$L^{2}({\mathbb{T}}^{d};{\mathbb{R}}^{d})$ into ${\mathcal{H}}_{-p}^{d}$ for
$p>0$, Ascoli-Arzelà theorem, and Chebyshev inequality, it is enough to prove
the following bounds. For each $T_{0}>0$ and each smooth vector field
$H\colon{\mathbb{T}}^{d}\to{\mathbb{R}}^{d}$ there exists constants
$C_{0}=C_{0}(T_{0})$ and $C_{1}=C_{1}(T_{0},H)$ such that for any $T\geq 1$
and $\delta\in(0,1)$

$\displaystyle
E_{P}\Big{[}\sup_{t\in[-T_{0},T_{0}]}\big{\|}{\boldsymbol{J}}(t)\big{\|}_{L^{2}({\mathbb{T}}^{d};{\mathbb{R}}^{d})}^{2}\Big{]}\leq
C_{0}\,\Big{[}\frac{1}{T}{\boldsymbol{I}}_{m,T,\rho}(P)+1\Big{]}$ (5.4)
$\displaystyle E_{P}\bigg{[}\sup_{\begin{subarray}{c}t,s\in[-T_{0},T_{0}]\\\
|t-s|<\delta\end{subarray}}\big{|}\langle{\boldsymbol{J}}(t),H\rangle-\langle{\boldsymbol{J}}(s),H\rangle\big{|}^{2}\bigg{]}\leq
C_{1}\delta\,\Big{[}\frac{1}{T}{\boldsymbol{I}}_{m,T,\rho}(P)+1\Big{]}.$ (5.5)

As already remarked right after the statement of Lemma 4.5, by choosing
$H=\nabla g$, the bound (5.5) provides indeed also a control on the continuity
modulus of the map $t\mapsto{\boldsymbol{\pi}}(t)$.

By the stationarity of $P$ and the argument below (2.26), if
${\boldsymbol{I}}_{m,T,\rho}(P)<+\infty$, then there exists a constant $C$
depending only on $E$ such that

$\frac{1}{2T}\,E_{P}\bigg{[}\int_{-T}^{T}dt\int_{{\mathbb{T}}^{d}}dx\,\frac{|{\boldsymbol{j}}\,|^{2}}{4\,\sigma({\boldsymbol{\rho}})}\bigg{]}\leq
C\Big{[}\frac{1}{T}\,{\boldsymbol{I}}_{m,T,\rho}(P)+1\Big{]}\;,$

where ${\boldsymbol{J}}(t)=\int_{0}^{t}{\boldsymbol{j}}(s)\,ds$. By Cauchy-
Schwarz inequality and condition (b’) on the current ${\boldsymbol{J}}$ stated
at the beginning of this section,

$\sup_{t\in[-T_{0},T_{0}]}\big{\|}{\boldsymbol{J}}(t)\big{\|}_{L^{2}({\mathbb{T}}^{d};{\mathbb{R}}^{d})}^{2}\leq
T_{0}\int_{-T_{0}}^{T_{0}}\\!dt\int\\!dx\big{|}{\boldsymbol{j}}(t,x)\big{|}^{2}\leq
T_{0}\int_{-T_{0}}^{T_{0}}\\!dt\int\\!dx\frac{\big{|}{\boldsymbol{j}}(t,x)\big{|}^{2}}{4\sigma({\boldsymbol{\rho}}(t,x))}\;\cdot$

Since ${\boldsymbol{I}}_{m,T,\rho}(P)<+\infty$ implies that $P$ is a
$T$-holonomic probability measure,

$E_{P}\bigg{[}\int_{-T_{0}}^{T_{0}}\\!dt\int\\!dx\frac{\big{|}{\boldsymbol{j}}(t,x)\big{|}^{2}}{4\sigma({\boldsymbol{\rho}}(t,x))}\bigg{]}=\frac{T_{0}}{T}E_{P}\bigg{[}\int_{-T}^{T}\\!dt\int\\!dx\frac{\big{|}{\boldsymbol{j}}(t,x)\big{|}^{2}}{4\sigma({\boldsymbol{\rho}}(t,x))}\bigg{]}$

which completes the proof of (5.4).

For $s<t$, Cauchy-Schwarz inequality and condition (b’) yield

$\big{|}\langle{\boldsymbol{J}}(t),H\rangle-\langle{\boldsymbol{J}}(s),H\rangle\big{|}^{2}\leq(t-s)\|H\|_{L^{2}({\mathbb{T}}^{d};{\mathbb{R}}^{d})}^{2}\int_{s}^{t}\\!du\int\\!dx\,\big{|}{\boldsymbol{j}}(u,x)\big{|}^{2}\;,$

so that (5.5) is obtained by the same argument as before.

_$\Gamma$ -liminf._ Denote by
$\color[rgb]{.2,0.2,.8}{\mathscr{S}}^{T}_{m,\mathrm{ac}}$ the subset of the
paths $({\boldsymbol{\pi}},{\boldsymbol{J}})\in{\mathscr{S}}^{T}$ satisfying
only conditions (a’), (b’) and let $A_{m,T}$ be the action defined in (5.1)
with the constraint
$({\boldsymbol{\pi}},{\boldsymbol{J}})\in{\mathscr{S}}^{T}_{m,\mathrm{ac},\rho}$
replaced by
$({\boldsymbol{\pi}},{\boldsymbol{J}})\in{\mathscr{S}}^{T}_{m,\mathrm{ac}}$.
Accordingly, let
${\boldsymbol{I}}_{m,T}\colon{\mathscr{P}}_{\mathrm{stat}}\to[0,+\infty]$ be
the functional defined by

${\boldsymbol{I}}_{m,T}(P):=\inf\big{\\{}A_{m,T}({\boldsymbol{\pi}},{\boldsymbol{J}})\,,\;({\boldsymbol{\pi}},{\boldsymbol{J}})\in\chi_{T}^{-1}\big{(}P\big{)}\big{\\}}\;.$

By the translation invariance of $A_{m,T}$, if $\chi_{T}^{-1}(P)$ is not empty
(i.e., if $P$ is $T$-holonomic) then

${\boldsymbol{I}}_{m,T}(P)=E_{P}\big{[}A_{m,T}\big{]}\;.$

Hence, in view of the translation invariance of $P$,

${\boldsymbol{I}}_{m,T,\rho}(P)\geq{\boldsymbol{I}}_{m,T}(P)\;=\;E_{P}\big{[}A_{m,T}\big{]}\;=\;TE_{P}\big{[}A_{m,1}\big{]}\;.$

Let $(\rho_{T}\colon T>0)$ be an arbitrary sequence and $\big{(}P_{T}\colon
T>0\big{)}\subset{\mathscr{P}}_{\mathrm{stat}}$ be a sequence converging to
$P$. By (2.26), the previous displayed bound and the lower semi-continuity of
$A_{m,1}$,

$\liminf_{T\to\infty}\frac{1}{T}{\boldsymbol{I}}_{m,T,\rho_{T}}(P_{T})\geq\liminf_{T\to\infty}E_{P_{T}}\big{[}A_{m,1}\big{]}\geq
E_{P}\big{[}A_{m,1}\big{]}={\boldsymbol{I}}_{m}(P)\;.$

_$\Gamma$ -limsup._ By Theorem 4.10, it suffices to consider the case in which
$P$ is an $S$-holonomic measure with smooth density. More precisely, we may
assume that

$P=\frac{1}{S}\int_{0}^{S}\delta_{\vartheta_{s}({{\boldsymbol{\pi}}}^{*},{{\boldsymbol{J}}}^{*})}\,ds$
(5.6)

for some $S>0$, where the $S$-periodic path
$({{\boldsymbol{\pi}}}^{*},{{\boldsymbol{J}}}^{*})$ has smooth densities
$({{\boldsymbol{\rho}}}^{*},{{\boldsymbol{j}}}^{*})$ with
${{\boldsymbol{\rho}}}^{*}$ bounded away from $0$ and $1$. Given the sequence
$(\rho_{T},\,T>0)\subset{\mathcal{M}}_{m}({\mathbb{T}}^{d})$, let
$\big{(}\bar{{\boldsymbol{\pi}}}(t),\bar{{\boldsymbol{J}}}(t)\big{)}$,
$t\in[0,\bar{T}]$ be the path provided by Lemma 4.12 with
$\pi_{0}=\rho_{T}\,dx$ and $\pi_{1}={{\boldsymbol{\pi}}}^{*}(0)$. Let also
$\big{(}{\boldsymbol{\pi}}(t),{\boldsymbol{J}}(t)\big{)}$, $t\in[0,+\infty)$
be the path defined by

$\big{(}{{\boldsymbol{\pi}}}(t),{\boldsymbol{J}}(t)\big{)}:=\begin{cases}\big{(}\bar{{\boldsymbol{\pi}}}(t)\,,\,\bar{{\boldsymbol{J}}}(t)\big{)}&\textrm{
if }t\in[0,\bar{T}],\\\
\big{(}{{\boldsymbol{\pi}}}^{*}(t-\bar{T})\,,\,\bar{{\boldsymbol{J}}}(\bar{T})+{{\boldsymbol{J}}}^{*}(t-\bar{T})\big{)}&\textrm{
if }t>\bar{T}.\end{cases}$

Note that, although not explicit in the notation, the path
$({{\boldsymbol{\pi}}},{{\boldsymbol{J}}})$ depends on $T$ via the sequence
$\rho_{T}$. Denote finally by
$\big{(}{{\boldsymbol{\pi}}}^{T},{{\boldsymbol{J}}}^{T}\big{)}$ the
$T$-periodization, as defined in (2.23), of
$\big{(}{{\boldsymbol{\pi}}},{{\boldsymbol{J}}}\big{)}$ and by $P_{T}$ the
associated $T$-holonomic probability, i.e.,

$P_{T}=\frac{1}{T}\int_{0}^{T}\delta_{\vartheta_{s}({{\boldsymbol{\pi}}}^{T},{{\boldsymbol{J}}}^{T})}\,ds\;.$

Since $\bar{T}/T\to 0$, as $T\to\infty$ the sequence $P_{T}$ converges to $P$
given by (5.6). Moreover, in view of (5.3) and (4.21),

${\boldsymbol{I}}_{m,T,\rho_{T}}(P_{T})=A_{m,\bar{T},\rho_{T}}\big{(}\bar{{\boldsymbol{\pi}}},\bar{{\boldsymbol{J}}}\big{)}+A_{m,T-\bar{T}}({{\boldsymbol{\pi}}}^{*},{{\boldsymbol{J}}}^{*})\leq
C_{0}+A_{m,T-\bar{T}}({{\boldsymbol{\pi}}}^{*},{{\boldsymbol{J}}}^{*})\;.$

Hence, by the $S$-periodicity of
$({{\boldsymbol{\pi}}}^{*},{{\boldsymbol{J}}}^{*})$,

$\limsup_{T\to\infty}\frac{1}{T}{\boldsymbol{I}}_{m,T,\rho_{T}}(P_{T})=\frac{1}{S}A_{m,S}({{\boldsymbol{\pi}}}^{*},{{\boldsymbol{J}}}^{*})=\frac{1}{S}E_{P}\big{[}A_{m,S}\big{]}={\boldsymbol{I}}_{m}(P)\;,$

which concludes the proof of Theorem 5.3. ∎

##  6\. Large deviations and projections

In this section, relying on the variational convergence proven before, we
discuss the large deviations asymptotics in the joint limit $N\to\infty$,
$T\to\infty$. In particular, we conclude the proof of Theorem 2.1 and discuss
the corresponding projections.

###### Proof of Theorem 2.1.

We start by considering the case in which we first perform the limit as
$T\to\infty$ and then take limit as $N\to\infty$. The asymptotic as
$T\to\infty$ follows directly from the Donsker-Varadhan large deviation
principle for the empirical process, see Corollary 3.2. By [3, Lemma 4.1] or
[18, Corollary 4.3] the limit as $N\to\infty$ is accomplished by the
$\Gamma$-convergence of the family
$\big{(}N^{-d}{\boldsymbol{I}}_{N,K_{N}},\,N\geq 1\big{)}$ that has been
proven in Theorem 4.1. Actually, the statements in [3, 18] give the upper
bound only for compact sets. However, the goodness of the functional
${\boldsymbol{I}}_{N,K}$ together with the equi-coercivity in Theorem 4.1
allow to deduce the upper bound for closed sets.

The proof of the statement when the limit as $T\to\infty$ is carried out after
the limit as $N\to\infty$ is accomplished by the similar argument. Indeed, the
asymptotic as $N\to\infty$ follows directly from the hydrodynamical large
deviations, see Corollary 5.2, while the $\Gamma$-convergence of the family
$\big{(}T^{-1}{\boldsymbol{I}}_{m,T,\rho},\,T\geq 1\big{)}$ has been proven,
uniformly with respect to $\rho$, in Theorem 5.3. ∎

We now discuss the level two projection and the hydrostatic limit.

###### Proof of Corollary 2.2.

Recall (2.28) and that $\wp_{T,N}={\mathfrak{R}}_{T,N}\circ\imath_{t}^{-1}$
where $\imath_{t}\colon{\mathscr{S}}\to{\mathcal{M}}_{+}({\mathbb{T}}^{d})$ is
the map $({\boldsymbol{\pi}},{\boldsymbol{J}})\mapsto{\boldsymbol{\pi}}(t)$.
Note that $i_{t}$ is not continuous since we are using the Skorohod topology.
However, the map ${\mathscr{P}}_{\rm stat}\ni P\mapsto
P\circ\imath_{t}^{-1}\in{\mathscr{P}}\big{(}{\mathcal{M}}_{+}({\mathbb{T}}^{d})\big{)}$
is continuous since, by stationary, the $P$-probability of a jump at a time
$t$ is zero. The large deviations asymptotic thus follows from Theorem 2.1 by
the contraction principle.

We now show that the zero level set of ${\mathscr{I}}_{m}$ is equal to the set
of invariant probability measures for the flow $\Phi^{m}$ associated to the
hydrodynamic evolution (2.18). By the goodness of ${\boldsymbol{I}}_{m}$, if
$\wp$ lies in the zero level set of ${\mathscr{I}}_{m}$ then the infimum in
(2.28) is achieved, i.e. there exists $P\in{\mathscr{P}}_{\textrm{stat}}$
satisfying ${\boldsymbol{I}}_{m}(P)=0$ and $\wp=P\circ\imath_{t}^{-1}$. As
follows from (2.26) and (2.25) ${\boldsymbol{I}}_{m}(P)=0$ implies that $P$
almost surely $({\boldsymbol{\pi}},{\boldsymbol{J}})$ have densities
$({\boldsymbol{\rho}},{\boldsymbol{j}})$ that satisfy
${\boldsymbol{j}}=-\nabla{\boldsymbol{\rho}}+\sigma({\boldsymbol{\rho}})E$.
Hence, the marginal of $P$ on the first variable is concentrated on the set of
${\boldsymbol{\pi}}$ whose density ${\boldsymbol{\rho}}$ solves (1.2) with
$D=1$. By stationarity of $P$, this implies that $P\circ\imath_{t}^{-1}$ is an
invariant probability of $\Phi^{m}$ as claimed.

It remains to show that if $E$ is orthogonally decomposable then
${\mathscr{I}}_{m}(\wp)=0$ implies $\wp=\delta_{\bar{\rho}\,dx}$ where
$\bar{\rho}$ is the unique stationary solution to (1.2) with $D=1$. As already
remarked, if $E$ is orthogonally decomposable then the quasi-potential of the
WASEP dynamics can be explicitly computed and it is a Lypuanov functional for
the hydrodynamic evolution. The argument in [5, Thm. 7.7] then implies that
there exists a unique stationary solution of mass $m$ to the hydrodynamic
equation that is globally attractive, hence a unique stationary probability
for the flow $\Phi^{m}$ that is concentrated on the stationary solution. ∎

We consider now the level one projection.

###### Proof of Corollary 2.3.

Let
$\psi\colon{\mathscr{P}}_{\mathrm{stat}}\to{\mathcal{M}}_{+}({\mathbb{T}}^{d})\times{\mathcal{H}}_{-p}^{d}$
be the map defined by

$\psi(P):=\Big{(}E_{P}\big{[}{\boldsymbol{\pi}}(t)\big{]}\;,\;\frac{1}{t}E_{P}\big{[}{\boldsymbol{J}}(t)\big{]}\Big{)}$

where we understand that $\psi$ is defined only for the probabilities $P$ such
that for any $t\in{\mathbb{R}}$ we have
$E_{P}\big{[}\|{\boldsymbol{J}}(t)\|_{{\mathcal{H}}_{-p}^{d}}\big{]}<+\infty$.
Note that $\psi$ does not depend on $t\neq 0$ by the stationarity of $P$.

Recall the definitions of $\pi_{T,N}$, $J_{T,N}$, and ${\mathfrak{R}}_{T,N}$,
in (2.29), (2.30), and (2.24). Recall also the relation (2.31) according to
which for each ${\boldsymbol{\eta}}\in D({\mathbb{R}}_{+},\Sigma_{N})$ and
$t\in{\mathbb{R}}$, $t\neq 0$, we have

$\psi\big{(}{\mathfrak{R}}_{T,N}\big{)}=\int\\!{\mathfrak{R}}_{T,N}(d{\boldsymbol{\pi}},d{\boldsymbol{J}})\,\Big{(}{\boldsymbol{\pi}}(t),\frac{{\boldsymbol{J}}(t)}{t}\Big{)}=\big{(}\pi_{T,N},J_{T,N}\big{)}-\frac{1}{T}\big{(}0,\mathcal{E}_{T,N}\big{)}\,.$
(6.1)

Since by (2.20) the error term $\mathcal{E}_{T,N}$ is uniformly bounded in $N$
and $T$, it is irrelevant in the large deviations asymptotics for
$T\to+\infty$. We can therefore deduce the large deviations for the pair
$\big{(}\pi_{T,N},J_{T,N}\big{)}$ from the large deviations for
$\psi\big{(}{\mathfrak{R}}_{T,N}\big{)}$.

Since $\psi$ is not continuous the result does not follow directly from
Theorem 2.1 and the contraction principle. However, in the terminology of [12,
§ 4.2.2], it possible to approximate $\psi$ by a sequence of continuous
functions and construct exponentially good approximations of the family
$\big{(}{\mathbb{P}}^{N}_{\eta}\circ\big{(}\pi_{T,N},J_{T,N}\big{)}^{-1}\colon
T>0,\,N\geq 1\big{)}$. We obtain in this way the result and observe that the
rate functional is given by $I_{m}(\pi,J)=\inf\left\\{{\bf
I}_{m}(P):P\in{\mathscr{P}}_{\rm stat},\psi(P)=(\pi,J)\right\\}$. ∎

The next result concerns the projection on the density for the level one large
deviations functional.

###### Proof of Theorem 2.4.

In the case $E=-\nabla U$, developing the square in formula (2.25) we have
that the cross term

$\int_{0}^{T}dt\int_{\mathbb{T}^{d}}dx\,\frac{{{\boldsymbol{j}}}\cdot\left(\nabla{{\boldsymbol{\rho}}}+\sigma({{\boldsymbol{\rho}}})\nabla
U\right)}{2\sigma({{\boldsymbol{\rho}}})}$

after an integration by parts and using the continuity equation coincides with

$\frac{1}{2}\int_{\mathbb{T}^{d}}dx\,\left[h({\boldsymbol{\rho}}(T))-h({\boldsymbol{\rho}}(0))-\left({\boldsymbol{\rho}}(T)-{\boldsymbol{\rho}}(0)\right)U\right]\,,$
(6.2)

where $h(\rho)=\rho\log\rho+(1-\rho)\log(1-\rho)$. By stationarity the
expected value of (6.2) with respect to any $P\in{\mathscr{P}}_{\rm stat}$ is
zero. We have therefore that when $E=-\nabla U$

${\boldsymbol{I}}_{m}(P)=\frac{1}{T}\,E_{P}\bigg{[}\int_{0}^{T}dt\int_{{\mathbb{T}}^{d}}dx\,\Big{(}\frac{|\,\nabla{\boldsymbol{\rho}}+\sigma({\boldsymbol{\rho}})\nabla
U\,|^{2}}{4\,\sigma({\boldsymbol{\rho}})}+\frac{|\,{\boldsymbol{j}}\,|^{2}}{4\,\sigma({\boldsymbol{\rho}})}\Big{)}\bigg{]}\;.$
(6.3)

Consider a $P\in{\mathscr{P}}_{\rm stat}$ and call
$\wp\in{\mathscr{P}}\left(\mathcal{M}_{m}(\mathbb{T}^{d})\right)$ its
1-marginal $\wp=P\circ\imath_{t}^{-1}$ (see notation above formula (2.28)).
Let $A\colon\mathcal{M}_{+}(\mathbb{T}^{d})\to{\mathscr{S}}$ be the map that
associate to $\pi\in\mathcal{M}_{+}(\mathbb{T}^{d})$ the element
$({\boldsymbol{\pi}},{\boldsymbol{J}})\in{\mathscr{S}}$ defined by
${\boldsymbol{\pi}}(t)=\pi$ and ${\boldsymbol{J}}(t)=0$ for any $t\in[0,T]$.
Finally let us define $\tilde{P}\in{\mathscr{P}}_{\rm stat}$ as $\wp\circ
A^{-1}$.

Since the second term in the right hand side of (6.3) is non negative and
since $P\circ\imath_{t}^{-1}=\tilde{P}\circ\imath_{t}^{-1}=\wp$, we deduce

${\boldsymbol{I}}_{m}(P)\geq{\boldsymbol{I}}_{m}(\tilde{P})=E_{\wp}\bigg{[}\int_{{\mathbb{T}}^{d}}dx\,\Big{(}\frac{|\,\nabla\rho+\sigma(\rho)\nabla
U\,|^{2}}{4\,\sigma(\rho)}\Big{)}\bigg{]}=E_{\wp}\left[{\mathscr{V}}_{m}\right]\,.$

We have therefore that

$I^{(1)}_{m}(\pi)=\inf\left\\{E_{\wp}({\mathscr{V}}_{m})\;\
\wp\in{\mathscr{P}}\left(\mathcal{M}_{m}(\mathbb{T}^{d})\right)\,E_{\wp}(\pi^{\prime})=\pi\right\\}=\mathrm{co}({\mathscr{V}}_{m})(\pi)\,,$

the last equality follows since in the middle we have one of the possible
definitions of convex hull. Since $\sigma$ is concave, in the case $\nabla
U=0$ we have that ${\mathscr{V}}_{m}$ is convex and therefore
$\mathrm{co}({\mathscr{V}}_{m})={\mathscr{V}}_{m}$. ∎

Finally, we give a sketch of the proof of Theorem 2.5. This is based on
analysis in [2, 3, 8] and we just show how to deduce the result based on the
arguments there.

###### Proof of Theorem 2.5.

Let us call $({\boldsymbol{\pi}}^{*},{\boldsymbol{J}}^{*})$ the element of
${\mathscr{S}}_{m}$ defined by ${\boldsymbol{\pi}}^{*}(t)=m$ and
${\boldsymbol{J}}^{*}(t)=jt$. The result is obtained by the analysis of the
action functional $A_{m,T}$ (2.25). In the case $(i)$ for $E\leq E_{0}$ by [2,
3, 8] we have that for any
$({\boldsymbol{\pi}},{\boldsymbol{J}})\in{\mathscr{S}}_{m}$ such that
${\boldsymbol{J}}(T)=jT$ it holds
$A_{m,T}({\boldsymbol{\pi}}^{*},{\boldsymbol{J}}^{*})\leq
A_{m,T}({\boldsymbol{\pi}},{\boldsymbol{J}})$ and this allows to deduce that
$\delta_{({\boldsymbol{\pi}}^{*},{\boldsymbol{J}}^{*})}$ is the minimizer in
(2.33).

In the case $(ii)$ for $E>E_{1}$ it is possible to construct [2, 3, 8] a time
dependent $({\boldsymbol{\pi}},{\boldsymbol{J}})\in{\mathscr{S}}_{m}$, that
has indeed the structure of a traveling wave, such that
${\boldsymbol{J}}(T)=jT$ and
$A_{m,T}({\boldsymbol{\pi}}^{*},{\boldsymbol{J}}^{*})>A_{m,T}({\boldsymbol{\pi}},{\boldsymbol{J}})$.
Considering $P\in{\mathscr{P}}_{\rm stat}$ defined by
$P=\frac{1}{T}\int_{0}^{T}dt\delta_{\theta_{t}({\boldsymbol{\pi}},{\boldsymbol{J}})}$
we have therefore that
${\boldsymbol{I}}_{m}(P)<{\boldsymbol{I}}_{m}(\delta_{({\boldsymbol{\pi}}^{*},{\boldsymbol{J}}^{*})})$.
∎

##  7\. Uniqueness of periodic solutions

Fix $T>0$. Throughout this section,
$F:{\mathbb{R}}\times{\mathbb{T}}^{d}\to{\mathbb{R}}^{d}$ is a smooth,
$T$-periodic vector field. We investigate in this section the asymptotic
behavior of solutions to the Cauchy problem

$\left\\{\begin{aligned}
&\partial_{t}{\boldsymbol{u}}\;=\;\Delta{\boldsymbol{u}}\,+\,\nabla\cdot[\,\sigma({\boldsymbol{u}})\,F\,]\\\
&{\boldsymbol{u}}(0,\cdot)=u_{0}(\cdot)\;,\end{aligned}\right.$ (7.1)

where the initial condition $u_{0}:{\mathbb{T}}^{d}\to[0,1]$ is such that
$0\leq u_{0}(x)\leq 1$ for all $x\in{\mathbb{T}}^{d}$.

Existence of weak solutions is provided by the hydrodynamic limit of WASEP.
This argument shows that the solution takes value in the interval $[0,1]$.
These bounds can be derived also from the maximum principle and the
observation that $\sigma(1)=\sigma(0)=0$. By parabolic regularity, a weak
solution is smooth in $(0,\infty)\times{\mathbb{T}}^{d}$. Uniqueness is
derived as in [14, Lemma 7.2]. The proof of this lemma yields that the
$L^{1}({\mathbb{T}}^{d})$ norm of the difference of two weak solutions does
not increase in time. The main result of this section strengthen this lemma
and asserts that the $L^{1}({\mathbb{T}}^{d})$ distance of two different weak
solutions decreases in time. It reads as follows.

######  Theorem 7.1.

Let $F:{\mathbb{R}}\times{\mathbb{T}}^{d}\to{\mathbb{R}}^{d}$ be a smooth,
$T$-periodic vector field. For each $m\in[0,1]$, the equation

$\partial_{t}{\boldsymbol{u}}\;=\;\Delta{\boldsymbol{u}}\,+\,\nabla\cdot[\,\sigma({\boldsymbol{u}})\,F\,]\;.$

admits a unique $T$-periodic solution
${\boldsymbol{u}}:{\mathbb{R}}\times{\mathbb{T}}^{d}\to[0,1]$ such that
$\int_{{\mathbb{T}}^{d}}{\boldsymbol{u}}(t,x)\,dx=m$ and
$0\leq{\boldsymbol{u}}(t,x)\leq 1$ for all $t$. This solution is represented
by ${\boldsymbol{u}}^{(m)}$. Moreover, for each
$u_{0}:{\mathbb{T}}^{d}\to[0,1]$ such that
$\int_{{\mathbb{T}}^{d}}u_{0}(x)\,dx=m$, $0\leq u_{0}(x)\leq 1$ for all
$x\in{\mathbb{T}}^{d}$, the unique weak solution of (7.1) converges to
${\boldsymbol{u}}^{(m)}$ in $L^{1}({\mathbb{T}}^{d})$ as $t\to\infty$.

The proof of this result relies on a method of coupling of two diffusions due
to Lindvall and Rogers [17].

###  7.1. Coupling diffusions.

Let $G:{\mathbb{R}}_{+}\times{\mathbb{T}}^{d}\to{\mathbb{R}}^{d}$ be a smooth
vector field uniformly bounded: there exists $C_{0}<\infty$ such that
$\sup_{(t,x)\in{\mathbb{R}}_{+}\times{\mathbb{T}}^{d}}\|\,G(t,x)\,\|\leq
C_{0}$.

Denote by ${\mathscr{L}}_{t}$ the time-dependent generator

${\mathscr{L}}_{t}f\;=\;\Delta f\,+\,\nabla f\cdot G_{t}\;,\quad
f\,\in\,C^{2}({\mathbb{T}}^{d})\;.$ (7.2)

Let $\color[rgb]{.2,0.2,.8}(Z^{x}_{t}:t\geq 0)$, $x\in{\mathbb{T}}^{d}$, be
the ${\mathbb{T}}^{d}$-valued, continuous-time Markov process whose generator
is ${\mathscr{L}}_{t}$ and which starts from $x$.

Recall that a coupling between $Z^{x}_{t}$ and $Z^{y}_{t}$ is a process
$(\widetilde{Z}^{x}_{t},\widetilde{Z}^{y}_{t})$ whose first (resp. second)
coordinate evolves as $Z^{x}_{t}$ (resp. $Z^{y}_{t}$). The coupling time,
denoted by $\tau^{Z}_{x,y}$, is the first time at which the processes meet:

$\tau^{Z}_{x,y}\;:=\;\inf\big{\\{}\,t>0:\widetilde{Z}^{x}_{t}\,=\,\widetilde{Z}^{y}_{t}\,\big{\\}}\;.$

Next result relies on the Lindvall–Rogers coupling [17].

######  Proposition 7.2.

There exist constants $A<\infty$ and $\lambda>0$, which depends only on
$\sup_{(t,x)\in{\mathbb{R}}_{+}\times{\mathbb{T}}^{d}}\|\,G(t,x)\,\|$, and,
for each $x$, $y\in{\mathbb{T}}^{d}$, a coupling between $Z^{x}_{t}$,
$Z^{y}_{t}$, such that

$\sup_{x,y\in{\mathbb{T}}^{d}}P\big{[}\,\tau^{Z}_{x,y}\,\geq\,t\,\big{]}\;\leq\;A\,e^{-\lambda
t}$

for all $t\geq 0$.

Let $\color[rgb]{.2,0.2,.8}{\mathbb{P}}_{x}$ be the probability measure on
$C({\mathbb{R}}_{+},{\mathbb{T}}^{d})$ induced by the diffusion associated to
the generator ${\mathscr{L}}_{t}$ starting from $x\in{\mathbb{T}}^{d}$.
Expectation with respect to ${\mathbb{P}}_{x}$ is represented by
$\color[rgb]{.2,0.2,.8}{\mathbb{E}}_{x}$.

######  Corollary 7.3.

There exist constants $A<\infty$ and $\lambda>0$, which depends only on
$\sup_{(t,x)\in{\mathbb{R}}_{+}\times{\mathbb{T}}^{d}}\|\,G(t,x)\,\|$, such
that

$\sup_{x,y\in{\mathbb{T}}^{d}}\Big{|}\,{\mathbb{E}}_{x}\big{[}\,f(Z_{t})\,\big{]}\,-\,{\mathbb{E}}_{y}\big{[}\,f(Z_{t})\,\big{]}\,\Big{|}\;\leq\;A\,e^{-\lambda
t}\,\|f\|_{\infty}$

for every $f\in C({\mathbb{T}}^{d})$.

###### Proof.

Since the difference may be written as

$\Big{|}\,E\big{[}\,f(\widetilde{Z}^{x}_{t})\,-\,f(\widetilde{Z}^{y}_{t})\,\big{]}\,\Big{|}\;\leq\;2\,\|f\|_{\infty}\,P\big{[}\,\tau^{Z}_{x,y}\,\geq\,t\,\big{]}\;,$
(7.3)

the assertion is an elementary consequence of Proposition 7.2. ∎

###### Proof of Proposition 7.2.

Denote by $\color[rgb]{.2,0.2,.8}(W_{t}:t\geq 0)$, the standard Brownian
motion on ${\mathbb{R}}^{d}$. Let
$\color[rgb]{.2,0.2,.8}b:{\mathbb{R}}_{+}\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$
be the ${\mathbb{T}}^{d}$-periodic vector field whose restriction to
${\mathbb{T}}^{d}$ coincides with $G$. Denote by $X^{x}_{t}$ the solutions of
the SDE

$\left\\{\begin{aligned} &dX^{x}_{t}\;=\;c(t,X^{x}_{t})\,dt\,+\,dW_{t}\;,\\\
&X^{x}_{0}=x\;,\end{aligned}\right.$

where $\color[rgb]{.2,0.2,.8}c(t,x)=(1/2)\,b(t,x)$. For each
$x\in{\mathbb{R}}^{d}$, $X^{x}_{t}$ is a diffusion on ${\mathbb{R}}^{d}$ whose
time-dependent generator, denoted by ${\mathscr{A}}_{t}$, is given by

${\mathscr{A}}_{t}f\;=\;(1/2)\,\Delta f\,+\,\nabla f\cdot c_{t}\;,\quad
f\,\in\,C^{2}_{0}({\mathbb{R}}^{d})\;,$

where $C^{2}_{0}({\mathbb{R}}^{d})$ stands for the twice continuously
differentiable functions with compact support. We replaced $b(t,x)$ by
$c(t,x)$ in order to have a simple relation between the generators
${\mathscr{A}}_{t}$ and ${\mathscr{L}}_{t}$.

Fix $x$, $y\in{\mathbb{T}}^{d}$. Lindvall and Rogers [17] provide a coupling
between $X^{x}_{t}$ and $X^{y}_{t}$, represented by
$(\widetilde{X}^{x}_{t},\widetilde{X}^{y}_{t})$, such that, before hitting the
origin, $D_{t}:=\|\widetilde{X}^{x}_{t}-\widetilde{X}^{y}_{t}\|$ evolves as

$dD_{t}\;=\;2\,dB_{t}\;+\;\big{\langle}\frac{\widetilde{X}^{x}_{t}-\widetilde{X}^{y}_{t}}{D_{t}}\,,\,c(t,\widetilde{X}^{x}_{t})-c(t,\widetilde{X}^{y}_{t})\,\big{\rangle}\,dt\;,$
(7.4)

where $\langle\,\cdot\,,\,\cdot\,\rangle$ stands for the scalar product in
${\mathbb{R}}^{d}$, and $B_{t}$ is a one-dimensional Brownian motion. Note
that the drift term is bounded.

Denote by $(\widetilde{Z}^{x}_{t},\widetilde{Z}^{y}_{t})$ the projection of
the process $(\widetilde{X}^{x}_{t},\widetilde{X}^{y}_{t})$ on
${\mathbb{T}}^{d}\times{\mathbb{T}}^{d}$. Each coordinate of the pair
$(\widetilde{Z}^{x}_{t},\widetilde{Z}^{y}_{t})$ is a Markov process whose
generator is equal to $(1/2){\mathscr{L}}_{t}$. Hence,
$(\widetilde{Z}^{x}_{2t},\widetilde{Z}^{y}_{2t})$ is a coupling to
$Z^{x}_{t}$, $Z^{y}_{t}$, and, to prove the proposition, it is enough to show
that there exist constants $A<\infty$ and $\lambda>0$, which depends only on
$\sup_{(t,x)\in{\mathbb{R}}_{+}\times{\mathbb{T}}^{d}}\|\,G(t,x)\,\|$, such
that

$\sup_{x,y\in{\mathbb{T}}^{d}}P\big{[}\,\tau^{\widetilde{Z}}_{x,y}\,\geq\,t\,\big{]}\;\leq\;(1/2)\,A\,e^{-\lambda
t}$ (7.5)

for all $t\geq 0$, where $\tau^{\widetilde{Z}}_{x,y}$ is the first time the
processes $(\widetilde{Z}^{x}_{t},\widetilde{Z}^{y}_{t})$ meet.

By construction, before hitting $0$,
$\widetilde{D}_{t}:=\|\widetilde{Z}^{x}_{t}-\widetilde{Z}^{y}_{t}\|$ evolves
as $D_{t}$, except when $D_{t}$ attains the maximal distance between two
points in ${\mathbb{T}}^{d}$, that is, when $D_{t}$ hits
$\color[rgb]{.2,0.2,.8}L:=\sqrt{d}/2$, in which case $\widetilde{D}_{t}$ is
reflected, while $D_{t}$ evolves according to (7.4).

Let
$\color[rgb]{.2,0.2,.8}M:=2\sup_{(t,x)\in{\mathbb{R}}_{+}\times{\mathbb{R}}^{d}}\|\,c(t,x)\,\|=\sup_{(t,x)\in{\mathbb{R}}_{+}\times{\mathbb{R}}^{d}}\|\,b(t,x)\,\|$.
By definition of $b$,
$M=\sup_{(t,x)\in{\mathbb{R}}_{+}\times{\mathbb{T}}^{d}}\|\,G(t,x)\,\|$.
Moreover, for all $z$ such that $\|z\|=1$, $|\,\langle
z\,,\,c(t,\widetilde{X}^{x}_{t})-c(t,\widetilde{X}^{y}_{t})\,\big{\rangle}\,|\leq
M$.

Let $\widehat{D}_{t}$ the diffusion on $[0,L+1]$ which is absorbed at the
origin, reflected at $L+1$ and which evolves according to the SDE

$d\widehat{D}_{t}\;=\;2\,dB_{t}\;+\;M\,dt\;.$

By the previous bound on the drift term of $D_{t}$, we may couple
$\widetilde{D}_{t}$ and $\widehat{D}_{t}$ in such a way that
$\widetilde{D}_{t}\leq\widehat{D}_{t}$ for all $t\geq 0$, almost surely,
provided $\widetilde{D}_{0}\leq\widehat{D}_{0}$. In particular,
$\widetilde{D}_{0}$ hits the origin before $\widehat{D}_{0}$. Therefore, the
coupling time of $(\widetilde{Z}^{x}_{t},\widetilde{Z}^{y}_{t})$ is bounded
above by the absorption time of $\widehat{D}_{t}$, represented by $H^{r}_{0}$,
where $r$ stands for the initial state. An elementary computation yields that
there exists a finite constant $T_{0}$, depending only on $M$ and $L$ such
that

$\sup_{r\in[0,L+1]}E[\,H^{r}_{0}\,]\;\leq\;T_{0}\quad\text{so
that}\quad\sup_{r\in[0,L+1]}P[\,H^{r}_{0}>2T_{0}\,]\;\leq\;\frac{1}{2}\;\cdot$

In consequence,

$\sup_{x,y\in{\mathbb{T}}^{d}}P\big{[}\,\tau^{\widetilde{Z}}_{x,y}\,\geq\,2T_{0}\,\big{]}\;\leq\;\frac{1}{2}\;\cdot$

To complete the proof of (7.5) [and the one of the proposition], it remains to
apply the Markov property. ∎

###  7.2. Asymptotic behavior of linear parabolic equations.

Let $G:{\mathbb{R}}_{+}\times{\mathbb{T}}^{d}\to{\mathbb{R}}^{d}$ be a smooth
vector field satisfying the hypotheses stated in the previous subsection.

######  Proposition 7.4.

Fix two probability densities $w_{1}$, $w_{2}$ on ${\mathbb{T}}^{d}$,
$w_{j}\colon{\mathbb{T}}^{d}\to{\mathbb{R}}_{+}$,
$\int_{{\mathbb{T}}^{d}}w_{j}(x)\,dx=1$, $j=1$, $2$. Denote by
${\boldsymbol{w}}_{j}\colon{\mathbb{R}}_{+}\times{\mathbb{T}}^{d}\to{\mathbb{R}}_{+}$
the unique weak solution of the linear parabolic equation

$\left\\{\begin{aligned}
&\partial_{t}{\boldsymbol{w}}_{j}\;=\;\Delta{\boldsymbol{w}}_{j}\,-\,\nabla\cdot[\,{\boldsymbol{w}}_{j}\,G\,]\\\
&{\boldsymbol{w}}_{j}(0,\cdot)=w_{j}(\cdot)\end{aligned}\right.$ (7.6)

Then, there are $A<\infty$ and $\lambda>0$, which depends only on
$\sup_{(t,x)\in{\mathbb{R}}_{+}\times{\mathbb{T}}^{d}}\|\,G(t,x)\,\|$, such
that

$\int\big{|}\,{\boldsymbol{w}}_{2}(t,x)\,-\,{\boldsymbol{w}}_{1}(t,x)\,\big{|}\,dx\;\leq\;A\,e^{-\lambda
t}$

for all $t\geq 0$.

###### Proof.

Recall the definition of the diffusions $(Z^{x}_{t}:t\geq 0)$,
$x\in{\mathbb{T}}^{d}$, introduced in the previous subsection. Denote its
transition probability by $p_{t}(x,dy)=p_{t}(x,y)dy$ so that

${\mathbb{E}}_{x}[\,f(Z_{t})\,]\;=\;\int p_{t}(x,y)\,f(y)\,dy$

for all functions $f\in C({\mathbb{T}}^{d})$.

Since ${\mathscr{L}}_{t}$ is the generator of the diffusion $Z_{t}$, for every
function $f$ in $C^{2}({\mathbb{T}}^{d})$ and $t>0$,

${\mathbb{E}}_{x}[\,f(Z_{t})\,]\;=\;f(x)\;+\;\int_{0}^{t}{\mathbb{E}}_{x}[\,({\mathscr{L}}_{s}f)(Z_{s})\,]\;ds\;.$

integrating both sides of this identity with respect to $w_{j}(x)dx$ and
integrating by parts yields that
${\boldsymbol{v}}_{j}(t,x):=\int_{{\mathbb{T}}^{d}}w_{j}(y)p_{t}(y,x)dy$
solves (7.6) with initial condition ${\boldsymbol{v}}_{j}(0,x)=w_{j}(x)$. By
the uniqueness of weak solutions,

${\boldsymbol{w}}_{j}(t,x)\;=\;\int_{{\mathbb{T}}^{d}}w_{j}(y)\,p_{t}(y,x)\,dy\;.$

Since $\int{\boldsymbol{w}}_{j}(t,x)\,f(x)\,dx=\int
w_{j}(x)\,{\mathbb{E}}_{x}[\,f(Z_{t})\,]\,dx$ for every continuous function
$f:{\mathbb{T}}^{d}\to{\mathbb{R}}$, as $w_{j}(x)$ is a probability density,

$\displaystyle\int_{{\mathbb{T}}^{d}}{\boldsymbol{w}}_{2}(t,x)\,f(x)\,dx\;-\;\int_{{\mathbb{T}}^{d}}{\boldsymbol{w}}_{1}(t,x)\,f(x)\,dx$
$\displaystyle\quad\;=\;\int_{{\mathbb{T}}^{d}}dx\,w_{1}(x)\,\int_{{\mathbb{T}}^{d}}dy\,w_{2}(y)\,\Big{\\{}{\mathbb{E}}_{x}\big{[}\,f(Z_{t})\,\big{]}\;-\;{\mathbb{E}}_{y}\big{[}\,f(Z_{t})\,\big{]}\,\Big{\\}}\;.$

Therefore, by (7.3), for every $t\geq 0$,

$\sup_{f}\,\Big{|}\,\int{\boldsymbol{w}}_{2}(t,x)\,f(x)\,dx\;-\;\int{\boldsymbol{w}}_{1}(t,x)\,f(x)\,dx\,\Big{|}\;\leq\;2\,\sup_{x,y\in{\mathbb{T}}^{d}}\,P\big{[}\,\tau^{Z}_{x,y}\,\geq\,t\,\big{]}\;,$

where the supremum is carried over all continuous function $f$ such that
$\|f\|_{\infty}\leq 1$. Hence,

$\int\big{|}\,{\boldsymbol{w}}_{2}(t,x)\,-\,{\boldsymbol{w}}_{1}(t,x)\,\big{|}\,dx\;\leq\;2\,\sup_{x,y\in{\mathbb{T}}^{d}}\,P\big{[}\,\tau^{Z}_{x,y}\,\geq\,t\,\big{]}\;,$

and the assertion of the proposition follows from Proposition 7.2. ∎

We turn to Theorem 7.1 whose proof relies on the following estimate.

######  Proposition 7.5.

There exist constants $A<\infty$ and $\lambda>0$, which depends only on
$\sup_{(t,x)\in[0,T]\times{\mathbb{T}}^{d}}\|\,F(t,x)\,\|$, with the following
property. Fix $0<m<1$ and $u_{j}\colon{\mathbb{T}}^{d}\to[0,1]$, $j=1$, $2$,
such that $\int_{{\mathbb{T}}^{d}}u_{j}(x)\,dx=m$. Denote by
${\boldsymbol{u}}_{j}\colon{\mathbb{R}}_{+}\times{\mathbb{T}}^{d}\to{\mathbb{R}}_{+}$
the unique weak solution of (7.1) with initial condition $u_{j}$. Then,

$\int\big{|}\,{\boldsymbol{u}}_{2}(t,x)\,-\,{\boldsymbol{u}}_{1}(t,x)\,\big{|}\,dx\;\leq\;A\,e^{-\lambda
t}$

for all $t\geq 0$.

###### Proof.

Let
${\boldsymbol{v}}(t,x)={\boldsymbol{u}}_{2}(t,x)-{\boldsymbol{u}}_{1}(t,x)$,
so that $\int_{{\mathbb{T}}^{d}}{\boldsymbol{v}}(t,x)\,dx=0$ for all $t\geq
0$. Since $\sigma(b)-\sigma(a)=(b-a)(1-a-b)$, ${\boldsymbol{v}}(t,x)$ solves
the linear equation

$\partial_{t}{\boldsymbol{w}}\;=\;\Delta{\boldsymbol{w}}\,+\,\nabla\cdot[\,{\boldsymbol{w}}\,G\,]\;,$
(7.7)

where $G$ is the vector field
$G=(1-{\boldsymbol{u}}_{1}-{\boldsymbol{u}}_{2})F$.

Let $v_{0}\colon{\mathbb{T}}^{d}\to{\mathbb{R}}$ be given by
$v_{0}(x)=u_{2}(x)-u_{1}(x)$. Denote by $v^{+}$, $v^{-}$, the positive,
negative part of $v_{0}$, respectively. Note that
$\int_{{\mathbb{T}}^{d}}v^{+}(x)dx=\int_{{\mathbb{T}}^{d}}v^{-}(x)dx=:m^{\prime}\in[0,m]$.
If $m^{\prime}=0$, $0=v_{0}(x)=u_{2}(x)-u_{1}(x)$, and there is nothing to
prove. Assume that $m^{\prime}>0$ and let $w_{2}(x)=v^{+}(x)/m^{\prime}$,
$w_{1}(x)=v^{-}(x)/m^{\prime}$ so that $w_{j}$ is the density of a probability
measure on ${\mathbb{T}}^{d}$.

Denote by ${\boldsymbol{w}}_{j}(t,x)$ the solution of (7.7) with initial
condition $w_{j}(x)$. By linearity
$m^{\prime}[{\boldsymbol{w}}_{2}(t,x)-{\boldsymbol{w}}_{1}(t,x)]$ solves (7.7)
with initial condition $m^{\prime}[w_{2}(x)-w_{1}(x)]=v_{0}(x)$. Since
${\boldsymbol{v}}(t,x)$ solves the same Cauchy problem,
${\boldsymbol{v}}(t,x)=m^{\prime}[{\boldsymbol{w}}_{2}(t,x)-{\boldsymbol{w}}_{1}(t,x)]$.
Thus, as $m^{\prime}\leq 1$,

$\int\big{|}\,{\boldsymbol{u}}_{2}(t,x)\,-\,{\boldsymbol{u}}_{1}(t,x)\,\big{|}\,dx\;=\;\int\big{|}\,{\boldsymbol{v}}(t,x)\,\big{|}\,dx\;\leq\;\int\big{|}\,{\boldsymbol{w}}_{2}(t,x)\,-\,{\boldsymbol{w}}_{1}(t,x)\,\big{|}\,dx\;.$

To complete the proof, it remains to recall the statement of Proposition 7.4,
and to observe that
$\sup_{(t,x)\in{\mathbb{R}}_{+}\times{\mathbb{T}}^{d}}\|\,G(t,x)\,|\|\leq\|\,F\,\|_{\infty}$
because ${\boldsymbol{u}}_{j}(t,x)$ takes value in the interval $[0,1]$. ∎

From this result we can deduce the first assertion of Theorem 7.1.

######  Corollary 7.6.

For each $m\in(0,1)$, the equation (7.1) admits a unique $T$-periodic solution
${\boldsymbol{u}}:{\mathbb{R}}\times{\mathbb{T}}^{d}\to[0,1]$.

###### Proof.

Fix $m\in(0,1)$ and denote by $L^{1}_{m}({\mathbb{T}}^{d})$ the closed
subspace of $L^{1}({\mathbb{T}}^{d})$ defined by
$\color[rgb]{.2,0.2,.8}L^{1}_{m}({\mathbb{T}}^{d})=\\{u\in
L^{1}({\mathbb{T}}^{d}):\int_{{\mathbb{T}}^{d}}u(x)\,dx=m\,,\,0\leq u(x)\leq
1\,\\}$.

Define the operator ${\mathfrak{P}}:L_{m}^{1}({\mathbb{T}}^{d})\to
L_{m}^{1}({\mathbb{T}}^{d})$ given by
${\mathfrak{P}}(u)={\boldsymbol{u}}(T,\cdot)$, where ${\boldsymbol{u}}(t,x)$
is the weak solution of (7.1) with initial condition $u(\cdot)$. Let
$u_{0}:{\mathbb{T}}^{d}\to[0,1]$ be given by $u_{0}(x)=m$ for all $x$, and set
$u_{j+1}={\mathfrak{P}}u_{j}$, $j\geq 0$. We claim that the sequence
$(u_{j}:j\geq 1)$ is Cauchy in $L^{1}({\mathbb{T}}^{d})$. Fix $n$, $j\geq 1$.
Since ${\mathfrak{P}}_{n+j}u={\mathfrak{P}}_{n}{\mathfrak{P}}_{j}u$, by
Proposition 7.5,

$\big{\|}\,{\mathfrak{P}}_{n+j}u\,-\,{\mathfrak{P}}_{n}u\,\big{\|}_{1}\;=\;\big{\|}\,{\mathfrak{P}}_{n}[\,{\mathfrak{P}}_{j}u\,-\,u\,]\,\big{\|}_{1}\;\leq\;Ae^{-\lambda
nT}\;.$

Denote by $w$ the limit in $L^{1}$ of the sequence $u_{j}$, and observe that
${\mathfrak{P}}w=w$. This proves that the solution of equation (7.1) with
initial condition $w$ is $T$-periodic. By Proposition 7.5, such $T$-periodic
solution is unique. ∎

###### Proof of Theorem 7.1.

Fix $m\in[0,1]$. As the result is trivial for $m=0$ or $1$, we may assume that
$0<m<1$. In this range, the assertions of the theorem corresponds to the ones
of Proposition 7.5 and Corollary 7.6. ∎

##  Appendix A Dynamical bounds

We present in this section some estimates used in the article. Let

$c_{x,y}(\eta)\;=\;\eta_{x}\,[1-\eta_{y}]\,e^{(1/2)\,E_{N}(x,y)}\;,\quad(x,y)\,\in\,{\mathbb{B}}_{N}\;$

and recall the notation introduced in (2.18).

######  Lemma A.1.

Given a set of bounded functions
$\phi^{x,y}_{s}:D([0,\infty),\Sigma_{N})\to{\mathbb{R}}$,
$(x,y)\in{\mathbb{B}}_{N}$, progressively measurable, the process

${\mathbb{M}}^{\phi}_{t}\;=\;\exp\sum_{(x,y)\in{\mathbb{B}}_{N}}\Big{\\{}\int_{0}^{t}\phi^{x,y}_{s}\,{\mathscr{N}}^{x,y}_{(s,s+ds]}({\boldsymbol{\eta}})\,-\,N^{2}\int_{0}^{t}c_{x,y}({\boldsymbol{\eta}}(s))\\{e^{\phi^{x,y}_{s}}-1\\}\,ds\Big{\\}}$
(A.1)

is a mean one, positive martingale with respect to ${\mathbb{P}}_{\eta}^{N}$
for any configuration $\eta\in\Sigma_{N}$.

The proof of this lemma is similar to the one of Proposition A.2.6 in [15] and
left to the reader. This martingale corresponds to the Radon-Nikodym
derivative (restricted to the interval $[0,t]$) of the law of a jump process
with rates $c_{x,y}(\eta)e^{\phi^{x,y}}$ with respect
${\mathbb{P}}_{\eta}^{N}$.

Recall that the symmetric simple exclusion process is the Markov chain on
$\Sigma_{N}$ whose generator is $L_{N}$, introduced in (2.1), with $E\equiv
0$. Denote by $\color[rgb]{.2,0.2,.8}\nu_{\alpha}$, $0\leq\alpha\leq 1$, the
Bernoulli product measure on $\Sigma_{N}$ with density $\alpha$, and by
$\color[rgb]{.2,0.2,.8}{\mathbb{P}}^{0}_{\nu_{\alpha}}$ the probability
measure on $D({\mathbb{R}}_{+},\Sigma_{N})$ induced by the symmetric simple
exclusion process starting from $\nu_{\alpha}$. Expectation with respect to
${\mathbb{P}}^{0}_{\nu_{\alpha}}$ is represented by
$\color[rgb]{.2,0.2,.8}{\mathbb{E}}^{0}_{\nu_{\alpha}}$.

######  Lemma A.2.

For all $1<p<\infty$ and $E\in C^{1}({\mathbb{T}}^{d};{\mathbb{R}}^{d})$,
there exists a constant $C_{p}$ such that

$\log\,{\mathbb{E}}^{0}_{\nu_{1/2}}\Big{[}\,\Big{(}\frac{d{\mathbb{P}}^{N}_{\mu_{N,K}}}{d{\mathbb{P}}^{0}_{\nu_{1/2}}}\Big{|}_{[0,T]}\,\Big{)}^{p}\,\Big{]}\;\leq\;C_{p}\,(1\,+\,T)\,N^{d}$

for all $T>0$, $N\geq 1$, $0\leq K\leq N^{d}$.

###### Proof.

The proof reduces to a standard computation of exponential martingales. To
emphasize the dependence of the measure ${\mathbb{P}}^{N}_{\mu_{N,K}}$ on the
external field $E$, in this proof, we represent the measure
${\mathbb{P}}^{N}_{\mu_{N,K}}$ by ${\mathbb{P}}^{E}_{\mu_{N,K}}$. Clearly,

${\mathbb{E}}^{0}_{\nu_{1/2}}\Big{[}\,\Big{(}\frac{d{\mathbb{P}}^{E}_{\mu_{N,K}}}{d{\mathbb{P}}^{0}_{\nu_{1/2}}}\Big{|}_{[0,T]}\,\Big{)}^{p}\,\Big{]}\;=\;{\mathbb{E}}^{0}_{\mu_{N,K}}\Big{[}\,\Big{(}\frac{d\mu_{N,K}}{d\nu_{1/2}}\Big{)}^{p-1}\,\Big{(}\frac{d{\mathbb{P}}^{E}_{\mu_{N,K}}}{d{\mathbb{P}}^{0}_{\mu_{N,K}}}\Big{|}_{[0,T]}\,\Big{)}^{p}\,\,\Big{]}\;.$

As $\nu_{1/2}(\eta)=(1/2)^{N^{d}}$, this expression is bounded by

$2^{(p-1)N^{d}}\,{\mathbb{E}}^{0}_{\mu_{N,K}}\Big{[}\,\Big{(}\frac{d{\mathbb{P}}^{E}_{\mu_{N,K}}}{d{\mathbb{P}}^{0}_{\mu_{N,K}}}\Big{|}_{[0,T]}\,\Big{)}^{p}\,\,\Big{]}\;.$

On the other hand,

${\mathbb{E}}^{0}_{\mu_{N,K}}\Big{[}\,\Big{(}\frac{d{\mathbb{P}}^{E}_{\mu_{N,K}}}{d{\mathbb{P}}^{0}_{\mu_{N,K}}}\Big{|}_{[0,T]}\,\Big{)}^{p}\,\Big{]}\;=\;{\mathbb{E}}^{pE}_{\mu_{N,K}}\Big{[}\,\Big{(}\frac{d{\mathbb{P}}^{E}_{\mu_{N,K}}}{d{\mathbb{P}}^{0}_{\mu_{N,K}}}\Big{|}_{[0,T]}\,\Big{)}^{p}\;\frac{d{\mathbb{P}}^{0}_{\mu_{N,K}}}{d{\mathbb{P}}^{pE}_{\mu_{N,K}}}\Big{|}_{[0,T]}\,\,\Big{]}\;.$

Note that in the last expectation the external field is $pE$. A direct
computation based on the explicit formula for the Radon-Nikodym derivatives
provided by Lemma A.1 yields that

$\Big{\|}\,\Big{(}\frac{d{\mathbb{P}}^{E}_{\mu_{N,K}}}{d{\mathbb{P}}^{0}_{\mu_{N,K}}}\Big{|}_{[0,T]}\,\Big{)}^{p}\;\frac{d{\mathbb{P}}^{0}_{\mu_{N,K}}}{d{\mathbb{P}}^{pE}_{\mu_{N,K}}}\Big{|}_{[0,T]}\,\Big{\|}_{\infty}\;\leq\;e^{C_{p}\,T\,N^{d}}$

for some finite constant $C_{p}$, see [5, Lemma 4.5]. ∎

Until the end of the appendix, fix $T>0$, $m\in(0,1)$ and a sequence
$(K_{N}:N\geq 1)$ such that $K_{N}/N^{d}\to m$. Consider a progressively
measurable, continuous function
$w\colon{\mathbb{R}}\times{\mathbb{T}}^{d}\times{\mathcal{M}}_{+}({\mathbb{T}}^{d})\times
D({\mathbb{R}};\mathcal{H}^{d}_{-p})\to{\mathbb{R}}^{d}$ with support on
$[0,T]\times{\mathbb{T}}^{d}\times{\mathcal{M}}_{+}({\mathbb{T}}^{d})\times
D({\mathbb{R}};\mathcal{H}_{-p}^{d})$. Recall from (4.5) the definition of the
progressively measurable function
$G_{w}\colon{\mathbb{R}}\times{\mathbb{T}}^{d}\times{\mathscr{S}}_{m,{\rm
ac}}\to{\mathbb{R}}^{d}$. For $\varepsilon>0$ and a cylinder function $\Psi$,
let

$F^{w,\Psi}_{N,\varepsilon}(t,{\boldsymbol{\eta}})\;=\;\frac{1}{N^{d}}\sum_{x\in{\mathbb{T}}^{d}_{N}}G_{w}\Big{(}t\,,\,x\,,\,{\boldsymbol{\pi}}_{N}\,,\,{\boldsymbol{J}}_{N}\Big{)}\,\big{\\{}\,(\tau_{x}\Psi)(\eta(t))-\widehat{\Psi}(({\boldsymbol{\pi}}_{N}^{\varepsilon})(t,x))\,\big{\\}}\;,$

where
$({\boldsymbol{\pi}}^{\varepsilon}_{N})(t,x)=(2\varepsilon)^{-d}{\boldsymbol{\pi}}_{N}(t,[x-\varepsilon,x+\varepsilon]^{d})$,
and $\widehat{\Psi}:[0,1]\to{\mathbb{R}}$ is the function given by

$\widehat{\Psi}(\alpha)\;=\;E_{\nu_{\alpha}}\big{[}\,\Psi\,\big{]}\;.$

######  Lemma A.3.

For all $\delta>0$

$\lim_{\varepsilon\to
0}\limsup_{N\to\infty}\frac{1}{N^{d}}\log{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\Big{[}\,\Big{|}\int_{0}^{T}F^{w,\Psi}_{N,\varepsilon}(t,{\boldsymbol{\eta}})\,dt\,\Big{|}>\delta\,\Big{]}\;=\;-\infty\;.$

###### Proof.

First, we claim that for all $\delta>0$,

$\lim_{\varepsilon\to
0}\limsup_{N\to\infty}\frac{1}{N^{d}}\log{\mathbb{P}}^{0}_{\nu_{1/2}}\Big{[}\,\Big{|}\int_{0}^{T}F^{w,\Psi}_{N,\varepsilon}(t,{\boldsymbol{\eta}})\,dt\,\Big{|}>\delta\,\Big{]}\;=\;-\infty\;.$

We refer to [15, Theorem 10.3.1] for the proof in the case in which $w$ does
not depend on $\pi$ and ${\boldsymbol{J}}$. The arguments to include this
dependence are tedious, but straightforward and left to the reader. The
extension to the measure ${\mathbb{P}}^{N}_{\mu_{N,K_{N}}}$ follows from
Schwarz inequality and Lemma A.2. ∎

Next result is a consequence of the entropy inequality, [15, Proposition
A1.8.2], and the previous lemma.

######  Corollary A.4.

Let $({\mathbb{Q}}_{N}:N\geq 1)$ be a sequence of probability measures in
${\mathscr{P}}^{N,K_{N}}_{\rm stat}$. Assume that there exists a finite
constant $C_{0}$ such that for all $S>0$,

$\limsup_{N\to\infty}\frac{1}{N^{d}}\,{\mathbb{H}}^{(S)}\big{(}\,{\mathbb{Q}}_{N}\,|\,{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\,\big{)}\;\leq\;C_{0}\,S\;.$

Then, for all $\delta>0$

$\lim_{\varepsilon\to
0}\limsup_{N\to\infty}{\mathbb{Q}}_{N}\Big{[}\,\Big{|}\int_{0}^{T}F^{w,\Psi}_{N,\varepsilon}(t,{\boldsymbol{\eta}})\,dt\,\Big{|}>\delta\,\Big{]}\;=\;0\;.$

Fix a vector field $F$ in
$C^{1}({\mathbb{R}}\times{\mathbb{T}}^{d};{\mathbb{R}}^{d})$ with compact
support in $(0,T)\times{\mathbb{T}}^{d}$, $a>0$, and recall the definition of
${\mathcal{E}}_{a,\varepsilon}(F,{\boldsymbol{\pi}})$,
${\mathcal{V}}_{a,\varepsilon}(F,{\boldsymbol{\pi}},{\boldsymbol{J}})$,
$\varepsilon>0$ in (4.4).

######  Lemma A.5.

There exist finite, positive constants $a$ and $C_{0}$ such that for all
vector fields $F$ in
$C^{1}({\mathbb{R}}\times{\mathbb{T}}^{d};{\mathbb{R}}^{d})$ with compact
support in $(0,T)\times{\mathbb{T}}^{d}$,

$\displaystyle\limsup_{\varepsilon\to
0}\limsup_{N\to\infty}\frac{1}{N^{d}}\,\log{\mathbb{E}}^{N}_{\mu_{N,K_{N}}}\Big{[}\exp\big{\\{}N^{d}\,{\mathcal{E}}_{a,\varepsilon}(F,{\boldsymbol{\pi}}_{N})\,\big{\\}}\,\Big{]}\;\leq\;C_{0}\,(1+T)\;,$
$\displaystyle\limsup_{\varepsilon\to
0}\limsup_{N\to\infty}\frac{1}{N^{d}}\,\log{\mathbb{E}}^{N}_{\mu_{N,K_{N}}}\Big{[}\,\exp\big{\\{}N^{d}\,{\mathcal{V}}_{a,\varepsilon}(F,{\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})\big{\\}}\,\Big{]}\;\leq\;C_{0}\,(1+T)\;.$

###### Proof.

We claim that there exists a finite constant $a$ such that for any $T>0$ and
any $F$ in $C^{1}({\mathbb{R}}\times{\mathbb{T}}^{d};{\mathbb{R}}^{d})$ with
compact support in $(0,T)\times{\mathbb{T}}^{d}$,

$\displaystyle\limsup_{\varepsilon\to
0}\limsup_{N\to\infty}\frac{1}{N^{d}}\,\log{\mathbb{E}}^{0}_{\nu_{1/2}}\Big{[}\exp
N^{d}\,\big{\\{}\,{\mathcal{E}}_{a,\varepsilon}(F,{\boldsymbol{\pi}}_{N})\,\big{\\}}\,\Big{]}\;\leq\;0\;.$

This statement is proven in [6, Section 3.2] (see the proof of the bound
presented in the last displayed equation at page 2367). To deduce the first
assertion of the lemma from this result, it suffices to apply Schwarz
inequality, to recall the statement of Lemma A.2, and to observe that
$2{\mathcal{E}}_{a,\varepsilon}(F,{\boldsymbol{\pi}}_{N})={\mathcal{E}}_{a/2,\varepsilon}(2F,{\boldsymbol{\pi}}_{N})$.

We turn to the second assertion of the lemma. By Lemma A.1,

${\mathbb{E}}^{N}_{\mu_{N,K_{N}}}\Big{[}\exp\Big{\\{}\,2\,N^{d}\,{\boldsymbol{J}}_{N}(F)\,-\,{\boldsymbol{W}}_{N}(T)\,\Big{\\}}\,\Big{]}\;=\;1\;,$

provided

${\boldsymbol{W}}_{N}(T)\;=\;N^{2}\sum_{(x,y)\in{\mathbb{B}}_{N}}\int_{0}^{T}{\boldsymbol{\eta}}_{x}(s)\,[1-{\boldsymbol{\eta}}_{y}(s)]\,\\{e^{2\,F_{N}(s,x,y)}-1\\}\,ds\;.$

Recalling (4.4), by adding and subtracting $(1/2N^{d}){\boldsymbol{W}}_{N}(T)$
and applying Schwarz inequality we get

$\displaystyle{\mathbb{E}}^{N}_{\mu_{N,K_{N}}}\Big{[}\,\exp
N^{d}\,\Big{\\{}\,{\boldsymbol{J}}_{N}(F)\;-\;a\,\int_{{\mathbb{R}}}ds\int_{{\mathbb{T}}^{d}}dx\,\sigma({\boldsymbol{\pi}}_{N}^{\varepsilon})\,|F|^{2}\Big{\\}}\,\Big{]}$
$\displaystyle\quad\;\leq\;{\mathbb{E}}^{N}_{\mu_{N,K_{N}}}\Big{[}\,\exp\Big{\\{}\,{\boldsymbol{W}}_{N}(T)\;-\;2\,N^{d}\,a\,\int_{{\mathbb{R}}}ds\int_{{\mathbb{T}}^{d}}dx\,\sigma({\boldsymbol{\pi}}_{N}^{\varepsilon})\,|F|^{2}\Big{\\}}\,\Big{]}^{1/2}\;.$

Expanding the exponential $\exp\\{2\,F_{N}(s,x,y)\\}$ which appears in the
definition of ${\boldsymbol{W}}_{N}(T)$, summing by parts, using Lemma A.3,
and the first part of the proof yields the desired bound. ∎

Consider a continuous function
$w\colon[0,T]\times{\mathbb{T}}^{d}\times{\mathcal{M}}_{+}({\mathbb{T}}^{d})\times
D({\mathbb{R}};{\mathcal{H}}_{-p}^{d})\to{\mathbb{R}}^{d}$ that is
continuously differentiable in $x$ and such that for each
$(x,\pi)\in{\mathbb{T}}^{d}\times{\mathcal{M}}_{+}({\mathbb{T}}^{d})$ and
$t\in[0,T]$ the map $[0,t]\times
D({\mathbb{R}};{\mathcal{H}}_{-p}^{d})\ni(s,{\boldsymbol{J}})\to
w(s,x,\pi,{\boldsymbol{J}})$ is measurable with respect to the Borel
$\sigma$-algebra on $[0,t]\times D([0,t];{\mathcal{H}}^{d}_{-p}\big{)}$. Let
$\phi^{x,y}\colon[0,T]\times D([0,t],\Sigma_{N})\to{\mathbb{R}}$,
$(x,y)\in{\mathbb{B}}_{N}$, be given by

$\phi^{x,y}(t)\;=\;\int_{x}^{y}w(t,\cdot\,,{\boldsymbol{\pi}}_{N}(t),{\boldsymbol{J}}_{N})\,\cdot
d\ell\;,$

and let ${\mathbb{M}}^{\phi}_{T}$ be the martingale introduced in (A.1),

######  Lemma A.6.

Let $({\mathbb{Q}}_{N}:N\geq 1)$ be a sequence of probability measures in
${\mathscr{P}}^{N,K_{N}}_{\rm stat}$ such that
${\mathbb{Q}}_{N}\circ({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})^{-1}\to P$
for some $P\in{\mathscr{P}}_{\rm stat}$ satisfying (2.27). Assume that there
exists a finite constant $C_{0}$ such that for all $S>0$,

$\limsup_{N\to\infty}\frac{1}{N^{d}}\,{\mathbb{H}}^{(S)}\big{(}\,{\mathbb{Q}}_{N}\,|\,{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\,\big{)}\;\leq\;C_{0}\,S\;.$

Then, for each $w$ as above,

$\frac{1}{T}\,\lim_{N\to\infty}\frac{1}{N^{d}}\,E_{{\mathbb{Q}}_{N}}\big{[}\,\log{\mathbb{M}}^{\phi}_{T}\,\big{]}\;=\;E_{P}\big{[}\,V_{T,w}\,\big{]}\;,$

where $V_{T,w}$ has been introduced in (4.6)

###### Proof.

On the one hand, by definition of $\phi^{x,y}$ and of the current
${\boldsymbol{J}}_{N}$, and since
${\mathbb{Q}}_{N}\circ({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})^{-1}\to P$
for some measure $P\in{\mathscr{P}}_{\rm stat}$ satisfying (2.27),

$\lim_{N\to\infty}\frac{1}{N^{d}}\,E_{{\mathbb{Q}}_{N}}\Big{[}\,\sum_{(x,y)\in{\mathbb{B}}_{N}}\int_{0}^{T}\phi^{x,y}_{s}\,{\mathscr{N}}^{x,y}_{(s,s+ds]}\,\Big{]}\;=\;E_{P}\Big{[}\,\int_{0}^{T}ds\int_{{\mathbb{T}}^{d}}dx\;G_{w}\cdot{\boldsymbol{j}}\,\Big{]}\;.$

On the other hand, a straightforward computation yields that

$\displaystyle
N^{2}\sum_{(x,y)\in{\mathbb{B}}_{N}}c_{x,y}(\eta)\\{e^{\phi^{x,y}}-1\\}\;=\;N^{d}\,\langle\pi_{N}\,,\,\nabla\cdot
G_{w}\rangle$
$\displaystyle\quad\;+\;\frac{1}{2}\sum_{j=1}^{d}\sum_{x\in{\mathbb{T}}^{d}_{N}}[\eta_{x+{\mathfrak{e}}_{j}}-\eta_{x}]^{2}\,w_{j}(x)\,[w_{j}(x)+E_{j}(x)]\;+\;o(N^{d})\;.$

Therefore, by Corollary A.4 and since
${\mathbb{Q}}_{N}\circ({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})^{-1}\to P$
for some measure $P\in{\mathscr{P}}_{\rm stat}$ satisfying (2.27),

$\displaystyle\lim_{N\to\infty}\frac{1}{N^{d}}\,E_{{\mathbb{Q}}_{N}}\Big{[}\,N^{2}\sum_{(x,y)\in{\mathbb{B}}_{N}}\int_{0}^{T}ds\,c_{x,y}({\boldsymbol{\eta}}(s))\\{e^{\phi^{x,y}_{s}}-1\\}\,\Big{]}$
$\displaystyle\quad\;=\;E_{P}\Big{[}\int_{0}^{T}dt\int_{{\mathbb{T}}^{d}}dx\,\Big{\\{}\,G_{w}\cdot\big{[}\,-\,\nabla{\boldsymbol{\rho}}\,+\,\sigma({\boldsymbol{\rho}})\,E\,\big{]}\,+\,\sigma({\boldsymbol{\rho}})\,|G_{w}|^{2}\,\Big{\\}}\,\Big{]}\;.$

The assertion of the lemma follows from the two previous estimates. ∎

### Acknowledgments

We are grateful to M. Mariani and L. Rossi for useful discussions.

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of true Class 1 or 2 clusters meeting the magnitude/footprint constraints, and
if systematic identification/misclassification exists then this figure will
drop or rise accordingly depending on which survey has the problem.

Blue and red shaded portions of the outer rings of the doughnut plots in Fig.
14 aim to better understand the types of sources for which the surveys find
different classifications. We begin with objects identified as Class 1 or 2 by
PHANGS-HST but not by LEGUS (shown in blue). The wedges labelled L3 and L4
were classified as Class 3 and Class 4, respectively, by LEGUS (but Class 1 or
2 by PHANGS-HST). Approximately one quarter to one third of the PHANGS-HST-
only Class 1 or 2 population falls into this category. We note that this
happens more often in NGC 3351 than for other targets (42% for ML, 58% for
human). In the case of human classification, this may be due to the greater
exposure depth (which was not used in ML), in the overlapping LEGUS and
PHANGS-HST fields. An even larger contribution ($\sim 40{-}70$%) to this
misclassification category comes from sources which were somehow not even
considered by LEGUS (labelled ‘no ID’). These result from intrinsic
differences in the types of sources detected by SExtractor versus DOLPHOT, or
due to automated cuts made by LEGUS after the detection stage (CI, number of
bands with good photometry). A minority of sources, belonging to doughnut
wedges labelled with asterisks, were at least considered as LEGUS candidates
and seem to have met the conditions for LEGUS classification given by Adamo et
al. (2017) but did not receive an eventual class for some reason we have not
been able to determine. Next, we examine the sources deemed Class 1 or 2 by
LEGUS but not by PHANGS-HST (red region in the outer doughnut). A first
category is comprised of sources we flag as ‘doubles’ (see Sec. 4.4), where
two objects are detected within 2.5 pixels of one another. Extended sources at
such small separations, even if real and not a redundant DOLPHOT detection of
the wings of the brighter extended object, would be challenging to classify
accurately. We find that $10{-}20$% of the LEGUS Class 1 or 2 objects not
included by PHANGS-HST fall into this category. Whitmore et al. (2021) also
comment on the preferential inclusion of ‘doubles’ by LEGUS, and find they are
typically assigned to Class 2. Differences in classification between surveys
can also be seen as many PHANGS-HST Class 1 or 2’s are called Class 3 or 4 by
LEGUS. In terms of Class 3 (see ML3, H3 doughnut wedges), they amount to
$\sim$15%. One comment applicable to both survey classification efforts is the
difficulty in classifying objects in crowded environments – this is
particularly relevant to the distinction between Class 3 and Class 1+2. For
example, imagine taking a confident Class 1 or 2 cluster and placing it in a
region with many point sources or even other clusters. Based on the relative
location of objects, it is possible that such a Class 1 or 2 is frequently
judged by the classifier (either ML or human) as an apparent Class 3 compact
association because the only evidence remaining of cluster-nature is the very
inner profile of the source, which can be overridden or overlooked due to
environment. We have no reason to believe this happens more often in either
survey, and the ML3/H3 fractions are indeed similar to H3 in Fig. 14. A far
more significant (and systematic) classification trend between LEGUS and
PHANGS-HST is represented by the ML4/H4 wedges which dominate the LEGUS-only
Class 1 or 2 population. In most targets more than 50% (and up to $>70$%) of
LEGUS Class 1 or 2 sources not called 1 or 2 by PHANGS-HST are instead
classified as contaminants by PHANGS-HST. We return to this point below.
Finally, a small percentage of LEGUS Class 1 or 2 clusters are omitted from
the PHANGS-HST Class 1 or 2 census for reasons we did not track down (red
outer wedge labelled with asterisk), but possibly these sources failed cuts on
MCI error.

In the case of the human comparison set, we have more detailed information on
the appearance of sources LEGUS called Class 1 or 2 and PHANGS-HST classified
as contaminants (that is, Class 4 in the original LEGUS system, and now
subdivided by BCW into Classes 4.1, 4.2, …, 4.12 in order to specify the
variety of contaminant cluster candidate, e.g. single star, stellar pair,
stellar triple, saturated star, diffraction spike, galaxy nucleus, background
galaxy, etc. – see Whitmore et al. (2021) for details). In Fig. 15, we show
doughnut charts for these human Class 4 (‘H4’) sources, splitting them into up
to six BCW subclasses (others of the 12 were not encountered). Specifically,
the H4 sources are classified as a single star, pair or triple stars,
redundant (another accepted Class 1, 2, 3 within 5 pixels), edge artefact, or
too faint to classify. This categorical assessment of contributors to the H4
LEGUS-only population is most informative/reliable in cases having many
sources. For this reason NGC 628-E and NGC 1433 should be viewed cautiously,
as they each have $\leq 15$ H4 sources (only 8 in NGC 628-E).

The first observation to be made from Fig. 15 is that sources appearing to BCW
as single stars are a minority in this population, amounting to less than 25%
in all cases. The bulk of H4 sources are thought to be pairs or triples of
stars according to the PHANGS-HST human classification. In general pairs
constitute $\sim 45\%$, and triples $\sim 33\%$, of the H4 LEGUS-only
population. Combined with the already flagged ‘doubles’ from the analysis
above, we conclude that the majority of objects called Class 1 or 2 by LEGUS
and not by PHANGS-HST are actually multiple stellar sources not meeting the
source surface density needed to qualify as Class 3 associations. Whitmore et
al. (2021) describes how this distinction is set by counting peaks within a 5
pixel radius. This effect is probably accentuated by situations in which
SExtractor does not deblend such multiples and places a source between the
actual peaks. This is a particular strength of using the PSF-fitting source
detection algorithm (DOLPHOT) for our work, though it comes with the price of
spurious detections in the wings of bright objects (largely eliminated by our
doubles check, Sec. 4.4) and the need to explicitly eliminate redundant
counting of associations due to proper deblending of subclumps (last paragraph
of Sec. 4.4). The subclasses not yet described (redundant, edge, too faint)
remain in the minority across the entire H4 population, with faint objects
only cropping up in NGC 1566 and edge artefacts in NGC 1566 and NGC 628-E. The
slightly more numerous ‘redundant’ population can be attributed to differences
in source detection between surveys.

Lastly, we emphasise one final point from Fig. 14 concerning exact
classification agreement. In the outer ring of the doughnut plots, in two
shades of green, we indicate the fraction of Class 1 or 2 clusters in common
between surveys that were given the same Class (e.g. 1 and 1; 2 and 2) with
the darker wedge, and those that disagreed (e.g. 1 and 2; 2 and 1) with the
lighter colour. Quite commonly the survey classification agreed on specific
Class for this subset of the the overall comparison set. We direct the reader
to Whitmore et al. (2021) for a detailed, extensive discussion of confusion
between classes, including a discussion of confusion matrices for cross-survey
(PHANGS-HST versus LEGUS) and internal (VGG versus ResNet, human versus VGG,
etc.) comparisons.

Complementing the limited PHANGS-HST-ML to LEGUS-human classification results
(top row of Fig. 14) of this Section, we direct the reader to the more
extensive analysis of similar issues by Whitmore et al. (2021). In particular,
Whitmore et al. (2021) introduce use of the colour-colour diagram as a ’figure
of merit’ to further test the relative performances of the different methods.

###  6.5 Resulting PHANGS-HST cluster catalogues

This paper is not intended to provide astrophysical analysis of the cluster
populations recovered by our methods, but rather to focus on the adopted
procedures. Nevertheless, without presentation of some scientifically relevant
plots, it would be impossible to fully appreciate the improvement we have
attained relative to previous works, except in the sense of the comparison
described in Sec. 6.4. In Fig. 16, we show the PHANGS-HST Class 1+2 clusters
identified by human and ML classification for NGC 628-C. Fig. 17 shows the
spatial distribution of clusters for the same field (trimmed to the
WFC3/UVIS+ACS/WFC footprint) colour-coded by age in both contexts. These
figures displaying our results for one target are both internally instructive,
emphasising the more expansive cluster population recovered using ML
classification relative to human-based efforts, and also can be compared
versus the results of Adamo et al. (2017) who used the same data. Appendix A
contains equivalent figures for all the fields analysed in our paper.

##  7 Discussion

###  7.1 General comments on our overall strategy

A limiting step in most studies of extragalactic star clusters is the manual
effort needed to remove artefacts from candidate lists. It is difficult to
automatically identify candidate star clusters with a high degree of certainty
due to environmental complexity (effects such as crowding and high
background), in addition to resolution limits. The MCI approach we introduced
was originally designed to reduce the number of candidate star clusters that
would need to be manually examined to remove artefacts such as pairs of stars,
background galaxies, edge effects, etc. The basic strategy was to leverage
photometry from several apertures rather than just the two used to determine
the concentration index (CI), which has been the standard way to filter out
artefacts. In this way, selection criteria could be enforced that insured that
the profile followed a typical profile of a cluster rather than a star, pair
of star, or other more complicated morphologies. Original trials using this
approach were largely successful, reducing the size of the candidate list by
about 50%.

During implementation, several other considerations reduced this percentage
somewhat, while enabling other improvements to the overall strategy. One
example is that the selection routine for both stars and cluster candidates
has been unified to use a single primary method (DOLPHOT), rather than using
two completely different methods employed by many other projects. This
eliminates the necessity for subsequently cross matching the stellar and
cluster catalogues after totally independent selection/classification
decisions have been made, which would otherwise be a significant source of
noise and complexity at a later stage of a project. However, a side-effect of
the use of DOLPHOT is that it introduces a much larger initial candidate list
since DOLPHOT’s selection routine uses an iterative method to optimally
deblend neighbouring sources and find fainter, inconspicuous sources in the
wings of bright objects. This is the main source of the growth of our
candidate cluster lists. Another similar, but less important effect from a
growth standpoint is the development of a more sophisticated method of
removing point-like objects, allowing us to include proportionately more (of
particularly compact) cluster candidates than possible using only the
concentration index, hence improving the completeness levels.

The MCI approach also enables other potential improvements, many of which will
be more evident in later stages of the PHANGS-HST project. For example, the
development of cluster models needed to define selection regions for the MCI
values will be useful in the analysis of various physical characteristics of
the clusters, such as whether the objects are likely to be bound or unbound,
whether certain kinds of potential profiles (e.g. very diffuse) are not
actually found in nature, and how the profiles of clusters change as a
function of age.

In summary, while some of the initial goals of the MCI method have not been
fully realised (i.e. the reduction in the candidate list), the overall
improvements to the project is evident and is expected to have long range
benefits relative to the PHANGS-HST science goals and cluster studies in
general.

###  7.2 Realised scientific advantages of our method

Our fully integrated pipeline for cluster detection, identification, and ML
classification produces reliable catalogues of clusters with minimal human
effort. These catalogues have clear advantages over prior efforts, even using
the same observational data and neglecting the relative ease with which they
can be generated. This subsection is devoted to highlighting such scientific
benefits in broad terms.

The advantage of reproducibility, and presumed replicability, should not be
understated. Even within LEGUS, the premier cluster study prior to PHANGS-HST,
attaining reproducibility was challenging (Pérez et al., 2021). The burden of
human classification relative to the limited pool of trained team member
classifiers ($\sim 12$) led to compromises in the uniformity of the adopted
procedure. For instance, in general for each galaxy only a small subset (3
people) of classifiers examined the candidates (Adamo et al., 2017). This
could impart systematic differences in the effective classification scheme
between targets, if individual volunteers gave more weight to different
aspects of the classification procedure. In cases where the number of
candidates per target were very high, the classification task was divided
across the galaxy, assigning stripes of coverage to different subsets of the
classification group. It should be said that membership in each subset was
shuffled, and the mode was adopted as final judgement amongst the
classification votes per candidate, in hopes of minimising systematic effects.
The adopted LEGUS method represents the best that can be achieved with a small
group of human volunteers. 151515Experimentation with citizen science based
classification was undertaken by LEGUS as well, to potentially overcome the
limits of person power, but the lack of interactive inspection tools (image
contrast, radial profiles) on current platforms led to the conclusion this was
not yet feasible. We note that in the more resolved case of the Magellanic
Clouds, and M31 / M33 (with $HST$), citizen science cluster identification has
proven rather successful (e.g. Johnson et al., 2012).

For PHANGS-HST, at no stage in the ML classification branch of our pipeline
does human judgement enter the procedure in real time, so in one sense our
analysis is reproducible. However, the training of the ML network embodies the
non-reproducible161616Whitmore et al. (2021) highlight the fact that even an
expert reviewer, conducting classifications of the same objects separated by a
period of several years, will assign a small fraction of clusters to differing
classes. expert knowledge of cluster classification of co-author BCW, as well
as the collective efforts of LEGUS classifiers (used for training by Wei et
al. 2020 alongside BCW-only classifications). Because there are humans
involved in the construction of the training sets, the outcome of the
procedure may change if the training set is constructed by a different
human[s], though Wei et al. (2020) conclude that LEGUS and BCW training sets
seem to produce comparable accuracies. We expect ML classification accuracy to
improve with further training method experimentation and as the database of
human classified objects grows and becomes fully representative, both factors
described by Whitmore et al. (2021). Systematics can still exist in our ML
classification, dependent upon the overall suitability of the training sets to
the actual objects being classified, and possibly on the degree to which the
training sets are balanced among classes (clusters versus stars versus
artefacts) or even among cluster morphologies within a class (physically large
clusters versus compact). In regard to overall suitability, the majority of
galaxies in the full PHANGS-HST sample are more distant than the LEGUS targets
that were used by Wei et al. (2020). This is one of the main reasons that we
still undertake human classification in our project, with the goal of
accumulating a training sample that probes a more fully representative range
of distance spanning the Local Volume (e.g. LEGUS) out to objects beyond the
Virgo cluster (some of the PHANGS-HST targets). In essence, the construction
of a training set should be seen as an integral part of the ML procedure, not
as something that precedes it. To summarise, we hope that full adoption of ML
classification by the star cluster community (see also important work by Pérez
et al. 2021 and Bialopetravičius et al. 2019) will lead to replicability
amongst groups that often have somewhat different data sets, and such
replicability can build consensus regarding our understanding of cluster
formation / evolution / destruction.

Secondly, and in a more practical sense, another methodological advantage is
that by relying on ML classification we enable complete evaluation of all
candidates down to our detection (selection) limit. Naturally, classifications
of fainter objects can be expected to individually be less confident
(distinguishing between Class 1 and Class 2 for example) than for bright
analogues in the same environment (see Whitmore et al. 2021). However,
integrated over an ensemble of many objects the cluster / not-a-cluster ML
evaluation (that is, Class 1 or 2, or not) is still reliable near the faint
limit of our candidate selection ($V$-band $\mathrm{S/N}\sim 10$), based on
spot checks of deeper human classification versus the ML classification. We
emphasise further that by classifying our complete list of candidates we are
able to include the often substantial, easy to classify population of faint
candidates occupying isolated, low surface brightness, low confusion
environments. These typically get missed by human classification efforts,
unless specifically remedied post facto (e.g. over 100 faint diffuse sources
were added manually to the NGC 4449 sample in Whitmore et al. 2020).

Unfortunately, improved recovery of faint clusters enabled by ML is
counteracted by losses in our ability to determine accurate photometric ages
for young star clusters with mass $\lessapprox 10^{4}$ M⊙ due to stochasticity
(e.g. Fouesneau et al., 2014; Krumholz et al., 2015; Hannon et al., 2019;
Whitmore et al., 2020). In some science use cases requiring single object
characterisation, this may limit the usefulness of clusters at faint apparent
magnitude, particularly as galaxy distance decreases.

Even so, ML classification for complete candidate samples dramatically
increases the number of recovered clusters, translating into statistical
improvement in the age–mass diagram, reaching significantly lower mass at
fixed age, or alternatively greater age at fixed mass. In Fig. 16, we show the
PHANGS-HST Class 1+2 clusters identified by human (open circles) and ML
classification (dots). Class 2 clusters are colour-coded green. Notice how the
ML population reaches masses about 0.5 dex ($\sim 3\times$) lower than the
human classification. In the alternative perspective, at fixed mass (at least
for intermediate age clusters), the ML catalogue probes up to a factor ten
older in age. Appendix A includes age–mass diagrams for all the fields
analysed in this paper. The degree of human-to-ML improvement varies within
the five fields, owing to differences in depth of the human classifications.

There are numerous trickle-down scientific gains to expect coming from
expansion of the identified cluster population. For example, improvements in
the age–mass diagram will impact studies of the cluster mass function by
allowing the bins in age that are typically used to each extend to lower mass
(e.g. R. Chandar et al. in prep, note that the intermediate age clusters are
often examined separately as they have survived removal from their natal
environment). This will generally boost the statistical significance of mass
function fit results through increased numbers, and may perhaps allow
sensitivity to detect broken power law mass functions. Far better populated
age–mass diagrams will yield improvement in modelling of cluster disruption
(mass dependent versus mass independent) as it provides additional statistics
for low mass clusters in an evolved, intermediate age state. We anticipate
$\Gamma$ (the fraction of star formation occurring in bound clusters, e.g.
Bastian, 2008; Kruijssen, 2012) determinations will increase, given that some
of the newly recovered clusters are of Class 1 and have ages significantly
larger than their crossing time (thus are likely bound at high confidence).

Another trickle-down scientific gain from the increased number of clusters is
improvement of studies that use clusters as ‘clocks’ to estimate timescales
for physical processes via cross-correlation analyses (such as cloud
disruption, via comparison to the ALMA CO catalogues unique to PHANGS-HST).
This is due to the increased surface density of the recovered cluster
population in the ML census versus human classification outcome, as best
appreciated in Fig. 17. Note the improved clarity of patterns in age in the ML
cluster distribution. Appendix A presents the full set of ML classified
catalogue spatial distributions. In the timescale ‘clock’ application, having
a more complete cluster catalogue means that the power of apparent physical
association between cluster and cloud will contribute to the cross-correlation
function at the ‘correct’ length scale more frequently than otherwise in the
limited human catalogue. Viewed in another way, thinking in terms of nearest
neighbour analysis, incompleteness in a cluster catalogue might lead to the
‘wrong’ neighbour being selected. This being said, measurements (e.g. Cook et
al., 2019) of $\Gamma$ in main sequence galaxies like those in the PHANGS-HST
sample are generally $\lessapprox 30$%, and will almost certainly remain less
than 50% even with expanded cluster catalogues. Therefore, in order to capture
the majority of the star formation activity, we stress the importance of using
other localized tracers such as the most compact multi-scale associations
(e.g. Larson et al. 2021) as well for timescale ’clock’ applications.

Lastly, although discussed at length in the previous Section, we end the
presentation of scientific benefits associated to our methods by reiterating
the improvement we achieved with respect to excluding (often random) pairs and
triples of point-like sources from the Class 1 and 2 census. This reduction in
contamination is partially due to our specific attention to this issue but
also due to our adoption of DOLPHOT rather than SExtractor as source detection
code. Our choice to use DOLPHOT as a starting point also strategically
facilitates a self-consistent inventory of point sources and clusters, and
enables determination of multi-scale stellar associations (e.g. Larson et al.,
2021). Fig. 18 illustrates the combined census of star formation products
attained by PHANGS-HST by displaying the associations and clusters of NGC
628-C together. It is abundantly clear that a complete view of recent (few 107
yr) star formation requires methods that capture structures spanning from very
likely bound ($\Pi>$ many), possibly but indeterminately bound
($\Pi\lessapprox$ few), to unbound and already dissolved/disrupted. Another
illustration of the innovative PHANGS-HST approach to combined recovery of
clusters and associations can be found in Fig. 9 of Lee et al. (2021),
highlighting a region of NGC 3351 with all components of our multi-observatory
PHANGS program displayed. With the union of PHANGS cluster, association, CO
cloud (from ALMA), and H ii region (from MUSE and traditional narrowband
imaging) catalogues we will be able to place unprecedented constraints on the
cyclic process of star formation in nearby galaxies.

Figure 18: For a small portion of the NGC 628-C field, located in a spiral arm
west of the galaxy center, we show PHANGS-HST Class 1 and 2 ML clusters (as
green circles) and the hierarchical associations of Larson et al. (2021)
determined on 32 and 64 pc scales using NUV point-sources (as purple and
orange polygons, respectively). The HST image is a colour composite
constructed from $I,V,B$ as red, green, and blue.

##  8 Concluding remarks

In this paper we have described a new analysis pipeline developed for the
PHANGS-HST survey, designed to detect, measure, optimally select, and
morphologically classify star cluster candidates.

Our pipeline employs innovative strategies such as: (1) source detection and
deblending using PSF photometry code DOLPHOT, leading to self-consistency
between cluster and association catalogues, (2) cluster candidate selection
based on multiple concentration indices (MCIin and MCIout) that more fully
characterise the source morphology than possible with a single CI, (3)
selection criteria optimised with guidance from synthetic clusters and
previously confirmed clusters, and with special allowance for potential
candidates more luminous than the Humphreys-Davidson (H-D) limit, and lastly
(4) fully-pipelined ML classification of cluster candidates.

To test the performance of our method, we apply the pipeline to four galaxies:
NGC 628, NGC 1433, NGC 1566, and NGC 3351, with assumed distances between 8.3
and 18 Mpc. Across the four galaxies, the median candidate sample size
selected for ML (human) classification is 3389 (2366) with range of 1989–8822
(1230–5695). The median number of candidates in each galaxy classified by a
human (limited by practicality) is 1039 with range of 174–1442; the median
number of these sources which are verified to be single-peaked star clusters
(Class 1 or 2) is 298 with range of 91–634. For NGC 1566, at 18 Mpc which is
close to the median distance of the PHANGS-HST sample, for such human verified
clusters, the median stellar mass is $3.1\times 10^{4}~{}M_{\odot}$ with a
25–75 percentile interval of $\sim 1.5{-}7.0\times 10^{4}~{}M_{\odot}$. These
values drop by about $3{-}5\times$ for the other targets in the current paper
(at 8–10 Mpc). All candidates have been classified using ML (deep transfer
learning CNN models of Wei et al., 2020), yielding Class 1 or 2 cluster
classifications for 4292 objects (for VGG, 17% of the ML sample). The ML-
classified Class 1 or 2 samples reach $\sim$1 magnitude fainter, $\sim
2\times$ lower mass, and are $2{-}5\times$ larger in number, than the human
classified samples. The resulting age-mass diagrams and spatial distributions
are provided in Appendix A.

These four galaxies were selected for study because they have star cluster
catalogues which have been independently produced by the LEGUS program, using
a standard approach for selecting candidates. Comparison between the LEGUS
catalogues and the ones resulting from our PHANGS-HST pipeline allow for one
test of the effectiveness of our new approach. Based on the union of the two
catalogues, 48–74% (56% median, see lower portion of Fig. 14 and note ($b$) of
Tab. 7) of human verified Class 1 and 2 star clusters appear in both
catalogues. The remainder, clusters which are unique to one of the catalogues,
are discovered in roughly equal proportion by the two methods. Interestingly,
we find that the majority of clusters unique to LEGUS are classified as
contaminants in the PHANGS-HST catalogues, and the majority of those unique to
PHANGS-HST appear to be likely clusters not captured by the LEGUS pipeline.

Overall, we find that our new selection method is effective, and furthermore,
yield a substantially larger cluster population than previously identified in
each of our target galaxies. These cluster catalogues, and the forthcoming
ones from the other 34 galaxies in the PHANGS-HST program, will transform our
understanding of the link between cluster populations and environmental
properties of the ISM (CO measured with ALMA; H i with the JVLA, Karl G.
Jansky Very Large Array; and optical nebular emission lines with MUSE IFU
spectroscopy), especially when combined with our upcoming JWST observations
recovering embedded clusters while also clarifying PAH and small dust grain
properties. The complete potential of our PHANGS-HST data set would not have
been accessible without successful pipeline implementation of ML
classification, owing to the very large number of cluster candidates. This
seamless integration of automated classification has not been achieved, and
deployed to an entire galaxy sample, previously in the field of star clusters.
As noted in Section 5.2 and Whitmore et al. (2021), we expect that the
accuracy of ML classification will further improve as more extensive
experimentation with varied training strategies is undertaken. Even so, the
current accuracy is on par with an expert human classifier.

Our analysis of synthetic clusters provides a database with measured
properties for a rather diverse set of parametric Moffat cluster morphologies.
We indicated such linkage between cluster morphology and metric values could
be used to further simulate the distribution of observed characteristics for a
mock cluster population. Folding in completeness estimates as a function of
cluster properties would allow us to interpret the ensemble measured
properties of a detected cluster population in terms of assumed (cluster
formation rate, destruction $d^{2}N/dM\,dt$, and the distribution of $A_{V}$
and morphology). This forward-modelling approach will be explored in future
work. Aspects of this idea, omitting information on cluster morphology, have
been published in the context of hierarchical Bayesian analysis by Krumholz et
al. (2019b). By incorporating cluster structural measurements, either MCI-
based or via Moffat fitting, and ideally also the population of OB
associations that were once clusters, we can expect the utility of such work
to grow considerably.

## Acknowledgements

Based on observations made with the NASA/ESA Hubble Space Telescope, obtained
from the data archive at the Space Telescope Science Institute. STScI is
operated by the Association of Universities for Research in Astronomy, Inc.
under NASA contract NAS 5-26555. Support for Program number 15654 was provided
through a grant from the STScI under NASA contract NAS5-26555.

Most of the plots in this paper were generated with TOPCAT (Taylor, 2005)
and/or its sister command-line package STILTS (Taylor, 2006), both developed
and generously released/maintained for public use by Mark Taylor. A
significant amount of interactive data exploration and testing was conducted
using TOPCAT.

Our pipeline makes extensive use of the following software packages: DOLPHOT,
photutils, astropy, matplotlib, numpy, IMFIT, pytorch, CIGALE. We extend our
appreciation to their respective developers.

This research has made use of the NASA/IPAC Extragalactic Database (NED) which
is operated by the Jet Propulsion Laboratory, California Institute of
Technology, under contract with NASA.

JMDK gratefully acknowledges funding from the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) through an Emmy Noether Research Group
(grant number KR4801/1-1) and the DFG Sachbeihilfe (grant number KR4801/2-1),
as well as from the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme via the ERC Starting Grant
MUSTANG (grant agreement number 714907).

SCOG and RSK acknowledge support from the DFG via SFB 881 “The Milky Way
System” (sub-projects A1, B1, B2 and B8) and from the Heidelberg cluster of
excellence EXC 2181-390900948 “STRUCTURES: A unifying approach to emergent
phenomena in the physical world, mathematics, and complex data”, funded by the
German Excellence Strategy. They also acknowledge funding from the European
Research Council via the ERC Synergy Grant “ECOGAL” (grant 855130).

TGW acknowledges funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No. 694343).

##  9 Data Availability

The imaging observations underlying this article can be retrieved from the
Mikulski Archive for Space Telescopes at
https://archive.stsci.edu/hst/search_retrieve.html under proposal GO-15654.
High level science products, including science ready mosaicked imaging,
associated with HST GO-15654 are provided at
https://archive.stsci.edu/hlsp/phangs-hst with digital object identifier
10.17909/t9-r08f-dq31

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##  Appendix A Summary Figures

This appendix presents the age–mass and age-coded spatial distribution plots
for all the fields we processed as part of this paper.

Figure 19: The cluster population of the NGC 628-C field as displayed in Fig.
16 and Fig. 17, duplicated here for easy comparison to Figs. 20–23.

Figure 20: As in Fig. 19, except for NGC 628-E. In the lower panel, clusters
are only plotted if they have WFC3/UVIS+ACS/WFC coverage. Spiral arms features
extending outward from the centre of the galaxy (lying off plot to the right)
are marked.

Figure 21: As in Fig. 19, except for NGC 1433. We show the ring-like
structural features identified by the environmental masks of Querejeta et al.
(2021). Note that the $HST$ field does not fully cover the NW and SW portions
of the outer ring.

Figure 22: As in Fig. 19, except for NGC 1566. The spiral arms of Querejeta et
al. (2021) are marked.

Figure 23: As in Fig. 19, except for NGC 3351. Querejeta et al. (2021) ring-
like structural features are drawn.

##  Appendix B DOLPHOT parameters

DOLPHOT serves as the primary source detection algorithm for PHANGS-HST,
feeding both the cluster identification pipeline and the multi-scale
association finder of Larson et al. (2021). As such, even though many choices
are similar to usage of DOLPHOT by prior studies (e.g. LEGUS, Sabbi et al.,
2018), PHAT (Williams et al., 2014, 2021), we provide a listing of relevant
parameters here so as to allow for reproducible research.

We note that some parameters may be further tweaked before the final PHANGS-
HST data products are released. In particular, we anticipate reducing
SecondPass in order to help mitigate ‘doubles’ (redundant detections of
extended objects) in the cluster context and over-decomposition of diffuse
emission into point sources in the association context. If this, and any other
changes are made, they will be reported in the catalogue release papers.

Table 8: DOLPHOT parameters adopted for PHANGS-HST Parameter | Meaning
---|---
Exposure-specific parameters:
img*_RAper = 8.0 | photometry aperture size (float)
img*_RChi = 3.0 | aperture size for determining Chi (float); if <=0 use RAper
img*_RSky = 15 35 | radii defining sky annulus (float>=RAper+0.5)
img*_RSky2 = 4 10 | radii defining sky annulus (for FitSky=2 option)
img*_RPSF = 13 | PSF size (integer>0)
img*_aprad = 10 | radius for aperture correction $HST$-DISABLED (0.5" for WFC3/UVIS and ACS/WFC)
img*_apsky = 15 25 | sky annulus for aperture correction
The following parameters affect the finding and measurement of stars:
RCentroid = 2 | centroid box size (integer>0)
SigFind = 3.0 | sigma detection threshold (float)
SigFindMult = 0.85 | Multiple for quick-and-dirty photometry (float>0)
SigFinal = 3.5 | sigma output threshold (float)
MaxIT = 25 | maximum iterations (integer>0)
PSFPhot = 1 | photometry type (integer/0=aper,1=psf,2=wtd-psf)
PSFPhotIt = 3 | number of iterations in PSF-fitting photometry (integer>=0)
FitSky = 3 | fit sky? (integer/0=no,1=yes,2=small,3=with-phot)
SkipSky = 2 | spacing for sky measurement (integer>0)
SkySig = 2.25 | sigma clipping for sky (float>=1)
NegSky = 1 | allow negative sky values? (0=no,1=yes)
NoiseMult = 0.10 | noise multiple in imgadd (float)
FSat = 0.999 | fraction of saturate limit (float)
PosStep = 0.1 | search step for position iterations (float)
dPosMax = 2.5 | maximum single-step in position iterations (float)
RCombine = 1.5 | minimum separation for two stars for cleaning (float)
SigPSF = 5.0 | min $\mathrm{S/N}$ for psf parameter fits (float)
Settings to enable/disable features:
UseWCS = 1 | use WCS header info in alignment (integer 0=no, 1=use to est. shift/rotate/scale, 2=use to est. full distortion sol.)
Align = 2 | align images? (integer 0=no,1=const(x/yoff),2=lin(x/yoff+scale),3=cube(x/yoff+distortion,4=full3rdorderpolynom)
AlignIter = 2 | number of iterations on alignment? (integer>0)
AlignTol = 0 | number of pixels to search in preliminary alignment (float>=0)
AlignStep = 1.0 | stepsize for preliminary alignment search (float>0)
AlignOnly = 0 | exit after alignment
Rotate = 1 | allow cross terms in alignment? (integer 0=no, 1=yes)
SubResRef = 1 | subpixel resolution for reference image (integer>0)
SecondPass = 5 | second pass finding stars (integer 0=no,1=yes,>1=multiple passes)
SearchMode = 1 | algorithm for astrometry (0=max SNR/chi, 1=max SNR)
Force1 = 1 | force type 1/2 (stars)? (integer 0=no,1=yes)
PSFres = 1 | make PSF residual image? (integer 0=no,1=yes)
psfstars = | Coordinates of PSF stars
psfoff = 0.0 | coordinate offset (PSF system - DOLPHOT system)
ApCor = 1 | find/make aperture corrections? (integer 0=no,1=yes)
VerboseData = 1 | to write all displayed numbers to a .data file
Flags for $HST$ modes:
ForceSameMag = 0 | force same count rate in images with same filter? (integer 0=no, 1=yes)
FlagMask = 4 | photometry quality flags to reject when combining magnitudes
CombineChi = 0 | combined magnitude weights uses chi? (integer 0=no, 1=yes)
ACSuseCTE = 0 | apply CTE corrections on ACS data? (integer 0=no, 1=yes)
WFC3useCTE = 0 | apply CTE corrections on WFC3 data? (integer 0=no, 1=yes)
ACSpsfType = 0 | use Anderson PSF cores? (integer 0=no, 1=yes)
WFC3UVISpsfType = 0 | use Anderson PSF cores? (integer 0=no, 1=yes)
WFC3IRpsfType = 0 | use Anderson PSF cores? (integer 0=no, 1=yes)
InterpPSFlib = 1 | interpolate PSF library spatially