# On the Core-Collapse Supernova Explanation for LAMOST J1010+2358

S. K. Jeena,1 Projjwal Banerjee1, and Alexander Heger2,3,4
1Department of Physics, Indian Institute of Technology Palakkad, Kerala, India
2School of Physics and Astronomy, Monash University, Vic 3800, Australia
3OzGrav: The ARC Centre of Excellence for Gravitational Wave Discovery,
Australia
4ARC Centre of Excellence for Astrophysics in Three Dimensions (ASTRO-3D),
Australia
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Low-metallicity very massive stars with an initial mass of $\sim
140$–$260\,\mathrm{M}_{\odot}$ are expected to end their lives as pair-
instability supernovae (PISNe). The abundance pattern resulting from a PISN
differs drastically from regular core-collapse supernova (CCSN) models and is
expected to be seen in very metal-poor (VMP) stars of ${\rm[Fe/H]}\lesssim-2$.
Despite the routine discovery of many VMP stars, the unique abundance pattern
expected from PISNe has not been unambiguously detected. The recently
discovered VMP star LAMOST J1010+2358, however, shows a peculiar abundance
pattern that is remarkably well fit by a PISN, indicating the potential first
discovery of a bonafide star born from gas polluted by a PISN. In this paper,
we study the detailed nucleosynthesis in a large set of models of CCSN of Pop
III and Pop II star of metallicity ${\rm[Fe/H]}=-3$ with masses ranging from
$12$–$30\,\mathrm{M}_{\odot}$. We find that the observed abundance pattern in
LAMOST J1010+2358 can be fit at least equally well by CCSN models of $\sim
12$–$14\,\mathrm{M}_{\odot}$ that undergo negligible fallback following the
explosion. The best-fit CCSN models provide a fit that is even marginally
better than the best-fit PISN model. We conclude the measured abundance
pattern in LAMOST J1010+2358 could have originated from a CCSN and therefore
cannot be unambiguously identified with a PISN given the set of elements
measured in it to date. We identify key elements that need to be measured in
future detections in stars like LAMOST J1010+2358 that can differentiate
between CCSN and PISN origin.

###### keywords:

stars: massive – stars: Population III – stars: Population II – stars:
abundances – stars: chemically peculiar – nuclear reactions, nucleosynthesis,
abundances

††pubyear: 2023††pagerange: On the Core-Collapse Supernova Explanation for
LAMOST J1010+2358–LABEL:lastpage

##  1 Introduction

Very metal-poor (VMP) stars with $[\text{Fe}/\text{H}]\leq-2$ are crucial for
exploring the chemical evolution of the early Galaxy within the first Gyr from
the Big Bang. Until that time, core-collapse supernovae (CCSNe) resulting from
the death of massive stars of $\gtrsim 8\,\mathrm{M}_{\odot}$ are the dominant
contributors to nucleosynthesis. In particular, the surface composition of low
mass VMP stars of $\lesssim 0.8\,\mathrm{M}_{\odot}$ are fossil records of the
composition of the interstellar medium present in the early Galaxy. Current
chemical evolution models suggest that many VMP stars have formed from a gas
cloud predominantly polluted by the explosion of a single massive star (Ryan
et al., 1996; Ritter et al., 2012; Chiaki et al., 2018). This, in turn, can be
used to infer detailed information about the nucleosynthesis from individual
massive stars, which can be used to gain insight into their masses and the
associated initial mass function (IMF) of first-generation (Pop III) and early
(Pop II) massive stars. We therefore expect to observe the abundance pattern
that results from the explosion of very massive stars of $\sim
140$–$260\,\mathrm{M}_{\odot}$ that end their life as pair-instability
supernovae (PISNe) (Ober et al., 1983; Heger & Woosley, 2002) in at least some
of the low-mass VMP stars. In particular, for Pop III stars, simulations
suggest a top-heavy IMF with many very massive stars (Abel et al., 1998; Abel
et al., 2002; Hirano et al., 2015). The abundance patterns produced by Pop III
PISNe, however, are markedly different from those produced by CCSNe that
result from typical massive stars with initial masses of $\lesssim
100\,\mathrm{M}_{\odot}$ (Heger & Woosley, 2002). PISNe produce abundance
patterns that have a large deficit of odd $Z$ elements such as Na, Al, P, Cl,
and K relative to even $Z$ elements when compared to regular CCSN. Thus, the
abundance pattern in VMP stars formed from gas polluted by a single PISN
should be easily identifiable. Despite the discovery of several hundreds of
VMP stars with sufficiently detailed abundance patterns and the claim of a
potential VMP star with PISN signature (Aoki et al., 2014), until recently, no
clear candidate VMP stars have been identified that show a clear signature
arising from a PISN.

This situation, however, has changed with the recent discovery of the VMP star
LAMOST J1010+2358 (hereafter J1010+2358) by Xing et al. (2023). This star has
a peculiar abundance pattern that has been shown to be well fit by a PISN
resulting from a Pop III star of $260\,\mathrm{M}_{\odot}$ with a He core of
$130\,\mathrm{M}_{\odot}$. The peculiar features observed in the abundance
pattern of J1010+2358 that make this star stand out compared to other VMP
stars in the halo are the very low upper limit of Na abundance of
$[\text{Na}/\text{Fe}]<-2.02$ along with the highly sub-solar value of Mg of
$[\text{Mg}/\text{Fe}]=-0.66$ as well as a sub-solar value of
$[\text{Ca}/\text{Fe}]=-0.13$. The sub-solar value of $\alpha$ elements such
as Mg and Ca is usually attributed to contributions from SN 1a (Iwamoto et
al., 1999; Ohshiro et al., 2021). This star, however, also has sub-solar
values [X/Fe] for elements from Ca to Zn and, in particular, of
$[\text{Cr}/\text{Fe}]$ and $[\text{Mn}/\text{Fe}]$. Xing et al. (2023) found
that this pattern was incompatible with an abundance pattern arising from the
mixture of SN 1a and CCSN yields, leaving PISN origin as the most likely
explanation. Xing et al. (2023) also explored a large range of "classical"
massive star CCSN models from the literature, however, none of these models
provided a better match.

In this paper, we calculate the nucleosynthesis in massive stars of initial
mass ranging from $12$–$30\,\mathrm{M}_{\odot}$ that undergo CCSN with a
standard explosion energy of $1.2\times 10^{51}\,\mathrm{erg}$, with
primordial (Pop III) and $[Z]=-3$ (Pop II) initial composition. We find that
the observed abundance pattern of J1010+2358 can be fit remarkably well using
regular CCSN models provided they do not undergo fallback of material
containing Fe group elements. The quality of fit is even better or at least
comparable to those found by Xing et al. (2023) using a
$260\,\mathrm{M}_{\odot}$ of Heger & Woosley (2002).

The layout of the paper is as follows: In Section 2, we briefly describe the
methods used for the models. The details of the evolution and nucleosynthesis
in CCSN models and the best-fit models for J1010+2358, and the comparison with
PISN models are discussed in Section 3. Finally, we conclude with a summary of
the paper in Section 4.

##  2 Methods

We simulate the evolution and nucleosynthesis of non-rotating stars of initial
mass ranging from $12\,\mathrm{M}_{\odot}$ to $30\,\mathrm{M}_{\odot}$, with
an initial composition corresponding to the primordial Big Bang
nucleosynthesis that is adopted from Cyburt et al. (2002). In the mass range
of $12$–$15\,\mathrm{M}_{\odot}$ we use intervals of
$0.1\,\mathrm{M}_{\odot}$, for $15$–$20\,\mathrm{M}_{\odot}$ we use
$0.2\,\mathrm{M}_{\odot}$ intervals, and for $20$–$30\,\mathrm{M}_{\odot}$ we
use $0.5\,\mathrm{M}_{\odot}$ intervals. We designate models with primordial
metallicity as z models and label the models with their initial mass. For
example a z model of $12\,\mathrm{M}_{\odot}$ is referred to as z12. We also
simulate models on the same mass grid but with an initial metallicity of
$10^{-3}$ of the solar metallicity where we use the abundances from Big Bang
nucleosynthesis for elements up to Li and scaled solar abundances from Asplund
et al. (2009) for all elements from Be to Zn. We refer to these as the v
models which are also labelled with their initial progenitor mass. We use the
1D hydrodynamic stellar evolution code Kepler (Weaver et al., 1978; Rauscher
et al., 2003) to follow the evolution of the star from its birth to its death
via CCSN and calculate the detailed nucleosynthesis using a large adaptive co-
processing network with reaction rates based on Rauscher et al. (2002). The
explosion is modelled by a spherically symmetric piston starting from the base
of the oxygen shell that coincides with the radius where the entropy per
baryon exceeds $4\,k_{\rm B}$ similar to earlier studies such as Heger &
Woosley (2010). We label the mass coordinate corresponding to this radius as
$M_{\rm cut,ini}$, which we assume to collapse and form the proton-neutron
star.

##  3 Results and Discussion

The evolution of single massive stars and the resulting nucleosynthesis has
been studied extensively over the last several decades and our current
understanding is discussed in detail in the review by Woosley et al. (2002).
Broadly speaking, for Pop III and Pop II stars of low metallicity, stars of
progenitor mass $\sim 10$–$30\,\mathrm{M}_{\odot}$ undergo collapse of the
central Fe core that can, in many cases, result in a successful explosion via
the neutrino-driven mechanism, leading to a CCSN that leaves behind either a
neutron star or a black hole. Stars of $30$–$100\,\mathrm{M}_{\odot}$ also
undergo core collapse but are unlikely to undergo a successful explosion and
collapse into a black hole (Müller et al., 2016). Stars of $\sim
100$–$140\,\mathrm{M}_{\odot}$ have He cores of $\sim
40$–$64\,\mathrm{M}_{\odot}$ result in instability caused by electron-positron
pair creation at the core and are referred to as pulsation PISN (PPISN). Such
stars undergo several pulsations following central C burning that can eject
the entire H envelope and even some of the material from the He shell. Stars
in this mass range eventually undergo core collapse to form a black hole.
Stars of $140$–$260\,\mathrm{M}_{\odot}$ have He cores of $\sim
64$–$130\,\mathrm{M}_{\odot}$ undergo a single pulse due to pair-instability
following central C burning that completely disrupts the stars leading to an
energetic explosion resulting in a PISN.

Below, we briefly summarise some of the key features of nucleosynthesis in
non-rotating Pop III and Pop II massive stars of initial mass
$12$–$30\,\mathrm{M}_{\odot}$ that result in CCSN with a typical explosion
energy of $\sim 10^{51}\,\mathrm{erg}$. We focus on the major isotopes of key
elements. The purpose of this review is to put in context the nucleosynthesis
site for which later ejection or fallback determines the resulting abundance
patterns.

Figure 1: Elemental pattern relative to Fe from the CCSN ejecta for z models
of $12$–$30\,\mathrm{M}_{\odot}$ in steps of 2 for standard SN explosion
energy of $1.2\times 10^{51}\,\mathrm{erg}$ without mixing and fallback.

###  3.1 Nucleosynthesis up to core collapse

A massive star first undergoes core H burning followed by core He burning. The
primary product following core He burning is 12C and 16O. The next burning
stage is core C burning which primarily produces 20Ne, 24Mg, 23Na and
${}^{27\\!}$Al. Next 20Ne is burned via $(\gamma,\alpha)$, leaving behind 16O,
24Mg, and 28Si as the main product. During this stage, the $\alpha$ particles
released via $(\gamma,\alpha)$ also burn 23Na to ${}^{27\\!}$Al. Following
20Ne depletion, the core contracts and as the temperature reaches $\sim
2\times 10^{9}\,$K, core O burning ignites resulting in 28Si and 32S as the
main products. During early oxygen burning as the core grows into the O-Ne-Mg
shell that surrounds it, the Mg in the core is destroyed as well. As the core
contracts even further, core Si burning first results in a quasi-equilibrium
of isotopes of several $\alpha$ elements and odd $Z$ elements. When the core
contracts further, nuclear statistical equilibrium (NSE) is established that
is dominated by iron group isotopes which form the Fe core that ultimately
collapses into either a neutron star or a black hole. Material outside the Fe
core comprises concentric shells of progressively lighter elements that are
left behind in earlier burning stages. In addition, while the core is burning
heavier fuel in the centre, partial or even complete shell burning of lighter
fuel also takes place. This is particularly relevant during the final phase of
the star’s life after core O depletion. At that stage, a star may exhibit
concurrent shell burning of C, Ne, and even O. In particular, convective
_shell_ Ne burning usually leads to the destruction of 23Na via
23Na$(\alpha,\gamma)^{27\\!}$Al. If, however, the convective Ne shell grows
and mixes in material from the C-O shell, then the destruction of Na can be
mitigated by Na production due to C burning. Shell O burning can be an
important source of large amounts of isotopes of Si to Sc. Otherwise, these
are created mostly by explosive O burning during the CCSN (Ritter et al.,
2018). If, however, shell O burning takes place before the collapse, it can
produce isotopes of Si to Sc in amounts comparable to or in excess of what is
produced during the explosion.

Figure 2: Elemental pattern relative to Fe resulting from PISN models from
Heger & Woosley (2002) that are labelled with the corresponding He core mass
ranging from $70$–$130\,\mathrm{M}_{\odot}$. Data is adapted from StarFit
(Heger & Woosley, 2010).

In some cases, the convective O-burning shell can merge with the convective
O-Ne-Mg shell, resulting in a mixing of products of O burning further out into
the star, which can result in a large enhancement of isotopes of Si to Sc in
the final ejecta and reduced the amplitude of odd-even abundance pattern. The
mixing, however, can also lead to the destruction of Na because the material
of the O-Ne-Mg shell is mixed into the hotter regions of the O-burning shell.
In many cases we have studied, however, the combined convectively-mixed
O-burning and O-Ne-Mg shell grows and mixes in material from the C-O shell,
which can mitigate the destruction of Na. In this case, the destruction of Na
depends sensitively on the temperature at the base of the convective O burning
shell.

###  3.2 Explosive nucleosynthesis following core collapse

Explosive nucleosynthesis occurs when the CCSN shock travels through the
mantle. The high temperature in the post-shock region processes the innermost
parts of the ejecta, which comprises the O-Si shell into Fe peak elements
ranging from Ti to Zn via complete O and Si burning. The outer regions of the
O-Si and O-Ne-Mg shells undergo incomplete Si burning as well as O burning
that primarily produces isotopes of elements from Si to Sc. Further out, the
shock can also burn some of the Ne in the O-Ne-Mg shell resulting in the usual
Ne burning product, i.e., 16O and 24Mg. Notably, similar to Ne shell burning
during the pre-SN stage, explosive Ne burning destroys 23Na. The enormous
amount of neutrinos emitted by the proto-neutron star can also lead to
neutrino-induced spallation reactions ($\nu$-process) leading to substantial
production of 7Li, 11B, and 19F (Woosley et al., 1990; Heger et al., 2005).

###  3.3 Abundance Pattern from CCSN

The final abundance pattern emerging from the ejecta of a single CCSN depends
not only on the details of nucleosynthesis but also on the mixing of material
from different parts of the ejected core and the amount of material that falls
back. An exact calculation of mixing and fallback requires full 3D
hydrodynamic simulations that model the explosion using full-neutrino
transport in order to get a self-consistent explosion via the neutrino-driven
mechanism. Such computations, however, are much too expensive to be employed
on a large set of models. Usually, mixing and fallback are treated in 1D
explosion models in a parametric fashion by treating them as free parameters
to fit an abundance pattern such as Nomoto et al. (2013); Tominaga et al.
(2014). Alternatively, fallback from a spherically symmetric explosion can be
calculated by studying the long-term behaviour of the ejecta that depends on
the explosion energy, which is again a free parameter. Then the mixing is
treated in a uniform manner for all models that is only calibrated to fit the
light curve for SN 1987A for a specific progenitor Heger & Woosley (2010).

Both approaches require some amount of fallback of the innermost ejecta that
contains the Fe group elements in order to fit the abundance pattern observed
in VMP stars, in particular, to match the high $[\text{X}/\text{Fe}]\gtrsim
0.3$ for the alpha elements.

Figure 3: (a) The chemical abundances of J1010+2358 compared with the top
three best-fit models from mixing and fallback models of CCSN of
$12$–$30\,\mathrm{M}_{\odot}$ with explosion energy of $1.2\times
10^{51}\,\mathrm{erg}$. (b) Same as (a) but without any fallback.

###  3.4 CCSN Ejecta from z Models

It is instructive to look at the abundance pattern resulting from the ejecta
without any mixing and fallback. Figure 1 shows the abundance pattern from the
CCSN ejecta with fiducial explosion energy of $1.2\times
10^{51}\,\mathrm{erg}$ without any fallback for z models ranging from
$12$–$30\,\mathrm{M}_{\odot}$ where only the models with even integer masses
are shown for clarity. In addition, we also plot the z13.2 that has one of the
lowest levels of $[\text{Na}/\text{Fe}]$ and $[\text{Mg}/\text{Fe}]$ and z19.4
that undergoes merger of O burning shell with convective O-Ne-Mg shell. The
value of $[\text{X}/\text{Fe}]$ for Fe peak elements from Ti to Zn is sub-
solar in all the models and shows a clear odd-even pattern. For $\alpha$
elements from Si to Ca, a large fraction of our models have
$[\text{X}/\text{Fe}]\sim 0$. We find that such models do not undergo shell O
burning prior to collapse and all of these elements are made predominantly
during explosive O burning. On the other hand, models that undergo O shell
burning exhibit a large production of elements from Si to Sc, much more than
what is produced during explosive burning. This can be clearly seen in models
such as z16 and z18 that have super-solar vales of
$[\text{X}/\text{Fe}]\gtrsim+0.3$ for the $\alpha$ elements from Si to Ca
along with enhanced production of odd $Z$ elements from P to Sc. Models that
undergo the merger of O shell burning with the convective O-Ne-Mg shell can
have a dramatic increase in the abundance of elements from Si to Sc as seen in
the z19.4 model. These models also have enhanced odd $Z$ elements from P to Sc
with a clearly diminished odd-even pattern.

Figure 4: Same as Fig. 1 but for v models.

Mg is distinct from the heavier $\alpha$ elements such as Si and Ca as it is
almost exclusively produced during C and Ne burning prior to collapse;
explosive burning mostly leads to the destruction of Mg in the innermost part
of the O-Ne-Mg shell. The yield of Mg generally scales with the amount of mass
in the O-Ne-Mg, which on average increases with progenitor mass. As can be
seen from Fig. 1, $[\text{Mg}/\text{Fe}]\lesssim 0$ from most models with
values as low as $-0.7$ for the lower mass models. Only in models $\gtrsim
20\,\mathrm{M}_{\odot}$, $[\text{Mg}/\text{Fe}]>0$ reaches values of up to
$\sim+0.3$ due to large O-Ne-Mg shell. The trend of Al is very similar to Mg
as it is also created during C and Ne burning and typically has sub-solar
values of $[\text{Al}/\text{Fe}]$ that correlate with $[\text{Mg}/\text{Fe}]$.
Ne also behaves similarly to Mg. Almost all of the ejected Ne is produced
during C burning prior to collapse. Thus, $[\text{Ne}/\text{Fe}]$ typically
also has values $\lesssim 0$ for most of the models of
$<20\,\mathrm{M}_{\odot}$, with values as low as $-0.7$ for lower mass models
of $13$–$14\,\mathrm{M}_{\odot}$. For models of $\gtrsim
20\,\mathrm{M}_{\odot}$, the $[\text{Ne}/\text{Fe}]$ can reach super-solar
values of up to $+0.6$ due to that much larger O-Ne-Mg shells. F is
predominantly produced by neutrino spallation on Ne (Woosley et al., 1990),
which is also partially destroyed by the SN shock heating. Overall,
$[\text{F}/\text{Fe}]$ correlates with $[\text{Ne}/\text{Fe}]$ and can have
values as low as $-1.5$ in lower mass models that have very low [Ne/Fe].

Na is similar to Ne as it is produced during C burning and
$[\text{Na}/\text{Fe}]$ overall correlates with $[\text{Ne}/\text{Fe}]$ and
thus also with $[\text{F}/\text{Fe}]$. Na, however, is more susceptible to
destruction in the O-Ne-Mg shell when there is Ne shell burning. Even if Ne
barely burns, it is sufficient to destroy Na via 23Na$(\alpha,\gamma)^{27}$Al.
This occurs at the final stages of models of $\gtrsim
12.7$–$16.4\,\mathrm{M}_{\odot}$ where convective Ne burning just prior to
core-collapse burns substantial amounts of Na, leading to
$[\text{Na}/\text{Fe}]\lesssim-1$. $[\text{Na}/\text{Fe}]$ is lowest in lower
mass models of $\sim 13\,\mathrm{M}_{\odot}$, where the O-Ne-Mg shell is less
massive and a large fraction of the Na is also destroyed by the passage of the
supernova shock wave that heats the O-Ne-Mg shell to higher temperatures due
to the compact structure of the lower mass models. This can result in
$[\text{Na}/\text{Fe}]$ values as low as $\sim-2.2$. Due to Na burning into
Al, lower mass models that have a low $[\text{Na}/\text{Fe}]$ do not have
similarly low values of $[\text{Al}/\text{Fe}]$. Interestingly, the low mass
models from $12$–$12.6\,\mathrm{M}_{\odot}$ do not have a very low
$[\text{Na}/\text{Fe}]$ as they do not undergo convective Ne burning.
Furthermore, in these models, the O-Ne-Mg shell extends out to a larger radius
where the shock wave becomes less energetic, such that the destruction of Na
by shock heating is lower.

Figure 5: Same as Fig. 3 but for v models.

###  3.5 PISN Yield Pattern

Considering that Xing et al. (2023) found that their best-fit for J1010+2358
was a PISN, we briefly discuss how the PISN abundance pattern compare to CCSN
yields from our z models discussed above. Figure 2 shows the PISN abundance
pattern from Pop III stars of He core mass ranging from
$70$–$130\,\mathrm{M}_{\odot}$ that are adapted from the StarFit database
(Heger & Woosley, 2010) based on the calculations by Heger & Woosley (2002).
These models cover the mass range of stars that can undergo PISN. The figure
shows that the PISN yield pattern varies substantially with the mass of the He
core. In particular, the yield of Fe peak elements increases rapidly relative
to the light and intermediate elements up to Sc. For example, the amount of Fe
produced increases from $\sim 0.1\,\mathrm{M}_{\odot}$ for the
$70\,\mathrm{M}_{\odot}$ He core model to $\sim 40\,\mathrm{M}_{\odot}$ for
the $130\,\mathrm{M}_{\odot}$ model (Woosley et al., 2002). This results in a
large variation of $[\text{X}/\text{Fe}]$ for C to Sc ranging from highly
super-solar for lower mass models to sub-solar for the heaviest PISN model. In
particular, the abundance ratios of $\alpha$ elements from Si to Ca,
$[\text{X}/\text{Fe}]\sim 0$ found in J1010+2358 are only produced in models
with He core masses of $\gtrsim 120\,\mathrm{M}_{\odot}$. The high-mass models
are also the only ones which have large sub-solar
$[\text{Mg}/\text{Fe}]\lesssim-0.5$ along with very low
$[\text{Na}/\text{Fe}]\lesssim-2$ similar to what is found in J1010+2358.
Thus, it is clear that only the most massive PISN can match the yield pattern
found in J1010+2358. The abundance pattern from the most massive models has
similar features to some of the lower mass z models of $\sim
13$–$14\,\mathrm{M}_{\odot}$ with respect to the observed elements in
J1010+2358. Except for Na, the higher mass PISN models have large differences
in odd $Z$ elements such as F, Al, P, Cl, and K compared to the CCSN models.
Additionally, C and O are extremely deficient in higher mass PISN models
relative to Fe, which is distinct from CCSN models.

###  3.6 Best-fit CCSN from z Model for J1010+2358

Several of our z models of $12$–$30\,\mathrm{M}_{\odot}$ without fallback
produce abundance pattern with $[\text{X}/\text{Fe}]\sim 0$ for $\alpha$
elements from Si to Ca, as well as $[\text{X}/\text{Fe}]<0$ for Fe peak
elements. Additionally, $[\text{Mg}/\text{Fe}]$ and $[\text{Na}/\text{Fe}]$
can have highly sub-solar values in models of $\sim
13$–$14\,\mathrm{M}_{\odot}$. These are very similar to the key features in
the abundance pattern seen in J1010+2358. In order to find the best-fit model
for J1010+2358, however, we need to consider the abundance pattern emerging
from the ejecta from all the models from $12$–$30\,\mathrm{M}_{\odot}$
including the possibility that the ejecta from CCSN can undergo both mixing
and fallback from aspherical explosions. We model this using a prescription
similar to Tominaga et al. (2007); Ishigaki et al. (2014); Jeena et al. (2023)
where an additional mass cut $M_{\rm cut,fin}$ is introduced above which all
material is ejected whereas only a fraction $f_{\rm cut}$ of the mass $\Delta
M_{\rm cut}=M_{\rm cut,fin}-M_{\rm cut,ini}$ is ejected. In this case, $f_{\rm
cut}$ and $M_{\rm cut,fin}$ are free parameters the former varies from $0$ to
$1$ and we vary $M_{\rm cut,fin}$ in steps of $0.1\,\mathrm{M}_{\odot}$ from a
minimum value of $M_{\rm cut,ini}$ to a maximum value corresponding to the
enclosed mass of the base of the H envelope. The abundance yield, $Y_{{\rm
X}_{i}}$, of any element ${\rm X}_{i}$, defined as the sum over all isotopes
of the ejecta mass fractions divided by their corresponding mass numbers,
depends on $M_{\rm cut,fin}$ and $f_{\rm cut}$. Thus, the ratio of the total
abundance of any element $X_{i}$ relative to a reference element $X_{\rm R}$
can be written as

$\frac{N_{{\rm X}_{i}}}{N_{\rm X_{R}}}=\frac{Y_{{\rm X}_{i}}(M_{\rm
cut,fin},f_{\rm cut})}{Y_{\rm X_{R}}(M_{\rm cut,fin},f_{\rm cut})}$ (1)

The best-fit model is then found by minimizing deviation from the observed
value to the model using a $\chi^{2}$ prescription that also takes into
account the observed uncertainty $\sigma_{i}$ for each element as described in
detail in Heger & Woosley (2010) and more recently in Jeena et al. (2023).
Here we set $\sigma_{i}=\max(\sigma_{i},0.1)$ in order to avoid making
$\chi^{2}$ overly sensitive to elements that have a very low value of
$\sigma_{i}$.

Figure 3a shows the top three best-fit z models with standard explosion energy
of $1.2\times 10^{51}\,\mathrm{erg}$. We also plot the best-fit PISN model
from Pop III star $260\,\mathrm{M}_{\odot}$ reported in Xing et al. (2023),
where the data is adapted from StarFit database (Heger & Woosley, 2010). All
of the top three best-fit models provide an excellent fit to the observed
abundance pattern. The top two best-fit models, z12.8 and z12.9, have a
slightly lower $\chi^{2}$ than the best-fit PISN model whereas z12.8 has
marginally higher $\chi^{2}$. It is important to note that although the
abundance patterns from best-fit CCSN models differ dramatically from the
best-fit PISN model, which has a huge deficiency of odd $Z$ elements, for the
limited set of elements that are measured in J1010+2358, there are only very
minor differences. Among these minor differences,
$[\text{Ti}/\text{Cr}]\sim-0.1$ to $-0.2$ in our best-fit CCSN models is more
consistent with the observed value of -0.07 compared to $-0.55$ in the PISN
model. $[\text{Co}/\text{Ni}]$ is about $-0.35$ and $-0.65$ in the CCSN and
PISN model, respectively, compared to the observed value of $-0.55$. In this
case, values from both models are consistent with the observed value within
the observational uncertainties. Another difference is the lower
$[\text{Na}/\text{Fe}]$ in the PISN model compared to the CCSN model. Although
the observed upper limit is consistent with the best-fit CCSN model, it
definitely fits better with the PISN model. We find, however, that the
$[\text{Na}/\text{Fe}]$ in CCSN models can be easily reduced by $0.2$–$0.3$
when the explosion energy is increased slightly to $1.5\times
10^{51}\,\mathrm{erg}$ without affecting the other yields. We find that in all
of the best-fit CCSN models, the amount of matter that undergoes fallback,
$(1-f_{\rm cut})\Delta M_{\rm cut}$, is minimal ranging from
$0.02-0.09\,\mathrm{M}_{\odot}$. Figure 3b shows the corresponding top three
best-fit models without any fallback. As can be seen from the figure, even
without any fallback and the associated adjustable free parameters, the top
three models provide an excellent fit with slightly lower or comparable
$\chi^{2}$ compared to the best-fit PISN model.

###  3.7 CCSN Ejecta from v Models and Best-fit for J1010+2358

Figure 4 shows the abundance pattern of selected v models from
$12$–$30\,\mathrm{M}_{\odot}$ with a standard explosion energy of $1.2\times
10^{51}\,\mathrm{erg}$ without any mixing and fallback. The overall features
are very similar to those of the z models discussed earlier. Again, we find
$[\text{X}/\text{Fe}]\lesssim 0$ for Fe peak elements for all models. Also,
stars that do not undergo shell O burning have $[\text{X}/\text{Fe}]\sim 0$
for $\alpha$ elements from Si to Ca, whereas models that do undergo O burning
have a large enhancement of elements from Si to Sc. Interestingly, in the v
models, stars such as v16.4 that undergo a merger of the O burning shell with
the convective O-Ne-Mg shell, which leads to a large destruction of Na that is
not seen in the z models. The primary reason for this is that the temperature
at the base of the O burning shell in v models is slightly higher than the z
models which leads to a large destruction of Na. Such models, however, cannot
provide a good fit for J1010+2358 as they have elevated, super-solar
abundances of $\alpha$ elements from Si to Ca.

Similar to our z models, lower mass v models have sub-solar
$[\text{X}/\text{Fe}]$ for Mg, Na, Ne, and F. In slight contrast, however, the
lowest mass v models starting from $12\,\mathrm{M}_{\odot}$ undergo shell Ne
burning prior to collapse and have a structure similar to $\sim
13\,\mathrm{M}_{\odot}$ z models. Thus, these are again ideal candidates for
matching the abundance pattern for J1010+2358. Figure 5a shows the top three
best-fit v models compared to the best-fit PISN model from Xing et al. (2023).
Again, the three best-fit v models v12.1, v12.2, and v12 provide excellent
fits to the observed abundance pattern of J1010+2358 with slightly lower
$\chi^{2}$ values compared to the best-fit PISN model. Similar to the z
models, the level of fallback in the best-fit models is negligible, with
fallback masses ranging from $0$–$0.04\,\mathrm{M}_{\odot}$. Figure 5b shows
the corresponding best-fit without any fallback where, again, the top three
best-fit models provide an excellent fit with slightly lower or comparable
$\chi^{2}$ value than the best-fit PISN model.

##  4 Summary and Conclusions

We find that the peculiar abundance pattern observed in J1010+2358 can be very
well reproduced by CCSN models $\sim 12$–$13\,\mathrm{M}_{\odot}$ of both Pop
III z and Pop II v stars. We find that the quality of the best-fit from CCSN
models is marginally better than the best-fit PISN model. Remarkably, the
best-fit CCSN models are characterised by negligible fallback and thus have
essentially no free parameters to fit the abundance pattern. Judging by the
quality of fit quantified by the $\chi^{2}$, both the CCSN models and the PISN
models provide an equally good fit for the observed abundance pattern for
J1010+2358.

One of the striking differences between CCSN and PISN is indeed the large
deficiency of odd $Z$ elements F, Al, P, Cl, and K that could be used to
clearly differentiate the two scenarios. None of these elements, however, are
observed or have strong upper limits in J1010+2358. Future detection or strong
upper limits on any of these elements will allow us to distinguish between
CCSN and PISN models. Among the odd $Z$ elements, Na is a notable and crucial
exception, where the large observed deficiency of Na with an upper limit of
$[\text{Na}/\text{Fe}]<-2.02$ is not unique to PISN but can also be found in
CCSN models. This is due to the distinct origin of Na, which is produced
entirely due to C burning compared to heavier odd $Z$ elements from P to Sc
that are produced during O burning. Moreover, because Na has fewer protons
than heavier odd $Z$ elements, it is much more susceptible to being destroyed
by CCSN shock heating. Among other elements that are clearly different in the
best-fit CCSN and PISN models are C and O. Because the best-fit CCSN models do
not undergo significant fallback, they have the lowest value form
$[\text{C}/\text{Fe}]$ and $[\text{O}/\text{Fe}]$ of $\sim-0.5$ that is
possible from a CCSN model. The best-fit PISN model, however, has a much lower
value of $[\text{C}/\text{Fe}]=-1.4$ and $[\text{O}/\text{Fe}]=-0.75$. Thus, a
detection or a strong upper limit of C or O could help distinguish between
CCSN and PISN as the source of elements in J1010+2358. We note here that the
lack of heavy elements found in J1010+2358 as indicated by the strong upper
limit of $[\text{Sr}/\text{Fe}]<-2.5$ and $[\text{Ba}/\text{Fe}]<-1.17$ cannot
be used to distinguish between PISN and CCSN because neither PISN nor our
regular Pop III CCSN models produce any heavy elements. _The observational
determination of these key odd- $Z$ elements is equally critical to other
future or past PISN-origin candidates._

Another important feature of J1010+2358 is its highly sub-solar
$[\text{Mg}/\text{Fe}]=-0.66$. Such low values of Mg are usually not
attributed to CCSN, but we find that it is in fact a general feature in many
of the models of $\lesssim 20\,\mathrm{M}_{\odot}$ provided fallback is
negligible. Specifically, as long as the innermost part of the ejecta that
contains the Fe peak elements is ejected, CCSN can result in sub-solar values
of $[\text{Mg}/\text{Fe}]$. Traditionally, Mg-poor VMP stars have been
associated with pollution by SN Ia rather than a CCSN (Ivans et al., 2003; Li
et al., 2022). Our study indicated that some of the Mg-poor stars could be the
result of CCSN that do not undergo substantial fallback of Fe peak elements.
We find that the same models that do not undergo fallback and that have sub-
solar Mg also have $[\text{X}/\text{Fe}]\sim 0$ for alpha elements from Si to
Ca which is also observed in J1010+2358. In future, we plan to explore other
VMP stars that have sub-solar $[\text{Mg}/\text{Fe}]$ to see if some of them
have their origins from CCSN instead of SN 1a.

The peculiar abundance pattern measured in J1010+2358 compared to other VMP
stars measured in the Galactic halo and the rarity of such abundance patterns
potentially have important ramifications. On one hand, it could point to the
low chance of detecting a star directly formed from a gas polluted by a PISN.
This can be due partly to the rarity of the most massive He core progenitors
required for PISN to produce the Mg and Na poor pattern observed in
J1010+2358. This is in contrast to the high fraction of low-mass CCSNe
progenitors among all stars that make CCSNe in a standard initial mass
function, which, though, may not apply to Pop III stars. Another factor that
could reduce the chance of such stars occurring in nature is that the chance
of forming a star from a gas polluted exclusively by a PISN is very rare due
to the large explosion energies associated with PISN. Such energetic
explosions lead to gas outflows from the host minihalo resulting in large-
scale dilution and mixing (Chiaki et al., 2018). On the other hand, if CCSNe
are indeed the source of the elements measured in J1010+2358, the rarity of
such patterns implies that CCSN that do not undergo fallback is quite rare,
even among low-mass CCSN progenitors. This is supported by the fact that the
majority of VMP stars measured in the Galactic halo have super-solar values
for $\alpha$ elements even though many CCSN without fallback can result in
subsolar $[\text{X}/\text{Fe}]$ for Mg and solar values for Si and Ca. In
fact, in previous studies by Heger & Woosley (2010) and (Tominaga et al.,
2007), substantial fallback of the innermost ejecta containing the Fe peak
elements was usually adopted in the 1D CCSN mixing and fallback models in
order to match the abundance pattern in VMP stars. The level of fallback in
the best-fit CCSN models of $12$–$14\,\mathrm{M}_{\odot}$ and the fraction of
models that do not undergo fallback can only be clarified by a larger set of
3D simulations of CCSN explosion in future.

## Data Availability

Data is available upon reasonable request.

## Acknowledgements

This work was supported by the Science and Engineering Research Board Grant no
SRG/2021/000673. A.H. was supported by the Australian Research Council (ARC)
Centre of Excellence (CoE) for Gravitational Wave Discovery (OzGrav) through
project number CE170100004, by the ARC CoE for All Sky Astrophysics in 3
Dimensions (ASTRO 3D) through project number CE170100013, and by ARC LIEF
grants LE200100012 and LE230100063.

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# The Langevin Monte Carlo algorithm in the non-smooth log-concave case

Joseph Lehec

###### Abstract

We prove non-asymptotic polynomial bounds on the convergence of the Langevin
Monte Carlo algorithm in the case where the potential is a convex function
which is globally Lipschitz on its domain, typically the maximum of a finite
number of affine functions on an arbitrary convex set. In particular the
potential is not assumed to be gradient Lipschitz, in contrast with most
existing works on the topic.

Keywords: Statistical Sampling, Markov Chain Monte Carlo, Convexity.
MS Classification: 62D05 (68W20, 65C05, 52A23)

##  1 Introduction

### Setting.

Sampling from a high-dimensional log-concave probability measure is a problem
dating back to the early nineties and the seminal work of Dyer, Frieze and
Kannan [13] and which has many applications to various fields such as Bayesian
statistics, convex optimization and statistical inference. This problem is
always addressed via Markov Chain Monte Carlo methods, but there is a large
variety of those: Metropolis-Hastings type random walks (ball walk), Glauber
like dynamics (hit and run) or Hamiltonian Monte Carlo. In this article, we
will consider the so-called Langevin algorithm, which is defined as follows.
Given a probability measure $\mu$ on $\mathbb{R}^{n}$ we let $\varphi$ be its
potential, namely $\mu$ has density $\mathrm{e}^{-\varphi}$ with respect to
the Lebesgue measure. The Langevin diffusion associated to $\mu$ is the
solution $(X_{t})$ of the following stochastic differential equation

$dX_{t}=dB_{t}-\frac{1}{2}\nabla\varphi(X_{t})\,dt,$ (1)

where $(B_{t})$ is a standard $n$-dimensional Brownian motion. The Langevin
algorithm is the Euler scheme associated to this diffusion: Given a time step
parameter $\eta$ we let $(\xi_{k})_{k\geq 1}$ be a sequence of i.i.d. centered
Gaussian vectors with covariance $\eta\,\text{Id}$ and set

$x_{k+1}=x_{k}+\xi_{k+1}-\frac{\eta}{2}\nabla\varphi(x_{k}).$ (2)

We shall focus on the _log-concave_ case, namely the case where the potential
$\varphi$ is convex. One originality of this work is that we will consider the
_constrained_ case, allowing the measure $\mu$ to be supported on a set $K$
different from $\mathbb{R}^{n}$. In other words the potential $\varphi$ is
allowed to take the value $+\infty$ outside some set $K$. Notice that the log-
concavity assumption implies that $K$ is convex. In the constrained case the
Langevin diffusion (1) becomes

$dX_{t}=dB_{t}-\frac{1}{2}\nabla\varphi(X_{t})\,dt-d\Phi_{t},$ (3)

where $(\Phi_{t})$ is a process that repels $(X_{t})$ inward when it reaches
the boundary of $K$, see the next section for a precise definition. The
discretization then becomes

$x_{k+1}=\mathcal{P}\left(x_{k}+\xi_{k+1}-\frac{\eta}{2}\nabla\varphi(x_{k})\right),$
(4)

where $\mathcal{P}$ is the projection on $K$: For $x\in\mathbb{R}^{n}$ the
point $\mathcal{P}_{K}(x)$ is the closest point to $x$ in $K$. This is the
algorithm we will study throughout the article. It was first introduced in our
joint paper with Bubeck and Eldan [5] and to the best of our knowledge it has
not been investigated since.

The second originality of this work is that we do not assume the potential
$\varphi$ to be smooth. More precisely we will assume that the gradient of
$\varphi$ (or rather its subdifferential) is uniformly bounded on $K$, but we
do not assume it to be Lipschitz or even continuous. Let us point out that
this is by no means an exotic situation, the reader could think for instance
of $\varphi$ being the maximum of a finite number of affine functions on $K$.
We do not make any assumption whatsoever on the convex set $K$.

The drawback of this very generic situation and of our approach is that we are
only able to get convergence estimates in Wasserstein distance. Recall that
the Wasserstein distance $W_{2}$ between two probability measures $\mu$ and
$\nu$ is defined as

$W_{2}^{2}(\mu,\nu)=\inf_{X\sim\mu,Y\sim\nu}\left\\{\mathbb{E}[|X-Y|^{2}]\right\\}.$

By a slight abuse of notation, if $X,Y$ are random vectors we also write
$W_{2}(X,Y)$ for the Wasserstein distance between the law of $X$ and that of
$Y$.

### Main results.

Our main result is the following bound between the Langevin algorithm (4)
after $k$ steps and its corresponding point $X_{k\eta}$ in the true Langevin
diffusion (3).

######  Theorem 1.

Assume that $\mu$ is log-concave, with globally Lipschitz potential $\varphi$
on its support $K$ and let $L$ be the Lipschitz constant. Assume that the time
step $\eta$ satisfies $\eta<nL^{-2}$ and suppose that the Langevin algorithm
and diffusion are initiated at the same point $x_{0}$. Then for every integer
$k$ we have

$\frac{1}{n}W_{2}^{2}(X_{k\eta},x_{k})\leq A\,k\,\eta^{3/2}$ (5)

where

$A=(2\mathrm{e}^{1/2}+1)\frac{(1+\sigma_{0})(n+2\log
k)^{1/2}}{r_{0}}+\frac{7}{6}\,\frac{L}{n^{1/2}},$ (6)

and

$r_{0}=d(x_{0},\partial
K)\quad\text{and}\quad\sigma_{0}=\frac{1}{n}\left(\varphi(x_{0})-\inf_{K}\\{\varphi(y)\\}\right).$
(7)

######  Remark.

The transport cost $W_{2}^{2}$ behaves additively when taking tensor products,
so the Wasserstein distance between any two probability measures on
$\mathbb{R}^{n}$ is typically of order $\sqrt{n}$. Therefore
$\frac{1}{n}W_{2}^{2}$ is of order $1$, which explains why we wrote the
theorem this way. The reader should thus have in mind that the theorem
provides some non trivial information as soon as the right-hand side of (5) is
smaller than some small constant $\varepsilon$.

The result depends on the starting point via the parameters $r_{0}$ and
$\sigma_{0}$. In order to get a meaningful bound the algorithm should not be
initiated too close to the boundary of $K$ or at a point where the potential
$\varphi$ is too large. Let us also point out that the theorem is also valid
when there is no support constraint, namely when $K=\mathbb{R}^{n}$. One just
replaces $r_{0}$ by $+\infty$, so that $A=O(Ln^{-1/2})$ in this case. Let us
comment also on the parameter $\sigma_{0}$. Obviously $\sigma_{0}=0$ if the
potential is minimal at $x_{0}$. Fradelizi’s theorem [16, Theorem 4] asserts
that if $\mu$ is log-concave on $\mathbb{R}^{n}$ with density $f$ and has its
barycenter at $x_{0}$ then

$\sup_{x\in\mathbb{R}^{n}}\\{f(x)\\}\leq e^{n}f(x_{0}).$

In terms of the parameter $\sigma_{0}$ this means that if $\mu$ has its
barycenter at $x_{0}$ then $\sigma_{0}\leq 1$. Since $\varphi$ is assumed to
be Lipschitz with constant $L$, if $x_{0}$ is at $O(nL^{-1})$ distance from
the barycenter then again $\sigma_{0}$ is order $1$. In general we shall think
of $\sigma_{0}$ as a parameter of order $1$. Also we are never going to take
more than $poly(n)$ steps so $\log k$ will always be negligible compared to
$n$. Under the previous assumptions the parameter $A$ thus satisfies

$A=O\left(\max\left(\frac{n^{1/2}}{r_{0}};\frac{L}{n^{1/2}}\right)\right).$

In order to estimate the complexity of the Langevin algorithm, we need to
combine the previous theorem with some estimate on the speed of convergence of
the Langevin diffusion $(X_{t})$ towards its equilibrium measure $\mu$. For
this we shall use two functional inequalites, the Poincaré inequality and the
logarithmic Sobolev inequality. Recall that the measure $\mu$ is said to
satisfy the logarithmic Sobolev inequality if for every probability measure
$\nu$ on $\mathbb{R}^{n}$ we have

$D(\nu\mid\mu)\leq\frac{C_{LS}}{2}\,I(\nu\mid\mu)$ (8)

where $D(\nu\mid\mu)$ and $I(\nu\mid\mu)$ denote respectively the relative
entropy and the relative Fisher information of $\nu$ with respect to $\mu$:

$D(\nu\mid\mu)=\int_{\mathbb{R}^{n}}\log\left(\frac{d\nu}{d\mu}\right)\,d\nu\quad\text{and}\quad
I(\nu\mid\mu)=\int_{\mathbb{R}^{n}}\left|\nabla\log\left(\frac{d\nu}{d\mu}\right)\right|^{2}\,d\nu.$

The smallest constant $C_{LS}$ for which (8) holds true is called the log-
Sobolev constant of $\mu$. The factor $\frac{1}{2}$ is just a matter of
convention, with this normalization the log-Sobolev constant of the standard
Gaussian measure is $1$, in any dimension. It is well-known that the log-
Sobolev inequality is stronger than the Poincaré inequality. More precisely,
letting $C_{P}$ be the best constant in the Poincaré inequality:

$\mathrm{var}_{\mu}(f)\leq C_{P}\int_{\mathbb{R}^{n}}|\nabla f|^{2}\,d\mu,$

we have $C_{P}\leq C_{LS}$.

######  Theorem 2.

Again assume that $\mu$ is log-concave with globally Lipschitz potential on
its support, with Lipschitz constant $L$. Let $x_{0}$ be a point in the
support of $\mu$ and recall the definition (7) of $\sigma_{0}$ and $r_{0}$.
Assume in addition that the measure $\mu$ satisfies the log-Sobolev inequality
with constant $C_{LS}$. Then after $k$ steps of the Langevin algorithm started
at $x_{0}$ with time step parameter $\eta<nL^{-2}$ we have

$\frac{1}{n}W_{2}^{2}(x_{k},\mu)\leq
2B\,\mathrm{e}^{-k\eta/2C_{LS}}+2A\,k\eta^{3/2}$

where again $A$ is given by (6) and

$B=4C_{LS}\left(1+\log\left(\frac{\max(C_{LS},1)\,n}{\min(r_{0},1)}\right)+\sigma_{0}+\frac{L}{n}\right).$

######  Remark.

Note that we initiate the Langevin algorithm at a Dirac point mass, we do not
need any _warm start_ hypothesis. The starting point only plays a role through
the parameters $r_{0}$ and $\sigma_{0}$.

Let us describe what the theorem gives in terms of the complexity of the
Langevin algorithm. Say we want
$\frac{1}{n}W_{2}^{2}(x_{k},\mu)\leq\varepsilon$ for some small $\varepsilon$.
The first term of the right-hand side is a little bit intricate, so let us
assume that all parameters of the problem are at most polynomial in the
dimension. Then that term is just $poly(n)\exp(-k\eta/2C_{LS})$, which is
negligible as soon as $k\eta=\Omega(C_{LS}\log n)$. Let us also assume that
$\sigma_{x}=O(1)$ (see the discussion above). Then the theorem shows that
choosing

$\eta=\Theta^{*}\left(\frac{\varepsilon^{2}}{C_{LS}^{2}}\min\left(\frac{r_{0}^{2}}{n},\frac{n}{L^{2}}\right)\right)$

and running the algorithm for

$k=\Theta^{*}\left(\frac{C_{LS}^{3}}{\varepsilon^{2}}\max\left(\frac{n}{r_{0}^{2}};\frac{L^{2}}{n}\right)\right)$

steps produces a point $x_{k}$ satisfying
$\frac{1}{n}W_{2}^{2}(x_{k},\mu)\leq\varepsilon$. The notation $\Theta^{*}$
hides universal constants as well as possible $polylog(n)$ dependencies, this
is a common practice in this field.

Note in particular that if we treat all parameters other than the dimension as
constants, then we already get a non trivial information after a number of
steps of the algorithm which is nearly linear in the dimension.

Of course not every log-concave measure satisfies log-Sobolev, simply because
log-Sobolev implies sub-Gaussian tails. However there are a number of
interesting cases in which the log-Sobolev inequality is known to hold true,
which we list below.

  1. 1.

If the potential $\varphi$ is $\alpha$-uniformly convex for some $\alpha>0$,
in the sense that $x\mapsto\varphi(x)-\frac{\alpha}{2}|x|^{2}$ is convex, then
$\mu$ satisfies $\log$-Sobolev with constant $1/\alpha$. This is the
celebrated Bakry-Émery criterion, see [1]. See also [3] for an alternate proof
based on the Prékopa-Leindler inequality.

  2. 2.

If $\mu$ is log-concave and is supported on a ball of radius $R$, then $\mu$
satisfies log-Sobolev with constant $R^{2}$, up to a universal constant. This
follows trivially from E. Milman’s result that, within the class of log-
concave measures, Gaussian concentration and the log-Sobolev inequality are
equivalent, see [21, Theorem 1.2.], or [18, Theorem 2].

  3. 3.

If $\mu$ is log-concave, supported on a ball of radius $R$ and isotropic, in
the sense that its covariance matrix is the identity matrix, then Lee and
Vempala [19] have shown that $\mu$ satisfies log-Sobolev with constant $R$, up
to a universal factor. Note that the isotropy condition implies that
$R\geq\sqrt{n}$, so this improves greatly upon the previous result in the
isotropic case.

In the first case, notice that since the potential is at the same time
globally Lipschitz and uniformly convex, the support of $\mu$ must be bounded.
Actually if the potential is globally Lipschitz it cannot grow fast enough at
infinity to insure log-Sobolev. So if we insist on assuming that the potential
is Lipschitz and on using log-Sobolev then we have to assume that the support
is bounded.

One way around this issue is to use the Poincaré inequality rather than log-
Sobolev. Indeed every log-concave measure satisfies the Poincaré inequality.
Kannan, Lovasz and Simonovits [17] proved that the Poincaré constant of an
isotropic log-concave measure on $\mathbb{R}^{n}$ is $O(n)$ and conjectured
that it should actually be bounded. This conjecture, which was the major open
problem in the field of asymptotic convex geometry, was recently nearly solved
by Yuansi Chen [8], who proved an $n^{o(1)}$ bound for the Poincaré constant
of an isotropic log-concave vector in dimension $n$. The result of Chen relies
on a technique invented by Eldan [14] which was also used by Lee an Vempala
[19] to prove a $O(\sqrt{n})$ bound for the KLS constant, as well as the
aforementioned log-Sobolev result. Recall that if $\nu$ is a probability
measure, absolutely continuous with respect to $\mu$, the chi-square
divergence of $\nu$ with respect to $\mu$ is defined as

$\chi^{2}(\nu\mid\mu)=\int_{\mathbb{R}^{n}}\left(\frac{d\nu}{d\mu}-1\right)^{2}\,d\mu.$

Our next theorem then states as follows.

######  Theorem 3.

Assume that $\mu$ is a log-concave probability measure with globally Lipschitz
potential $\varphi$ on its domain, with constant $L$. Then after $k$ steps of
the Langevin algorithm initiated at a random point $x_{0}$ taking values in
the domain, and with time step parameter $\eta$ satisfying $\eta\leq nL^{-2}$,
we have

$\frac{1}{n}W_{2}^{2}(x_{k},\mu)\leq\frac{4}{n}\,C_{P}\chi^{2}(x_{0}\mid\mu)\mathrm{e}^{-k\eta/C_{P}}+2\,Ak\eta^{3/2}.$

where $C_{P}$ is the Poincaré constant of $\mu$ and where

$A=(2\mathrm{e}^{1/2}+1)(n+2\log
k)^{1/2}\,\mathbb{E}\left[\frac{1+\sigma_{0}}{r_{0}}\right]+\frac{7}{6}\,\frac{L}{n^{1/2}}.$

Note that $r_{0}$ and $\sigma_{0}$ are random here.

So the price we have to pay for using Poincaré rather than log-Sobolev is a
warm start hypothesis: the algorithm must be initiated at a random point
$x_{0}$ whose chi-square divergence to $\mu$ is finite. In the unconstrained
case, namely when $\mu$ is supported on the whole $\mathbb{R}^{n}$, a natural
choice for a warm start is an appropriate Gaussian measure. One can indeed get
the following estimate.

######  Lemma 4.

Suppose $\mu$ is log-concave, supported on the whole $\mathbb{R}^{n}$, with
globally Lipschitz potential, with Lipschitz constant $L$. Let $\gamma$ be the
Gaussian measure centered at a point $x_{0}$ and with covariance
$\frac{n}{L^{2}}Id$. Then

$\log\chi^{2}(\gamma\mid\mu)\leq
n(1+\sigma_{0})+\frac{n}{2}\log\left(\frac{L^{2}C_{P}}{n}\right),$

where $C_{P}$ is the Poincaré constant of $\mu$.

In particular when $\sigma_{0}=O(1)$ and all other parameters of the problem
are at most polynomial in $n$, we get $\log\chi_{2}^{2}(\gamma\mid\mu)\leq
O(n\log n)$. With this choice of a warm start, and observing that in the
unconstrained case the parameter $A$ is just $O(L/\sqrt{n})$, the previous
theorem gives $\frac{1}{n}W_{1}^{2}(x_{k},\mu)\leq\varepsilon$ after

$k=\Theta^{*}\left(\frac{C_{P}^{3}L^{2}n^{2}}{\varepsilon^{2}}\right)$

steps, with $\eta$ chosen appropriately. Also in the constrained case, one can
get a non trivial complexity estimate from Theorem 3 by choosing the uniform
measure on a ball contained in the support as a warm start. We leave this
annoying computation to the reader.

Finally let us point out that it is also possible to obtain interesting bounds
from our result when the potential is not globally Lipschitz, simply by
restricting the measure to a large ball. For simplicity let us only spell out
the argument when the measure $\mu$ is supported on the whole $\mathbb{R}^{n}$
and when we have a linear control on the gradient of the potential, but the
method could give non trivial bounds in more general situations. So let $\mu$
be a log-concave measure supported on the whole $\mathbb{R}^{n}$, let
$\varphi$ be its potential, and consider the Langevin algorithm associated to
the measure $\mu$ conditioned on the ball of radius $R$:

$x_{k+1}=\mathcal{P}\left(x_{k}+\sqrt{\eta}\xi_{k+1}-\frac{\eta}{2}\nabla\varphi(x_{k})\right),$
(9)

where $\mathcal{P}$ is the orthogonal projection on the ball of radius $R$:

$\mathcal{P}(x)=\begin{cases}x&\text{if }|x|\leq R\\\
\frac{Rx}{|x|}&\text{otherwise}\end{cases}$

In this special case, Theorem 2 yields the following complexity for the
Langevin algorithm.

######  Theorem 5.

Assume that $\mu$ is log-concave, supported on the whole $\mathbb{R}^{n}$ and
that the gradient of the potential $\varphi$ grows at most linearly:

$|\nabla\varphi(x)|\leq\beta(|x|+1),$

for all $x\in\mathbb{R}^{n}$ and for some $\beta>0$. Assume that the Langevin
algorithm is initiated at $0$, that $\sigma_{0}=O(1)$, that
$\int|x|^{2}\,d\mu=O(n)$, and let $C_{LS}$ be the log-Sobolev constant of
$\mu$, with the convention that it equals $+\infty$ if $\mu$ does not satisfy
log-Sobolev. Then choosing $R=\Theta^{*}(\sqrt{n})$,
$\eta=\Theta^{*}\left(\varepsilon^{2}\max(\beta,1)^{-2}\min(C_{LS},n)^{-2}\right)$
and running the algorithm (9) initiated at $0$ for

$k=\Theta^{*}\left(\frac{\min(C_{LS},n)^{3}\max(\beta,1)^{2}}{\varepsilon^{2}}\right)$

steps produces a point $x_{k}$ satisfying
$\frac{1}{n}W_{2}^{2}(x_{k},\mu)\leq\varepsilon$.

Note in particular that in the case where $C_{LS}$ and $\beta$ are of constant
order the complexity does not depend on the dimension.

### Related works.

We end this introduction with a discussion on a short selection of related
works. Let us first mention that as far as we know, the Langevin algorithm
with support constraint was only investigated in our previous paper with
Bubeck and Eldan [5]. In this paper the potential was assumed to be gradient
Lipschitz. In all the works that we could find on the Langevin Monte Carlo
algorithm the potential is always assumed to be smooth, most of the time
gradient Lipschitz. This hypothesis is somewhat relaxed in the recent article
[7], but the authors analyze the Langevin algorithm for a smoothed out
approximation of $\mu$, and in any case they still require the gradient of the
potential to be Hölder continuous. The present work appears to be the first
were $\nabla\varphi$ is allowed to be discontinuous.

Let us give the state of the art convergence bounds for the Langevin algorithm
in the smooth, unconstrained case. The first quantitative result appears to be
Dalalyan’s article [9]. The result is in total variation distance rather than
Wasserstein but as in the present work the strategy consists in writing

$TV(x_{k},\mu)\leq TV(x_{k},X_{k\eta})+TV(X_{k\eta},\mu)$ (10)

and estimating both terms separately. His assumption is that the potential
$\varphi$ satisfies

$\alpha Id\leq\nabla^{2}\varphi\leq\beta Id$

pointwise on the whole $\mathbb{R}^{n}$, where $\alpha$ and $\beta$ are
positive constants. Actually a closer look at his argument shows that he does
not really use log-concavity. Indeed, his main contribution is a bound for the
relative entropy of the Langevin algorithm at time $k$ with respect to the
corresponding point in the Langevin diffusion. That part of the argument is a
nice application of Girsanov’s formula and does not use log-concavity at all,
only the fact that $\nabla\varphi$ is Lipschitz is needed. Dalalyan only uses
strict log-concavity to estimate how fast the diffusion $(X_{t})$ converges to
$\mu$. But that only requires Poincaré for an exponentially fast decay in chi-
square divergence or log-Sobolev for a decay in relative entropy. Dalalyan’s
theorem can thus be rewritten as follows: if $d\mu=\mathrm{e}^{-\varphi}\,dx$
is supported on the whole $\mathbb{R}^{n}$, if $\nabla\varphi$ is Lipschitz
with constant $\beta$ and if $\mu$ satisfies the log-Sobolev inequality with
constant $C_{LS}$ then after $k$ steps of the Langevin algorithm with times
step parameter $\eta$ we have

$TV(x_{k},\mu)\leq D(x_{0}\mid\mu)^{1/2}\mathrm{e}^{-k\eta/2C_{LS}}+\beta
n^{1/2}(1+\mathbb{E}[\sigma_{0}])^{1/2}k^{1/2}\eta,$

where again
$\sigma_{0}=\frac{1}{n}\left(\varphi(x_{0})-\min_{\mathbb{R}^{n}}\\{\varphi\\}\right)$.
The result depends on a warm start hypothesis, the algorithm must be initiated
from a random point $x_{0}$ whose relative entropy to the target measure is
finite. On the other hand, it is not hard to see that one can find a Gaussian
measure whose relative entropy to $\mu$ is $O^{*}(n)$. As a result, it follows
from the previous bound that if $\eta$ is chosen appropriately then one has
$TV(x_{k},\mu)\leq\varepsilon$ after

$k=\Theta^{*}\left(\frac{C_{LS}^{2}\beta^{2}n}{\varepsilon^{2}}\right)$

steps of the algorithm.

Durmus and Moulines [12] have the same set of hypothesis as Dalalyan but they
prove a result in Wasserstein distance rather than total variation. As opposed
to Dalayan they really use the hypothesis $\nabla^{2}\varphi\geq\alpha Id$ for
some positive $\alpha$. Also their approach is a bit different from that of
Dalalyan: instead of bounding $W_{2}(x_{k},X_{k\eta})$ and
$W_{2}(X_{k\eta},\mu)$ separately they directly obtain a recursive inequality
for $W_{2}(x_{k},\mu)$. Their approach essentially yields the following
result: Suppose that $\alpha Id\leq\nabla^{2}\varphi\leq\beta Id$ pointwise on
the whole $\mathbb{R}^{n}$ for some positive constants $\alpha,\beta$. Assume
also that the time step parameter $\eta$ satisfies $\eta\leq\frac{1}{2\beta}$.
Then

$W_{2}(x_{k},\mu)\leq\left(1-\frac{\alpha\eta}{2}\right)^{k}W_{2}(x_{0},\mu)+\frac{2\beta}{\alpha}n^{1/2}\eta^{1/2}.$
(11)

Actually, the result of Durmus and Moulines is a bit more involved, for a
(short) proof of that very statement see [10, Theorem 1.1].

This result implies that $\frac{1}{n}W_{2}^{2}(x_{k},\mu)\leq\varepsilon$
after a number of steps
$k=O^{*}\left(\frac{\beta^{2}}{\alpha^{3}\varepsilon}\right)$, with time step
parameter of order $\varepsilon\alpha^{2}/\beta^{2}$. This should be compared
to the complexity given by Theorem 5 in this case. Indeed, observe that the
hypothesis $\alpha Id\leq\nabla^{2}\varphi\leq\beta Id$ implies that the log-
Sobolev constant is $1/\alpha$ at most and that $\nabla\varphi$ grows
linearly. Therefore Theorem 5 applies, and it gives the following complexity:
$k=\Theta^{*}\left(\frac{\beta^{2}}{\alpha^{3}\varepsilon^{2}}\right)$. The
dependence in $\varepsilon$ is thus worse ($\varepsilon^{2}$ rather
$\varepsilon$) but the dependence in the other parameters is the same, which
is quite remarkable given the fact that Theorem 5 holds under considerably
weaker assumptions.

Lastly, let us also mention Vempala and Wibisono’s work [25] whose approach is
similar in spirit to that of Durmus and Moulines but gives a result closer to
Dalalyan’s. They prove that if $\nabla\varphi$ is Lipschitz with constant
$\beta$, if $\mu$ satisfies log-Sobolev with constant $C_{LS}$, and if the
time step parameter satisfies $\eta\leq 1/(4C_{LS}\beta^{2})$ then after $k$
steps of the algorithm one has

$D(x_{k}\mid\mu)\leq\mathrm{e}^{-k\eta/C_{LS}}D(x_{0}\mid\mu)+8n\beta^{2}C_{LS}\eta.$
(12)

As in the result of Dalalyan the measure $\mu$ is not assumed to be log-
concave, only log-Sobolev is required. Let us note that this result recovers
Dalalyan’s by Pinsker’s inequality. Let us also remark that combining it with
the transport inequality $W_{2}^{2}(x_{k},\mu)\leq 2C_{LS}\,D(x_{k}\mid\mu)$,
which is a consequence of log-Sobolev, one gets

$W_{2}^{2}(x_{k},\mu)\leq
2C_{LS}D(x_{0}\mid\mu)\mathrm{e}^{-k\eta/C_{LS}}+16\,n\,\beta^{2}C_{LS}^{2}\eta.$

This pretty much recovers (11) under a weaker hypothesis: Log-Sobolev rather
than uniform convexity of the potential, with two caveats: The hypothesis on
the time step parameter is a bit more restrictive, and this is from a warm
start in the relative entropy sense.

### Acknowledgments.

The author is grateful to Sébastien Bubeck and Ronen Eldan for a number of
useful discussions related to this work. We are also grateful to Andre
Wibisono who brought to our attention the $W_{2}$/chi-square inequality used
in the proof of Theorem 3. In the first version of the paper the formulation
of that theorem was slightly weaker, with $W_{1}$ in place $W_{2}$.

##  2 The Langevin diffusion with reflected boundary condition

In section we define properly the Langevin diffusion with reflection at the
boundary of $K$:

$dX_{t}=dB_{t}-\frac{1}{2}\nabla\varphi(X_{t})-d\Phi_{t}.$

Recall that $d\mu=\mathrm{e}^{-\varphi}dx$ is assumed to be log-concave, which
means that $\varphi\colon\mathbb{R}^{n}\to\mathbb{R}\cup\\{+\infty\\}$ is
convex. Actually, it will be slightly more convenient for our purposes to
assume that the domain of $\varphi$ is the whole $\mathbb{R}^{n}$ and that the
measure $\mu$ is given by
$\mu(dx)=\mathbf{1}_{K}(x)\mathrm{e}^{-\varphi(x)}\,dx$ where $K$ is a convex
subset of $\mathbb{R}^{n}$ with non empty interior. Since we do not assume the
potential $\varphi$ to be everywhere differentiable, the expression
$\nabla\varphi(X_{t})$ needs to be clarified. Let us agree on the convention
that in the sequel $\nabla\varphi(x)$ stands for the element of the
subdifferential of $\varphi$ at point $x$ whose Euclidean norm is minimal.
Since $\varphi$ is assumed to be a convex function whose domain is the whole
$\mathbb{R}^{n}$, for every $x\in\mathbb{R}^{n}$ the subdifferential of
$\varphi$ at $x$ is a non empty closed convex set, so that the Euclidean norm
does uniquely attain its minimum on this set.

According to Tanaka [24, Theorem 3.1], given a continuous semi-martingale
$(W_{t})$ taking values in $\mathbb{R}^{n}$ and satisfying $W_{0}\in K$, there
exists a unique couple $(X_{t},\Phi_{t})$ of continuous semi-martingales such
that, almost surely

  1. 1.

$X_{t}\in K$ for all $t\in\mathbb{R}_{+}$,

  2. 2.

$X_{t}=W_{t}-\Phi_{t}$ for all $t\in\mathbb{R}_{+}$,

  3. 3.

$(\Phi_{t})$ is of the form $\Phi_{t}=\int_{0}^{t}\nu_{s}\,d\ell_{s}$ where
$\ell$ is a measure on $\mathbb{R}_{+}$ which is finite on bounded intervals
and supported on the set $\\{t\in\mathbb{R}_{+}\colon X_{t}\in\partial K\\}$,
and for any such $t$ the vector $\nu_{t}$ is an outer unit normal to the
boundary of $K$ at $X_{t}$.

In words the process $(\Phi_{t})$ is a finite variation and continuous process
which repels $(X_{t})$ inwards when it reaches the boundary of $K$. In the
sequel we shall say that the process $(X_{t})$ is the _reflection_ of
$(W_{t})$ at the boundary of $K$ and that the process $(\Phi_{t})$ is
_associated_ to $(X_{t})$. The process $(\ell_{t})$ is called the _local time_
of $(X_{t})$ at the boundary of $K$.

Now given a standard Brownian motion $(B_{t})$ and a starting point $x\in K$,
we want to argue that there exists a unique process $(X_{t})$ such that
$(X_{t})$ is the reflection at the boundary of $K$ of the semi-martingale

$x+B_{t}-\frac{1}{2}\int_{0}^{t}\nabla\varphi(X_{s})\,ds.$

In other words we want

$dX_{t}=dB_{t}-\frac{1}{2}\nabla\varphi(X_{t})\,dt-d\Phi_{t},$

where $(\Phi_{t})$ is associated to $(X_{t})$. This is a stochastic
differential equation with reflected boundary condition. If $\nabla\varphi$ is
Lipschitz continuous then again Tanaka [24, Theorem 4.1] shows that this
equation admits a unique strong solution. Let us now explain why Tanaka’s
result remains valid in the present context, even though $\nabla\varphi$ is
not assumed to be continuous.

First of all, notice that since $\nabla\varphi$ is the gradient of a convex
function, it is a monotone map, in the sense that

$\langle x-y,\nabla\varphi(x)-\nabla\varphi(y)\rangle\geq 0,\quad\forall
x,y\in\mathbb{R}^{n}.$

This property immediately implies pathwise uniqueness for the equation. Indeed
suppose that $X$ and $\widetilde{X}$ are two solutions of the equation and
that $\Phi$ ant $\widetilde{\Phi}$ are the associated processes. Then

$d|X_{t}-\widetilde{X}_{t}|^{2}=-\frac{1}{2}\langle
X_{t}-\widetilde{X}_{t},\nabla\varphi(X_{t})-\nabla\varphi(\widetilde{X}_{t})\rangle\,dt-\langle
X_{t}-\widetilde{X}_{t},d\Phi_{t}\rangle-\langle\widetilde{X}_{t}-X_{t},d\widetilde{\Phi}_{t}\rangle.$

The first term of the right-hand side is non positive by monotony of
$\nabla\varphi$. The second term is also non positive. Indeed the fact that
$\Phi$ is associated to $X$ and $\widetilde{X}$ takes values in $K$ imply that
$\langle X_{t}-\widetilde{X}_{t},d\Phi_{t}\rangle\geq 0$ for all $t$.
Similarly $\langle\widetilde{X}_{t}-X_{t},d\widetilde{\Phi}_{t}\rangle\geq 0$.
Therefore the quantity $|X_{t}-\widetilde{X}_{t}|$ is almost surely non
increasing, which obviously implies that the equation has the pathwise
uniqueness property. To get existence, one option is to approximate $\varphi$
by a smooth convex function and pass to the limit, as in [6]. That paper is a
little involved and is written French so let us give an alternative argument
for completeness. This argument only works when $\nabla\varphi$ is bounded,
but this is the only case we shall consider here. When $\nabla\varphi$ is
bounded, it is well known that an application of Girsanov yields the existence
of a solution to the equation. Indeed, let $X$ be the reflection at the
boundary of $K$ of the process $x+B$. We have

$dX_{t}=dB_{t}-d\Phi_{t},$

where $\Phi$ is associated to $X$. Since $\nabla\varphi$ is bounded the
process $(D_{t})$ given by

$D_{t}=\exp\left(-\frac{1}{2}\int_{0}^{t}\langle\nabla\varphi(X_{s}),dB_{s}\rangle-\frac{1}{8}\int_{0}^{t}|\nabla\varphi(X_{s})|^{2}\,ds\right)$

is a positive martingale with expectation $1$. If we fix a time horizon $T>0$
and define a new probability measure by $d\mathbb{Q}=D_{T}\,d\mathbb{P}$, then
by Girsanov, the process $(\widetilde{B}_{t})_{t\in[0,T]}$ given by

$\widetilde{B}_{t}=B_{t}+\frac{1}{2}\int_{0}^{t}\nabla\varphi(X_{s})\,ds,$

is a standard Brownian motion under the new measure $\mathbb{Q}$. Since

$dX_{t}=d\widetilde{B}_{t}-\frac{1}{2}\nabla\varphi(X_{t})\,dt+d\Phi_{t},$

where $\Phi$ is associated to $X$ this shows that under $\mathbb{Q}$ the
process $X$ solves the equation driven by $\widetilde{B}$. This proves _weak_
existence of a solution, in the sense that we had to change the probability
space and the Brownian motion. However it is well known that weak existence
and pathwise uniqueness altogether imply strong existence, see for instance
[15, Chapter IV, Theorem 1.1]. Strictly speaking this only shows strong
existence on a finite time interval $[0,T]$. But we can eventually let $T$
tend $+\infty$ and use pathwise uniqueness again to get strong existence of a
solution defined for all time. Details are left to the reader.

The solution $(X_{t})$ of the equation is a Markov process, whose semigroup is
denoted $(P_{t})$ in the sequel: For every test function
$f\colon\mathbb{R}^{n}\to\mathbb{R}$ and every $x\in K$

$P_{t}f(x)=\mathbb{E}_{x}[f(X_{t})]$

where the subscript $x$ next to the expectation denotes the starting point of
$X_{t}$. By Itô’s formula, if $f$ is $\mathcal{C}^{2}$-smooth in a
neighborhood of $K$ then

$df(X_{t})=\langle\nabla f(X_{t}),dB_{t}\rangle-\frac{1}{2}\langle\nabla
f(X_{t}),\nabla\varphi(X_{t})\rangle\,dt+\frac{1}{2}\Delta
f(X_{t})\,dt-\langle\nabla f(X_{t}),d\Phi_{t}\rangle.$

Here we are using Itô’s formula for a continuous semi-martingale having a
finite variation part, see for instance [23, Chapter IV, Corollary 32.10]. If
$f$ satisfies the Neumann boundary condition: $\langle\nabla f(x),\nu\rangle$
for every $x\in\partial K$ and every $\nu$ that is normal to the boundary of
$K$ at $x$ then the last term of the right-hand side vanishes. Taking
expectation we then see that the generator of the semigroup $(P_{t})$ is

$\frac{1}{2}Lf:=\frac{1}{2}\left(\Delta f-\langle\nabla
f,\nabla\varphi\rangle\right)$

with Neumann boundary condition. Also, an integration by part then shows that

$\int_{K}(Lf)g\,d\mu=-\int_{K}\langle\nabla f,\nabla g\rangle\,d\mu,$

for every $f,g$ in the domain of $L$. In particular the operator $L$ is
symmetric in $L^{2}(\mu)$, which implies that $\mu$ is a reversible measure
for the semigroup $(P_{t})$.

##  3 Discretization of the Langevin diffusion

In this section we prove Theorem 1. This is the main contribution of the
article. We begin with a bound on the local time $(\ell_{t})$ of the diffusion
$(X_{t})$ at the boundary of $K$. We need to show that $\ell_{t}=O(t)$. That
lemma is essentially taken from our previous work with Bubeck and Eldan [5],
except that we have simplified the proof and improved the result a bit.

######  Lemma 6.

Assume that the Langevin diffusion $(X_{t})$ is initiated at point $x_{0}$ in
the interior of $K$ and recall the definition (7) of $r_{0}$ and $\sigma_{0}$.
Then for every $t>0$ we have

$\mathbb{E}[\ell_{t}^{2}]^{1/2}\leq\frac{n(1+\sigma_{0})t}{r_{0}}.$

###### Proof.

By Itô’s formula we have

$d|X_{t}-x_{0}|^{2}=2\langle X_{t}-x_{0},dB_{t}\rangle-\langle
X_{t}-x_{0},\nabla\varphi(X_{t})\rangle dt-2\langle
X_{t}-x_{0},d\Phi_{t}\rangle+n\,dt.$

Recall that $d\Phi_{t}=\nu_{t}\,d\ell_{t}$ where $\nu_{t}$ is an outer unit
normal at $X_{t}$. By definition of $(\Phi_{t})$, $r_{0}$ and $\ell_{t}$ we
have

$\langle X_{t}-x_{0},d\Phi_{t}\rangle\geq\sup_{x\in K}\langle
x-x_{0},d\Phi_{t}\rangle\geq r_{0}d\ell_{t}.$

Also by convexity of $\varphi$

$-\langle
X_{t}-x_{0},\nabla\varphi(X_{t})\rangle\leq\varphi(x_{0})-\varphi(X_{t})\leq
n\sigma_{0}.$

We thus obtain

$|X_{t}-x_{0}|^{2}+2r_{0}\ell_{t}\leq n(1+\sigma_{0})t+\int_{0}^{t}\langle
X_{s}-x_{0},dB_{s}\rangle.$ (13)

Taking expectation already gives a bound on the first moment of $\ell_{t}$. To
get a bound on the second moment observe that (13) implies that

$4r_{0}^{2}\mathbb{E}[\ell_{t}^{2}]\leq
n^{2}(1+\sigma_{0})^{2}t^{2}+\mathbb{E}\left[\left(\int_{0}^{t}\langle
X_{s}-x_{0},dB_{s}\rangle\right)^{2}\right].$

By Itô’s isometry and using (13) again, this time to bound
$\mathbb{E}[|X_{t}-x_{0}|^{2}]$, we get

$\mathbb{E}\left[\left(\int_{0}^{t}\langle
X_{s}-x_{0},dB_{s}\rangle\right)^{2}\right]=\mathbb{E}\left[\int_{0}^{t}|X_{s}-x_{0}|^{2}\,ds\right]\leq
n(1+\sigma_{0})\frac{t^{2}}{2}.$

Plugging this into the previous display yields the desired inequality. ∎

We also need the following elementary bound on the maximum of Gaussian
vectors. We provide a proof for completeness.

######  Lemma 7.

Let $G_{1},\dotsc,G_{k}$ be standard Gaussian vectors on $\mathbb{R}^{n}$.
Then

$\mathbb{E}\left[\max_{i\leq
k}\left\\{|G_{i}|^{2}\right\\}\right]\leq\mathrm{e}(n+2\log k).$

###### Proof.

Set $\chi_{i}=|G_{i}|^{2}$ for every $i$. The $p$-th moment of $\chi_{i}$
satisfies

$\mathbb{E}[\chi_{i}^{p}]=\frac{2^{p}\Gamma\left(\frac{n}{2}+p\right)}{\Gamma\left(\frac{n}{2}\right)}\leq(n+2(p-1))^{p},$

at least when $p$ is an integer. Therefore

$\mathbb{E}\left[\max_{i\leq k}\\{\chi_{i}\\}\right]\leq\left[\sum_{i\leq
k}\mathbb{E}[\chi_{i}^{p}]\right]^{1/p}\leq k^{1/p}(n+2(p-1))$

Choosing $p$ to be the smallest integer larger than $\log k$ yields the
result. ∎

We are now in a position to prove the main result.

###### Proof of Theorem 1.

We first couple the diffusion $(X_{t})$ and its discretization $(x_{k})$ in
the most natural way one could think of, by choosing the sequence $(\xi_{k})$
as follows:

$\xi_{k}=B_{k\eta}-B_{(k-1)\eta},\quad k\geq 1.$ (14)

Observe that for any $x\in K$ and $y\in\mathbb{R}^{n}$ we have
$|x-\mathcal{P}_{K}(y)|\leq|x-y|$. Therefore

$\begin{split}|X_{(i+1)\eta}-x_{i+1}|^{2}&=\left|X_{(i+1)\eta}-\mathcal{P}\left(x_{i}+\xi_{i+1}-\frac{\eta}{2}\nabla\varphi(x_{i})\right)\right|^{2}\\\
&\leq\left|X_{(i+1)\eta}-x_{i}-\xi_{i+1}+\frac{\eta}{2}\nabla\varphi(x_{i})\right|^{2}.\end{split}$

Let $(\widetilde{X}_{t})$ be the process defined by

$\widetilde{X}_{t}=x_{i}+B_{t}-B_{i\eta}-\frac{t-i\eta}{2}\nabla\varphi(x_{i})$

for all $t$ between $i\eta$ and $(i+1)\eta$. Then
$\widetilde{X}_{i\eta}=x_{i}$ and the previous display can be rewritten as

$|X_{(i+1)\eta}-x_{i+1}|^{2}\leq|X_{(i+1)\eta}-\widetilde{X}_{(i+1)\eta}|^{2}.$

The process $(X_{t}-\widetilde{X}_{t})$ is continuous with finite variation on
$[i\eta,(i+1)\eta]$ (the Brownian part cancels out). Therefore, on that
interval we have

$\begin{split}d|X_{t}-\widetilde{X}_{t}|^{2}&=2\langle
X_{t}-\widetilde{X}_{t},dX_{t}-d\widetilde{X}_{t}\rangle\\\ &=-\langle
X_{t}-\widetilde{X}_{t},\nabla\varphi(X_{t})-\nabla\varphi(x_{i})\rangle\,dt-2\langle
X_{t}-\widetilde{X}_{t},d\Phi_{t}\rangle.\end{split}$

Again by monotony of $\nabla\varphi$ we have $\langle
X_{t}-x_{i},\nabla\varphi(X_{t})-\nabla\varphi(x_{i})\rangle\geq 0$. Also
since $x_{i}\in K$ and $(\Phi_{t})$ is associated to $(X_{t})$ we have
$\langle X_{t}-x_{i},d\Phi_{t}\rangle\geq 0$. Plugging this back in the
previous display yields

$d|X_{t}-\widetilde{X}_{t}|^{2}\leq\langle\widetilde{X}_{t}-x_{i},(\nabla\varphi(X_{t})-\nabla\varphi(x_{i}))\,dt+2d\Phi_{t}\rangle.$

Now we replace $\widetilde{X}_{t}$ by its value, and we integrate between
$i\eta$ and $(i+1)\eta$. We get

$\begin{split}|X_{(i+1)\eta}-x_{i+1}|^{2}&\leq|X_{i\eta}-x_{i}|^{2}\\\
&+\int_{i\eta}^{(i+1)\eta}\langle
B_{t}-B_{i\eta}-\frac{t-i\eta}{2}\nabla\varphi(x_{i}),(\nabla\varphi(X_{t})-\nabla\varphi(x_{i}))\,dt+2d\Phi_{t}\rangle.\end{split}$

We now take expectation. Note that the martingale property of the Brownian
motion implies that $\mathbb{E}[\langle
B_{t}-B_{i\eta},\nabla\varphi(x_{i})\rangle]=0$ and that

$\mathbb{E}[\langle B_{t}-B_{i\eta},d\Phi_{t}\rangle]=\mathbb{E}[\langle
B_{(i+1)\eta}-B_{i\eta},d\Phi_{t}\rangle].$

We also use the hypothesis $|\nabla\varphi|\leq L$ and the inequality
$\mathbb{E}[|B_{t}-B_{i\eta}|]\leq n^{1/2}(t-i\eta)^{1/2}$. We obtain

$\begin{split}\mathbb{E}\left[|X_{(i+1)\eta}-x_{i+1}|^{2}\right]&\leq\mathbb{E}\left[|X_{i\eta}-x_{i}|^{2}\right]+\frac{2}{3}L\eta^{3/2}n^{1/2}\\\
&+2\,\mathbb{E}\left[\langle\xi_{i+1},\Phi_{(i+1)\eta}-\Phi_{i\eta}\rangle\right]+\frac{1}{2}L^{2}\eta^{2}+L\eta\,\mathbb{E}[\ell_{(i+1)\eta}-\ell_{i\eta}],\end{split}$
(15)

where $\xi_{i+1}=B_{(i+1)\eta}-B_{i\eta}$. By Cauchy-Schwarz and Lemma 7

$\begin{split}\mathbb{E}\left[\sum_{i=0}^{k-1}\langle\xi_{i+1},\Phi_{(i+1)\eta}-\Phi_{i\eta}\rangle\right]&\leq\mathbb{E}\left[\max_{i\leq
k}\left\\{|\xi_{i}|\right\\}\,\ell_{k\eta}\right]\\\
&\leq\mathrm{e}^{1/2}(n+2\log
k)^{1/2}\eta^{1/2}\mathbb{E}[\ell_{k\eta}^{2}]^{1/2}.\end{split}$

Summing (15) over $i$ thus yields

$\begin{split}\mathbb{E}\left[|X_{k\eta}-x_{k}|^{2}\right]&\leq
2\mathrm{e}^{1/2}(n+2\log
k)^{1/2}\eta^{1/2}\mathbb{E}[\ell_{k\eta}^{2}]^{1/2}\\\
&+L\eta\,\mathbb{E}[\ell_{k\eta}]+\frac{2}{3}Ln^{1/2}k\eta^{3/2}+\frac{1}{2}L^{2}k\eta^{2}.\end{split}$

Now recall from Lemma 6 that

$\mathbb{E}[\ell_{k\eta}^{2}]^{1/2}\leq\frac{n(1+\sigma_{0})k\eta}{r_{0}}.$

Lastly, we use the assumption $\eta<n/L^{2}$ to simplify the inequality a bit.
We finally obtain

$\mathbb{E}\left[|X_{k\eta}-x_{k}|^{2}\right]\leq(2\mathrm{e}^{1/2}+1)(n+2\log
k)^{1/2}\frac{n(1+\sigma_{0})}{r_{0}}k\eta^{3/2}+\frac{7}{6}Ln^{1/2}k\eta^{3/2},$

which is the result. ∎

##  4 Convergence of the algorithm under log-Sobolev

In this section we prove Theorem 2. Observe first that the Wasserstein
distance is indeed a distance, so it satisfies the triangle inequality and we
have

$\frac{1}{n}W_{2}^{2}(x_{k},\mu)\leq\frac{2}{n}W_{2}^{2}(x_{k},X_{k\eta})+\frac{2}{n}W_{2}^{2}(X_{k\eta},\mu).$

The first term of the right-side is handled by Theorem 1, so we only need to
bound the second term. This is the purpose of the following lemma. In this
lemma $(P_{t})$ stands for the semi-group of the Langevin diffusion (3). In
other words $\nu P_{t}$ denotes the law of $X_{t}$ when $X_{0}$ has law $\nu$.

######  Lemma 8.

Assume that $\mu$ is log-concave with globally Lipschitz potential on its
support, with Lipschitz constant $L$. Assume also that $\mu$ satisfies log-
Sobolev with constant $C_{LS}$. Then for every $x_{0}$ in the interior of the
support of $\mu$ and every $t>0$ we have

$\frac{1}{n}W_{2}^{2}(\delta_{x_{0}}P_{t},\mu)\leq
4C_{LS}\left(1+\log\left(\frac{\max(C_{LS},1)\,n}{\min(r_{0},1)}\right)+\sigma_{0}+\frac{L}{n}\right)\mathrm{e}^{-t/2C_{LS}},$

where again the parameters $\sigma_{0}$ and $r_{0}$ are defined by (7).

###### Proof.

If $\mu$ satisfies the logarithmic Sobolev inequality then it satisfies the
transport inequality:

$W_{2}^{2}(\nu;\mu)\leq 2C_{CLS}D(\nu\mid\mu)$

for every probability measure $\nu$. This is due to Otto and Villani [22], see
also [2]. The log-Sobolev inequality also implies that the relative entropy
decays exponentially fast along the semigroup $(P_{t})$:

$D(\nu P_{t}\mid\mu)\leq\mathrm{e}^{-t/C_{LS}}D(\nu\mid\mu).$

This is really folklore, one just need to observe that the derivative of the
entropy is the Fisher information, and combine log-Sobolev with a Gronwall
type argument. Combining the two inequalities yields

$W_{2}^{2}(\nu P_{t},\mu)\leq 2C_{LS}\mathrm{e}^{-t/C_{LS}}D(\nu\mid\mu).$

This cannot be applied directly to a Dirac point mass. However, observe that
the convexity of $\varphi$ implies that the Wasserstein distance is non
increasing along the semigroup: For any two probability measures
$\nu_{0},\nu_{1}$ and every time $t$ we have

$W_{2}^{2}(\nu_{0}P_{t},\nu_{1}P_{t})\leq W_{2}^{2}(\nu_{0},\nu_{1}).$

This is a well-known fact, which is easily seen using parallel coupling. See
the proof of the pathwise uniqueness property in section 2. Combining this
with the triangle inequality for $W_{2}$ we thus get

$\begin{split}W_{2}^{2}(\delta_{x_{0}}P_{t},\mu)&\leq
2W_{2}^{2}(\delta_{x_{0}}P_{t},\nu P_{t})+2W_{2}^{2}(\nu P_{t},\mu)\\\ &\leq
2W_{2}^{2}(\delta_{x_{0}},\nu)+4C_{LS}\mathrm{e}^{-t/C_{LS}}D(\nu\mid\mu).\end{split}$
(16)

This is valid for every $\nu$ and it is natural to take $\nu$ to be the
measure $\mu$ conditioned to the ball $B(x_{0},\delta)$ for some small
$\delta>0$. Then $W^{2}_{2}(\delta_{x_{0}},\nu)\leq\delta^{2}$ and

$D(\nu\mid\mu)=\log\left(\frac{1}{\mu(B(x_{0},\delta))}\right).$

If $\delta\leq r_{0}$ then $B(x_{0},\delta)$ is included in the support of
$\mu$ and we have

$\log\left(\frac{1}{\mu(B(x_{0},\delta))}\right)\leq\max_{B(x_{0},\delta)}\\{\varphi\\}+n\log\left(\frac{1}{\delta}\right)+\log\left(\frac{1}{v_{n}}\right),$

where $v_{n}$ is the Lebesgue measure of the unit ball in dimension $n$.
Recall that $\log\left(\frac{1}{v_{n}}\right)\leq n\log n$. Also

$\max_{B(x_{0},\delta)}\\{\varphi\\}\leq\varphi(x_{0})+L\delta\leq\min_{K}\\{\varphi\\}+n\sigma_{0}+L\delta.$

Moreover

$\min_{K}\\{\varphi\\}\leq\int_{\mathbb{R}^{n}}\varphi\,d\mu=S(\mu),$

where $S$ denotes the Shannon entropy. It is well-known that among measures of
fixed covariance the Gaussian measure maximizes the Shannon entropy (this is
just Jensen actually). Therefore

$\begin{split}S(\mu)&=\frac{n}{2}\log(2\pi
e)+\frac{1}{2}\log\mathrm{det}(\mathrm{cov}(\mu))\\\ &\leq\frac{n}{2}\log(2\pi
eC_{LS}).\end{split}$

The last inequality is just a consequence of the fact that the log-Sobolev
inequality implies Poincaré, which in turn implies a bound on the covariance
matrix. Plugging everything back in (16) we get

$\frac{1}{n}W_{2}^{2}(\delta_{x_{0}}P_{t},\mu)\leq\frac{2\delta^{2}}{n}+4C_{LS}\left(\frac{3}{2}+\log\left(\frac{n}{\delta}\right)+\sigma_{0}+\frac{L}{n}\right)\mathrm{e}^{-t/C_{LS}}$

for every $\delta\leq\min(r_{0},1)$. Choosing
$\delta=\min\left((2nC_{LS})^{1/2}\mathrm{e}^{-t/2C_{LS}},r_{0},1\right)$ and
using the inequality $x\mathrm{e}^{-x}\leq\mathrm{e}^{-x/2}$ yields the
result. ∎

##  5 A convergence result using Poincaré only

In this section we prove Theorem 3. Again the idea is to write

$\frac{1}{n}W_{2}^{2}(x_{k},\mu)\leq\frac{2}{n}W_{2}^{2}(x_{k},X_{k\eta})+\frac{2}{n}W_{2}^{2}(X_{k\eta},\mu),$

and to bound the first term using Theorem 1. Actually, note that here we allow
$x_{0}=X_{0}$ to be random, so we rather condition on $x_{0}$, apply Theorem 1
and then take expectation again.

Therefore it is enough to bound the second term. This is where the Poincaré
inequality enters the picture. Note that this part of the argument does not
rely on the log-concavity of $\mu$. We shall use the following transport/chi-
square divergence inequality: If $\mu$ satisfies Poincaré with constant
$C_{P}$ then for every probability measure $\nu$ on $\mathbb{R}^{n}$ we have

$W_{2}^{2}(\nu,\mu)\leq 2C_{P}\chi^{2}(\nu\mid\mu).$

It seems that this was first proved by Ding [11], with a worst constant. The
result with constant $2$ is due to Liu [20]. His argument is combines together
the Langevin diffusion and the Hamilton-Jacobi semigroup.

On the other hand it is well-known that under Poincaré the chi-square
divergence decays exponentially fast along the Langevin diffusion. Letting
$(P_{t})$ be the semigroup of the Langevin diffusion associated to $\mu$ we
have

$\chi^{2}(\nu P_{t}\mid\mu)\leq\mathrm{e}^{-t/C_{P}}\chi^{2}(\nu\mid\mu).$

We thus get the following:

######  Lemma 9.

Suppose that $\mu$ satisfies Poincaré with constant $C_{P}$. Then for every
probability measure $\nu$ on $\mathbb{R}^{n}$ and every $t>0$ we have

$W_{2}^{2}(\nu P_{t},\mu)\leq
2C_{P}\,\chi^{2}(\nu\mid\mu)\,\mathrm{e}^{-t/C_{P}}.$

This finishes the proof of Theorem 3.

We end this section with a simple estimate of the chi-square divergence of an
appropriate Gaussian measure to $\mu$ in the unconstrained case.

###### Proof of Lemma 4.

Recall that $\mu$ is assumed to be supported on the whole $\mathbb{R}^{n}$
with convex and globally Lipschitz potential $\varphi$. Let $\gamma$ be the
Gaussian measure centered at a point $x_{0}$ and with covariance $\alpha Id$
for some $\alpha>0$. Then

$\begin{split}\chi^{2}(\gamma\mid\mu)&\leq(2\pi\alpha)^{-n}\int_{\mathbb{R}^{n}}\mathrm{e}^{-\frac{1}{\alpha}|x-x_{0}|^{2}+\varphi(x)}\,dx\\\
&\leq(2\pi\alpha)^{-n}\int_{\mathbb{R}^{n}}\mathrm{e}^{-\frac{1}{\alpha}|x-x_{0}|^{2}+\varphi(x_{0})+L|x-x_{0}|}\,dx\\\
&\leq(2\pi\alpha)^{-n}\int_{\mathbb{R}^{n}}\mathrm{e}^{-\frac{1}{2\alpha}|x-x_{0}|^{2}+\varphi(x_{0})+\frac{1}{2}L^{2}\alpha}\,dx=(2\pi\alpha)^{-n/2}\mathrm{e}^{\varphi(x_{0})+\frac{1}{2}L^{2}\alpha}.\end{split}$

Also, reasoning along the same lines as in the previous section we get

$\varphi(x_{0})\leq\min_{\mathbb{R}^{n}}\\{\varphi\\}+n\sigma_{0}\leq\frac{n}{2}\log(2\pi\mathrm{e})+\frac{n}{2}\log
C_{P}+n\sigma_{0}.$

Putting everything together and choosing $\alpha=n/L^{2}$ yields

$\begin{split}\log\chi^{2}(\gamma\mid\mu)&\leq\frac{n}{2}\left(-\log\alpha+1+\log
C_{P}+2\sigma_{0}\right)+\frac{1}{2}L^{2}\alpha\\\
&=n\left(1+\sigma_{0}+\frac{1}{2}\log\left(\frac{L^{2}C_{P}}{n}\right)\right),\end{split}$

which is the result. ∎

##  6 An extension to the non-globally Lipschitz case

We begin this section with a simple lemma about the Wassertein distance of
$\mu$ to $\mu$ restricted to a large ball.

######  Lemma 10.

Let $\mu$ be a log-concave measure on $\mathbb{R}^{n}$, and let $\mu_{R}$ be
the measure $\mu$ conditioned on the ball centered at $0$ of radius $R$. There
exists a universal constant $C$ such that

$W_{2}^{2}(\mu,\mu_{R})\leq
CM\exp\left(-\frac{R}{C\sqrt{M}}\right),\quad\forall R\geq C\sqrt{M}$

where $M=\int_{\mathbb{R}^{n}}|x|^{2}\,d\mu$.

###### Proof.

Let $X$ have law $\mu$. Note that by Borell’s lemma [4, Lemma 3.1] we have
$\mathbb{P}(|X|\geq t)\leq\mathrm{e}^{-t/C_{0}\sqrt{M}}$ for all $t\geq
C_{0}\sqrt{M}$ for some universal constant $C_{0}$. This also implies that
$\mathbb{E}[|X|^{4}]\leq C_{1}M^{2}$ for some $C_{1}$. Now assume that $R\geq
C_{0}\sqrt{M}$, let $\widetilde{X}$ have law $\mu_{R}$ and be independent of
$X$ and let

$Y=\begin{cases}X&\text{if }|X|\leq R\\\
\widetilde{X}&\text{otherwise}.\end{cases}$

Then $Y$ also has law $\mu_{R}$, so that

$\begin{split}W_{2}^{2}(\mu,\mu_{R})&\leq\mathbb{E}[|X-Y|^{2}]=\mathbb{E}[|X-\widetilde{X}|^{2};|X|>R]\\\
&\leq 4\mathbb{E}[|X|^{4}]^{1/2}\,\mathbb{P}(|X|>R)^{1/2}\leq
C_{1}M\mathrm{e}^{-R/(C_{0}M^{1/2})},\end{split}$

which is the result. ∎

###### Proof of Theorem 5.

Assuming that $\int_{\mathbb{R}^{n}}|x|^{2}\,d\mu=O(n)$, the previous lemma
shows that $\frac{1}{n}W_{2}^{2}(\mu,\mu_{R})$ will be negligible as soon as
$R$ is a sufficiently large multiple of $\sqrt{n}\log n$. Now we apply Theorem
2 to $\mu_{R}$ and initiating the Langevin algorithm at $0$. Then the
parameter $r_{0}$ is of order $\sqrt{n}\log n$. Moreover the hypothesis

$|\nabla\varphi(x)|\leq\beta(|x|+1)$

show that the potential of $\mu_{R}$ is Lipschitz with constant
$O^{*}(\beta\sqrt{n})$. Therefore the constant $A$ defined by (6) satisfies
$A=O^{*}(\max(\beta,1))$ in this case. On the other hand since $\mu_{R}$ is
log-concave and supported on a ball of radius $O^{*}(\sqrt{n})$ its log-
Sobolev constant is $O^{*}(n)$ at most. Also, if $\mu$ satisfies log-Sobolev,
then the log-Sobolev constant of $\mu_{R}$ cannot be larger than a constant
factor times that of $\mu$. This follows easily from the fact that within log-
concave measures log-Sobolev and Gaussian concentration are equivalent, see
[21, Theorem 1.2]. To sum up, the log-Sobolev constant of $\mu_{R}$ is
$O^{*}(\min(n,C_{LS}))$ where $C_{LS}$ is the log-Sobolev constant of $\mu$
(which is possibly infinite). Applying Theorem 2 we see that after

$k=\Theta^{*}\left(\frac{\min(C_{LS},n)^{3}\max(\beta,1)^{2}}{\varepsilon^{2}}\right)$

of the Langevin algorithm for $\mu_{R}$ initiated at $0$, and with appropriate
time step parameter we have
$\frac{1}{n}W_{2}^{2}(x_{k},\mu_{R})\leq\varepsilon$. Since
$\frac{1}{n}W_{2}^{2}(\mu_{R},\mu)$ is negligible this implies
$\frac{1}{n}W_{2}^{2}(x_{k},\mu)\leq 2\varepsilon$. ∎

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