# Data Release of the AST3-2 Automatic Survey from Dome A, Antarctica

Xu Yang1,2, Yi Hu1,Zhaohui Shang1,Bin Ma3, Michael C.B. Ashley4, Xiangqun
Cui5,6, Fujia Du5,6, Jianning Fu7, Xuefei Gong5,6, Bozhong Gu5,6, Peng
Jiang6,8, Xiaoyan Li5,6, Zhengyang Li5,6, Charling Tao9,10, Lifan Wang6,11,12,
Lingzhe Xu5, Shi-hai Yang5,6, Ce Yu13, Xiangyan Yuan5,6, Ji-lin Zhou14, Zhenxi
Zhu6,12

1National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100101, China
2University of Chinese Academy of Sciences, Beijing 100049, China
3School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai 519082, China
4School of Physics, University of New South Wales, Sydney NSW 2052, Australia
5Nanjing Institute of Astronomical Optics and Technology, Nanjing 210042,
China
6Chinese Center for Antarctic Astronomy, Nanjing 210008, China
7Department of Astronomy, Beijing Normal University, Beijing, 100875, China
8Key Laboratory for Polar Science, MNR, Polar Research Institute of China,
Shanghai, 200136, China
9Aix Marseille Univ., CNRS/IN2P3, CPPM, Marseille, France
10Physics Department and Tsinghua Center for Astrophysics (THCA), Tsinghua
University, Beijing, 100084, China
11Department of Physics and Astronomy, Texas A&M University, College Station,
TX 77843, USA
12Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008,
China
13College of Intelligence and Computing, Tianjin University, Tianjin 300350,
China
14School of Astronomy and Space Science, Nanjing University, Nanjing 210023,
China Contact e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

AST3-2 is the second of the three Antarctic Survey Telescopes, aimed at wide-
field time-domain optical astronomy. It is located at Dome A, Antarctica,
which is by many measures the best optical astronomy site on the Earth’s
surface. Here we present the data from the AST3-2 automatic survey in 2016 and
the photometry results. The median 5$\sigma$ limiting magnitude in $i$-band is
17.8 mag and the light curve precision is 4 mmag for bright stars. The data
release includes photometry for over 7 million stars, from which over 3,500
variable stars were detected, with 70 of them newly discovered. We classify
these new variables into different types by combining their light curve
features with stellar properties from surveys such as StarHorse.

###### keywords:

surveys – catalogues – stars:variables:general

††pubyear: 2023††pagerange: Data Release of the AST3-2 Automatic Survey from
Dome A, Antarctica–A

##  1 Introduction

Time-domain astronomy has led to many astronomical discoveries through
exploring the variability of astronomical objects over time. Transient targets
such as supernovae (SNe), gamma-ray bursts, and tidal disruption events (TDEs)
give valuable insights in astronomy and fundamental physics. Many survey
projects have been undertaken to search for variable sources by repeatedly
scanning selected sky areas. Deep surveys over wide areas of sky require
specialized telescopes such as the Large Binocular Telescope (LBT; Hill &
Salinari 2000) and the Large Synoptic Survey Telescope (LSST; Ivezic et al.
2008), and results from such surveys will doubtless make revolutionary
discoveries in coming years. High cadence is also important for time-domain
surveys when searching for transients such as exoplanets, rapidly-changing
objects, and short-term events. The Wide Angle Search for Planets (WASP;
Pollacco et al. 2006) consortium has discovered numerous exoplanets with its
high cadence. The Zwicky Transient Facility (ZTF; Bellm et al. 2019) has
discovered over 3,000 supernovae from its first year of operations with a
cadence as rapid as 3 days.

The Antarctic plateau is an ideal site for ground-based time-domain astronomy
with its long clear polar nights that can provide long-term continuous
observing time as well as other excellent observing conditions (Storey 2005,
2007; Ashley 2013). The clean air can minimize the scattering of light, the
cold air is good for infrared observations due to the low thermal background,
and the stable atmosphere provides remarkably good seeing.

As the highest location on the Antarctic ice cap, Dome A was first reached by
the 21st CHInese National Antarctic Research Expedition (CHINARE) in 2005. It
is also the place where the Chinese Kunlun station was established. Many site
testing studies have been conducted here during the past decade, and the
results have confirmed that Dome A is an excellent site for astronomical
observations. A complete summary of the astronomy-related work at Dome A can
be found in Shang (2020). We present some important results briefly below.

The Chinese Small Telescope ARray (CSTAR) showed that the median $i$-band sky
background of moonless clear nights is 20.5 mag arcsec-2 (Zou et al., 2010).
The KunLun Cloud and Aurora Monitor (KLCAM) showed that the nighttime clear
sky rate is 83 per cent, which is better than most ground-based sites (Yang et
al., 2021). Moreover, the Surface layer NOn-Doppler Acoustic Radar (SNODAR;
Bonner et al. 2010) showed a very shallow atmospheric turbulent boundary layer
at Dome A, with a median thickness of only 13.9 m. The multilayer Kunlun
Automated Weather Station (KLAWS) showed that a temperature inversion often
occurs near the ground, which leads to a stable atmosphere where cooler air is
trapped under warmer air (Hu et al., 2014, 2019). The results from SNODAR and
KLAWS suggest that extremely good seeing is relatively easy to obtain at Dome
A since the telescope only has to be above the shallow turbulent boundary
layer to achieve free-atmosphere conditions. This is impractical at
traditional observatory sites where the boundary layer is typically many
hundreds of metres above the ground. In 2019, the two KunLun Differential
Image Motion Monitors (KL-DIMMs) directly confirmed these ideas by measuring
the seeing at Dome A from an 8 m tall tower. Superb night-time seeing as good
as 0.13″was recorded. The median free-atmosphere seeing was 0.31″and the KL-
DIMMs reached the free atmosphere from the 8m tower 31% of the time (Ma et
al., 2020b). In summary, the studies described above have demonstrated that by
many measures Dome A has the best optical observational conditions from the
Earth’s surface.

With such exceptional observing conditions, telescopes were planned and
constructed to operate at Dome A for time-domain astronomy. The first-
generation optical telescope, CSTAR, was installed in 2008 January (Yuan et
al., 2008; Zhou et al., 2010). It observed a 20 deg2 sky area centred at the
South Celestial Pole with four co-aligned 14.5cm telescopes. CSTAR obtained
data for three years and has contributed to many studies on stellar
variability (Wang et al., 2011; Yang et al., 2015; Zong et al., 2015; Liang et
al., 2016; Oelkers et al., 2016). The three Antarctic Survey Telescopes (AST3;
Cui et al. 2008) were later planned as the second-generation optical
telescopes at Dome A, with larger apertures and the ability to point and track
over the sky, as opposed to CSTAR’s conservative engineering approach of
having a fixed altitude.

The first AST3 telescope (AST3-1) was installed at Dome A in 2012 by the 28th
CHINARE. AST3-1 surveyed a sky area of roughly 2000 deg2 and the data have
been released (Ma et al., 2018). AST3-1 also monitored some specific sky
regions such as the Large and Small Magellanic Clouds. These data were used
for research on exoplanets and variable stars. For example, AST3-1 detected
about 500 variable stars around the Galactic disk centre, with 339 of them
being newly discovered (Wang et al., 2017a).

The AST3 telescopes were originally conceived as multi-band survey telescopes
operating together, but the goal has not been achieved due to various logistic
difficulties, such as the required amount of electrical power. The second AST3
telescope (AST3-2) was installed in 2015 by the 31st CHINARE. This work is
based on the data from AST3-2. The third AST3 (AST3-3) has been constructed
and will be equipped with a K-dark infrared camera (Burton et al., 2016; Li et
al., 2016).

Here we present the data and photometry from the AST3-2 sky survey as well as
an analysis of the light curves. We first present the basic design of AST3-2
in section 2 and go on to discuss the survey parameters and operational
strategy in section 3. In section 4 we discuss the data reduction process and
results. In section 5 we present the light curves, the result of period
searches, and the classification of objects. The overall statistics of the
catalogue and data access are discussed in section 6. Finally, we summarize
the results in section 7.

##  2 Instrument

The details of the AST3 system have been presented in previous works (Yuan et
al., 2010; Yuan & Su, 2012; Yuan et al., 2014). Here we briefly describe the
basic features of the AST3-2, the second telescope of AST3.

AST3-2 has the same modified Schmidt optical design as the AST3-1. It has a
680mm primary mirror, an entrance pupil diameter of 500mm, a 3.73 f-ratio, and
an SDSS $i$ filter. The AST3 telescopes were designed specially to work in the
harsh environment of Dome A where the ambient temperature in the observation
season ranges from $-80\degr$C to $-50\degr$C. The AST3 telescopes and the
mounting system were built with low thermal expansion materials such as Invar
to minimize the thermal effects. This design enables the AST3-2 to work in
extremely low temperatures, but we still had occasional problems with gears
being stuck or jammed by ice. To cope with optical element frosting problems
that are common in Antarctica, a defrosting system was designed with an
indium-tin-oxide (ITO) coating on the entrance aperture to the telescope and a
warm blower inside the tube. However, in the first year of operation, the
frosting problem on the first surface was not completely solved. The ITO
coating was sometimes insufficient to defrost the ice and the blow heater had
to work frequently, resulting in significant tube seeing and poor image
quality. To solve this problem, an external defrosting blower system was
installed in front of the telescope in 2016.

AST3-2 is equipped with a 10K $\times$ 10K STA1600FT CCD with a pixel size of
9µm. There are 16 read-out channels for the CCD to reduce the read-out time,
which is 2.5s in fast read-out mode and 40s in slow read-out mode. To prevent
shutter failure in cold weather, the camera works without a mechanical
shutter, instead relying on frame-transfer mode and dedicating half of the CCD
area to a buffer that is not exposed to light. The astronomically usable area
of the CCD is therefore 10K $\times$ 5K pixels, with a scale of 1″/pixel over
a FOV of 2.93°$\times$ 1.47°. Since the CCD camera is installed inside the
telescope tube, it also faced some heat dissipation problems, causing the CCD
to often operate at temperatures as warm as -50$\degr$C to -40$\degr$C,
leading to a significant dark current. Since we could not take dark frames on-
site and the previously-taken laboratory dark images have different patterns,
a new method was developed to derive a dark frame from the science images and
will be discussed in section 4.1.2. There was also a problem with the AST3 CCD
in that the photon transfer curve became non-linear at a level around 25000
ADU, leading to the brighter-fatter effect (Ma et al., 2014d). Fig. 1 shows a
raw image taken by AST3-2. Detailed laboratory tests of the CCD performance
can be found in Ma et al. (2012) and Shang et al. (2012).

Figure 1:  An example of a raw image that is taken from the survey fields by
AST3-2. There are 16 readout channels with different bias levels and overscan
regions. The lower 8 channels are read out towards the bottom of the CCD, and
the upper 8 channels are read out towards the top. Each of the readouts has an
area of 1500 pixels $\times$ 2660 pixels including overscan. The overscan
regions have 180 columns on the right of each readout and 20 horizontal rows
in the middle of the image.

The AST3-2 is powered by the PLATeau Observatory for Dome A (PLATO-A; Ashley
et al. 2010). PLATO-A is a self-contained automated platform providing an
average power of 1kW for at least 1 year. It also provides Internet access
through the Iridium satellite constellation. The hardware and software of the
control, operation, and data system (CODS) of AST3-2 were designed to be
responsible for the automated sky survey (Shang et al., 2012; Hu et al.,
2016a; Shang et al., 2016; Ma et al., 2020a). The CODS consists of the main
control system, the data storage array, and the pipeline system. To ensure the
success of the sky survey, we developed the CODS to be stable and reliable
under the conditions of low power availability (1 kW), low data bandwidth (a
maximum of about 2 GB over the course of the year), and the unattended
situation in the harsh winter of Dome A. The supporting software provides a
fully automatic survey control and a real-time data processing pipeline on-
site.

##  3 Observations and Data

The observing season at Dome A starts in mid-March when the Sun reaches 13
degrees below the horizon, i.e., at the end of twilight (Zou et al., 2010).
The automated and unattended AST3 sky survey strategy was designed to optimize
the available observing time and was realized with a survey scheduler in the
CODS software (Liu et al., 2018). The scheduler provides three different
survey modes depending on the scientific requirements. The SN survey mode
mainly focuses on a survey for SNe and other transients, the exoplanet search
mode aims at discovering and monitoring short-period exoplanets, and an
additional special mode mainly targets the follow-up of transients.

Following twilight, the AST3-2 was initially dedicated to the SN survey mode,
lasting from 2016 March 24 to May 16, at which point the long continuous polar
night began and the survey switched to exoplanet mode. The SN survey was
designed for the early discovery of SNe as well as other transients, and for
time-domain astronomy of variable stars. It surveyed sky areas of 2200 deg2,
covering 565 fields with about 30 visits each in a cadence of a half to a few
days based on the fraction of dark time within a day. Fig. 2 shows the sky
coverage of this survey. The real-time pipeline from CODS performed onsite
data reduction and sent the SN or other transient candidates back to China for
further confirmation and follow-up observations. For example, the real-time
pipeline discovered the SN 2016ccp (Hu et al., 2016b) and the Type IIP SN
2017fbq (Wang et al., 2017b). During the test observations in Mohe, China, the
AST3-2 recorded the SN 2014J in M82(Ma et al., 2014b) and discovered the type
Ia SN 2014M (Ma et al., 2014a). The real-time pipeline is also capable of
detecting other variables such as dwarf novae (Ma et al., 2016), although most
of the variables were not reported in the real-time pipeline. So in this work,
we mainly use the SN survey mode data when the hard disks were physically
returned from Dome A to obtain the photometric catalogue and light curves of
other variables.

Figure 2: The sky coverage of survey observations from AST3-2 in 2016. Each
rectangle region is a target sky field based on the survey scheduler.

The AST3-2 exoplanet project is named the CHinese Exoplanet Searching Program
from Antarctica (CHESPA). To search for short-period exoplanets rapidly and
continuously, the exoplanet search mode started during the period of
continuous dark polar nights: from May 16 to June 22. The exoplanet search
covered a smaller sky area than the SN survey, with 10 to 20 fields in each
target region. The target region during 2016 contained 10 adjacent fields from
the southern continuous viewing zone of TESS (Ricker et al., 2009). This part
of the data has been analysed in previous works (Zhang et al., 2019a, b; Liang
et al., 2020).

Finally, a special mode was designed for the rapid follow-up of observations
of interesting transients from the AST3-2 SN or exoplanet surveys, or from
surveys by other telescopes. This mode has the highest priority. When an
interesting target triggers the alert, it will pause other observations and
resume them after the special observation is finished. In 2017, AST3-2
successfully detected the first optical counterpart of the gravitational wave
source GW170817 (Hu et al., 2017).

##  4 Data Reduction

The 2016 data of AST3-2 was retrieved by the 33rd CHINARE. We focus on the SN
survey data for this work. First, we carried out the image corrections for CCD
image pre-processing, cross-talk, image trimming, overscan, dark current,
flat-field, and an unusual diagonal stripey noise described below. Then we
performed photometric and astrometric calibration to obtain the source
catalogue. Finally, we cross-matched the catalogues to obtain the light
curves. Details of the data reduction process are discussed in the subsections
below.

###  4.1 Preprocessing

####  4.1.1 Image trimming and overscan subtraction

The AST3 raw image has 12000 $\times$ 5300 pixels including overscan regions
and is divided into 16 channels with a size of 1500 $\times$ 2650 each. As
described in section 2, the AST3 CCD works in frame-transfer mode, which means
it does not have a shutter. Since the zero-second exposure is not a true zero
because the frame transfer period takes time, photons would be gathered in the
0s bias frame when there is no shutter. This design makes it hard to take a
bias frame on-site. Instead, we used the overscan regions to remove the effect
of the bias voltage. As Fig. 1 shows, the overscan regions are the right 180
columns of each channel and 20 rows in the middle of the full raw image.

Because the top and bottom rows of the CCD are insensitive to light, we
removed another 80 rows each from the top and bottom of the CCD full images.
After overscan correction and image trimming, the final raw images have a size
of 10560 $\times$ 5120 pixels.

####  4.1.2 Dark current subtraction

As described in section 2, the CCD temperature was not very stable and could
be above $-50^{\circ}$C sometimes, making the dark current non-negligible.
Moreover, the laboratory dark images had different patterns and were not
usable for dark correction in practice. Additionally, we could not take dark
frames on site because the CCD does not have a shutter and the AST3 was
unattended for at least one year. Therefore, a new method was developed to
derive the dark frame from scientific images to solve this problem and it had
been successfully applied to the AST3-1 images (Ma et al., 2014c, 2018). Here
we briefly describe this method and how we utilized it in the AST3-2
preprocessing.

The brightness $I$ of a pixel $(x,y)$ can be described as follow:

$I(x,y)=S_{T}+D(T)+\Delta d(T,x,y),$ (1)

where $S_{T}$ is the sky background, $D(T)$ is the median dark current level
at temperature $T$, and $\Delta d(T,x,y)$ is the deviation from the median
dark current in pixel $(x,y)$ at temperature $T$. The stars can be ignored by
a median algorithm if we combine large numbers of images from different sky
fields. For a single image, $D(T)$ can be considered constant. Also, the sky
brightness can be considered a constant because it is spatially flat enough
after twilight (Yang et al., 2017). The first two terms on the right-hand side
of the equation (1) can be considered constant for a single image. To derive
the distribution of the deviation from the median dark current level $\Delta
d(T,x,y)$, we need to take two scientific images that were taken at the same
temperature but with different sky brightnesses. By scaling the two images to
an equivalent median level and subtracting one from another, we can derive a
$\Delta d(T,x,y)$ image at a specific temperature $T$. We repeated this
process for different pairs of images at the same $T$ and combined the dark
images to construct a master dark image for the specific temperature $T$. Fig.
3 shows the master dark image derived from the 2016 observations.

For different temperatures, the dark current level doubles as the temperature
increases every 7.3∘ for the AST3 CCD between $-80^{\circ}$ and $-40^{\circ}$
(Ma et al., 2012). We used this relation to scale the master dark image to
different temperatures and correct the dark current for all images.

Figure 3: The dark frame generated from observation. The difference between
bright and dark regions is obvious.

####  4.1.3 Flat field correction

During the beginning of the observing season, we took numerous twilight sky
images and produced a master flat-field image. The large FOV of AST3 led to a
non-uniform large-scale gradient of the twilight images, which varied with the
Sun elevation and angular distance from the field. The method of brightness
gradient correction was studied in Wei et al. (2014). Two-dimensional fitting
was applied to each flat image to correct the brightness gradient. Finally, we
median combined the corrected flat images to construct a master flat-field
image. After the correction, the mean root-mean-square (RMS) of the master
flat was far below 1 per cent.

####  4.1.4 Cross-talk and stripey noise corrections

Due to the simultaneous CCD readouts, when one amplifier reads a saturated
pixel, other amplifiers will be affected. There are significant CCD cross-talk
effects in the raw images. As Fig. 4 shows, when one saturated pixel is read
in one readout channel, the other 15 channels will have a negative ghost image
at the exact position of the saturated pixel presenting as a dark spot. To
remove the effect, we initially planned to locate all the saturated pixels,
find the position of the related ghost pixels, and add the appropriate
negative values back. However, the unsaturated pixels around the saturated
ones also have cross-talk effects, making the ghost images hard to locate. So
we developed a method to correct the cross-talk effect during the correction
for the stripey noise, described below.

Figure 4: An example of the cross-talk effect in the AST3 raw image. The left
panel shows a saturated star in one channel that would cause the cross-talk
effect in other channels. The middle panel shows the mirror pixels where the
saturated pixels are in another CCD channel. The stripey noise can also be
seen in this area. The right panel shows the same image as in the middle panel
but after the cross-talk effect and the stripey noise are corrected.

As Fig. 5 shows, the raw images of AST3 in 2016 have shown an unusual kind of
stripey noise. After careful investigation, we found that the diagonal stripes
were due to electromagnetic interference at 16 kHz caused by a broken ground
shield in the cables for the telescope’s DC motor drives. Because this noise
lies in exactly the same positions in each of the 16 CCD channels, and is
extremely reproducible, for each channel we constructed a filtered image from
the other 15 channels by median combining the star-removed images of single
channels. By subtracting from each channel the filtered image, we can remove
the stripey noise to the point where it is not detectable, as Fig. 5 shows.
The pattern of the noise is similar to the cross-talk effect, which also lies
at the same position of different readout channels. So, the above method also
helped to correct the cross-talk problem.

Figure 5: Upper: An example of the stripey noise in the AST3 raw image due to
a break in a cable shield. Lower: The same image area but with the noise
removed through the method discussed in section 4.1.4.

###  4.2 Photometry and astrometry

####  4.2.1 Photometry

We performed aperture photometry using the Source Extractor (Bertin & Arnouts,
1996). Considering the changing full width at half maximum (FWHM) of our
images, we used multiple apertures to adapt to the varying image quality. The
aperture radii were set to 3, 5, and 7 pixels. Because the median FWHM of our
data is 5 pixels, we set the default aperture radius as 5 pixels, or 5″at our
pixel scale of 1″/pixel. An additional Kron-like elliptical aperture magnitude
MAG_AUTO was adopted for galaxies. Fig. 6 shows the photometric accuracy of
two consecutive images by comparing the magnitude differences between them.

Figure 6:  The upper panel presents the magnitude differences between two
consecutive exposures b160505.000122 and b160505.000123 as a function of
$i$-band magnitude measured with a circular aperture of 3″in radius. The lower
panel presents the magnitude RMS calculated in 0.2 magnitude bins. The
different colours represent different circular apertures. The solid lines are
the expected error from photon noise of 5″radius. The trend of magnitude
$\sigma$ changing with aperture goes contrary at different magnitude ends. At
the brighter end, the $\sigma$ is lower when the aperture radius is higher. It
means for brighter stars larger apertures are appropriate when the photon
noise of the star itself is dominant. The stars brighter than 11 magnitudes
are saturated and have higher $\sigma$. At the fainter end, the $\sigma$ is
lower when the aperture radius is lower. This is because the sky background
dominated the noise for fainter stars and a smaller aperture is more suitable.
The number of stars is insufficient at the very faint end, thus the $\sigma$
is not real and seems to be smaller than the ideal photon noise.

####  4.2.2 Astrometry

For astrometry, we used SCAMP to solve the World Coordinate System (Bertin,
2006). We adopted the Position and Proper Motions eXtended catalogue (PPMX) as
a reference, which contains 440 sources per deg2 with the one-dimensional
precision of 40 mas (Röser et al., 2008). As a result, the external precision
of our astrometric calibration is 0.1″and the internal precision is 0.06″, in
both Right Ascension (RA) and Declination (Dec.).

####  4.2.3 Flux calibration

We adopted the SkyMapper catalogue as the $i$-band magnitude reference for the
flux calibration (Wolf et al., 2018). The SkyMapper Southern Survey is a
southern hemispheric survey carried out with the SkyMapper Telescope at Siding
Spring Observatory in Australia. It covers an area of 17,200 deg2 and the
limiting magnitude reaches a depth of roughly 18 magnitudes in $uvgriz$ pass
band.

We first chose our best frame of each survey field for absolute calibration.
The “best” refers to the images that have the best image quality in one field
considering the number of detected sources, background brightness, FWHM, and
elongation. Then we calculate the zero point in $i$-band magnitude for
calibration. We only chose the stars that are between 11 and 14 magnitudes in
the $i$-band for calibration to balance high accuracy with a sufficient number
of stars.

After the absolute calibration, we used these calibrated images as references
to relatively calibrate the other images of each survey field. However, we
found that the zero point changes with position and the cause still requires
further investigation. To avoid large field non-uniformity of the zero point,
we decide to do the flux calibration in each readout channel.

For the AST3-2 survey, we only have $i$-band data. To investigate the colour
term, we compared the AST3-2 $i$-band data with SkyMapper $i$\- and $g$-band
data as Fig. 7 shows. The colour coefficient is 0.02, much smaller than that
for AST3-1 reported by Ma et al. (2018). However, we used a different
reference catalogue from AST3-1, which adopted the AAVSO Photometric All-Sky
Survey catalogue (APASS; Henden et al. (2016)). Our $i$-band magnitude matches
relatively well with the SkyMapper catalogue, but to compare with other
catalogues observed in the same band we need to be cautious.

Figure 7: Left panel: The magnitude difference between AST3-2 and SkyMapper in
the AST3-2 image b160505.000122. Right panel: The difference between the
$i$-band catalogue of SkyMapper and AST3-2 versus the SkyMapper $r-i$
magnitude.

###  4.3 Data quality

Fig. 8 displays the distributions of data quality, showing the median value of
elongation of $\sim$1.17, FWHM of 5″, background of 670 ADU, and limiting
magnitude of 17.8 mag. Some issues with tracking stability AST3-2 led to the
elongated star profiles. We also see this problem from the range of FWHM,
which varies from 3 to 7 arcseconds. Another cause of the wide FWHM
distribution was the changing tube seeing. In the extremely cold and high
relative humidity conditions at Dome A, there can be frost on the first
surface of the optical system that reduces the transmission and changes the
point-spread function through scattering. As described in section 2, a heater
and a blower were used to prevent the frosting problem, and the tube seeing
would be unstable when they were working. As a result, the limiting magnitude
is not as good as that of the first AST3 telescope AST3-1 (Ma et al., 2018).

Figure 8: The statistics of the star elongation, FWHM, sky background, and
limiting magnitude of the data, which had median values of 1.17, 5″, 670 ADU,
and 17.8 mag, respectively.

##  5 Stellar Variability and Statistics

###  5.1 Time series

Images with poor quality were first excluded to ensure the quality of the
light curves. Such images could be due to heavy frost, or doubling of stars
images from tracking problems. We also excluded images with a background
brightness larger than 10,000 ADU, median FWHM larger than 8″, fewer than 2000
stars, and median elongations larger than 2. In this way, we excluded about 30
per cent of the images. We then cross-matched the targets in each field and
obtained light curves. Finally, an additional outlier elimination was
performed to remove the false targets with obvious anomalous magnitudes and
FWHMs. Fig. 9 shows a typical light curve dispersion with an aperture radius
of 5″.

Figure 9: The light curve rms as a function of magnitude. Each data point
represents a light curve from the region b160505.000122.

###  5.2 Period search

On average, we observed each survey field 30 times during the year. Some of
the targets might not be detected in some images due to poor image quality
etc. Thus, together with the image selections in section 5.1, for each target,
the total number of epochs could be less than 30. To analyse the stellar
variability with enough detection and better image quality, we restricted
ourselves to sky fields with more than 30 observations. There are about one-
third of the observations were exposed 3 times continuously, originally for
image combination. Due to the tracking problem discussed in section 4.3 and
section 5.1, some of the multi-images would be excluded in the image
selections and we tend not to combine them. For the remaining multi-images, we
did not count them as individual observations, but we used them as independent
data points in light curve analysis. Then, we rejected the targets that were
only detected in less than 50 per cent of the images. Finally, in the period
analysis, we chose the light curves with a significant variability of more
than 2.5$\sigma$.

For our survey data, the sampling in the light curves with time is not uniform
and thus we used the Lomb-Scargle (LS) method for period search (Lomb, 1976;
Scargle, 1982). Light curves with a signal-to-noise ratio (SNR) larger than 5
are considered eligible candidates. Then we cross-matched the candidate light
curves with the International Variable Star Index (VSX; Watson et al. 2006)
and found 3,551 known variables. For candidates that were not in the VSX
catalogue, we visually inspected whether their periodicities were significant
or not. For candidates that were significantly variable and periodic, we then
checked whether it was a false signal. For example, Fig.10 shows a comparison
of the true and false EA-type variable candidates. The former is an EA-type
variable candidate included in the VSX catalogue. The latter shows a similar
light curve pattern but turned out to be a false signal affected by an
outlier. We manually excluded these kinds of false signals and we take the
true ones as variable candidates. In total, we found 70 new variables.

Figure 10: Upper: An example light curve of an EA-type variable star folded in
2 phases. Lower: An example false signal showing an EA-type variable pattern.

###  5.3 New variables

For the newly discovered variable candidates, we tried to visually classify
them into different classes by their periods, amplitudes, and light curve
patterns. We also obtained their effective temperature, surface gravity, and
metallicity from StarHorse (Anders et al., 2019) to help the classification.
Moreover, we obtained their B-V from the UCAC4 (Zacharias et al., 2013),
APASS9 (Henden et al., 2016), NOMAD (Zacharias et al., 2004), and SPM4.0
(Girard et al., 2011) catalogues. However, due to insufficient observations,
it was still hard to classify them such as the example in section 5.2. The
insufficient observations at the minimum luminosity make it hard to classify.
Many under-sampled candidates were excluded from the candidate list if they
were not a known variable.

For this reason, we were only able to classify the candidates into 5 different
classes. There are 17 candidates classified as long-period variables (LPV)
either because they were observed in less than one period or because we could
hardly distinguish one periodic signal from its light curve. We found 5
candidates with Cepheid-like signals and we classified them as pulsating stars
(PUL). We found 4 candidates as eclipsing binary (EC) candidates by their
periods and patterns. Of the remaining candidates, 24 of them have small
amplitudes ($<0.1$) and long periods (a few days to a dozen days), and they
are likely to be rotational variables (ROT). The final 20 candidates have
periods shorter than 2 days and some of their periods are even shorter than
0.2 days. Most of these candidates have strange phase diagram patterns and we
are not sure whether they are real or a result of a lack of data points. Under
this circumstance, we classified them as possible rotational variables (pROT).
Fig. 11 shows the typical phased or time-series light curves of each class.

As mentioned in section 5.2, when we try to classify the light curves, we only
consider the ones with 30 epochs or more to ensure there are enough
observations for a reliable period. We can confirm some of them that have
obvious and distinctive light curve patterns. But for many variables that met
the 30 epoch threshold, the absence of critical data points in the light
curves might lead to a false period, and an incorrect pattern in their phase
diagrams. In such cases we erred on the side of not claiming them as newly
discovered variables.

We cross-matched our variable candidates with the VSX catalogue version
2022-10-31. Interestingly, we initially used an earlier version of the VSX
catalogue and our count of new candidates was 126; 56 of these were listed in
the latest version, which gave us the opportunity of comparing our
classifications with VSX. The classifications agreed well, with disagreements
mainly with LPVs and ECs. Some stars were identified as rotational variables
in the VSX that we classified as LPVs because we have relatively time coverage
and we considered all the light curves with less than one period as LPVs. As
for the ECs in VSX, we classified some of them as ROTs or pROTs since we did
not have enough critical data points to confirm them.

Figure 11: Example light curves of the variables in different classes folded
in 2 phases. Periods and types are marked on the light curves.

##  6 Data Availability

The AST3-2 data is available through the Chinese Astronomical Data Center
(CADC)111https://cstr.cn/11379.11.100669, 222https://doi.org/10.12149/100669.
The data contains an $i$-band catalogue, a light curve catalogue, and
preprocessed images.

The $i$-band catalogue contains over 7 million sources with a median limiting
magnitude of 17.8 mag. For objects with multiple observations, we adopted
their median positions and median magnitudes. Table 1 shows the Database
Schema of the catalogue.

Table 2 details the information in the light curve catalogue. The light curves
are presented as time series and the catalogue contains information from every
observation after quality filtering. The periodic variables discussed in this
work are also presented and listed in Appendix A.

There are also 22576 images in the format of Flexible Image Transport System
(FITS) presented in the data set. These are the preprocessed FITS images
discussed in section 4 with observing information such as date, exposure time,
and WCS coordinates.

Table 1: AST3-2 survey catalogue Table Schema Column Name | Description
---|---
ID | Source index
RA | Right Ascension in J2000 (deg)
Dec. | Declination in J2000 (deg)
MAG | Median aperture magnitudes (mag)
MAGERR | Standard deviation of magnitudes (mag)
COUNT | Number of observations
Table 2: AST3-2 light curve catalogue Table Schema Column Name | Description
---|---
DATE | The beginning time of observation in
| ISO time
MJD | The beginning time of observation in
| Modified Julian date
X | Windowed X position in CCD (pixel)
Y | Windowed Y position in CCD (pixel)
RA | Right Ascension in J2000 (deg)
DEC | Declination in J2000 (deg)
MAG | Aperture magnitudes in 5″radius (mag)
MAGERR | Aperture magnitude errors in 5″radius
| (mag)
FLUX | Flux (ADU)
FLUXERR | Flux error (ADU)
MAG$\\_$AUTO | Magnitude in Kron aperture (mag)
MAGERR$\\_$AUTO | Magnitude error in Kron aperture (mag)
BACKGROUND | Background brightness (ADU)
FWHM | Full width at half-maximum in Gaussian
| profile (pixel)
ELONGATION | Ratio of semi-major to semi-minor axis
A | Semimajor axis length (pixel)
B | Semiminor axis length (pixel)
THETA | Position angle of semimajor axis (degrees
| east from north)
MAG$\\_$3 | Aperture magnitudes in 3″radius (mag)
MAGERR$\\_$3 | Aperture magnitude errors in 3″radius
| (mag)
MAG$\\_$7 | Aperture magnitudes in 7″radius (mag)
MAGERR$\\_$7 | Aperture magnitude errors in 7″radius
| (mag)

##  7 Summary

The second AST3 telescope AST3-2 was deployed at Dome A, Antarctica in 2015\.
In 2016, it worked fully automatically on a sky survey for SNae and semi-
automatically on an exo-planet search. In this work, we report on the 2016 SN
survey data observed between Mar. 23 and May 16. We surveyed 2200 deg2 fields
with about 30 visits each in a cadence of a half to a few days. After the raw
data was retrieved, we preprocessed the data, performed aperture photometry,
calibrated the magnitudes, obtained the light curves of the 565 sky fields,
and briefly studied the variability of the light curves. In this paper, we
present the data release of the photometric data from the AST3-2 SN survey in
2016. It consists of 22000 scientific images, 7 million sources brighter than
$i\sim$18 with photometry, astrometry, and light curves.

The 5$\sigma$ limiting magnitude of this dataset is 17.8 mag with 4 mmag
precision in the light curves of bright stars. The median FWHM, elongation,
and background brightness are 5.0″, 1.17, and 670 ADU, respectively. We found
70 new variable candidates out of $\sim$ 3,500 variable stars. We check the
stellar properties from surveys such as StarHorse to help us classify these
variables into 5 types.

## Acknowledgements

We thank the CHINARE for their great efforts in installing AST3-2, maintaining
AST3-2 and PLATO-A, and retrieving data. This work has been supported by the
National Natural Science Foundation of China under Grant Nos. 11873010,
11733007, 11673037, 11403057, and 11403048, the Chinese Polar Environment
Comprehensive Investigation and Assessment Programmes under grant No.
CHINARE2016-02-03, and the National Basic Research Program of China (973
Program) under Grant No. 2013CB834900. PLATO-A is supported by the Australian
Antarctic Division. Data Publishing is supported by China National
Astronomical Data Center (NADC), CAS Astronomical Data Center and Chinese
Virtual Observatory (China-VO).

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##  Appendix A The Light curves of New Candidates

Fig.12 shows the phased new variable candidates, and Table 3 shows the list of
them.

Figure 12: Light curves of the new candidates in different classes folded in 2
phases except the LPVs show no apparent periods and are in the form of time
series. The dots with error bars are observed data and the red curve is the
fitted line. Periods and types are marked on the light curves. The pulsating
stars are marked as PUL and the possible rotational variables are marked as
pROT.

Figure 13: continued

Light curves of the new candidates in different classes folded in 2 phases
except the LPVs show no apparent periods and are in the form of time series.
The dots with error bars are observed data and the red curve is the fitted
line. Periods and types are marked on the light curves. The pulsating stars
are marked as PUL and the possible rotational variables are marked as pROT.

Figure 14: continued

Light curves of the new candidates in different classes folded in 2 phases
except the LPVs show no apparent periods and are in the form of time series.
The dots with error bars are observed data and the red curve is the fitted
line. Periods and types are marked on the light curves. The pulsating stars
are marked as PUL and the possible rotational variables are marked as pROT.

Figure 15: continued

Light curves of the new candidates in different classes folded in 2 phases
except the LPVs show no apparent periods and are in the form of time series.
The dots with error bars are observed data and the red curve is the fitted
line. Periods and types are marked on the light curves. The pulsating stars
are marked as PUL and the possible rotational variables are marked as pROT.

Figure 16: continued

Light curves of the new candidates in different classes folded in 2 phases
except the LPVs show no apparent periods and are in the form of time series.
The dots with error bars are observed data and the red curve is the fitted
line. Periods and types are marked on the light curves. The pulsating stars
are marked as PUL and the possible rotational variables are marked as pROT.

Figure 17: continued

Light curves of the new candidates in different classes folded in 2 phases
except the LPVs show no apparent periods and are in the form of time series.
The dots with error bars are observed data and the red curve is the fitted
line. Periods and types are marked on the light curves. The pulsating stars
are marked as PUL and the possible rotational variables are marked as pROT.

Figure 18: continued

Light curves of the new candidates in different classes folded in 2 phases
except the LPVs show no apparent periods and are in the form of time series.
The dots with error bars are observed data and the red curve is the fitted
line. Periods and types are marked on the light curves. The pulsating stars
are marked as PUL and the possible rotational variables are marked as pROT.

Figure 19: continued

Light curves of the new candidates in different classes folded in 2 phases
except the LPVs show no apparent periods and are in the form of time series.
The dots with error bars are observed data and the red curve is the fitted
line. Periods and types are marked on the light curves. The pulsating stars
are marked as PUL and the possible rotational variables are marked as pROT.

Figure 20: continued

Light curves of the new candidates in different classes folded in 2 phases
except the LPVs show no apparent periods and are in the form of time series.
The dots with error bars are observed data and the red curve is the fitted
line. Periods and types are marked on the light curves. The pulsating stars
are marked as PUL and the possible rotational variables are marked as pROT.

Table 3: The list of new variable candidates. Namea | Magnitudeb | Period | Amplitude | Type | B-V | B-V ref. | Teffc | logg | [Fe/H] |
---|---|---|---|---|---|---|---|---|---|---
| (mag) | (days) | (mag) |  | (mag) |  | (K) | ([cm/s2]) |  |
AST3II004.7306-53.7192 | 10.7 | 2.04 | 0.05 | ROT? | 0.22 | UCAC4d | 7714 | 3.90 | -0.38 |
AST3II031.1576-53.1179 | 10.8 | 0.40 | 0.07 | ROT? | 0.76 | UCAC4 | 5888 | 4.09 | -0.18 |
AST3II049.6394-54.8318 | 11.7 | 1.63 | 0.09 | ROT? | 0.52 | UCAC4 | 5489 | 3.68 | -0.81 |
AST3II058.9192-57.4095 | 11.5 | 0.13 | 0.06 | ROT? | 0.85 | UCAC4 | 6201 | 4.49 | -0.78 |
AST3II012.5062-54.6717 | 12 | 0.73 | 0.03 | ROT? | 1.21 | UCAC4 | 5210 | 4.48 | 0.27 |
AST3II092.4386-56.3706 | 12 | 0.13 | 0.05 | ROT? | 0.79 | UCAC4 | 5703 | 4.33 | -0.02 |
AST3II043.1351-52.5872 | 12.2 | 0.12 | 0.03 | ROT? | 0.53 | UCAC4 | 6257 | 4.30 | -0.19 |
AST3II009.7028-54.3714 | 12.3 | 1.16 | 0.08 | ROT? | 0.44 | UCAC4 | 6039 | 3.94 | -0.79 |
AST3II026.3788-52.5052 | 12.6 | 3.27 | 0.08 | ROT? | 1.55 | APASS9e | 3141 | 4.70 | 0.06 |
AST3II341.7130-57.5785 | 12.9 | 1.73 | 0.13 | ROT? | 0.90 | UCAC4 | 5546 | 4.50 | -0.10 |
AST3II301.9477-49.6249 | 13.5 | 1.43 | 0.11 | ROT? | 1.28 | NOMADf | 3767 | 4.64 | 0.21 |
AST3II026.1444-55.8204 | 13.8 | 1.96 | 0.24 | ROT? | 0.88 | UCAC4 | 5177 | 3.45 | -0.53 |
AST3II036.2637-60.6502 | 13.8 | 0.10 | 0.07 | ROT? | 0.86 | UCAC4 | 5137 | 4.59 | -0.13 |
AST3II035.1183-54.4928 | 13.9 | 1.31 | 0.11 | ROT? | 0.85 | UCAC4 | 5329 | 4.57 | -0.22 |
AST3II082.1244-53.3142 | 13.9 | 2.21 | 0.10 | ROT? | 1.30 | UCAC4 | 5077 | 4.51 | 0.19 |
AST3II043.6166-52.4197 | 14.1 | 0.14 | 0.21 | ROT? | 0.92 | UCAC4 | 5826 | 4.32 | -0.12 |
AST3II288.8210-60.8878 | 14.1 | 1.15 | 0.15 | ROT? | 1.44 | UCAC4 | 4933 | 4.60 | 0.00 |
AST3II070.9519-50.9647 | 14.3 | 0.13 | 0.25 | ROT? | 1.02 | UCAC4 | 5433 | 4.41 | 0.12 |
AST3II009.0299-53.6067 | 14.8 | 0.14 | 0.60 | ROT? | 0.75 | NOMAD | 5482 | 4.40 | 0.12 |
AST3II031.0155-53.2111 | 16.5 | 0.12 | 0.89 | ROT? | 1.71 | NOMAD | 4145 | 4.62 | 0.40 |
Table 4: continued

The list of new variable candidates.  Namea Magb P Amp. Type B-V B-V ref.
Teffc logg [Fe/H] (mag) (days) (mag) (mag) (K) ([cm/s2])
AST3II290.4978-60.8112 13.5 0.41 0.08 EC 0.60 UCAC4 6397 3.97 -0.29
AST3II288.4890-60.4526 13.5 0.78 0.06 EC 0.43 UCAC4 6349 2.68 -1.38
AST3II288.6902-59.7634 13.7 0.37 0.07 EC 0.85 UCAC4 5801 4.13 0.00
AST3II339.6513-56.7764 14.1 0.40 0.33 EC 0.28 UCAC4 6630 4.08 -0.49
AST3II287.9363-60.0357 10.6 23.72 0.04 LPV 1.07 UCAC4 4480 1.14 -1.40
AST3II314.4222-46.9618 10.7 200 0.04 LPV 0.30 UCAC4 8526 4.25 -0.15
AST3II347.0135-60.9865 10.6 55.93 0.05 LPV 0.89 UCAC4 5023 3.55 -0.11
AST3II344.2833-48.7002 10.8 34.12 0.07 LPV 1.54 UCAC4 3728 0.79 -0.21
AST3II076.3264-53.1369 11 54.39 0.05 LPV 0.80 UCAC4 5410 3.30 -0.40
AST3II287.8703-60.8919 10.9 16.93 0.05 LPV 1.29 UCAC4 4611 2.41 0.10
AST3II076.9649-48.4256 11.1 28.69 0.1 LPV 1.10 UCAC4 4831 2.30 -0.64
AST3II289.0164-58.1376 11.3 15.52 0.03 LPV 1.53 UCAC4 4007 0.93 -0.58
AST3II301.4456-51.2031 11.4 46.78 0.08 LPV 0.78 NOMAD 4115 1.28 -0.39
AST3II018.3213-54.4865 11.7 62.96 0.05 LPV 1.67 APASS9 3018 4.94 0.24
AST3II336.7393-51.1822 11.8 16.35 0.05 LPV 0.96 UCAC4 5033 4.65 -0.51
AST3II073.3528-49.2170 11.9 65.51 0.16 LPV 0.92 UCAC4 4805 2.58 -0.47
AST3II289.4560-60.7725 12 23.72 0.06 LPV 0.82 UCAC4 5156 3.20 -0.49
AST3II042.7007-54.3642 12.8 98.65 0.09 LPV 0.59 SPM4.0g 6359 4.34 -0.41
AST3II051.0401-52.3076 13.3 123.73 0.10 LPV 1.53 UCAC4 4372 0.81 -1.55
AST3II070.2096-51.1571 13.7 200 0.11 LPV 1.77 UCAC4 * * *
AST3II290.1816-60.4481 13.8 43.79 0.10 LPV 0.91 UCAC4 5153 3.45 -0.49
AST3II093.0735-56.2041 12.3 2.66 0.07 PUL 0.24 UCAC4 5560 3.46 -1.42
AST3II039.9789-53.5451 13.3 4.49 0.07 PUL 0.77 UCAC4 5880 4.25 -0.14
AST3II070.3697-50.8891 13.5 1.01 0.05 PUL 0.62 UCAC4 10089 4.40 -0.65
AST3II037.8094-53.8255 14.6 1.41 0.21 PUL 1.33 UCAC4 4947 4.65 -0.48
AST3II317.2367-54.4383 15.4 0.36 0.66 PUL 0.27 UCAC4 6880 3.83 -0.79
AST3II041.7958-54.1112 11.1 9.19 0.07 ROT 0.50 UCAC4 8240 4.17 0.05
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AST3II024.5874-53.8901 12.3 4.7 0.08 ROT 1.55 APASS9 3272 4.84 0.09
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AST3II081.3298-46.8904 12.9 1.71 0.08 ROT 0.64 UCAC4 6093 4.03 -0.46
AST3II025.8764-53.9192 13 3.83 0.04 ROT 0.72 UCAC4 5562 4.41 -0.27
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AST3II084.1197-53.2223 13.2 7.42 0.07 ROT 0.76 UCAC4 5156 3.53 -0.36
AST3II290.0774-60.2915 13.1 10.29 0.08 ROT 1.14 UCAC4 6085 4.28 -0.38
AST3II288.7966-60.6563 13.2 12.15 0.07 ROT 1.10 UCAC4 4723 2.53 -0.22
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AST3II073.2523-44.4335 13.6 0.13 0.11 ROT 0.58 UCAC4 6146 4.30 -0.31
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AST3II288.1145-59.8068 13.5 3.08 0.07 ROT 0.26 NOMAD 4877 4.59 0.12
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AST3II290.2670-58.6500 14.7 6.15 0.20 ROT 1.62 UCAC4 4004 4.69 -0.29
AST3II288.3231-58.8371 14.9 12.61 0.26 ROT 0.74 NOMAD 4942 3.36 -0.18
AST3II329.6830-50.2914 15.1 3.61 0.33 ROT 0.77 NOMAD 5193 3.28 -0.68


# Stabilization by Noise for Chafee-Infante Equation in Perforated Domains

Hong Hai Ly  Department of Mathematics
University of Ostrava, Ostrava, Czech<EMAIL_ADDRESS>Bao Quoc Tang
Department of Mathematics and Scientific Computing, University of Graz, Graz,
<EMAIL_ADDRESS>
###### Abstract

The stabilization by noise for parabolic equations in perforated domains, i.e.
domains with small holes, is investigated. We show that when the holes are
small enough, one can stabilize the unstable equations using suitable
multiplicative Itô noise. The results are quantitative, in the sense that we
can explicitly estimate the size of the holes and diffusion coefficients for
which stabilization by noise takes place. This is done by using the asymptotic
behaviour of the first eigenvalue of the Laplacian as the hole shrinks to a
point.

Keywords— Stabilization by noise, Perforated domains, Asymptotic behaviour of
eigenvalues

##  1 Introduction and main result

Consider a bounded domain $\mathcal{O}$ in $\mathbb{R}^{n}$, $n\in\\{2,3\\}$,
with Lipschitz boundary $\partial\mathcal{O}$. Within this domain, there is a
family of small pairwise disjoint open holes
$\\{E^{i}_{\varepsilon}\\}_{i=1}^{\ell}$, $\ell\geq 1$, which have size of
order $\varepsilon>0$, i.e. for each $E^{i}_{\varepsilon}$, there is a ball
$\mathbb{B}^{i}_{\varepsilon}$ with radius $\varepsilon$ such that
$E^{i}_{\varepsilon}\subset\mathbb{B}^{i}_{\varepsilon}$. We assume that the
holes are properly inside the domain, i.e.
$\overline{E_{\varepsilon}^{i}}\subset\mathcal{O}$, and the boundary $\partial
E^{i}_{\varepsilon}$ is also Lipschitz for each $i=1,\ldots,\ell$. The open
perforated domain
$\mathcal{O}_{\varepsilon}=\mathcal{O}\backslash\bigcup\limits_{i=1}^{\ell}\overline{E^{i}_{\varepsilon}}$
is obtained by removing these holes from $\mathcal{O}$. We study in this work
a stochastic Chafee-Infante equation defined on this domain, described by the
following

$\begin{cases}du+(-\Delta u+u^{3}-\beta
u)dt=h(t,u)dW_{t},&x\in\mathcal{O}_{\varepsilon},\;t>0,\\\
u(x,t)=0,&x\in\partial\mathcal{O}_{\varepsilon},\;t>0,\\\
u(x,0)=u_{0}(x),&x\in\mathcal{O}_{\varepsilon},\end{cases}$ (1)

where $\beta>0$ and $W_{t}$ is a standard real-valued scalar Wiener process
defined on the probability space $(\Omega,\mathcal{F},P)$ with natural
filtration $(\mathcal{F}_{t})_{t\geq 0}$, $dW_{t}$ denotes the Itô
differential and $u_{0}\in L^{2}(\mathcal{O}_{\varepsilon})$ is given initial
data. It holds obviously
$\partial\mathcal{O}_{\varepsilon}=\cup_{j=1}^{\ell}\partial
E_{\varepsilon}^{i}\cup\partial\mathcal{O}$. The diffusion coefficient
$h:[0,\infty)\times\mathbb{R}\to\mathbb{R}$ is assumed to satisfy following
conditions:

  1. ($H_{1}$)

for all $t\geq 0$, $u\in\mathbb{R}$, $h(t,0)=0$,

  2. ($H_{2}$)

for all $t\geq 0$, $u\in\mathbb{R}$, $|h(t,u)|^{2}\leq\gamma(t)|u|^{2}$, where
$\gamma:[0,\infty)\to[0,\infty)$ is a continuous function such that

$\limsup_{t\to\infty}\frac{1}{t}\int_{0}^{t}\gamma(s)ds\leq\gamma_{0}\quad\text{
with }\gamma_{0}\text{ is a non-negative constant.}$

  3. ($H_{3}$)

for $t\geq 0$, $u\in\mathbb{R}$, $|h(t,u)u|^{2}\geq\rho(t)|u|^{4}$, where
$\rho:[0,\infty)\to[0,\infty)$ is a continuous function such that

$\liminf_{t\to\infty}\frac{1}{t}\int_{0}^{t}\rho(s)ds\geq\rho_{0}\quad\text{
with }\rho_{0}\text{ is a non-negative constant.}$

The assumption ($H_{1}$) is to ensure that $u\equiv 0$ is a trivial steady
state solution to (1). Conditions ($H_{2}$) and ($H_{3}$) are Lipschitz
continuity at zero and dissipative properties. Typical examples of the
diffusion coefficient $h(t,u)$ satisfying conditions ($H_{1}$)–($H_{3}$)
include

  * •

the linear function $h(t,u)=\alpha u$, $\alpha\in\mathbb{R}$, for which
$\gamma(t)=\rho(t)=\gamma_{0}=\rho_{0}=\alpha^{2}$ for all $t\in\mathbb{R}$,

  * •

or the nonlinear function
$h(t,u)=\dfrac{(2+t^{2})(1+u^{2})}{(1+t^{2})(2+u^{2})}u$ for which $\gamma(t)$
and $\rho(t)$ can be chosen as

$\gamma(t)=16\left(\frac{2+t^{2}}{1+t^{2}}\right)^{2},\quad\rho(t)=\left(\frac{2+t^{2}}{2(1+t^{2})}\right)^{2},$

and consequently one can choose $\gamma_{0}=64$ and $\rho_{0}=1$.

Note that when $h(t,u)\equiv 0$, (1) becomes the classical deterministic
Chafee-Infante equation, for which the stability of the zero steady state is
determined by the value of $\beta>0$ and the first eigenvalue of $-\Delta$
with homogeneous Dirichlet boundary condition.

To put our work into context, let us briefly review the literature on
stabilization by noise. The stabilization using multiplicative noise has been
investigated since the eighties firstly for finite dimensional systems, see
e.g. [GS82, ACW83, Arn90]. For the linear finite dimensional system
$\dot{x}=Ax$, this phenomenon is completely characterized in [ACW83] where it
was proved to be stabilizable by linear multiplicative noise if and only
$\text{Trace}(A)<0$. The case of nonlinear systems and nonlinear noise, still
in finite dimensional setting, is more complicated as it was shown in [AMR08]
that random noise can also destabilize stable systems. The first work dealing
with infinite dimensional system is [Kwi99], therein the author showed that
the multiplicative noise shifts the spectrum of Laplacian to the right, thus
stabilizing the system. The stabilization by noise for partial differential
equations has been then studied extensively, see e.g. [CLM01, Kwi02, CGAR03,
CR04, CCLR07]. All these works showed that the is an infinite range of noise
intensity that stabilizes the deterministic system. Notably, a recent work
[FSTT19] showed that when the noise is acting solely on the boundary, one
might stabilize the deterministic system only with a finite range of noise
intensities. It is worthwhile to emphasize that, up to our knowledge, all the
existing works considered solid domains, i.e. without holes. On the other
hand, the stabilization by deterministic control in perforated domains has
been also investigated by many authors since it appears frequently in
applications, see e.g. [Zhi92, CDZ91, CDZ94, BCCW11, BS22]. Motivated by these
developments, we investigate in this paper, for the first time, the
stabilizing effect of multiplicative noise in perforated domains for the
Chafee-Infante equation (1).

The main result of this paper is the following theorem.

######  Theorem 1.1.

Under ($H_{2}$)–($H_{3}$), there exists a unique global strong solution to the
stochastic equation (1).

If ($H_{1}$) is fulfilled and

$\rho_{0}-\frac{\gamma_{0}}{2}>-\lambda_{1}^{D}(\mathcal{O})+\beta,$

where $\lambda_{1}^{D}(\mathcal{O})$ is the first eigenvalue of the Dirichlet-
Laplacian on $\mathcal{O}$, then there exists $\varepsilon_{0}$ such that

$\varepsilon_{0}\sim\begin{cases}e^{-\left(\lambda^{D}_{1}(\mathcal{O})+\rho_{0}-(\beta+{\gamma_{0}}/{2})\right)^{-1}}&\text{
for }n=2,\\\
\lambda^{D}_{1}(\mathcal{O})+\rho_{0}-(\beta+{\gamma_{0}}/{2})&\text{ for
}n=3,\end{cases}$ (2)

and for all $0<\varepsilon\leq\varepsilon_{0}$, the zero steady state solution
to (1) is globally exponentially stable almost surely, i.e.

$\|u(t)\|_{L^{2}(\mathcal{O}_{\varepsilon})}\leq Ce^{-\delta
t}\quad\mathbb{P}-\text{a.s.}$

where the positive constants $C,\delta$ are independent of $\varepsilon$. Here
$A\sim B$ means that $C_{0}A\leq B\leq C_{1}A$ for positive constants
$C_{0},C_{1}$ independent of $A$ and $B$.

We remark that $\varepsilon_{0}$ defined in (2) depends explicitly and only on
the parameters $\beta,\rho_{0},\gamma_{0}$ and the domain $\mathcal{O}$
through the first eigenvalue $\lambda_{1}^{D}(\mathcal{O})$. It is known that
when $\beta>\lambda_{1}^{D}(\mathcal{O})$, the zero steady state to the
deterministic equation (1), i.e. $h\equiv 0$, is unstable (see e.g. [FSTT19,
Theorem 2.2] for the case of Robin boundary conditions, whose arguments can be
also applied to Dirichlet boundary conditions). By choosing the diffusion
coefficient $h(t,u)$ such that (2) holds, for instance, $h(t,u)=\alpha u$ with
$\alpha^{2}>2(-\lambda_{1}^{D}(\mathcal{O})+\beta)$, then the zero solution is
(exponentially) stabilized provided that the size $\varepsilon$ of the holes
do not exceed $\varepsilon_{0}$ given in (2). Our main idea is to use the
asymptotic behavior of the first eigenvalue
$\lambda_{1}^{D}(\mathcal{O}_{\varepsilon})$ of the Dirichlet-Laplacian in
$\mathcal{O}_{\varepsilon}$ as $\varepsilon\to 0$, which will be proved in
Section 2. We then utilize these results in Section 3 to prove our main
theorem.

##  2 Asymptotic behavior of the first eigenvalue

On the perforated domain $\mathcal{O}_{\varepsilon}$, we investigate the
eigenvalue problem

$\displaystyle\begin{cases}-\Delta\varphi(x)=\lambda(\varepsilon)\varphi(x),&x\in\mathcal{O}_{\varepsilon},\\\
\varphi(x)=0,&x\in\partial\mathcal{O}_{\varepsilon}.\end{cases}$ (3)

Here, $\varphi(x)$ represents the eigenfunction, and $\lambda(\varepsilon)$ is
the corresponding eigenvalue.

Let $\lambda_{1}^{D}(\mathcal{O}_{\varepsilon})$ denote the first eigenvalue
of $-\Delta$ in $\mathcal{O}_{\varepsilon}$ under the Dirichlet boundary
condition on $\partial\mathcal{O}_{\varepsilon}$, and let
$\lambda_{1}^{D}(\mathcal{O})$ be the first eigenvalue of $-\Delta$ in
$\mathcal{O}$ under the Dirichlet boundary condition on $\partial\mathcal{O}$.
It is classical that

$\|\nabla
u\|_{L^{2}(\mathcal{O})}^{2}\geq\lambda_{1}^{D}(\mathcal{O})\|u\|_{L^{2}(\mathcal{O})}^{2}\;\text{
and }\;\|\nabla
v\|_{L^{2}(\mathcal{O}_{\varepsilon})}^{2}\geq\lambda_{1}^{D}(\mathcal{O}_{\varepsilon})\|v\|_{L^{2}(\mathcal{O}_{\varepsilon})}^{2},$
(4)

for all $u\in H_{0}^{1}(\mathcal{O})$ and all $v\in
H_{0}^{1}(\mathcal{O}_{\varepsilon})$.

The asymptotic behavior of $\lambda_{1}^{D}(\mathcal{O}_{\varepsilon})$ as
$\varepsilon\to 0$ has been extensively investigated in the literature, see
e.g. [Oza81, Oza82, MNP85, ALM22].In the early works, the holes are often
considered to be balls, which makes the analysis easier. In order to allow
various shapes for the holes, later works utilize the concept of capacity
potential, which measures the ability of the set $A$ to hold electric charge
or conduct heat, see e.g. [Cho54]. The capacity potential of a set
$A\subset\mathcal{O}$ with respect to $\mathcal{O}$ is defined as follows:

$\displaystyle\textup{cap}(A):=\inf\left\\{\int_{\mathcal{O}}|\nabla
u|^{2}dx:\,u\in H^{1}_{0}(\mathcal{O}),\,u\geq 1\text{ on }A\right\\}.$

######  Lemma 2.1.

As $\varepsilon\to 0$, the capacity potential
$\textup{cap}(E^{i}_{\varepsilon})$ is estimated as

$\displaystyle\textup{cap}(E^{i}_{\varepsilon})\leq\begin{cases}\dfrac{2\pi}{-\log\varepsilon}+o(\log^{-1}(\varepsilon))&\text{if
}n=2,\\\ 4\pi\varepsilon+o(\varepsilon)&\text{if }n=3.\end{cases}$

###### Proof.

According to [Flu95], the capacity
$\textup{cap}(\mathbb{B}^{i}_{\varepsilon})$ is given by:

$\displaystyle\textup{cap}(\mathbb{B}^{i}_{\varepsilon})=\begin{cases}\dfrac{2\pi}{-\log\varepsilon}+o(\log^{-1}(\varepsilon))&\text{if
}n=2,\\\ 4\pi\varepsilon+o(\varepsilon)&\text{if }n=3.\end{cases}$

The monotonicity property of capacity [Cho54, Chapter II]) states that the
capacity of a smaller set is less than or equal to the capacity of any larger
set containing it. Consequently, since
$E^{i}_{\varepsilon}\subseteq\mathbb{B}^{i}_{\varepsilon}$, we have
$\text{cap}(E^{i}_{\varepsilon})\leq\text{cap}(\mathbb{B}^{i}_{\varepsilon})$,
which gives the desired bound and finishes the proof of this Lemma. ∎

As $\varepsilon\to 0$, each hole $E_{\varepsilon}^{i}$ is assumed to shrink to
a fixed point $x_{i}\in E_{\varepsilon}^{i}$ for each $i=1,\ldots,\ell$. Now,
we present the following theorem that establishes a relationship between the
first eigenvalues of the perforated and original domains.

######  Theorem 2.2.

The first eigenvalue of problem (3) has the following asymptotics as
$\varepsilon$ tends to zero

$\displaystyle\lambda^{D}_{1}(O_{\varepsilon})=\lambda^{D}_{1}(\mathcal{O})+\sum_{i=1}^{\ell}\varphi_{1}^{2}(x_{i})\,\textup{cap}(E^{i}_{\varepsilon})+o(\textup{cap}(\cup_{i=1}^{\ell}E^{i}_{\varepsilon})),\quad\ell\in\mathbb{N},$
(5)

where $\varphi_{1}$ represents the first eigenfunction corresponding to
$\lambda^{D}_{1}(\mathcal{O})$. Consequently

$\displaystyle\lambda_{1}^{D}(\mathcal{O}_{\varepsilon})=\begin{cases}\lambda_{1}^{D}(\mathcal{O})+O((\log\varepsilon)^{-1})&\text{
for }n=2,\\\ \lambda_{1}^{D}(\mathcal{O})+O(\varepsilon)&\text{ for }n=3.\\\
\end{cases}$ (6)

###### Proof.

By applying [Flu95, Theorem 6] and using sub-additivity of the capacity and
its asymptotic additivity for small disjoint sets, we obtain the asymptotic
expansion for each simple eigenvalue $\lambda_{k}(\mathcal{O}_{\varepsilon})$
as follows:

$\lambda_{k}(\mathcal{O}_{\varepsilon})=\lambda_{k}(\mathcal{O})+\sum_{i=1}^{\ell}\varphi_{k}^{2}(x_{i})\textup{cap}(E^{i}_{\varepsilon})+o(\textup{cap}(\cup_{i=1}^{\ell}E^{i}_{\varepsilon}))\text{
as }\varepsilon\to 0,$

where $\varphi_{k}$ are $L^{2}(\mathcal{O})$-orthonormal eigenfunctions
associated with $\lambda_{k}(\mathcal{O})$. It is classical that the first
eigenvalue is simple, and therefore we have

$\displaystyle\lambda^{D}_{1}(O_{\varepsilon})=\lambda^{D}_{1}(\mathcal{O})+\sum_{i=1}^{\ell}\varphi_{1}^{2}(x_{i})\textup{cap}(E^{i}_{\varepsilon})+o(\textup{cap}(\cup_{i=1}^{\ell}E^{i}_{\varepsilon})),$

which is the desired asymptotic expansion (5).

To prove (6), it is sufficient to show that $\varphi_{1}$ is in
$L^{\infty}(\mathcal{O})$. Since the boundary $\partial\mathcal{O}$ is only
Lipschitz continuous, using maximal regularity of elliptic equation for
$-\Delta\varphi_{1}=\lambda_{1}^{D}(\mathcal{O})\varphi_{1}$ only gives
$\varphi_{1}\in H^{3/2}(\mathcal{O})$ which is not enough to obtain
$L^{\infty}(\mathcal{O})$-bound in the case $n=3$. Therefore, we use the
results in [Dav89] (see Theorem 2.1.4 and Example 2.1.8 therein), where it was
shown that $\varphi_{k}$ are in $L^{\infty}(\mathcal{O})$ for all $k\geq 1$.
The asymptotic estimate (6) the follows directly from (5), Lemma 2.1 and the
boundedness of $\varphi_{1}$. ∎

##  3 Stabilization by noise

We start with the well-posedness for the stochastic equation (1).

######  Theorem 3.1 ([Par80, Theorem 3.1]).

Under assumptions ($H_{2}$)–($H_{3}$), the equation (1) has a unique global
strong solution in the sense

  1. (i)

$u\in L^{2}(\Omega;C([0,T];L^{2}(\mathcal{O}_{\varepsilon}))$, and

  2. (ii)

for $f(u)=u^{3}-\beta u$,

$u(t)+\int_{0}^{t}(-\Delta
u(s)+f(u(s)))ds=u_{0}+\int_{0}^{t}h(s,u(s)dW_{s},\quad\;\text{a.e.}\;t\in(0,T),\;\text{a.s.}\;\omega\in\Omega.$

We are now ready to prove our main result.

###### Proof of Theorem 1.1.

Fix $0\neq u_{0}\in L^{2}(\mathcal{O}_{\varepsilon})$ and let $u(t)$ be the
corresponding solution to problem (1) with initial $u_{0}$ obtained in Theorem
3.1. Applying Itô’s formula (see e.g. [GM10, Theorem 4.3] or, more general
[GM10, Theorem 6.10]), for
$\psi(u):=\log\|u(t)\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}$, and using

$\psi^{\prime}(u)=\frac{2\langle
u,\cdot\rangle_{L^{2}(\mathcal{O}_{\varepsilon})}}{\|u\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}},\quad\psi^{\prime\prime}(u)\xi=\frac{2\langle\xi,\cdot\rangle_{L^{2}(\mathcal{O}_{\varepsilon})}}{\|u\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}}-\frac{4\langle
u,\xi\rangle_{L^{2}(\mathcal{O}_{\varepsilon})}\langle
u,\cdot\rangle_{L^{2}(\mathcal{O}_{\varepsilon})}}{\|u\|^{4}_{L^{2}(\mathcal{O}_{\varepsilon})}},\quad
u,\xi\in L^{2}(\mathcal{O}_{\varepsilon}).$

we get

$\displaystyle\log\|u(t)\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}=$
$\displaystyle\log\|u_{0}\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}-\int_{0}^{t}\frac{2}{\|u(s)\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}}\times$
(7) $\displaystyle\times\left(\|\nabla
u(s)\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}+\int_{\mathcal{O}_{\varepsilon}}f(u(s))u(s)dx-\frac{1}{2}\|h(s,u(s))\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}\right)ds$
$\displaystyle-2\int_{0}^{t}\frac{\langle
u(s),h(s,u(s))\rangle^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}}{\|u(s)\|^{4}_{L^{2}(\mathcal{O}_{\varepsilon})}}ds+2\int_{0}^{t}\frac{\langle
u(s),h(s,u(s))\rangle_{L^{2}(\mathcal{O}_{\varepsilon})}}{\|u(s)\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}}dW_{s}.$

Since $f(u)=u^{3}-\beta u$, we have

$\displaystyle\int_{\mathcal{O}_{\varepsilon}}f(u(s))u(s)dx\geq-\beta\|u(s)\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}.$
(8)

Moreover, thanks to the Poincaré inequality in (4)

$\displaystyle\|\nabla
u(s)\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}\geq\lambda^{D}_{1}(\mathcal{O}_{\varepsilon})\|u(s)\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}.$
(9)

Using assumption ($H_{2}$), we get

$\displaystyle\int_{0}^{t}\frac{\|h(s,u(s))\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}}{\|u(s)\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}}ds\leq\int_{0}^{t}\gamma(s)ds.$
(10)

Let $\delta>0$ be arbitrary. To deal with the stochastic integral, we utilize
the exponential martingale inequality (see, e.g. [Liu97, Lemma 1.1]) to get

$\displaystyle\mathbb{P}\left\\{\omega:\sup_{0\leq t\leq
w}\left[\int_{0}^{t}\frac{\langle
u(s),h(s,u(s))\rangle_{L^{2}(\mathcal{O}_{\varepsilon})}}{\|u(s)\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}}dW_{s}-\frac{\alpha}{2}\int_{0}^{t}\frac{\langle
u(s),h(s,u(s))\rangle^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}}{\|u(s)\|^{4}_{L^{2}(\mathcal{O}_{\varepsilon})}}ds\right]>\frac{2\log
k}{\alpha}\right\\}\leq\frac{1}{k^{2}}$

where $0<\alpha<1$, $w=k\delta$ and $k=1,2,\dots$. We then apply the well-
known Borel-Cantelli lemma to obtain that there exists an integer
$k_{0}(\delta,\omega)>0$ for almost all $\omega\in\Omega$ such that

$\displaystyle\int_{0}^{t}\frac{\langle
u(s),h(s,u(s))\rangle_{L^{2}(\mathcal{O}_{\varepsilon})}}{\|u(s)\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}}dW_{s}\leq\frac{2\log
k}{\alpha}+\frac{\alpha}{2}\int_{0}^{t}\frac{\langle
u(s),h(s,u(s))\rangle^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}}{\|u(s)\|^{4}_{L^{2}(\mathcal{O}_{\varepsilon})}}ds$
(11)

for all $0\leq t\leq k\delta$, $k\geq k_{0}(\delta,\omega)$. Combining
(7)–(11), we see that there exists a positive random integer $k_{1}(\delta)$
such that almost surely

$\displaystyle\log\|u(t)\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}\leq\log\|u_{0}\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}$
$\displaystyle+\frac{4\log
k}{\alpha}+(2\beta-2\lambda_{1}^{D}(\mathcal{O}_{\varepsilon}))t$
$\displaystyle+\int_{0}^{t}\gamma(s)ds-(2-\alpha)\int_{0}^{t}\frac{\langle
u(s),h(s,u(s))\rangle^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}}{\|u(s)\|^{4}_{L^{2}(\mathcal{O}_{\varepsilon})}}ds,$

for all $(k-1)\delta\leq t\leq k\delta$, $k\leq k_{1}(\delta)$, which for the
preceding $\delta>0$ together with ($H_{3}$) immediately implies that

$\displaystyle\log\|u(t)\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}\leq\log\|u_{0}\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}+\frac{4\log
k}{\alpha}+(2\beta-2\lambda_{1}^{D}(\mathcal{O}_{\varepsilon}))t+\int_{0}^{t}\gamma(s)ds-(2-\alpha)\int_{0}^{t}\rho(s)ds.$

Thus,

$\displaystyle\frac{1}{t}\log\|u(t)\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}\leq\frac{1}{t}\log\|u_{0}\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}+\frac{1}{t}\frac{4\log
k}{\alpha}+(2\beta-2\lambda_{1}^{D}(\mathcal{O}_{\varepsilon}))-(2-\alpha)\frac{1}{t}\int_{0}^{t}\rho(s)ds+\frac{1}{t}\int_{0}^{t}\gamma(s)ds.$

Taking the limsup of both sides as $t\to\infty$, using
$\limsup(-y_{n})=-\liminf y_{n}$, and the assumptions ($H_{2}$) and ($H_{3}$)

$\displaystyle\limsup_{t\to\infty}\frac{1}{t}\log\|u(t)\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}\leq
2\beta-2\lambda_{1}^{D}(\mathcal{O}_{\varepsilon})-(2-\alpha)\rho_{0}+\gamma_{0}.$

Passing to the limit $\alpha\to 0$ yields

$\displaystyle\limsup_{t\to\infty}\frac{1}{t}\log\|u(t)\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}\leq
2\beta+\gamma_{0}-2\lambda_{1}^{D}(\mathcal{O}_{\varepsilon})-2\rho_{0}\quad\text{
a.s.}$

Due to Theorem 2.2, we have the asymptotic formula

$\displaystyle\lambda_{1}^{D}(\mathcal{O}_{\varepsilon})=\begin{cases}\lambda_{1}^{D}(\mathcal{O})+O((\log\varepsilon)^{-1})&\text{
for }n=2,\\\ \lambda_{1}^{D}(\mathcal{O})+O(\varepsilon)&\text{ for }n=3.\\\
\end{cases}$

Therefore, for all $\varepsilon\leq\varepsilon_{0}$ with $\varepsilon_{0}$ is
in (2), we arrive at the estimate

$\displaystyle\limsup_{t\to\infty}\frac{1}{t}\log\|u(t)\|^{2}_{L^{2}(\mathcal{O}_{\varepsilon})}\leq
2\beta+\gamma_{0}-2\lambda_{1}^{D}(\mathcal{O})-2\rho_{0}\quad\text{ a.s.}$

It follows that for
$\lambda_{1}^{D}(\mathcal{O})+\rho_{0}>\beta+\frac{1}{2}\gamma_{0}$, the zero
solution becomes exponentially stable. ∎

Acknowledgement.

This work is carried out during an internship of Hong-Hai Ly at University of
Graz, and the university’s hospitality is acknowledged. Hong-Hai Ly is also
supported by grant SGS09/PřF/2023.

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