x_{2},$

or anti-order-preserving in $\mathbb{R}^{d}$, i.e.,

$h(t,x_{1})\geq_{\mathbb{R}_{+}^{d}}h(t,x_{2})\ \ \text{whenever}\
x_{1}\leq_{\mathbb{R}_{+}^{d}}x_{2}.$

Here, $x\leq_{\mathbb{R}_{+}^{d}}y$ means that $y-x\in\mathbb{R}_{+}^{d}$ for
all $x,y\in\mathbb{R}^{d}$.

$\boldsymbol{(H_{2})}$ $L:=\max\\{\
\sup_{t\in\mathbb{R},x\in\mathbb{R}^{d}}|\frac{\partial h_{i}(t,x)}{\partial
x_{j}}|,i,j=1,...,d\\}<-\frac{\lambda}{d^{2}}$.

$\boldsymbol{(\Sigma)}$
$\sup_{t\in\mathbb{R}}||\sigma(t)||:=\sup_{t\in\mathbb{R}}\max\\{|\sigma_{ij}(t)|:i=1,...,d;j=1,...,m\\}<\infty$,
where $\sigma_{ij}(t),i=1,...,d,j=1,...,m$ are continuous functions from
$\mathbb{R}$ to $\mathbb{R}$.

If $A$ is stable, by ([16], Chapter 2, Proposition 2.10) we have

$\parallel\Phi(t)\parallel:=\max\\{|\Phi_{ij}(t)|:i,j=1,...,d\\}\leq
e^{\lambda t},\ t\geq 0.$ (2.3)

In this paper, we use the norm $|x|:=\max\\{|x_{i}|:i=1,...,d\\}$,
$x\in\mathbb{R}^{d}$.

By Theorem 4.3 in [11], there is a modification of the unique solution for
equation (1.5), denoted by $\varphi(t,s,\omega)x$ and the integral version of
(1.5)

$\displaystyle\varphi(t,s,\omega)x$ $\displaystyle=$
$\displaystyle\Phi(t-s)x+\Phi(t)\displaystyle\int^{t}_{s}{\Phi^{-1}(r)h(r,\varphi(r,s,\omega)x)dr}+\Phi(t)\displaystyle\int^{t}_{s}{\Phi^{-1}(r)\sigma(r)dW_{r}}$
(2.4) $\displaystyle=$
$\displaystyle\Phi(t-s)x+\displaystyle\int^{t}_{s}{\Phi(t-r)h(r,\varphi(r,s,\omega)x)dr}+\displaystyle\int^{t}_{s}{\Phi(t-r)\sigma(r)dW_{r}}.$

holds almost surely. In addition, it has the following properties:
(i) For each $(t,s)\in\triangle$ and $x$, $\varphi(t,s,\cdot)x$ is
${\mathcal{F}}_{s}^{t}$-measurable.
(ii) For almost all $\omega$, $\varphi(t,s,\omega)x$ is continuous in
$(t,s,x)$and satisfies $\lim\limits_{t\downarrow s}\varphi(t,s,x,\omega)=x$.
(iii) For almost all $\omega$,

$\varphi(t,s,\omega)=\varphi(t,r,\omega)\circ\varphi(r,s,\omega),$

for all $s\leq r\leq t$, $s,r,t\in\mathbb{R}$.

Denote the standard $\mathbb{P}$-preserving ergodic Wiener shift by
$\theta:\mathbb{R}\times\Omega\rightarrow\Omega$,

$\theta_{t}\omega(\cdot):=W(t+\cdot)-W(t),\ \ \ t\in\mathbb{R}.$

###### Remark 2.4

The solution $\varphi(t,s,\omega)x$ of non-autonomous stochastic differential
equation (1.5) with the initial value $X(s)=x\in\mathbb{R}^{d}$ does not
satisfy the cocycle property, but satisfies the forward stochastic flow
property (1.2). Furthermore, it has the periodicity (1.3). In fact, for $s\leq
t$, $s,t\in\mathbb{R}$, by the periodicity of $h(\cdot,x)$ we have

$\displaystyle\begin{split}&\ \ \ \ \varphi(t+T,s+T,\omega)x\\\
&=\Phi(t-s)x+\int^{t+T}_{s+T}{\Phi(t+T-v)h(v,\varphi(v,s+T,\omega)x)dv}+\displaystyle\int^{t+T}_{s+T}{\Phi(t+T-v)\sigma(v)dW_{v}}\\\
&=\Phi(t-s)x+\int^{t}_{s}{\Phi(t-v)h(v+T,\varphi(v+T,s+T,\omega)x)dv}+\displaystyle\int^{t}_{s}{\Phi(t-v)\sigma(v+T)dW_{v+T}}\\\
&=\Phi(t-s)x+\int^{t}_{s}{\Phi(t-v)h(v,\varphi(v+T,s+T,\omega)x)dv}+\displaystyle\int^{t}_{s}{\Phi(t-v)\sigma(v)dW_{v}(\theta_{T}\omega)}\end{split}$

We thus have found that the function

$\psi(v,s,\theta_{T}\omega)x=\varphi(v+T,s+T,\omega)x,\ \ s\leq v$

satisfies

$\psi(t,s,\theta_{T}\omega)x=\Phi(t-s)x+\int^{t}_{s}{\Phi(t-v)h(v,\psi(v,s,\theta_{T}\omega)x)dv}+\displaystyle\int^{t}_{s}{\Phi(t-v)\sigma(v)dW_{v}(\theta_{T}\omega)}.$

By the uniqueness of the solution

$\varphi(t,s,\theta_{T}\omega)x=\psi(t,s,\theta_{T}\omega)x=\varphi(t+T,s+T,\omega)x.$

which implies that (1.3) holds.

Now we are in position to introduce the concept of an operator
${\mathcal{K}}$. Denote by $\varphi(t,-nT,\omega)x$ the solution starting from
time $-nT$ with the initial value $X(-nT)=x$. Then for any
$n\in\mathbb{N}_{+}$, $t>-nT$,

$\varphi(t,-nT,\omega)x=\Phi(t+nT)x+\displaystyle\int^{t}_{-nT}{\Phi(t-s)h(s,\varphi(s,-nT,\omega)x)ds}+\displaystyle\int^{t}_{-nT}{\Phi(t-s)\sigma(s)dW_{s}},$
(2.5)

holds almost surely.

We define the input-to-state characteristic operator $\mathcal{K}$:

$[\mathcal{K}(u)](t,\omega)=\displaystyle\int^{t}_{-\infty}{\Phi(t-s)u(s,\omega)ds}+\displaystyle\int^{t}_{-\infty}{\Phi(t-s)\sigma(s)dW_{s}},\
\ t\in\mathbb{R},\ \omega\in\Omega.$ (2.6)

where the stochastic process
$u:\mathbb{R}\times\Omega\rightarrow\mathbb{R}_{+}^{d}$ is bounded.

We denote
$|x|_{2}:=(\sum\limits_{i=1}^{d}{|x_{i}|^{2}})^{\frac{1}{2}},x\in\mathbb{R}^{d}$
and
$\parallel\Phi\parallel_{2}:=(\sum\limits_{i,j=1}^{d}{|\Phi_{i,j}|^{2}})^{\frac{1}{2}}$
in what follows, where $\Phi$ is a $(d\times d)$-dimensional matrix.

###### Remark 2.5

It is noticed that for fixed $t\in\mathbb{R},\omega\in\Omega$, the operator
${\mathcal{K}}$ is well defined. In fact, for any bounded stochastic process
$u$, since $\parallel\Phi(t)\parallel\leq e^{\lambda t},t\geq 0$, we have
$\parallel\Phi\parallel_{2}\leq d\parallel\Phi(t)\parallel\leq de^{\lambda
t}$, and it is not hard to prove that

$\displaystyle\int^{t}_{-\infty}|{\Phi(t-s)u(s,\omega)|_{2}ds}<\infty$

which implies that
$\lim_{r\rightarrow\infty}\int^{t}_{-r}{\Phi(t-s)u(s,\omega)ds}$ exists for
all $t\in\mathbb{R},\omega\in\Omega$. For any $r_{1}>r_{2}>0$,

$\displaystyle\begin{split}&\ \ \
\mathbb{E}\Big{|}\displaystyle\int^{t}_{-r_{1}}{\Phi(-s)\sigma(s)dW_{s}}-\displaystyle\int^{t}_{-r_{2}}{\Phi(-s)\sigma(s)dW_{s}}\Big{|}_{2}^{2}\\\
&=\mathbb{E}\displaystyle\int^{-r_{2}}_{-r_{1}}{||\Phi(-s)\sigma(s)||_{2}^{2}ds}\\\
&\leq
dm\cdot\sup\limits_{t\in\mathbb{R}}||\sigma(t)||\cdot\displaystyle\int^{-r_{2}}_{-r_{1}}{||\Phi(-s)||_{2}^{2}ds}\end{split}$

which together with assumption $\boldsymbol{(A)}$ and $\boldsymbol{(\Sigma)}$
shows that $\int^{t}_{-r}{\Phi(-s)\sigma(s)dW_{s}}$ converges in $L^{2}$ as
$r\rightarrow\infty$. Furthermore, $\int^{t}_{-r}{\Phi(-s)\sigma(s)dW_{s}}$ is
a continuous martingale. Hence, it follows from ([10], Problem 3.20 in Chapter
1) that $\int^{t}_{-r}{\Phi(-s)\sigma(s)dW_{s}}$ converges $\mathbb{P}$-a.s.
to an integrable random variable
$X_{\infty}:=\int^{t}_{-\infty}{\Phi(-s)\sigma(s)dW_{s}}$ as
$r\rightarrow\infty$.

##  3 Measurability and asymptotic behaviour

In this section, we give some lemmas to describe the dynamics of the pull-back
trajectory which will be used in the proof of our main result.

###### Lemma 3.1

For each $n\in\mathbb{N}^{+}$, $x\in\mathbb{R}^{d}$, let

$a_{n}^{h}(t,\omega)=\inf{\\{h(t,\varphi(t,-mT,\omega)x):m\geq
n,m\in\mathbb{N}_{+}\\}},\ \ t\in\mathbb{R},\omega\in\Omega,$

and

$b_{n}^{h}(t,\omega)=\ \sup{\\{h(t,\varphi(t,-mT,\omega)x):m\geq
n,m\in\mathbb{N}_{+}\\}},\ \ t\in\mathbb{R},\omega\in\Omega,$

where $\inf$ and $\sup$ mean the greatest lower bound and the least upper
bound, respectively. Then $a_{n}^{h}(t,\omega)$ and $b_{n}^{h}(t,\omega)$ are
$({\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}},{\mathcal{B}}(\mathbb{R}^{d}))$-measurable
stochastic processes.

Proof. First, we show that $a_{n}^{h}(t,\omega)$ and $b_{n}^{h}(t,\omega)$ are
well defined. It is clear that

$D_{n}^{h}(t,\omega):={\\{h(t,\varphi(t,-mT,\omega)x):m\geq
n,m\in\mathbb{N}_{+}\\}}$

is a bounded set for fixed
$n\in\mathbb{N}_{+},t\in\mathbb{R},\omega\in\Omega$, which implies that
$D_{n}^{h}(t,\omega)$ is order-bounded. Since $\mathbb{R}_{+}^{d}$ is strongly
minihedral ([2], Definition 3.1.7), $a_{n}^{h}(t,\omega)$ and
$b_{n}^{h}(t,\omega)$ exist. Since
$\inf{D_{n}^{h}(t,\omega)}=-\sup\\{-D_{n}^{h}(t,\omega)\\}$, it is sufficient
to consider $b_{n}^{h}(t,\omega)$ only. Let

$\beta_{n,M}^{h}(t,\omega)=\sup\Big{\\{}h(t,\varphi(t,-nT,\omega)x),h(t,\varphi(t,-(n+1)T,\omega)x),\cdots,h(t,\varphi(t,-(n+M)T,\omega)x)\Big{\\}},$

by the continuity of $h$, the measurability of $\varphi$ and Corollary
3.1.1(ii) in [2], $\beta_{n,M}^{h}(t,\omega)$ is
$({\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}},{\mathcal{B}}(\mathbb{R}^{d}))$-measurable
for every $M=1,2,\cdots$. It is also clear that

$\beta_{n,1}^{h}(t,\omega)\leq\beta_{n,2}^{h}(t,\omega)\leq\cdots\leq\beta_{n,M}^{h}(t,\omega)\leq\cdots.$

Moreover, by the boundedness of $h$ in $\mathbb{R}_{+}^{d}$,
$b_{n}^{h}(t,\omega)=\lim\limits_{{M}\to{\infty}}\beta_{n,M}^{h}(t,\omega)$ is
$({\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}},{\mathcal{B}}(\mathbb{R}^{d}))$-measurable
stochastic process.

###### Lemma 3.2

Assume that conditions $\boldsymbol{(A)}$, $\boldsymbol{(H_{1})}$ and
$\boldsymbol{(\Sigma)}$ hold. Let $\varphi(t,s,\omega)x$ be a solution of
stochastic system (1.5) with initial value $X(s)=x\in\mathbb{R}^{d}$. Then we
have

${\mathcal{K}}(\underline{\lim}h(\cdot,\varphi))\leq\underline{\lim}\varphi\leq\overline{\lim}\varphi\leq{\mathcal{K}}(\overline{\lim}h(\cdot,\varphi)),\
\ \mathbb{P}-a.s.,$ (3.1)

where

$[\underline{\lim}\varphi](t,\omega)=\lim\limits_{{n}\to{\infty}}\inf\\{\varphi(t,-mT,\omega)x:m\geq
n,m\in\mathbb{N}_{+}\\},\ \
x\in\mathbb{R}^{d},t\in\mathbb{R},\omega\in\Omega.$

$[\overline{\lim}\varphi](t,\omega)=\lim\limits_{{n}\to{\infty}}\sup\\{\varphi(t,-mT,\omega)x:m\geq
n,m\in\mathbb{N}_{+}\\},\ \
x\in\mathbb{R}^{d},t\in\mathbb{R},\omega\in\Omega.$

$[\underline{\lim}h(\cdot,\varphi)](t,\omega)=\lim\limits_{{n}\to{\infty}}a_{n}^{h}(t,\omega),\
\ x\in\mathbb{R}^{d},t\in\mathbb{R},\omega\in\Omega.$

and

$[\overline{\lim}h(\cdot,\varphi)](t,\omega)=\lim\limits_{{n}\to{\infty}}b_{n}^{h}(t,\omega),\
\ x\in\mathbb{R}^{d},t\in\mathbb{R},\omega\in\Omega.$

where $a_{n}^{h}$ and $b_{n}^{h}$ are as defined in Lemma 3.1.

Proof. Here, we only prove the first inequality for the sake of convenience
and the rest of the inequalities can be proved analogously. Similar to Lemma
3.1, we can easily get that $\underline{\lim}\varphi$ and
$\underline{\lim}h(\cdot,\varphi)$ exist, which are also two
$({\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}},{\mathcal{B}}(\mathbb{R}^{d}))$-measurable
stochastic processes. Then by (2.6) and the Fubini Theorem,
${\mathcal{K}}(\underline{\lim}h(\cdot,\varphi))$ is well defined and
$({\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}},{\mathcal{B}}(\mathbb{R}^{d}))$-measurable.
By the definition of the $\underline{\lim}h(\cdot,\varphi)$, ${\mathcal{K}}$
and Lebesgue’s dominated convergence theorem, we have
${\mathcal{K}}(\underline{\lim}h(\cdot,\varphi))=\lim\limits_{{n}\to{\infty}}{\mathcal{K}}(a_{n}^{h})$.
For fixed $n\in\mathbb{N}_{+}$, it is enough to prove that

$\displaystyle\begin{split}&\ \ \ \ [{\mathcal{K}}(a_{n}^{h})](t,\omega)\\\
&=\displaystyle\int^{t}_{-\infty}{\Phi(t-s)\inf\\{h(s,\varphi(s,-mT,\omega)x):m\geq
n,m\in\mathbb{N}_{+}\\}ds}+\displaystyle\int^{t}_{-\infty}{\Phi(t-s)\sigma(s)dW_{s}}\\\
&=\lim\limits_{{\tilde{m}}\to{\infty}\atop{\tilde{m}}\geq
n}\Big{\\{}\Phi(t+{\tilde{m}}T)x+\displaystyle\int^{t}_{nT-{\tilde{m}T}}{\Phi(t-s)\inf\\{h(s,\varphi(s,-mT,\omega)x):m\geq
n,m\in\mathbb{N}_{+}\\}ds}\\\ &\ \ \ \
+\displaystyle\int^{t}_{-{\tilde{m}T}}{\Phi(t-s)\sigma(s)dW_{s}}\Big{\\}}\\\
&=\lim\limits_{\tilde{n}\to{\infty}\atop\tilde{n}\geq
n}\inf\Big{\\{}\Phi(t+{\tilde{m}}T)x+\displaystyle\int^{t}_{nT-{\tilde{m}T}}{\Phi(t-s)\inf\\{h(s,\varphi(s,-mT,\omega)x):m\geq
n,m\in\mathbb{N}_{+}\\}ds}\\\ &\ \ \ \
+\displaystyle\int^{t}_{-{\tilde{m}T}}{\Phi(t-s)\sigma(s)dW_{s}}:{\tilde{m}}\geq\tilde{n},{\tilde{m}}\in\mathbb{N}_{+}\Big{\\}}\\\
&\leq\lim\limits_{\tilde{n}\to{\infty}\atop\tilde{n}\geq
n}\inf\Big{\\{}\Phi(t+{\tilde{m}}T)x+\displaystyle\int^{t}_{nT-{\tilde{m}T}}{\Phi(t-s)h(s,\varphi(s,-{\tilde{m}}T,\omega)x)ds}\\\
&\ \ \ \
+\displaystyle\int^{t}_{-{\tilde{m}T}}{\Phi(t-s)\sigma(s)dW_{s}}:{\tilde{m}}\geq\tilde{n},{\tilde{m}}\in\mathbb{N}_{+}\Big{\\}}\\\
&\leq\lim\limits_{\tilde{n}\to{\infty}}\inf\Big{\\{}\Phi(t+{\tilde{m}}T)x+\displaystyle\int^{t}_{-{\tilde{m}T}}{\Phi(t-s)h(s,\varphi(s,-{\tilde{m}}T,\omega)x)ds}\\\
&\ \ \ \
+\displaystyle\int^{t}_{-{\tilde{m}T}}{\Phi(t-s)\sigma(s)dW_{s}}:{\tilde{m}}\geq\tilde{n},{\tilde{m}}\in\mathbb{N}_{+}\Big{\\}}\\\
&=[\underline{\lim}\varphi](t,\omega),\end{split}$

where the third equality has used Lemma A.2 in [13], while the second-to-last
inequality has applied the positivity of $\Phi(t)$ and $h$.

###### Lemma 3.3

Assume that conditions $\boldsymbol{(A)}$, $\boldsymbol{(H_{1})}$ and
$\boldsymbol{(\Sigma)}$ hold. Let $\varphi(t,s,\omega)x$ be a solution of
stochastic system (1.5) with initial value $X(s)=x\in\mathbb{R}^{d}$. Then we
have the following:
$(i)$ if $h$ is order-preserving in $\mathbb{R}^{d}$, then

$h(\cdot,\underline{\lim}\varphi)\leq\underline{\lim}h(\cdot,\varphi)\leq\overline{\lim}h(\cdot,\varphi)\leq
h(\cdot,\overline{\lim}\varphi),\ \ \mathbb{P}-a.s.,$ (3.2)

$(ii)$ if $h$ is anti-order-preserving in $\mathbb{R}^{d}$, then

$h(\cdot,\overline{\lim}\varphi)\leq\underline{\lim}h(\cdot,\varphi)\leq\overline{\lim}h(\cdot,\varphi)\leq
h(\cdot,\underline{\lim}\varphi),\ \ \mathbb{P}-a.s.,$ (3.3)

Proof. Indeed, the proof of the first inequality in (3.2) is adequate and the
rest of the results of this lemma can be obtained analogously. Observe that
$h$ is order-preserving in $\mathbb{R}^{d}$; then for fixed
$n\in\mathbb{N}_{+},m\geq n,m\in\mathbb{N}_{+}$, we have

$h(t,\inf\\{\varphi(t,-kT,\omega)x:k\geq n,k\in\mathbb{N}_{+}\\})\leq
h(t,\varphi(t,-mT,\omega)x)$

and

$h(t,\inf\\{\varphi(t,-mT,\omega)x:m\geq
n,m\in\mathbb{N}_{+}\\})\leq\inf\\{h(t,\varphi(t,-mT,\omega)x):m\geq
n,m\in\mathbb{N}_{+}\\}.$ (3.4)

Let $n\rightarrow\infty$ in (3.4). By the continuity of $h$, we have

$\displaystyle\begin{split}[h(\cdot,\underline{\lim}\varphi)](t,\omega)&=h(t,\lim\limits_{{n}\to{\infty}}\inf\\{\varphi(t,-mT,\omega)x:m\geq
n,m\in\mathbb{N}_{+}\\})\\\
&=\lim\limits_{{n}\to{\infty}}h(t,\inf\\{\varphi(t,-mT,\omega)x:m\geq
n,m\in\mathbb{N}_{+}\\})\\\
&\leq\lim\limits_{{n}\to{\infty}}\inf\\{h(t,\varphi(t,-mT,\omega)x):m\geq
n,m\in\mathbb{N}_{+}\\}\\\
&=[\underline{\lim}h(\cdot,\varphi)](t,\omega).\end{split}$

###### Lemma 3.4

Assume that conditions $\boldsymbol{(A)}$, $\boldsymbol{(H_{1})}$ and
$\boldsymbol{(\Sigma)}$ hold. Let $\varphi(t,s,\omega)x$ be a solution of
stochastic system (1.5) with initial value $X(s)=x\in\mathbb{R}^{d}$. Then we
have

${\mathcal{K}}(a_{n}^{h})\leq\underline{\lim}\varphi\leq\overline{\lim}\varphi\leq{\mathcal{K}}(b_{n}^{h})\
\ \mathbb{P}-a.s.,n\in\mathbb{N}_{+},$ (3.5)

where $a_{n}^{h}$ and $b_{n}^{h}$ are as defined in Lemma 3.1. Furthermore,
define the gain operator

${\mathcal{K}}^{h}(u)(t,\omega)=h(t,[{\mathcal{K}}(u)](t,\omega)).$

Then we have the following:
$(i)$ if $h$ is order-preserving in $\mathbb{R}^{d}$, then for fixed
$n\in\mathbb{N}_{+}$,

$({\mathcal{K}}^{h})^{k}(a_{n}^{h})\leq\underline{\lim}h(\cdot,\varphi)\leq\overline{\lim}h(\cdot,\varphi)\leq({\mathcal{K}}^{h})^{k}(b_{n}^{h}),\
\ \mathbb{P}-a.s.,\ \ k\in\mathbb{N}_{+}.$ (3.6)

$(ii)$ if $h$ is anti-order-preserving in $\mathbb{R}^{d}$, then for fixed
$n\in\mathbb{N}_{+}$,

$({\mathcal{K}}^{h})^{2k}(a_{n}^{h})\leq\underline{\lim}h(\cdot,\varphi)\leq\overline{\lim}h(\cdot,\varphi)\leq({\mathcal{K}}^{h})^{2k}(b_{n}^{h}),\
\ \mathbb{P}-a.s.,\ \ k\in\mathbb{N}_{+}.$ (3.7)

Proof. By the definition of $a_{n}^{h}$ and $b_{n}^{h}$, it is evident that

$a_{n}^{h}\leq\underline{\lim}h(\cdot,\varphi)\leq\overline{\lim}h(\cdot,\varphi)\leq
b_{n}^{h},\ \ n\in\mathbb{N}_{+}.$

By the positivity of $\Phi$, ${\mathcal{K}}(u)$ is monotone with respect to
$u$, and consequently

${\mathcal{K}}(a_{n}^{h})\leq{\mathcal{K}}(\underline{\lim}h(\cdot,\varphi))\leq{\mathcal{K}}(\overline{\lim}h(\cdot,\varphi))\leq{\mathcal{K}}(b_{n}^{h}),\
\ \mathbb{P}-a.s.,\ \ n\in\mathbb{N}_{+}.$

by (3.1), we have

${\mathcal{K}}(a_{n}^{h})\leq\underline{\lim}\varphi\leq\overline{\lim}\varphi\leq{\mathcal{K}}(b_{n}^{h}),\
\ \mathbb{P}-a.s.,\ \ n\in\mathbb{N}_{+}.$

which implies that (3.5) holds.

In what follows, we claim that (3.6) and (3.7) hold.

If $h$ is order-preserving in $\mathbb{R}^{d}$, then it deduces that $h$
preserves the inequalities in (3.5):

${\mathcal{K}}^{h}(a_{n}^{h})\leq h(\cdot,\underline{\lim}\varphi)\leq
h(\cdot,\overline{\lim}\varphi)\leq{\mathcal{K}}^{h}(b_{n}^{h}),\ \
\mathbb{P}-a.s.,\ \ n\in\mathbb{N}_{+}.$

which together with (3.2) implies

${\mathcal{K}}^{h}(a_{n}^{h})\leq\underline{\lim}h(\cdot,\varphi)\leq\overline{\lim}h(\cdot,\varphi)\leq{\mathcal{K}}^{h}(b_{n}^{h}),\
\ \mathbb{P}-a.s.,\ \ n\in\mathbb{N}_{+}.$

This proves that (3.6) is true for $k=1$.

Next we assume that, for some $k\in\mathbb{N}$, we have obtained

$({\mathcal{K}}^{h})^{k}(a_{n}^{h})\leq\underline{\lim}h(\cdot,\varphi)\leq\overline{\lim}h(\cdot,\varphi)\leq({\mathcal{K}}^{h})^{k}(b_{n}^{h}),\
\ \mathbb{P}-a.s.,\ \ n\in\mathbb{N}_{+}.$

From the monotonicity of ${\mathcal{K}}$ and (3.1), we have:

$\displaystyle\begin{split}{\mathcal{K}}[({\mathcal{K}}^{h})^{k}(a_{n}^{h})]&\leq{\mathcal{K}}(\underline{\lim}h(\cdot,\varphi))\leq\underline{\lim}\varphi\\\
&\leq\overline{\lim}\varphi\leq{\mathcal{K}}(\overline{\lim}h(\cdot,\varphi))\leq{\mathcal{K}}[({\mathcal{K}}^{h})^{k}(b_{n}^{h})].\end{split}$

By the monotonicity of $h$ in $\mathbb{R}^{d}$ and (3.2), we get that

$({\mathcal{K}}^{h})^{k+1}(a_{n}^{h})\leq\underline{\lim}h(\cdot,\varphi)\leq\overline{\lim}h(\cdot,\varphi)\leq({\mathcal{K}}^{h})^{k+1}(b_{n}^{h}),\
\ \mathbb{P}-a.s.,\ \ n\in\mathbb{N}_{+}.$

Therefore, we conclude that (3.6) holds by mathematical induction.

If $h$ is anti-order-preserving in $\mathbb{R}^{d}$, similar to $h$ is order-
preserving in $\mathbb{R}^{d}$; we deduce that

${\mathcal{K}}^{h}(b_{n}^{h})\leq h(\cdot,\overline{\lim}\varphi)\leq
h(\cdot,\underline{\lim}\varphi)\leq{\mathcal{K}}^{h}(a_{n}^{h}),\ \
\mathbb{P}-a.s.,\ \ n\in\mathbb{N}_{+}.$

by (3.3), we have

${\mathcal{K}}^{h}(b_{n}^{h})\leq\underline{\lim}h(\cdot,\varphi)\leq\overline{\lim}h(\cdot,\varphi)\leq{\mathcal{K}}^{h}(a_{n}^{h}),\
\ \mathbb{P}-a.s.,\ \ n\in\mathbb{N}_{+}.$

Combining the monotonicity of ${\mathcal{K}}$ and (3.1), it shows that

${\mathcal{K}}[{\mathcal{K}}^{h}(b_{n}^{h})]\leq\underline{\lim}\varphi\leq\overline{\lim}\varphi\leq{\mathcal{K}}[{\mathcal{K}}^{h}(a_{n}^{h})],\
\ \mathbb{P}-a.s.,\ \ n\in\mathbb{N}_{+},$

which together with the anti-monotonicity of $h$ in $\mathbb{R}^{d}$ and (3.3)
implies

$({\mathcal{K}}^{h})^{2}(a_{n}^{h})\leq\underline{\lim}h(\cdot,\varphi)\leq\overline{\lim}h(\cdot,\varphi)\leq({\mathcal{K}}^{h})^{2}(b_{n}^{h}),\
\ \mathbb{P}-a.s.,\ \ n\in\mathbb{N}_{+}.$

The rest of the proof of (3.7) can be obtained analogously to $h$ is order-
preserving in $\mathbb{R}^{d}$ by the mathematical induction.

##  4 Main results

In this section, we state our main result on the existence of random periodic
solution of nonlinear stochastic system (1.5) and present its proof. We begin
with a lemma.

Let
${\mathcal{M}}_{{\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}}^{b,T}(\mathbb{R}\times\Omega;[0,N])$
be the space of ${\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}$-measurable
functions $f:\mathbb{R}\times\Omega\rightarrow[0,N]$ with $f(t+T,\omega)\leq
f(t,\theta_{T}\omega),t\in\mathbb{R},\omega\in\Omega$ or $f(t+T,\omega)\geq
f(t,\theta_{T}\omega),t\in\mathbb{R},\omega\in\Omega$, where
$N=(N_{1},...,N_{d}),N_{i}=\sup_{t\in\mathbb{R},x\in\mathbb{R}^{d}}|h_{i}(t,x)|,i=1,...,d$.
We introduce a metric on
${\mathcal{M}}_{{\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}}^{b,T}(\mathbb{R}\times\Omega;[0,N])$
as follows:

$\varrho(f_{1},f_{2}):=|f_{1}-f_{2}|_{\infty}=\mathop{\sup}\limits_{t\in\mathbb{R},\omega\in\Omega}{|f_{1}(t,\omega)-f_{2}(t,\omega)|},\
\ \text{for}\ \text{all}\
f_{1},f_{2}\in{\mathcal{M}}_{{\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}}^{b,T}.$

###### Lemma 4.1

Assume that conditions $\boldsymbol{(A)}$, $\boldsymbol{(H_{1})}$,
$\boldsymbol{(H_{2})}$ and $\boldsymbol{(\Sigma)}$ hold. Then
$({\mathcal{M}}_{{\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}}^{b,T},\varrho)$
is a complete metric space and the gain operator
${\mathcal{K}}^{h}:{\mathcal{M}}_{{\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}}^{b,T}\rightarrow{\mathcal{M}}_{{\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}}^{b,T}$,
$u\mapsto{\mathcal{K}}^{h}(u)$ is a contractive mapping, where
${\mathcal{K}}^{h}(u)(t,\omega)=h(t,[{\mathcal{K}}(u)](t,\omega))$ and the
definition of the input-to-state characteristic operator ${\mathcal{K}}$ can
be chosen as an $\mathbb{R}^{d}$-value version for all
$t\in\mathbb{R},\omega\in\Omega$.

Proof. It is clear that
$({\mathcal{M}}_{{\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}}^{b,T},\varrho)$
is a metric space. We show that the metric space
${\mathcal{M}}_{{\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}}^{b,T}$ is
complete with respect to $\varrho$. To prove this, we choose a Cauchy sequence
$\\{f_{n},n\in\mathbb{N}\\}$ in
$({\mathcal{M}}_{{\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}}^{b,T},\varrho)$;
we denote a function $f$ as follows:

$f(t,\omega):=\lim\limits_{{n}\to{\infty}}{f_{n}(t,\omega)}\in[0,N]\ \
\text{for}\ \text{all}\ t\in\mathbb{R},\omega\in\Omega,$

which holds based on the fact that $\\{f_{n}(t,\omega),n\in\mathbb{N}\\}$ is a
Cauchy sequence in $\mathbb{R}^{d}$ for fixed
$t\in\mathbb{R},\omega\in\Omega$. It is noticed that the limit of a family of
${\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}$-measurable functions is an
${\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}$-measurable function. Without
loss of generality, we assume that there is a subsequence $\\{f_{n_{k}}\\}$ of
$\\{f_{n}\\}$ such that

$f_{n_{k}}(t+T,\omega)\leq f_{n_{k}}(t,\theta_{T}\omega),\ \ t\in\mathbb{R},\
\omega\in\Omega.$

Let $n_{k}\rightarrow\infty$ in the above inequality, then $f(t+T,\omega)\leq
f(t,\theta_{T}\omega),t\in\mathbb{R},\omega\in\Omega$. In what follows, we
will prove that $|f-f_{n}|_{\infty}\rightarrow 0$ as $n\rightarrow\infty$. It
is noticed that $\\{f_{n}(t,\omega),n\in N\\}$ is a Cauchy sequence, we know
that for any $\varepsilon>0$, there exists an
$N_{0}=N_{0}(\varepsilon)\in\mathbb{N}$ such that for $n,m\geq N_{0}$,

$\mathop{\sup}\limits_{t\in\mathbb{R},\omega\in\Omega}{|f_{m}(t,\omega)-f_{n}(t,\omega)|}<\varepsilon.$

Let $m\rightarrow\infty$; then

$\mathop{\sup}\limits_{t\in\mathbb{R},\omega\in\Omega}{|f(t,\omega)-f_{n}(t,\omega)|}\leq\varepsilon\
\ \text{for}\ \text{all}\ n\geq N_{0},$

which implies that $|f-f_{n}|_{\infty}\rightarrow 0$ as $n\rightarrow\infty$.
Thus
$({\mathcal{M}}_{{\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}}^{b,T},\varrho)$
is a complete metric space.

Next we claim that
${\mathcal{K}}^{h}:{\mathcal{M}}_{{\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}}^{b,T}\rightarrow{\mathcal{M}}_{{\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}}^{b,T}$
is a contractive mapping. First, we should show that
${\mathcal{K}}^{h}:{\mathcal{M}}_{{\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}}^{b,T}\rightarrow{\mathcal{M}}_{{\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}}^{b,T}$
is well defined. From $\boldsymbol{(H_{1})}$, it follows that
$h:\mathbb{R}\times\mathbb{R}^{d}\rightarrow[0,N]$ and
$h(t+T,\omega)=h(t,\omega),t\in\mathbb{R},\omega\in\Omega$. For any
$f\in({\mathcal{M}}_{{\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}}^{b,T},\varrho)$,
by the definition of ${\mathcal{K}}$, the measurability of $\theta$, and the
Fubini theorem, it is evident that ${\mathcal{K}}(f)$ is a
${\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}$-measurable function. Without
loss of generality, we assume that $f(t+T,\omega)\leq
f(t,\theta_{T}\omega),t\in\mathbb{R},\omega\in\Omega$, then

$\displaystyle\begin{split}[{\mathcal{K}}(f)](s,\theta_{T}\omega)&=\displaystyle\int^{s}_{-\infty}{\Phi(s-r)f(r,\theta_{T}\omega))dr}+\displaystyle\int^{s}_{-\infty}{\Phi(s-r)\sigma(r)dW_{r}({\theta_{T}}\omega)}\\\
&=\displaystyle\int^{s+T}_{-\infty}{\Phi(s+T-r)f(r-T,\theta_{T}\omega)dr}+\displaystyle\int^{s+T}_{-\infty}{\Phi(s+T-r)\sigma(r-T)dW_{r}}\\\
&\geq\displaystyle\int^{s+T}_{-\infty}{\Phi(s+T-r)f(r,\omega)dr}+\displaystyle\int^{s+T}_{-\infty}{\Phi(s+T-r)\sigma(r)dW_{r}}\\\
&=[{\mathcal{K}}(f)](s+T,\omega)\ \ \ \mathbb{P}-a.s.\end{split}$

If $h$ is order-preserving in $\mathbb{R}^{d}$, we have

$\displaystyle\begin{split}{\mathcal{K}}^{h}(f)(t+T,\omega)&=h(t+T,{\mathcal{K}}(f)(t+T,\omega))\\\
&\leq h(t,{\mathcal{K}}(f)(t,\theta_{T}\omega))\\\
&={\mathcal{K}}^{h}(f)(t,\theta_{T}\omega)\end{split}$

Otherwise, if $h$ is anti-order-preserving in $\mathbb{R}^{d}$, we have
${\mathcal{K}}^{h}(f)(t+T,\omega)\geq{\mathcal{K}}^{h}(f)(t,\theta_{T}\omega)$.
Which yields
${\mathcal{K}}^{h}:{\mathcal{M}}_{{\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}}^{b,T}\rightarrow{\mathcal{M}}_{{\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}}^{b,T}$.
Finally, we prove that ${\mathcal{K}}^{h}$ is a contractive mapping. By
$\boldsymbol{(H_{1})}$ and $\boldsymbol{(H_{2})}$, we have

$\mathop{\sup}\limits_{t\in\mathbb{R},x\in\mathbb{R}^{d}}{||D_{x}(h(t,x))||}=\mathop{\sup}\limits_{t\in\mathbb{R},x\in\mathbb{R}^{d}}{\Big{|}\Big{|}\Big{(}\frac{\partial
h_{i}(t,x)}{\partial x_{j}}\Big{)}_{d\times d}\Big{|}\Big{|}}\leq L,$

Let $f_{1},f_{2}$ be two elements in
$({\mathcal{M}}_{{\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}}^{b,T},\varrho)$.
By the fact that $|\Phi(x)|\leq d||\Phi||\cdot|x|$ for all
$x\in\mathbb{R}^{d}$ and $\Phi\in\mathbb{R}^{d\times d}$, we get

$\displaystyle\begin{split}|{\mathcal{K}}^{h}(f_{1})-{\mathcal{K}}^{h}(f_{2})|_{\infty}&=|h(\cdot,{\mathcal{K}}(f_{1}))-h(\cdot,{\mathcal{K}}(f_{2}))|_{\infty}\\\
&=|D_{x}h(\cdot,[{\mathcal{K}}(f_{2})+\mu({\mathcal{K}}(f_{1})-{\mathcal{K}}(f_{2}))])\cdot[{\mathcal{K}}(f_{1})-{\mathcal{K}}(f_{2})]|_{\infty}\\\
&\leq
d\mathop{\sup}\limits_{t\in\mathbb{R},x\in\mathbb{R}^{d}}{||D_{x}(h(t,x))||}\cdot|{\mathcal{K}}(f_{1})-{\mathcal{K}}(f_{2})|_{\infty}\\\
&\leq
Ld\Big{|}\displaystyle\int^{t}_{-\infty}{\Phi(t-s)f_{1}(s,\omega)}ds-\displaystyle\int^{t}_{-\infty}{\Phi(t-s)f_{2}(s,\omega))}ds\Big{|}_{\infty}\\\
&\leq
Ld^{2}\displaystyle\int^{t}_{-\infty}{||\Phi(t-s)||\cdot|f_{1}-f_{2}|_{\infty}}ds\\\
&\leq
Ld^{2}|f_{1}-f_{2}|_{\infty}\displaystyle\int^{t}_{-\infty}{e^{\lambda(t-s)}}ds\\\
&=-\frac{Ld^{2}}{\lambda}|f_{1}-f_{2}|_{\infty}\end{split}$

where $\mu\in(0,1)$,$-\frac{Ld^{2}}{\lambda}<1$.

###### Theorem 4.2

Assume that conditions $\boldsymbol{(A)}$, $\boldsymbol{(H_{1})}$,
$\boldsymbol{(H_{2})}$ and $\boldsymbol{(\Sigma)}$ hold. Then the gain
operator

${\mathcal{K}}^{h}:{\mathcal{M}}_{{\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}}^{b,T}\rightarrow{\mathcal{M}}_{{\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}}^{b,T},\
\ u\mapsto{\mathcal{K}}^{h}(u)$

possesses a unique nonnegative fixed point
$u\in{\mathcal{M}}_{{\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}}^{b,T}(\mathbb{R}\times\Omega;[0,N])$
such that for all $t\in\mathbb{R},x\in\mathbb{R}^{d}$,

$\lim\limits_{{n}\to{\infty}}{\varphi(t,-nT,\omega)x}=[{\mathcal{K}}(u)](t,\omega),\
\ \mathbb{P}-a.s.$ (4.1)

Moreover,
$\varphi(t,s,\omega)[{\mathcal{K}}(u)](s,\omega)=[{\mathcal{K}}(u)](t,\omega)$,
$[{\mathcal{K}}(u)](s+T,\omega)=[{\mathcal{K}}(u)](s,\theta_{T}\omega)$,
$\mathbb{P}-a.s.$, $(t,s)\in\triangle$; i.e., the image
$[{\mathcal{K}}(u)](\cdot,\cdot)$ at the fixed point $u$ for the input-to-
state characteristic operator ${\mathcal{K}}$ is a random periodic solution.

Proof. In view of Lemma 3.4, regardless of the monotonicity or anti-
monotonicity for h, for fixed $n\in\mathbb{N}_{+}$, we have

$({\mathcal{K}}^{h})^{2k}(a_{n}^{h})\leq\underline{\lim}h(\cdot,\varphi)\leq\overline{\lim}h(\cdot,\varphi)\leq({\mathcal{K}}^{h})^{2k}(b_{n}^{h})\
\ \mathbb{P}-a.s.,k\in\mathbb{N}$ (4.2)

where $a_{n}^{h}$ and $b_{n}^{h}$ are as defined in Lemma 3.1. By Lemma 3.1,
$a_{n}^{h}$ and $b_{n}^{h}$ are bounded
${\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}$-measurable functions. By (1.3)
and the definition of $a_{n}^{h}$ in Lemma 3.1, we have

$\displaystyle\begin{split}a_{n}^{h}(t+T,\omega)&=\inf{\\{h(t+T,\varphi(t+T,-mT,\omega)x):m\geq
n,m\in\mathbb{N}_{+}\\}}\\\
&=\inf{\\{h(t,\varphi(t,-(m+1)T,\theta_{T}\omega)x):m\geq
n,m\in\mathbb{N}_{+}\\}}\\\
&\geq\inf{\\{h(t,\varphi(t,-mT,\theta_{T}\omega)x):m\geq
n,m\in\mathbb{N}_{+}\\}}\\\ &=a_{n}^{h}(t,\theta_{T}\omega).\end{split}$

Similarly, we have $b_{n}^{h}(t+T,\omega)\leq b_{n}^{h}(t,\theta_{T}\omega)$.
So $a_{n}^{h}$ and $b_{n}^{h}$ are both in
$({\mathcal{M}}_{{\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}}^{b,T},\varrho)$.
Since ${\mathcal{K}}^{h}$ is a contractive mapping on the complete metric
space
$({\mathcal{M}}_{{\mathcal{B}}(\mathbb{R})\otimes{\mathcal{F}}}^{b,T},\rho)$,
by the Banach fixed point theorem ([18]), there exists a unique nonnegative
stochastic process $u:\mathbb{R}\times\Omega\rightarrow[0,N]$ for
${\mathcal{K}}^{h}$ such that

$[{\mathcal{K}}^{h}(u)](t,\omega)=u(t,\omega)\ \ \ \text{for}\ \text{all}\
t\in\mathbb{R},\omega\in\Omega.$

then

$\lim\limits_{{k}\to{\infty}}{[({\mathcal{K}}^{h})^{2k}(a_{n}^{h})](t,\omega)}=u(t,\omega)=\lim\limits_{{k}\to{\infty}}{[({\mathcal{K}}^{h})^{2k}(b_{n}^{h})](t,\omega)}\
\ \ \rm{for}\ \rm{all}\ t\in\mathbb{R},\omega\in\Omega.$ (4.3)

It is noticed that $b_{n}^{h}(t+T,\omega)\leq b_{n}^{h}(t,\theta_{T}\omega)$,
$h(t+T,x)=h(t,x)$ and $({\mathcal{K}}^{h})^{2k}$ are monotone, then

$[({\mathcal{K}}^{h})^{2k}(b_{n}^{h})](t+T,\omega)\leq[({\mathcal{K}}^{h})^{2k}(b_{n}^{h})](t,\theta_{T}\omega)$

Let $k\rightarrow\infty$ in the above inequality, then $u(t+T,\omega)\leq
u(t,\theta_{T}\omega)$. Similarly, for $a_{n}^{h}(t+T,\omega)\geq
a_{n}^{h}(t,\theta_{T}\omega)$ we have $u(t+T,\omega)\geq
u(t,\theta_{T}\omega)$. So for all $t\in\mathbb{R},\omega\in\Omega$

$u(t+T,\omega)=u(t,\theta_{T}\omega)$ (4.4)

Combining (4.2) and (4.3), we have

$[\underline{\lim}h(\cdot,\varphi)](t,\omega)=[\overline{\lim}h(\cdot,\varphi)](t,\omega)=u(t,\omega)\
\ \mathbb{P}-a.s.$

which together with (3.1) implies that

$[\underline{\lim}\varphi](t,\omega)=[\overline{\lim}\varphi](t,\omega)=[{\mathcal{K}}(u)](t,\omega)\
\ \mathbb{P}-a.s.$

In order to prove (4.1), it remains to show that

$[\underline{\lim}\varphi](t,\omega)=[\overline{\lim}\varphi](t,\omega)=\lim\limits_{{n}\to{\infty}}\varphi(t,-nT,\omega)x\
\ \mathbb{P}-a.s.,x\in\mathbb{R}^{d}.$ (4.5)

By the definition of infimum and supremum, it is clear that

$\displaystyle\begin{split}\inf\\{\varphi(t,-mT,\omega)x:m\geq
n\\}&\leq{\varphi(t,-nT,\omega)x}\\\ &\leq\
\sup\\{\varphi(t,-mT,\omega)x:m\geq n\\}\ \
\mathbb{P}-a.s.,x\in\mathbb{R}^{d}.\end{split}$

Let $n\rightarrow\infty$ in the above inequality; then (4.5) holds, and so
(4.1) holds.

By (4.1) and the continuity of $\varphi$ in $\mathbb{R}^{d}$, we can show that
for fixed $(t,s)\in\triangle$ and $x\in\mathbb{R}^{d}$,

$\displaystyle\begin{split}\varphi(t,s,\omega)[{\mathcal{K}}(u)](s,\omega)&=\varphi(t,s,\omega)\lim\limits_{{n}\to{\infty}}{\varphi(s,-nT,\omega)x}\\\
&=\lim\limits_{{n}\to{\infty}}\varphi(t,s,\omega){\varphi(s,-nT,\omega)x}\\\
&=\lim\limits_{{n}\to{\infty}}{\varphi(t,-nT,\omega)x}\\\
&=[{\mathcal{K}}(u)](t,\omega)\ \ \ \mathbb{P}-a.s.\end{split}$

Furthermore, by (2.6) and (4.4) we have

$\displaystyle\begin{split}[{\mathcal{K}}(u)](s,\theta_{T}\omega)&=\displaystyle\int^{s}_{-\infty}{\Phi(s-r)u(r,\theta_{T}\omega)dr}+\displaystyle\int^{s}_{-\infty}{\Phi(s-r)\sigma(r)dW_{r}({\theta_{T}}\omega)}\\\
&=\displaystyle\int^{s+T}_{-\infty}{\Phi(s+T-r)u(r-T,\theta_{T}\omega)dr}+\displaystyle\int^{s+T}_{-\infty}{\Phi(s+T-r)\sigma(r-T)dW_{r}}\\\
&=\displaystyle\int^{s+T}_{-\infty}{\Phi(s+T-r)u(r,\omega)dr}+\displaystyle\int^{s+T}_{-\infty}{\Phi(s+T-r)\sigma(r)dW_{r}}\\\
&=[{\mathcal{K}}(u)](s+T,\omega)\ \ \ \mathbb{P}-a.s.\end{split}$

###### Remark 4.3

Now we consider the nonlinear stochastic differential equation with the
additive white noise of the form

$dX_{t}=[AX_{t}+h(X_{t})]dt+\sigma dW_{t},$ (4.6)

Here $h$ and $\sigma$ do not depend on time $t$, that is to say, the periodic
$T$ in the Theorem 4.2 can be chosen as an arbitrary real number. The
difference is that for this equation, we have a random dynamical system
$\varphi:\mathbb{R}\times\Omega\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$,
which satisfies the cocycle property
$\varphi(t,\theta_{s}\omega)\varphi(s,\omega)=\varphi(t+s,\omega)$. Equation
(4.6) has a stationary solution. This result is given by J.F. Jiang and X. Lv
([9]).

##  5 Examples

In this section, we present several examples to illustrate the use of Theorem
4.2. For simplicity, we only consider three-dimensional systems in the
following. First, we give an example that $h$ is order-preserving in
$\mathbb{R}^{3}$.

###### Example 5.1

Consider stochastic differential equation

$dx_{i}=[(Ax)_{i}+h_{i}(t,x_{i})]dt+\sigma_{i}(t)dW_{t}^{i},\ \ \ \ i=1,2,3,$
(5.1)

where

$A=\begin{bmatrix}-1&1&0\\\ 1&-2&0\\\ 0&1&-1\end{bmatrix}$

with three eigenvalues $\lambda_{1}=-1$, $\lambda_{2}=\frac{-3+\sqrt{5}}{2}$,
$\lambda_{3}=\frac{-3-\sqrt{5}}{2}$,

$\displaystyle\begin{split}h_{i}(t,x_{i})&=\frac{1}{10}sint+\frac{1}{6+g_{i}(x_{i})},\
\ i=1,2,3\end{split}$

where $g_{i}(x_{i})=\frac{\pi}{2}-\arctan x_{i}$ is decreasing with respect to
$x_{i},i=1,2,3$ and $\sigma_{i}(t)=\cos t$. By direct calculation, we obtain

$\Phi(t)=\begin{bmatrix}\frac{5+\sqrt{5}}{10}e^{\frac{-3+\sqrt{5}}{2}t}+\frac{5-\sqrt{5}}{10}e^{\frac{-3-\sqrt{5}}{2}t}&\frac{\sqrt{5}}{5}e^{\frac{-3+\sqrt{5}}{2}t}-\frac{\sqrt{5}}{10}e^{\frac{-3-\sqrt{5}}{2}t}&0\\\
\frac{\sqrt{5}}{5}e^{\frac{-3+\sqrt{5}}{2}t}-\frac{\sqrt{5}}{5}e^{\frac{-3-\sqrt{5}}{2}t}&\frac{5-\sqrt{5}}{10}e^{\frac{-3+\sqrt{5}}{2}t}+\frac{5+\sqrt{5}}{10}e^{\frac{-3-\sqrt{5}}{2}t}&0\\\
-e^{-t}+\frac{5+\sqrt{5}}{10}e^{\frac{-3+\sqrt{5}}{2}t}+\frac{5-\sqrt{5}}{10}e^{\frac{-3-\sqrt{5}}{2}t}&\frac{\sqrt{5}}{5}e^{\frac{-3+\sqrt{5}}{2}t}-\frac{\sqrt{5}}{5}e^{\frac{-3-\sqrt{5}}{2}t}&e^{-t}\end{bmatrix}$

It is not difficult to estimate that for any $t\geq 0$,

$||\Phi(t)||:=\max\\{|\Phi_{ij}(t)|:i,j=1,2,3\\}\leq e^{\lambda_{2}t},$

which implies that (2.3) holds. Moreover, it is easy to see that $\max_{1\leq
i\leq 3}\text{Re}\lambda_{i}=\lambda_{2}\leq 0$, $L\leq\frac{1}{36}$. So

$-\frac{9L}{\lambda_{2}}\leq\frac{1}{2(3-\sqrt{5})}<1.$

By Theorem 4.2, stochastic differential equation (5.1) has a unique random
periodic solution of periodic $2\pi$.

Next, we give two examples that $h$ is anti-order-preserving in
$\mathbb{R}^{3}$.

###### Example 5.2

Consider stochastic differential equation

$dx_{i}=[a_{i}x_{i}+h_{i}(t,x_{i-1})]dt+\sigma_{i}(t)dW_{t}^{i},\ \ \ \
i=1,2,3,$ (5.2)

where $a_{1}=-1,a_{2}=-2,a_{3}=-3$,

$\displaystyle\begin{split}h_{i}(t,x_{i-1})&=\frac{1}{6+\cos t+{\rm th}\
x_{i-1}}:=\frac{1}{5+\cos t+g_{i}(x_{i-1})},\ \ i=1,2,3\end{split}$

where $g_{i}(x_{i-1})=1+{\rm th}\ x_{i-1}$ is increasing with respect to
$x_{i-1},i=1,2,3$ and $\sigma_{i}(t)=\cos t$. It is easy to see that
$\lambda=-1$ and $L\leq\frac{1}{16}$, it follows that for any $t\geq 0$,

$||\Phi(t)||:=\max\\{|\Phi_{ij}(t)|:i,j=1,2,3\\}=e^{-t}$

and

$-\frac{9L}{\lambda}\leq\frac{9}{16}<1.$

By Theorem 4.2, stochastic differential equation (5.2) has a unique random
periodic solution of periodic $2\pi$.

###### Example 5.3

Consider stochastic differential equation

$dx_{i}=[(Ax)_{i}+h_{i}(t,x_{i-1})]dt+\sigma_{i}(t)dW_{t}^{i},\ \ \ \
i=1,2,3,$ (5.3)

where $x_{0}=x_{3},x_{4}=x_{1}$ and

$A=\begin{bmatrix}-1&\sqrt[3]{2}&0\\\ 0&-2&\sqrt[3]{2}\\\
\sqrt[3]{2}&0&-4\end{bmatrix}$

with three eigenvalues $\lambda_{1}=-3$, $\lambda_{2}=-2+\sqrt{2}$,
$\lambda_{3}=-2-\sqrt{2}$,

$\displaystyle\begin{split}h_{i}(t,x_{i-1})&:=\frac{1}{8}sint+\frac{1}{4+g_{i}(x_{i-1})},\
\ i=1,2,3\end{split}$

where $g_{i}(x_{i-1})=\frac{\pi}{2}+\arctan x_{i-1}$ is increasing with
respect to $x_{i-1},i=1,2,3$ and $\sigma_{i}(t)=\sin t$. By direct
calculation, we obtain

$\begin{gathered}\begin{split}\Phi(t)=&\left[\begin{gathered}e^{-3t}+\frac{\sqrt{2}}{2}e^{(-2+\sqrt{2})t}-\frac{\sqrt{2}}{2}e^{-(2+\sqrt{2})t}\\\
-\sqrt[3]{4}e^{-3t}+\frac{\sqrt[3]{4}-\sqrt[6]{2}}{2}e^{(-2+\sqrt{2})t}+\frac{\sqrt[3]{4}+\sqrt[6]{2}}{2}e^{-(2+\sqrt{2})t}\\\
\sqrt[3]{2}e^{-3t}+\frac{\sqrt[6]{2^{5}}-\sqrt[3]{2}}{2}e^{(-2+\sqrt{2})t}-\frac{\sqrt[6]{2^{5}}+\sqrt[3]{2}}{2}e^{-(2+\sqrt{2})t}\end{gathered}\right.&\\\
&\begin{gathered}-\sqrt[3]{2}e^{-3t}+\frac{\sqrt[3]{2}}{2}e^{(-2+\sqrt{2})t}+\frac{\sqrt[3]{2}}{2}e^{-(2+\sqrt{2})t}\\\
2e^{-3t}+\frac{\sqrt{2}-1}{2}e^{(-2+\sqrt{2})t}-\frac{\sqrt{2}+1}{2}e^{-(2+\sqrt{2})t}\\\
-\sqrt[3]{4}e^{-3t}+\frac{\sqrt[3]{4}-\sqrt[6]{2}}{2}e^{(-2+\sqrt{2})t}+\frac{\sqrt[3]{4}+\sqrt[6]{2}}{2}e^{-(2+\sqrt{2})t}\end{gathered}&\\\
&\left.\begin{gathered}-\sqrt[3]{4}e^{-3t}+\frac{\sqrt[3]{4}-\sqrt[6]{2}}{2}e^{(-2+\sqrt{2})t}+\frac{\sqrt[3]{4}+\sqrt[6]{2}}{2}e^{-(2+\sqrt{2})t}\\\
2\sqrt[3]{2}e^{-3t}+(\frac{3\sqrt[6]{2^{5}}}{4}-\sqrt[3]{2})e^{(-2+\sqrt{2})t}-(\frac{3\sqrt[6]{2^{5}}}{4}+\sqrt[3]{2})e^{-(2+\sqrt{2})t}\\\
-2e^{-3t}+(\frac{3}{2}-\sqrt{2})e^{(-2+\sqrt{2})t}+(\frac{3}{2}+\sqrt{2})e^{-(2+\sqrt{2})t}\end{gathered}\right]&,\end{split}\end{gathered}$

It is not difficult to prove that for any $t\geq 0$,

$||\Phi(t)||:=\max\\{|\Phi_{ij}(t)|:i,j=1,2,3\\}\leq e^{\lambda_{2}t},$

which can be found in ([9], Example 5.3) for a detailed proof. Moreover, it is
easy to see that $\max_{1\leq i\leq 3}\text{Re}\lambda_{i}=\lambda_{2}\leq 0$,
$L\leq\frac{1}{16}$, and so

$-\frac{9L}{\lambda_{2}}\leq\frac{9}{16(2-\sqrt{2})}<1.$

By Theorem 4.2, stochastic differential equation (5.3) has a unique random
periodic solution of periodic $2\pi$.

Acknowledgements The authors would like to express his deep gratitude to
Professor Jifa Jiang, who gave numerous suggestions which significantly
improve the readability of the paper. This work was supported in part by
National Natural Science Foundation of China (Nos. 11431014, 11671382,
11931004, 12031020, 12090014), CAS Key Project of Frontier Sciences (No.
QYZDJ-SSW-JSC003), the Key Laboratory of Random Complex Structures and Data
Sciences, CAS(No. 2008DP173182) and National Center for Mathematics and
Interdisciplinary Sciences CAS.

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