2009 rnn999 M. Bauer, N. Charon, T. Needham, and M. Nishino M. Bauer, N.
Charon, T. Needham, and M. Nishino<EMAIL_ADDRESS>
# Path constrained unbalanced optimal transport

Martin Bauer  1  Nicolas Charon  2  Tom Needham  1  and Mao Nishino  1
11affiliationmark:  Florida State University, Tallahassee, Florida 32306
22affiliationmark:  University of Houston, Houston, Texas 77204

(1 Month 20XX; 11 Month 20XX; 21 Month 20XX)

###### Abstract

Dynamical formulations of optimal transport (OT) frame the task of comparing
distributions as a variational problem which searches for a path between
distributions minimizing a kinetic energy functional. In applications, it is
frequently natural to require paths of distributions to satisfy additional
conditions. Inspired by this, we introduce a model for dynamical OT which
incorporates constraints on the space of admissible paths into the framework
of unbalanced OT, where the source and target measures are allowed to have a
different total mass. Our main results establish, for several general families
of constraints, the existence of solutions to the variational problem which
defines this path constrained unbalanced optimal transport framework. These
results are primarily concerned with distributions defined on a Euclidean
space, but we extend them to distributions defined over parallelizable
Riemannian manifolds as well. We also consider metric properties of our
framework, showing that, for certain types of constraints, our model defines a
metric on the relevant space of distributions. This metric is shown to arise
as a geodesic distance of a Riemannian metric, obtained through an analogue of
Otto’s submersion in the classical OT setting.

##  1 Introduction

#### Background:

The field of optimal transport (OT) studies various methods for comparing
probability distributions, generally guided by the intuition that the
comparison should be performed by measuring the optimal energy required to
transport mass from one distribution to the other. It is a well-established
area of study in both pure and applied mathematics [33], and has more recently
seen a surge of interest due to its applications in, for example, geometry
processing [31] and machine learning [26]. The classical Kantorovich
formulation of OT compares probability distributions $\rho_{0}$ and $\rho_{1}$
by searching for a _coupling_ of the distributions—that is, a joint
distribution with marginals $\rho_{0}$ and $\rho_{1}$—which is optimal with
respect to some cost. There are natural choices of the cost function in
certain situations (for example, when the distributions are Borel measures on
the same metric space), and under these circumstances, the optimal cost leads
to a metric on the space of distributions referred to as _Wasserstein
distance_. Motivated by applications in which the pure transport assumption
can be an unrealistic and limiting factor, this framework has been recently
extended to an unbalanced setting, which allows for the creation or
destruction of mass via a local source term [9, 23, 7]. An immediate advantage
of these _unbalanced optimal transport problems_ , which play a central role
in the present paper, is their ability to facilitate the comparison of
densities with uneven mass.

Another perspective on OT which is important for this paper is the dynamical
one. When considering densities on $\mathbb{R}^{n}$, a famous result of
Benamou and Brenier [1] states that the Wasserstein distance can be realized
as the “kinetic energy” of an energy-minimizing path of densities (i.e. a
time-evolving density) from the source to the target, where paths are subject
to a certain PDE encoding constraints on the evolution.

#### Central Goal:

The goal of this paper is to understand basic analytical and geometrical
properties of a certain variant of this formulation—the constrained unbalanced
optimal transport problem—which is both _unbalanced_ , in the sense that it is
able to compare distributions with different total mass, and _constrained_ ,
in the sense that additional requirements are imposed on the evolving path of
densities. In particular, our main results establish the existence of
minimizers for the associated optimization problem and give conditions under
which it leads to a metric on a relevant space of measures.

#### Motivation:

The motivation for these investigations comes from the general idea that
practical OT problems can naturally come with additional constraints on the
space of measures being considered. In certain cases, those constraints are
preserved along the geodesics of the OT metrics. This is notably the case when
one is interested in comparing Gaussian distributions over $\mathbb{R}^{d}$
based on the Wasserstein metric, for which the space of Gaussian distributions
is totally geodesic. Yet this situation is rather exceptional, and there are
many examples of fairly simple and natural constraints which are not preserved
along optimal paths for balanced or unbalanced optimal transport. For
instance, in the unbalanced OT model described above, the subspace of
probability measures is not totally geodesic as evidenced by some of the
examples studied in [9], cf. Example 3.13. Another example, which in fact
initiated the present work following the preliminary analysis presented in [8]
for the one-dimensional case, is the situation of area measures of convex
domains in $\mathbb{R}^{d}$. These are measures defined on the unit sphere
$S^{d-1}$ representing the distribution of area of the domain’s boundary along
the different normal directions. As suggested in [8], building a Riemannian
metric on the space of area measures induces a Riemannian metric over the set
of all convex sets via the correspondence given by the Minkowski-Fenchel-
Jessen theorem, which in turn may be used to evaluate distances and recover
geodesics between such sets. Although natural at first sight, standard
balanced or unbalanced optimal transport metrics on $S^{d-1}$ are not directly
suited for that purpose since area measures must satisfy a specific “closure”
constraint, namely that the first moment vanishes; a condition that is not
preserved along optimal paths for the Wasserstein or WFR metrics, cf. Example
5.7. More generally, one may be interested in the comparison of measures with
some prescribed set of moments or covariance structures.

Moreover, beyond the case of static/stationary constraints, another plausible
scenario is to observe the time evolution of a specific quantity – such as the
total mass, cf. Example 4.2, of the measure or the expectation of some
function – and then attempt to find an optimal interpolating path enforcing
this set of constraints. Finally, the present framework would also allow to
model the situation of a moving obstacle, i.e., unbalanced optimal transport
with a (potentially) time evolving barrier within the domain, cf. Example 4.2.
The common problem in all such examples is coming up with a constrained
version of OT or UOT that can incorporate path constraints into the dynamic
formulation. We should emphasize that several past works have considered
related constrained variants of dynamic optimal transport models, in
particular [27, 22, 24]. Yet, the aforementioned references have primarily
focused on deriving numerical methods to tackle specific cases. A systematic
study of the theoretical properties of constrained optimal transport,
especially in the unbalanced setting remained missing and is the content of
the present article. Of particular importance is the question of the existence
of optimal paths, which, as we shall see, cannot be deduced straightforwardly
from the corresponding results in standard optimal transport and is one of the
central contributions of the present work.

We emphasize that we work in the setting of _dynamical_ OT, where constraints
are used to define admissible paths of measures which interpolate between the
target measures. An alternative approach is to apply constraints in the static
(Kantorovich) formulation of OT by imposing additional conditions on couplings
between the target measures. For example, this is a useful perspective in
mathematical finance, where it is natural to impose _martingale constraints_
on transport plans [5, 18, 13]. This type of framework is also used to compare
Gaussian mixture models, in which case one obtains a natural metric by also
restricting couplings to be themselves Gaussian mixtures [11, 34]. Although
these formulations of constrained static OT are quite useful, they are unable
to capture the more detailed dynamic constraints described in the previous
paragraph. The constraint model we consider in this paper is distinct from and
complementary to those static models.

#### Mathematical Model:

To describe our results in more detail, let us introduce some notation. Let
$\Omega\subset\mathbb{R}^{n}$ be a compact set with $C^{1}$ boundary and let
$u_{0},u_{1}:\Omega\to\mathbb{R}$ be (sufficiently smooth) probability
densities with respect to Lebesgue measure. The _Benamou-Brenier formulation
of OT_ solves the optimization problem

$\inf_{u,v}\int_{0}^{1}\int_{\Omega}\frac{\|v(t,x)\|^{2}}{u(t,x)}\;dx\,dt\quad\mbox{subject
to}\quad\partial_{t}u+\nabla\cdot v=0,\;u(0,\cdot)=u_{0},\;u(1,\cdot)=u_{1},$
(1)

where the infimum is taken over paths of densities
$u:[0,1]\times\Omega\to\mathbb{R}$ and vector fields
$v:[0,1]\times\Omega\to\mathbb{R}^{n}$. This dynamical formulation of OT can
be extended to compare densities $u_{0}$ and $u_{1}$ which do not necessarily
have the same mass. Following [9], one modifies the Benamou-Brenier
formulation of OT by introducing a _source term_
$w:[0,1]\times\Omega\to\mathbb{R}$; the _unbalanced OT problem_ is then
written as

$\inf_{u,v,w}\int_{0}^{1}\int_{\Omega}\frac{\|v(t,x)\|^{2}+\delta^{2}w(t,x)^{2}}{u(t,x)}\;dx\,dt\quad\mbox{subject
to}\quad\partial_{t}u+\nabla\cdot v=w,\;u(0,\cdot)=u_{0},\;u(1,\cdot)=u_{1},$
(2)

where $\delta\geq 0$ is a hyperparameter controlling the penalty for the
creation or destruction of mass. The PDE constraint is referred to as the
_continuity equation_.

To address the constrained situations described above, we consider the
unbalanced OT energy with additional constraints on the path of densities $u$.
To be more specific, we consider _affine constraints_ on $u$ of the form

$\int_{\Omega}H(t,x)u(t,x)dx=F(t)$ (3)

for some functions $H:[0,1]\times\Omega\to\mathbb{R}^{d}$ and
$F:[0,1]\to\mathbb{R}^{d}$ (for some $d\geq 1$). We say that an affine
constraint is _time-independent_ if $F$ is equal to a constant
$C\in\mathbb{R}^{d}$ and the value of $H(t,x)$ does not depend on $t$ (in
which case we consider it as a function $H:\Omega\to\mathbb{R}$), and _time-
dependent_ otherwise. The unbalanced OT problem with affine constraint
associated to $H$ and $F$ is then given by the optimization problem (2)
subject to the additional constraint (3)—in fact, the general problem we
consider has a slightly more technical formulation which allows comparison of
arbitrary positive Radon measures (see Definitions 3.2 and 4.1).

#### Main Contributions:

We now describe our main results. In the following we call $(u,v,w)$ a _finite
energy path_ from $u_{0}$ to $u_{1}$, with respect to $H$ and $F$, if
$u(0,\cdot)=u_{0}$, $u(1,\cdot)=u_{1}$, $u$ satisfies the affine constraint
(3) and the continuity equation in (2), and the objective of (2) takes a
finite value on $(u,v,w)$. We say that $(u,v,w)$ is a _minimal energy path_
with respect to $H$ and $F$ if, in addition, it realizes the infimum in (2).
These concepts generalize to positive Radon measures, and the statements of
our results are given in this context.

Our first main result gives general conditions which guarantee the existence
of a minimizer to the constrained unbalanced optimal transport problem:

######  Theorem (Existence of minimizers; Theorem 4.3).

Let $\rho_{0}$ and $\rho_{1}$ be positive Radon measures on $\Omega$, let
$H:[0,1]\times\Omega\to\mathbb{R}^{d}$ be a continuous function and let
$F=(F_{1},\ldots,F_{d}):[0,1]\to\mathbb{R}^{d}$ be a function with integrable
component functions $F_{i}$. If a finite energy path from $\rho_{0}$ to
$\rho_{1}$ with respect to $H$ and $F$ exists, then a minimal energy path
exists.

We give several classes of functions $H$ and $F$ which guarantee the existence
of finite (hence minimal) energy paths in Theorems 3.5, 4.7 and 4.8. The
statements of the results for time-dependent functions are somewhat technical,
so we only state here the theorem in the case of time-independent constraints.
For $H:\Omega\to\mathbb{R}^{d}$ and $C\in\mathbb{R}^{d}$, let
$\mathcal{M}^{+}_{H,C}(\Omega)$ denote the space of positive Radon measures
$\rho$ satisfying $\int_{\Omega}H\;d\rho=C$.

######  Theorem (Existence of finite energy paths; Theorem 3.5).

Let $H:\Omega\to\mathbb{R}^{d}$ be a continuous function, let
$C\in\mathbb{R}^{d}$ and let
$\rho_{0},\rho_{1}\in\mathcal{M}^{+}_{H,C}(\Omega)$. Assume, in addition, that
one of the following holds:

  1. 1.

$\Omega$ is convex, $d=1$ and $H\in C^{1}(\Omega,[c,\infty))$ for some $c>0$,

  2. 2.

$C=0$, or

  3. 3.

there exists a density $\bar{\rho}$ with respect to which the Radon-Nikodym
derivative of both $\rho_{0}$ and $\rho_{1}$ is a continous and positive
function, i.e., $\frac{\rho_{i}}{\bar{\rho}}\in C(\Omega,\mathbb{R}_{>0})$.

Then there is a finite energy path from $\rho_{0}$ to $\rho_{1}$ with respect
to $H$ and the constant function $F=C$.

In the time-independent case, we further show that the constrained unbalanced
OT problem leads to a metric and describe the corresponding geometric picture,
including an analogue of Otto’s Riemannian submersion formalism in Section
3.3:

######  Theorem (Theorem 3.9).

Let $h:\Omega\to\mathbb{R}^{d}$ and $C\in\mathbb{R}^{d}$ such that there is a
finite energy path between any two measures in
$\mathcal{M}^{+}_{H,C}(\Omega)$. Then, the constrained unbalanced OT problem
induces a metric on $\mathcal{M}^{+}_{H,C}(\Omega)$, and this metric can be
interpreted as the geodesic distance of a (formal) Riemannian metric.

Lastly, in Section 5, we extend both the model and the existence results to
the case, where the domain is instead a Riemannian manifold. For the proof of
the existence of minimizers result, cf. Theorem 5.5, we make the assumption
that the manifold is parallelizable; the corresponding existence of finite
energy path results translate without any additional assumption to the
manifold situation.

#### Future Directions:

This paper is primarily focused on the fundamental theoretical aspects of the
constrained model. Our framework suggests a computational scheme based on the
convexity of the formulation, and we intend to develop such a numerical
implementation in future work. We also plan to extend our model to incorporate
other variants of unbalanced transport. While we specifically focus on a
constrained version of the Wasserstein-Fisher-Rao distance (see Definition
2.2), there are other dynamic unbalanced optimal transport distances which
should be amenable to constrained formulations. These include the various
distances described in [9] or the recently introduced _simple unbalanced
optimal transport_ model of [21]. The latter model could be interesting in
that it may allow for a more explicit description of the corresponding
constrained geometric picture. Although we were primarily interested here in
the situation of constraints on the measure path only, one could more
generally introduce simlar affine constraints on the vector field and source
term instead. We leave to future investigations the study of how the existence
results and proofs of this work can adapt to this alternative setting. Lastly,
while affine equality constraints are being considered here, generalizing this
framework to convex inequality constraints is another interesting path with
the potential to greatly extend the scope of applications of this model.

#### Acknowledgements:

M. Bauer was partially supported by NSF grants DMS-1953244 and DMS-2324962. T.
Needham was partially supported by NSF grants DMS-2107808 and DMS-2324962. N.
Charon was partially supported by NSF grants DMS-1945224 and DMS-1953267.

##  2 Preliminaries

Let us begin by setting notation that will be used globally throughout the
paper; more specialized notation will be introduced as needed. We will always
use $\Omega\subset\mathbb{R}^{n}$ to denote a fixed compact set with $C^{1}$
boundary. For certain results, we also require $\Omega$ to be convex, but this
condition is not generally assumed. For a space $X$, we use $\mathcal{M}(X)$
to denote the set of Radon measures and $\mathcal{M}^{+}(X)$ to denote the set
of positive Radon measures on $X$. We generally use standard notation for
concepts from analysis and measure theory without additional explanation;
e.g., $\mu_{0}\ll\mu_{1}$ if a measure $\mu_{0}$ is absolutely continuous with
respect to a measure $\mu_{1}$.

The rest of this subsection fills in details on the background topics of
optimal transport that were described in the introduction.

###  2.1 Dynamic optimal transport and Fisher-Rao distance.

The standard Kantorovich formulation of optimal transport compares probability
measures $\rho_{0}$ and $\rho_{1}$ on $\Omega\subset\mathbb{R}^{n}$ by solving
the optimization problem

$\inf_{\gamma\in\Gamma(\rho_{0},\rho_{1})}\int_{\Omega\times\Omega}\|x-y\|^{2}d\gamma(x,y),$
(4)

where $\Gamma(\rho_{0},\rho_{1})$ is the set of probability measures on
$\Omega\times\Omega$ with marginals $\rho_{0}$ and $\rho_{1}$, respectively.
More generally, one can consider other costs in the integrand, or even work
with measures over general metric spaces, but the formulation provided here is
sufficient to illustrate the idea. The square root of (4) defines a metric on
the space of measures, referred to as the Wasserstein distance.

A classic result of optimal transport theory due to Benamou and Brenier [1]
explains how the Kantorovich problem (4) can be recast in a dynamical
framework. Suppose that each measure $\rho_{i}$ has density
$u_{i}:\Omega\to\mathbb{R}$ with respect to Lebesgue measure. We consider a
path of densities $u:[0,1]\times\Omega\to\mathbb{R}$, together with a time-
dependent vector field $v:[0,1]\times\Omega\to\mathbb{R}^{n}$; we assume that
all maps are sufficiently smooth in time and space. The Benamou-Brenier result
[1, Proposition 1.1] says that solving (4) is equivalent to solving

$\inf_{u,v}\int_{0}^{1}\int_{\Omega}\frac{\|v(t,x)\|^{2}}{u(t,x)}\;dx\,dt,$
(5)

subject to the continuity equation:

$\partial_{t}u+\nabla\cdot v=0,\qquad u(0,\cdot)=u_{0},\;u(1,\cdot)=u_{1}.$
(6)

Another approach to metrizing the space of densities is given by the Fisher-
Rao distance. This metric was originally introduced to compare elements of
finite-dimensional submanifolds of probability densities in the context of
statistics [29], but has since been extended to allow comparison of more
general densities (see, e.g., [16, 2]). Given two positive densities
$u_{0},u_{1}:\Omega\to\mathbb{R}$, the Fisher-Rao distance between them is
given by

$\inf_{u,w}\int_{0}^{1}\int_{\Omega}\frac{w(t,x)^{2}}{u(t,x)}\;dx\,dt,\qquad\mbox{subject
to}\quad\partial_{t}u=w,\quad u(0,\cdot)=u_{0},\;u(1,\cdot)=u_{1},$ (7)

where the infimum is over sufficiently smooth paths of densities $u$ and
functions $w:[0,t]\times\Omega\to\mathbb{R}$. There is an apparent similarity
of structure between the Fisher-Rao distance and the dynamical formulation of
OT; the connection between these metrics is elucidated below.

###  2.2 Unbalanced dynamic optimal transport and the Wasserstein-Fisher-Rao
distance.

A dynamical formulation of _unbalanced_ optimal transport—i.e., where the
measures being compared are allowed to have different total mass—was
introduced in [9]. The idea is to relax the continuity equation (6) in the
Benamou-Brenier formulation of optimal transport by including a _source term_
which allows the creation and destruction of mass, and to then incorporate the
source into the objective function (5). The modified continuity equation is

$\partial_{t}u+\nabla\cdot v=w,$ (8)

where $w:[0,1]\times\Omega\to\mathbb{R}$ is the source term, and the
modification of the objective function is to be defined. This not only allows
one to compare positive measures with different mass (a useful extension in
applications), but also frequently results in more natural interpolations
between densities—see the recent survey [32] for an in-depth discussion and
several examples.

Working with the continuity equation in the form (8) necessarily restricts our
focus to measures $\rho$ with a differentiable density. Following [9], this
constraint can be relaxed by working instead with a weak (i.e.,
distributional) form of the equation.

######  Definition 2.1 (Distributional Continuity Equation).

Let $\rho_{0},\rho_{1}\in\mathcal{M}^{+}(\Omega)$. We say that a triple
$(\rho,\omega,\zeta)\in\mathcal{M}^{+}([0,1]\times\Omega)\times\mathcal{M}([0,1]\times\Omega)^{n}\times\mathcal{M}([0,1]\times\Omega)$
satisfies the continuity equation in the distributional sense with boundary
conditions $\rho_{0}$ and $\rho_{1}$ if, for any $\phi\in
C^{1}([0,1]\times\Omega)$,

$\int_{[0,1]\times\Omega}\partial_{t}\phi
d\rho+\int_{[0,1]\times\Omega}\nabla\phi\cdot
d\omega+\int_{[0,1]\times\Omega}\phi
d\zeta=\int_{\Omega}\phi(1,\cdot)d\rho_{1}-\int_{\Omega}\phi(0,\cdot)d\rho_{0}$
(9)

We denote by $\mathcal{CE}(\rho_{0},\rho_{1})$ the set of triples
$(\rho,\omega,\zeta)$ satisfying the continuity equation with boundary
conditions $\rho_{0},\rho_{1}$.

The unbalanced optimal transport distance of [9] is then defined as follows.

######  Definition 2.2 (WFR distance).

For $\delta>0$, the Wasserstein-Fisher-Rao infinitesimal cost is the function
$f_{\delta}:\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}\to[0,+\infty]$ on
$\mathcal{M}^{+}(\Omega)$ defined by

$f_{\delta}(\rho,\omega,\zeta):=\begin{cases}\frac{\|\omega\|^{2}+\delta^{2}\zeta^{2}}{2\rho},&\rho>0\\\
0,&(\rho,\omega,\zeta)=(0,0,0)\\\ +\infty,&\textrm{otherwise.}\end{cases}$
(10)

The associated Wasserstein-Fisher-Rao (WFR) distance is then defined by

$\mathrm{WFR}_{\delta}(\rho_{0},\rho_{1})^{2}:=\inf\left\\{\int_{[0,1]\times\Omega}f_{\delta}\left(\frac{d\mu}{d\lambda}\right)d\lambda\middle|\begin{aligned}
\mu=(\rho,\omega,\zeta)\in\mathcal{CE}(\rho_{0},\rho_{1})\end{aligned}\right\\},$
(11)

where $\lambda$ is any nonnegative measure on $[0,1]\times\Omega$ such that
$\rho,\omega,\zeta\ll\lambda$ and
$\frac{d\mu}{d\lambda}:=(\frac{d\rho}{d\lambda},\frac{d\omega}{d\lambda},\frac{d\zeta}{d\lambda})$.
We note that the value does not depend on the choice of $\lambda$ due to the
1-homogeneity of $f_{\delta}$.

######  Remark 2.3.

The Wasserstein-Fisher-Rao infinitesimal cost belongs to a more general family
of infinitesimal costs defined in [10]; one can generalize the definition of
WFR distance by replacing $f_{\delta}$ with another infinitesimal cost. We
restrict our attention to that particular $f_{\delta}$ because it is natural
in the sense that it interpolates between well-known distances on the space of
measures: it is (a constant multiple of) the Wasserstein distance when
$\delta=0$ and $\frac{1}{\delta^{2}}\mathrm{WFR}_{\delta}$ limits to the
Fisher-Rao distance as $\delta\rightarrow\infty$.

Considering $\rho$ as an element of $\mathcal{M}^{+}([0,1]\times\Omega)$
apparently loses the dynamical interpretation of $\rho$ as a “path of
measures”, but this interpretation is recovered via the concept of
disintegration. Recall that a disintegration of a measure $\nu$ on
$[0,1]\times\Omega$ is a Borel family of measures $\\{\nu_{t}\\}_{t\in[0,1]}$
such that $d\nu(t,x)=dt\otimes d\nu_{t}(x)$. We will write in short
$\nu=dt\otimes\nu_{t}$. Then the following holds:

######  Proposition 2.4.

Let $(\rho,\omega,\zeta)\in\mathcal{CE}(\rho_{0},\rho_{1})$. Then $\rho$ is
disintegrable: $\rho=dt\otimes\rho_{t}$ for a Borel family of measures
$\\{\rho_{t}\\}_{t}$. If we assume further that $\omega\ll\rho$ (coordinate-
wise) and $\zeta\ll\rho$ hold, we have the disintegration of $\omega$ and
$\zeta$: $\omega=dt\otimes\omega_{t},\zeta=dt\otimes\zeta_{t}$. The absolute
continuity conditions hold whenever the objective of (11) is finite so that
disintegrability holds in our setting without loss of generality.

######  Proof 2.5.

This result can be deduced from the disintegration theorem found in [3,
Theorem A.1], but we still write the proof for completeness. Let us first take
a function $\phi:[0,1]\to\mathbb{R}$, which is $C^{1}$ and compactly supported
on $(0,1)$, and define $\tilde{\phi}=\phi\circ\pi\in C^{1}([0,1]\times\Omega)$
where $\pi:[0,1]\times\Omega\to[0,1]$ is the projection onto $[0,1]$.
Substitute $\tilde{\phi}$ to the continuity equation to obtain

$\int_{[0,1]\times\Omega}\partial_{t}\tilde{\phi}d\rho+\int_{[0,1]\times\Omega}\tilde{\phi}d\zeta=0$
(12)

By the standard fact about pushforward measures, we have

$\int_{0}^{1}\partial_{t}\phi d(\pi_{*}\rho)+\int_{0}^{1}\phi
d(\pi_{*}\zeta)=0$ (13)

This equation implies that the distributional derivative of $\pi_{*}\rho$
equals $\pi_{*}\zeta$. Now, we define a function on $[0,1]$ by
$u(t)=\int_{0}^{t}d(\pi_{*}\zeta)$. Then, for any open interval
$A\subset[0,1]$, by Fubini’s theorem,

$\displaystyle\int_{A}u(t)dt$
$\displaystyle=\int_{t=0}^{t=1}1_{A}(t)u(t)dt=\int_{t=0}^{t=1}\int_{s=0}^{s=1}1_{A}(t)1_{[0,t]}(s)d(\pi_{*}\zeta)dt=\int_{s=0}^{s=1}\int_{t=0}^{t=1}1_{A}(t)1_{[s,1]}(t)dtd(\pi_{*}\zeta)$
(14) $\displaystyle=\int_{s=0}^{s=1}\int_{t=s}^{t=1}1_{A}(t)dtd(\pi_{*}\zeta)$
(15)

The function $v(s)=\int_{t=s}^{t=1}1_{A}(t)dt$ is generally not a compactly
supported $C^{1}((0,1))$ function but can be approximated by one since smooth
bump functions can approximate the indicator. Therefore, by applying the fact
that the distributional derivative of $\pi_{*}\rho$ is $\pi_{*}\zeta$,

$\displaystyle\int_{s=0}^{s=1}\int_{t=s}^{t=1}1_{A}(t)dtd(\pi_{*}\zeta)=\int_{s=0}^{s=1}1_{A}(s)d(\pi_{*}\rho)=\pi_{*}\rho(A)$
(16)

Thus, $u$ is a density function of $\pi_{*}\rho$, and thus $\pi_{*}\rho$ is
absolutely continuous with respect to the Lebesgue measure. Applying the
disintegration theorem, we obtain the proposition.

In light of this result, we frequently abuse terminology and refer to $\mu$,
$\rho$, $\omega$ and $\zeta$ as _paths of measures_ , or simply as _paths_.

##  3 The constrained Wasserstein-Fisher-Rao distance

We will first focus on a version of constrained, unbalanced optimal transport,
which leads to a constrained variant of the Wasserstein-Fisher-Rao distance
(11). We start by defining the following constrained space of measures:

######  Definition 3.1 (Constrained space).

Given $H:\Omega\to\mathbb{R}^{d}$ and $C\in\mathbb{R}^{d}$, for some integer
$d\geq 1$, let $\mathcal{M}^{+}_{H,C}(\Omega)$ be the set of all measures
satisfying the affine constraint described by $H$ and $C$, i.e.,

$\mathcal{M}^{+}_{H,C}(\Omega):=\left\\{\rho\in\mathcal{M}^{+}(\Omega)\middle|\int_{\Omega}H\;d\rho=C\right\\}.$
(17)

Using this definition, it is straightforward to consider the corresponding
constrained, unbalanced optimal transport problem:

######  Definition 3.2 (Affine-constrained WFR problem).

For a function $H:\Omega\to\mathbb{R}^{d}$ and a constant
$C\in\mathbb{R}^{d}$, we say that a measure
$\rho\in\mathcal{M}^{+}([0,1]\times\Omega)$ satisfies the affine constraint
with respect to $H$ and $C$ if it disintegrates in time
($\rho=dt\otimes\rho_{t}$) and $\rho_{t}\in\mathcal{M}^{+}_{H,C}(\Omega)$ for
all $t$. In this case, we write $\rho\in\mathcal{A}(H,C)$. For
$\rho_{0},\rho_{1}\in\mathcal{M}^{+}(\Omega)$, we consider the constrained set
of measures

$\mathcal{CE}_{H,C}(\rho_{0},\rho_{1}):=\left\\{\mu=(\rho,\omega,\zeta)\in\mathcal{CE}(\rho_{0},\rho_{1}))\mid\rho\in\mathcal{A}(H,C)\right\\}.$

We define the affine-constrained Wasserstein-Fisher-Rao energy with respect to
$H$ and $C$ by

${\textrm{WFR}_{\delta}^{H,C}}(\rho_{0},\rho_{1})^{2}=\inf\left\\{\int_{[0,1]\times\Omega}f_{\delta}\left(\frac{d\mu}{d\lambda}\right)d\lambda\middle|\begin{aligned}
\mu=(\rho,\omega,\zeta)\in\mathcal{CE}_{H,C}(\rho_{0},\rho_{1})\end{aligned}\right\\},$
(18)

where $\lambda$ is any nonnegative Borel measure on $[0,1]\times\Omega$ such
that $\rho,\omega,\zeta\ll\lambda$. By the homogeneity of $f_{\delta}$, the
integral in the definition does not depend on the choice of $\lambda.$

We will later show that ${\textrm{WFR}_{\delta}^{H,C}}$ defines a metric on
$\mathcal{M}^{+}_{H,C}(\Omega)$ under certain conditions on $H$ and $C$
(Theorem 3.9). First, we will study the analytical properties of the
optimization problem (18).

###  3.1 Existence of solutions.

The main results of this paper concern the existence of solutions for the
constrained WFR optimization problems. Therefore we will need the following
notions of finite energy and minimal energy paths:

######  Definition 3.3.

For $\rho_{0},\rho_{1}\in\mathcal{M}^{+}(\Omega)$, we say that
$\mu=(\rho,\omega,\zeta)\in\mathcal{CE}_{H,C}(\rho_{0},\rho_{1})$ is a finite
energy path with respect to $H$ and $C$ (or just a finite energy path) if the
objective of (18) is finite; that is, if
$\int_{[0,1]\times\Omega}f_{\delta}\left(\frac{d\mu}{d\lambda}\right)d\lambda<\infty$.
If $\mu$ realizes the infimum of (18), we say that $\mu$ is a minimal energy
path with respect to $H$ and $C$ (or just a minimal energy path).

We now state our first main result.

######  Theorem 3.4 (Existence of minimizers).

Let $\rho_{0},\rho_{1}\in\mathcal{M}^{+}(\Omega)$, let
$H:\Omega\to\mathbb{R}^{d}$ be a continuous function and let
$C\in\mathbb{R}^{d}$. If a finite energy path with respect to $H$ and $C$
exists, then a minimal energy path exists.

We will postpone the proof of this theorem to Section 4.1, where we shall in
fact prove a more general version of this result, where $H$ and $C$ may also
depend on the time $t$ (Theorem 4.3).

Our next goal is to prove the existence of finite energy paths (and
consequently, by Theorem 3.4, the existence of minimizers) assuming certain
conditions on either $H$ or on the given measures $\rho_{0}$ and $\rho_{1}$:

######  Theorem 3.5 (Existence of finite energy paths).

Let $H:\Omega\to\mathbb{R}^{d}$ be a continuous function and let
$\rho_{0},\rho_{1}\in\mathcal{M}^{+}_{H,C}(\Omega)$. Assume, in addition, that
one of the following holds:

  1. 1.

$\Omega$ is convex, $d=1$, $C>0$ and $H\in C^{1}(\Omega,[c,\infty))$ for some
$c>0$.

  2. 2.

the constraint is linear, i.e., $C=0$;

  3. 3.

there exists a density $\bar{\rho}$ with respect to which the Radon-Nikodym
derivative of both $\rho_{0}$ and $\rho_{1}$ is a continous and positive
function, i.e., $\frac{\rho_{i}}{\bar{\rho}}\in C(\Omega,\mathbb{R}_{>0})$.

Then there exists a finite energy path in
$\mathcal{CE}_{H,C}(\rho_{0},\rho_{1})$.

######  Remark 3.6.

The continuity of $H$ in Theorem 3.5 is not actually necessary, and there is a
finite energy path as long as the integral is defined. However, we only
consider continuous functions here as this is the condition for $H$ in Theorem
3.4.

######  Proof 3.7.

Given assumption (1) or assumption (2), the proof follows directly from the
corresponding more general results with time-varying constraints; see Theorems
4.7 and 4.8 below, respectively. It remains to prove the result given that
assumption (3) holds. Consider the path $(\rho,\omega,\zeta)$ with
$\rho_{t}=(1-t)\rho_{0}+t\rho_{1}$, $\omega$ identically zero and
$\zeta=\partial_{t}\rho=\rho_{1}-\rho_{0}$. This path clearly satisfies the
affine constraint and the continuity equation. Moreover, the path is finite
energy: since $\omega=0$, the energy is equal to the Fisher-Rao energy (that
is, the objective function of the variational problem defining the distance
(7)), which must be finite because the Fisher-Rao metric is a smooth,
Riemannian metric on the space of all measures satisfying assumption (3),
i.e., in the space of all measures that have a continous and strictly positive
Radon-Nikodym derivative w.r.t. to some fixed measure $\bar{\rho}$ (see [16,
2]). Note that this not true on the space of all Radon-measures and thus the
assumption is really needed, cf. [9, Section 1.2].

###  3.2 The constrained WFR distance

In this subsection, we will show that the constrained WFR energy induces a
metric structure on the set of all measures satisfying the given constraint.
Furthermore, we will see that, similar to the unconstrained setting, this
metric structure can be realized as the geodesic distance of a (formal)
Riemannian metric.

To this end, we will need the following formula, which is analoguous to the
unconstrained case in [9] and [12]:

######  Lemma 3.8.

Let $H:\Omega\to\mathbb{R}^{d}$ and $C\in\mathbb{R}^{d}$. For any
$\rho_{0},\rho_{1}\in\mathcal{M}^{+}_{H,C}(\Omega)$, the following equivalent
formulation of the constrained WFR energy holds:

${{\textrm{WFR}_{\delta}^{H,C}}(\rho_{0},\rho_{1})}=\inf\left\\{\int_{0}^{1}\left\\{\int_{\Omega}f_{\delta}\left(\frac{d\mu_{t}}{d\lambda_{t}}\right)d\lambda_{t}\right\\}^{1/2}dt\middle|\begin{aligned}
\mu=(\rho,\omega,\zeta)\in\mathcal{CE}_{H,C}(\rho_{0},\rho_{1})\end{aligned}\right\\}$
(19)

where $\mu_{t}$ is the disintegration of $\mu$ in time. Here, $\lambda_{t}$ is
any negative Borel measure so that $\mu_{t}\ll\lambda_{t}$. Moreover, any
minimal energy path $\mu$, solution to (18), satisfies

${{\textrm{WFR}_{\delta}^{H,C}}(\rho_{0},\rho_{1})^{2}=\int_{\Omega}f_{\delta}\left(\frac{d\mu_{t}}{d\lambda_{t}}\right)d\lambda_{t}}$
(20)

for a.e. $t\in[0,1]$.

We skip the proof for brevity as this is a direct adaptation of that of
Theorem 5.4 in [12]. This lemma now allows to prove that the constrained WFR
energy indeed defines a distance function on $\mathcal{M}^{+}_{H,C}(\Omega)$.

######  Theorem 3.9 (WFR defines a metric).

Let $H:\Omega\to\mathbb{R}^{d}$ and $C\in\mathbb{R}^{d}$ such that there is a
finite energy path of the constrained WFR energy between any two measures in
$\mathcal{M}^{+}_{H,C}(\Omega)$. Then ${\textrm{WFR}_{\delta}^{H,C}}$ defines
a metric on $\mathcal{M}^{+}_{H,C}(\Omega)$, and this metric can be
interpreted as the geodesic distance of a (formal) Riemannian metric.

######  Proof 3.10.

To show that ${{\textrm{WFR}_{\delta}^{H,C}}}$ is a distance function, we
adapt the proof in [8] to our situation. For any
$\rho_{0},\rho_{1}\in\mathcal{M}^{+}_{H,C}(\Omega)$, we trivially have
${\textrm{WFR}_{\delta}^{H,C}}(\rho_{0},\rho_{1})\geq 0$. If
${\textrm{WFR}_{\delta}^{H,C}}(\rho_{0},\rho_{1})=0$, then so is the
unconstrained WFR distance between $\rho_{0}$ and $\rho_{1}$ and thus we have
$\rho_{0}=\rho_{1}$. Conversely, if $\rho_{0}=\rho_{1}=\rho$, the constant
path where both the momentum and the source term are zero shows that
${\textrm{WFR}_{\delta}^{H,C}}(\rho,\rho)=0$ for any
$\rho\in\mathcal{M}^{+}_{H,C}(\Omega)$.

To show that ${\textrm{WFR}_{\delta}^{H,C}}$ is symmetric, we observe that any
finite energy path
$\mu=(dt\otimes\rho_{t},dt\otimes\omega_{t},dt\otimes\zeta_{t})$ from
$\rho_{0}$ to $\rho_{1}$ in $\mathcal{A}(H,C)$ can be reversed in time to
construct a finite energy path from $\rho_{1}$ to $\rho_{0}$ still in
$\mathcal{A}(H,C)$:
$\mu^{\prime}=(dt\otimes\rho_{1-t},-dt\otimes\omega_{1-t},-dt\otimes\zeta_{1-t})$.
The time reversal operation is a bijection on the space of finite energy paths
as repeating the process yields the identity map. Moreover, the operation
preserves the energy, so reversing a minimal energy path is, again, a minimal
energy path. Thus, we have
${\textrm{WFR}_{\delta}^{H,C}}(\rho_{0},\rho_{1})={\textrm{WFR}_{\delta}^{H,C}}(\rho_{1},\rho_{0})$.

It remains to prove the triangle inequality. Let
$\rho_{0},\rho_{1},\rho_{2}\in\mathcal{M}^{+}_{H,C}(\Omega)$ and consider a
minimal energy path
$\mu^{01}=(dt\otimes\rho_{t}^{01},dt\otimes\omega_{t}^{01},dt\otimes\zeta^{01}_{t})$
from $\rho_{0}$ to $\rho_{1}$ and a minimal energy path
$\mu^{12}=(dt\otimes\rho_{t}^{12},dt\otimes\omega_{t}^{12},dt\otimes\zeta^{12}_{t})$
from $\rho_{1}$ to $\rho_{2}$. Note that the existence of minimal energy paths
is guaranteed by Theorem 3.4. We then define
$\mu^{02}=(dt\otimes\rho_{t}^{02},dt\otimes\omega^{02}_{t},dt\otimes\zeta_{t}^{02})$
by

$\displaystyle\rho^{02}_{t}=\begin{cases}\rho^{01}_{2t}&0\leq t\leq 1/2\\\
\rho^{12}_{2t-1}&1/2\leq t\leq 1\end{cases},\ \
\omega^{02}_{t}=\begin{cases}2\omega^{01}_{2t}&0\leq t\leq 1/2\\\
2\omega^{12}_{2t-1}&1/2\leq t\leq 1\end{cases},\ \
\zeta^{02}_{t}=\begin{cases}2\zeta^{01}_{2t}&0\leq t\leq 1/2\\\
2\zeta^{12}_{2t-1}&1/2\leq t\leq 1\end{cases}$ (21)

By direct calculation, we have
$\mu^{02}\in\mathcal{CE}_{H,C}(\rho_{0},\rho_{1})$. Using Equation (19), we
calculate

$\displaystyle{\textrm{WFR}_{\delta}^{H,C}}(\rho_{0},\rho_{2})$
$\displaystyle\leq\int_{0}^{1/2}\left\\{\int_{\Omega}f_{\delta}\left(\frac{d\mu_{t}^{02}}{d\lambda_{t}}\right)d\lambda_{t}\right\\}^{1/2}dt+\int_{1/2}^{1}\left\\{\int_{\Omega}f_{\delta}\left(\frac{d\mu_{t}^{02}}{d\lambda_{t}}\right)d\lambda_{t}\right\\}^{1/2}dt$
(22)
$\displaystyle\leq\int_{0}^{1}\left\\{\int_{\Omega}4f_{\delta}\left(\frac{d\mu_{s}^{01}}{d\lambda_{s}}\right)d\lambda_{s}\right\\}^{1/2}\frac{1}{2}ds+\int_{0}^{1}\left\\{\int_{\Omega}4f_{\delta}\left(\frac{d\mu_{s^{\prime}}^{12}}{d\lambda_{s^{\prime}}}\right)d\lambda_{s^{\prime}}\right\\}^{1/2}\frac{1}{2}ds^{\prime}$
(23)
$\displaystyle=\left\\{\int_{0}^{1}\int_{\Omega}f_{\delta}\left(\frac{d\mu_{s}^{01}}{d\lambda_{s}}\right)d\lambda_{s}ds\right\\}^{1/2}+\left\\{\int_{0}^{1}\int_{\Omega}f_{\delta}\left(\frac{d\mu_{s^{\prime}}^{12}}{d\lambda_{s^{\prime}}}\right)d\lambda_{s^{\prime}}ds^{\prime}\right\\}^{1/2}$
(24)
$\displaystyle={\textrm{WFR}_{\delta}^{H,C}}(\rho_{0},\rho_{1})+{\textrm{WFR}_{\delta}^{H,C}}(\rho_{1},\rho_{2})$
(25)

Here, the first line is just the definition of the infimum, and the inequality
in the second uses the fact that
$\frac{\|2\omega\|^{2}+\delta^{2}(2\zeta)^{2}}{\rho}=4\frac{\|\omega\|^{2}+\delta^{2}\zeta^{2}}{\rho}$
and the substitution $s=2t,s^{\prime}=2t-1$. The equality in the third line is
just (20). Finally, the last line follows from the fact that $\mu^{01}$ and
$\mu^{12}$ are minimal energy paths.

Note that $\mathcal{M}^{+}_{H,C}(\Omega)$ is an affine subspace and thus a
submanifold of the infinite-dimensional manifold of all densities and that the
definition of ${\textrm{WFR}_{\delta}^{H,C}}$ coincides with the definition of
the unconstrained WFR distance up to the fact that the set of allowable paths
is constrained to this submanifold. Consequently,
${\textrm{WFR}_{\delta}^{H,C}}$ is indeed the geodesic distance of a (formal)
Riemannian metric, namely of the restriction of the Riemannian metric
corresponding to the unconstrained WFR distance to the submanifold
$\mathcal{M}^{+}_{H,C}(\Omega)$.

###  3.3 Geometric aspects of the constrained WFR distance

Finally, we will study geometric aspects of the proposed constrained UOT
model; namely, we aim to extend Otto’s Riemannian submersion from optimal
transport to our situation: a classical result [25] in (balanced) optimal
transport asserts that the flat $L^{2}$-metric on the group of diffeomorphisms
of $\Omega$ descends to a Riemannian metric on the space of probability
densities, whose geodesic distance function is exactly the Wasserstein
distance. The simple nature of the geometry of the $L^{2}$-metric – geodesics
are given by straight lines – gives rise to a beautiful geometric
interpretation of the correspondence between the static and dynamic
formulation of optimal transport. More recently, this picture has been
extended to the unbalanced situation by Gallouet and Vialard [19], see also
[17]. Before we describe the extension of this geometric interpretation to the
constrained model of the present article, we will briefly recall the
constructions for the unconstrained situations. In the following, we will
restrict ourselves to the category of smooth functions (densities, resp.),
i.e., we consider the space of all smooth, positive densities

$\operatorname{Dens}(\Omega):=\left\\{\psi\rho_{0}\mid\psi\in
C^{\infty}(\Omega,\mathbb{R}_{>0})\right\\},$ (26)

where $\rho_{0}$ is a fixed element of $\mathcal{M}^{+}(\Omega)$. We will also
consider the subset of all all smooth probability densities,

$\operatorname{Dens}_{1}(\Omega):=\left\\{\psi\rho_{0}\in\operatorname{Dens}(\Omega)\mid\int_{\Omega}\psi\rho_{0}=1\right\\}.$
(27)

Next we consider the group of smooth diffeomorphisms
$\operatorname{Diff}(\Omega)$ and equip it with the (flat)
$L^{2}(\rho_{0})$-Riemannian metric $G^{L^{2}}$, i.e.,

$G^{L^{2}}_{\varphi}(\delta\varphi,\delta\varphi)=\int_{\Omega}\langle\delta\varphi,\delta\varphi\rangle\rho_{0},\qquad\text{
for }\varphi\in\operatorname{Diff}(\Omega),\quad\delta\varphi\in
T_{\varphi}\operatorname{Diff}(\Omega).$ (28)

In this setup, Otto’s Riemannian submersion result [25] asserts that

$\displaystyle\pi:\operatorname{Diff}(\Omega)$
$\displaystyle\to\operatorname{Dens}_{1}(\Omega)\text{ given by }$
$\displaystyle\varphi$ $\displaystyle\mapsto\varphi_{*}\rho_{0}$

is a formal Riemannian submersion from $\operatorname{Diff}(\Omega)$ with the
(flat) $L^{2}(\rho_{0})$ metric onto the space of smooth probability densities
equipped with the (formal) Riemannian metric corresponding to the Wasserstein
distance. To extend this construction to the constrained, unbalanced setting
we follow the work of Gallouët and Vialard [19]: let $\mathcal{C}(\Omega)$ be
the cone over $\Omega$ and consider the automorphism group
$\operatorname{Aut}(\mathcal{C}(\Omega))$, which can be viewed as the semi-
direct product of $\operatorname{Diff}(\Omega)$ and
$C^{\infty}(\Omega,\mathbb{R}^{+})$. Next we need to generalize the
$L^{2}(\rho_{0})$-metric to this setting. Therefore we equip the cone
$\mathcal{C}(\Omega)$ with the cone metric

$\mathfrak{g}_{x,m}((dx,dm),(dx,dm)):=mdx^{2}+\delta^{2}\frac{dm^{2}}{m},$
(29)

which in turn leads to a natural definion of the corresponding
$L^{2}(\rho_{0})$-metric on $\operatorname{Aut}(\mathcal{C}(\Omega))$. In this
setup it is shown in [19], that the projection
$\pi_{0}:\operatorname{Aut}(\mathcal{C}(\Omega))\to\operatorname{Dens}(\Omega)$
given by $(\varphi,f)\mapsto\varphi_{*}(f^{2}\rho_{0})$ is again a (formal)
Riemannian submersion, where $\operatorname{Aut}(\mathcal{C}(\Omega))$ is
equipped with the (flat) $L^{2}(\rho_{0})$ Riemannian metric and
$\operatorname{Dens}(\Omega)$ is equipped with the (formal) Riemannian metric
corresponding to the Wasserstein-Fisher-Rao distance. Mimicking these
constructions, we obtain the following result for the constrained and
unbalanced situation. For a detailed proof in the unconstrained situation,
which can be readily adapted to the constrained situation, we refer to [19].

######  Proposition 3.11.

Let $H:\Omega\to\mathbb{R}^{d}$ and $C\in\mathbb{R}^{d}$. Denote by
$\operatorname{Dens}_{H,C}(\Omega)=\operatorname{Dens}(\Omega)\cap\mathcal{M}^{+}_{H,C}(\Omega)$
the set of smooth densities satisfying the constraint prescribed by $H$ and
$C$ and by $\operatorname{Aut}_{H,C}(\mathcal{C}(\Omega))$ the corresponding
subset of $\operatorname{Aut}(\mathcal{C}(\Omega))$ such that
$\pi_{0}(\operatorname{Aut}_{H,C}(\mathcal{C}(\Omega)))=\operatorname{Dens}_{H,C}(\Omega)$.
Then the projection

$\displaystyle\pi_{0}:\operatorname{Aut}_{H,C}(\mathcal{C}(\Omega))$
$\displaystyle\to\operatorname{Dens}_{H,C}(\Omega)\text{ given by }$
$\displaystyle(\varphi,f)$ $\displaystyle\mapsto\varphi_{*}(f^{2}\rho_{0})$

is a (formal) Riemannian submersion, where
$\operatorname{Aut}_{H,C}(\mathcal{C}(N))$ is equipped with the restriction of
the (flat) $L^{2}(\rho_{0})$ Riemannian metric and
$\operatorname{Dens}_{H,C}(\Omega)$ is equipped with the restriction of the
(formal) Riemannian metric corresponding to the Wasserstein-Fisher-Rao
distance.

The above result allows us to introduce a static (Monge) formulation of the
constrained WFR metric:

######  Definition 3.12 (Monge-formulation).

Let $H:\Omega\to\mathbb{R}^{d}$ and $C\in\mathbb{R}^{d}$. For
$\rho_{0},\rho_{1}\in\operatorname{Dens}_{+}(M)_{H,C}(\Omega)$ the Monge
formulation of the constrained WFR is given by

$\operatorname{M}\text{-}{\textrm{WFR}_{\delta}^{H,F}}(\rho_{0},\rho_{1})=\inf_{\varphi,f}\left\\{d_{\operatorname{Aut}_{H,C}}\big{(}(\operatorname{id},1),(\varphi,f)\big{)}:\varphi_{*}(\lambda\rho_{0})=\rho_{1}\right\\},$
(30)

where $d_{\operatorname{Aut}_{H,C}}$ denotes the geodesic distance of the
$L^{2}$-metric on the automorphism group restricted to the submanifold
$\operatorname{Aut}_{H,C}(\mathcal{C}(N))$.

It is clear that we have
$\operatorname{M}\text{-}{\textrm{WFR}_{\delta}^{H,F}}(\rho_{0},\rho_{1})\geq{\textrm{WFR}_{\delta}^{H,F}}(\rho_{0},\rho_{1})$
as the corresponding infimum for the constrained WFR distance is taken over a
much larger space. For the unconstrained version it has been shown in [17],
that under certain assumption on $\rho_{0}$ and $\rho_{1}$ and the underlying
domain $\Omega$, the two formulations are indeed equivalent, i.e., that one
has
$\operatorname{M}\text{-}{\textrm{WFR}_{\delta}^{H,F}}(\rho_{0},\rho_{1})={\textrm{WFR}_{\delta}^{H,F}}(\rho_{0},\rho_{1})$.
In future work it would be interesting to obtain a similar statement for the
constrained version, but this seems difficult as the induced geodesic distance
on the constrained automorphism group is significantly more complicated than
in the unconstrained case, where this reduces to the standard $L^{2}$-metric
on the cone.

We conclude this section by specifying the above result for one important
special case, namely the restriction of the WFR metric to the space of
probabilty densities:

######  Example 3.13 (The WFR metric on the space of probabilty densities).

In the following, let $\operatorname{Prob}(\Omega)$ denote the space of all
probability densities. To consider the restriction of the WFR metric onto this
space, we observe that
$\operatorname{Prob}(\Omega)=\operatorname{Dens}_{1,1}(\Omega)$, i.e, we are
exactly in the situation of the present article with $H(x)=1$ and $C=1$. To
study Proposition 3.11 in this situation it remains to investigate the
constrained space of automorphisms
$\operatorname{Aut}_{1,1}(\mathcal{C}(\Omega))$. Therefore we observe that the
action of the group of diffeomorphisms on the space of densities preserves the
total volume. Thus for $\rho_{0}\in\operatorname{Prob}(\Omega)$ we have that
$\varphi_{*}(f^{2}\rho_{0})\in\operatorname{Prob}(\Omega)$ if and only if
$\int_{\Omega}f^{2}\rho_{0}=1$, i.e., if $f$ is an element of the
$L^{2}(\rho_{0})$ unit sphere in $C^{\infty}(\Omega)$, which we denote in the
following by $\mathcal{S}_{1}$. Consequently, we can represent the constrained
space as the semidirect product
$\operatorname{Aut}_{1,1}(\mathcal{C}(\Omega))=\operatorname{Diff}(\Omega)\ltimes\mathcal{S}_{1}$.
This represention suggests that it could be possible to derive an explicit
formula for the corresonding geodesic distance on
$\operatorname{Aut}_{1,1}(\mathcal{C}(\Omega))$, thereby opening up a path to
study the equivalence of the Monge and the dynamic formulation for this
special situation.

######  Example 3.14 (Higher order moment constraints).

Beyond imposing constraints on the total mass, one may also consider
situations in which some higher order moments may be fixed, for instance the
expected value or variance. In the former case, restricting to measures with a
prescribed expected value $m\in\mathbb{R}^{d}$ corresponds to a constraint of
the form $\int_{\Omega}xd\rho(x)=m$ which exactly fits the model of this
section by setting $H(x)=x$ and $C=m$. More generally, setting
$H(x)=(x_{i_{1}}-c_{1})\ldots(x_{i_{n}}-c_{n})$ for $n\in\mathbb{N}$,
$\\{i_{1},\ldots,i_{n}\\}$ a set of indices in $\\{1,\ldots,d\\}$ and
$c_{1},\ldots,c_{n}\in\mathbb{R}$ allows to constrain any specific moment of
order $n$ to a given value. The results of this section show that the
corresponding metrics ${\textrm{WFR}_{\delta}^{H,F}}$ can be used to compare
and interpolate between measure distributions in these constrained subspaces
(assuming finite energy paths exist, notably under the conditions of Theorem
3.5).

##  4 Unbalanced optimal transport with time-dependent constraints

Having previously examined the case of constraints which are stationary in
time, we seek to extend this analysis to time-dependent ones. We should point
out right away that, in this case, one cannot expect the metric setting of
Section 3.2 to generalize and our primary focus will be on the issue of
existence of solutions to the constrained unbalanced optimal transport
problem. We introduce the following variant of the constrained, unbalanced
optimal transport energy, which obviously generalizes the problem in
Definition 3.2.

######  Definition 4.1 (WFR problem with time-varying constraints).

For functions $H:[0,1]\times\Omega\to\mathbb{R}^{d}$ and
$F:[0,1]\to\mathbb{R}^{d}$, we say that a measure
$\rho\in\mathcal{M}^{+}([0,1]\times\Omega)$ satisfies an affine constraint
with respect to $H$ and $F$ if it disintegrates in time
($\rho=dt\otimes\rho_{t}$) and

$\int_{\Omega}H(t,x)d\rho_{t}(x)=F(t)\qquad\mbox{for a.e. $t\in[0,1]$}.$ (31)

In this case, we write $\rho\in\mathcal{A}(H,F)$. For
$\rho_{0},\rho_{1}\in\mathcal{M}^{+}(\Omega)$, we consider the constrained set
of measures

$\mathcal{CE}_{H,F}(\rho_{0},\rho_{1}):=\left\\{\mu=(\rho,\omega,\zeta)\in\mathcal{CE}(\rho_{0},\rho_{1})\mid\rho\in\mathcal{A}(H,F)\right\\}.$

We define the Wasserstein-Fisher-Rao energy with time-varying affine-
constraints by

${\textrm{WFR}_{\delta}^{H,F}}(\rho_{0},\rho_{1})^{2}=\inf\left\\{\int_{[0,1]\times\Omega}f_{\delta}\left(\frac{d\mu}{d\lambda}\right)d\lambda\middle|\begin{aligned}
\mu=(\rho,\omega,\zeta)\in\mathcal{CE}_{H,F}(\rho_{0},\rho_{1})\end{aligned}\right\\},$
(32)

where $\lambda$ is any nonnegative Borel measure on $[0,1]\times\Omega$ such
that $\rho,\omega,\zeta\ll\lambda$. By the homogeneity of $f_{\delta}$, the
integral in the definition does not depend on the choice of $\lambda.$

######  Example 4.2 (Total mass constraints and moving barrier).

Here we want to describe two specific examples, that can be formulated in our
setting. The first one, time dependent total mass constraints, is a direct
extension of Example 3.13 to the time dynamic situation, i.e., $H=1$ but $F$
can be an arbitrary time dependent function that describes the evolution of
the total mass. Such a situation naturally occurs in applications, e.g. when
modelling a population density over time, where one only observes the total
population but not its spatial distribution at intermediate times. Under mild
conditions on $F$ we will show the existence of minimizers for this model in
Theorems 4.3 and 4.8.

As a second example, we aim to consider the situation of a domain $\Omega$
with a time-dependent barrier, i.e., a region $\Gamma(t)\subset\Omega$ through
which no mass can be transported or created. Assuming that for each
$t\in[0,1]$, $\Gamma(t)$ is an open subdomain of $\Omega$ that varies
continuously in time, we can formulate this in our setting by choosing
$F(t)=0$ and $H:[0,1]\times\Omega$ a continous function with $H(t,x)>0$ for
$x\in\Gamma(t)$ and $H(t,x)=0$ if $x\in\Omega\setminus\Gamma(t)$. It is indeed
clear that the condition $\int_{\Omega}H(t,x)d\rho_{t}(x)=0$ is then
equivalent to $\rho_{t}(\Gamma(t))=0$, in other words that no mass appears in
$\Gamma(t)$. The existence of minimizers for such a model is again guaranteed
by Theorems 4.3 and 4.8 below as long as the initial and the terminal measures
$\rho_{0}$ and $\rho_{1}$ stays inside the barrier at almost all time, i.e.,
$\rho_{0}(\Gamma(t))=\rho_{1}(\Gamma(t))=0$ for a.e. $t$.

###  4.1 Existence of minimizers of the WFR energy with time-varying
constraints

We are now able to state our general results on existence of minimizers. The
finite and minimal energy path terminology is defined in analogy with the
previous section.

######  Theorem 4.3 (Existence of minimizers for time-varying affine-
constraints).

Let $\rho_{0},\rho_{1}\in\mathcal{M}^{+}(\Omega)$, let
$H:[0,1]\times\Omega\to\mathbb{R}^{d}$ be a continuous function and let
$F=(F_{1},\ldots,F_{d}):[0,1]\to\mathbb{R}^{d}$ be a function with integrable
component functions $F_{i}$. If a finite energy path with respect to $H$ and
$F$ exists, then a minimal energy path exists.

Note that this theorem also implies the corresponding result for stationary
constraints, cf. Theorem 3.4. The proof of the theorem uses the Fenchel-
Rockafellar theorem from convex analysis, which we state here for the
convenience of the reader.

######  Theorem 4.4 (Fenchel-Rockafellar strong duality [6]).

Let $X,Y$ be Banach spaces, let
$G:X\to\mathbb{R}\cup\\{+\infty\\},F:Y\to\mathbb{R}\cup\\{+\infty\\}$ be
convex functions and let $A:X\to Y$ be a bounded linear map. If there exists
$x_{0}\in X$ such that $G(x_{0})<\infty,F(Ax_{0})<\infty$ and $F$ is
continuous at $Ax_{0}$, we have

$\inf_{x\in X}\\{F(Ax)+G(x)\\}=\sup_{y^{*}\in
Y^{*}}\\{-F^{*}(y^{*})-G^{*}(-A^{*}y^{*})\\}$ (33)

and there exists a $y^{*}\in Y^{*}$ which attains the sup on the right-hand
side if it is finite.

######  Proof 4.5 (Proof of Theorem 4.3).

To apply the Fenchel-Rockafellar theorem, we introduce the primal problem

$\inf_{x=(\phi,\psi)\in X}\\{F(Ax)+G(x)\\},$ (34)

with the following spaces and functionals:

  * •

The Banach spaces $X$ and $Y$ are

$X=C^{1}([0,1]\times\Omega)\times C([0,1])^{d}\quad\mbox{and}\quad
Y=C([0,1]\times\Omega)\times C([0,1]\times\Omega)^{n}\times
C([0,1]\times\Omega),$

respectively.

  * •

The convex function $F:Y\to\mathbb{R}\cup\\{+\infty\\}$ is defined by

$F(\alpha,\beta,\gamma)=\int_{[0,1]\times\Omega}\iota_{B_{\delta}}(\alpha(t,\theta),\beta(t,\theta),\gamma(t,\theta))dtd\theta,$

where

$B_{\delta}=\left\\{(a,b,c)\in\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}\middle|a+\frac{1}{2}\left(\|b\|^{2}+\frac{c^{2}}{\delta^{2}}\right)\leq
0\right\\}$ (35)

and $\iota_{B_{\delta}}$ is the convex indicator function of $B_{\delta}$;
that is,

$\iota_{B_{\delta}}(a,b,c)=\begin{cases}0&(x,y,z)\in B_{\delta}\\\
+\infty&\textrm{otherwise.}\end{cases}$ (36)

Note that the Fenchel conjugate of $\iota_{B_{\delta}}$ is the Wasserstein-
Fisher-Rao infinitesimal cost $f_{\delta}$ used to define
${\textrm{WFR}_{\delta}^{H,F}}$.

  * •

The convex function $G:X\to\mathbb{R}\cup\\{+\infty\\}$ is defined by

$G(\phi,\psi)=-\sum_{i=1}^{d}\int_{0}^{1}\psi_{i}(t)F_{i}(t)dt+\int_{\Omega}\phi(0,\cdot)d\rho_{0}-\int_{\Omega}\phi(1,\cdot)d\rho_{1}.$

where $\psi_{i}$ and $f_{i}$ are the $i$-th component of $\psi$ and $f$,
respectively.

  * •

The bounded linear map $A:X\to Y$ is defined by

$A(\phi,\psi)=\left(\partial_{t}\phi+\sum_{i=1}^{d}H_{i}\psi_{i},\nabla\phi,\phi\right).$

Theorem 4.4 indeed applies to this problem as $x_{0}=(\phi_{0},\psi_{0})$,
where $\phi_{0}(t)=-\delta^{2}t$ and $\psi_{0}(t)=(0,\cdots,0)$, satisfies the
necessary conditions. Therefore,

$\inf_{x\in X}\\{F(Ax)+G(x)\\}=\sup_{y^{*}\in
Y^{*}}\\{-F^{*}(y^{*})-G^{*}(-A^{*}y^{*})\\}$ (37)

By the Riesz-Markov-Kakutani Theorem [15, Theorem 7.17], we can identify

$Y^{*}\approx\mathcal{M}([0,1]\times\Omega)\times\mathcal{M}([0,1]\times\Omega)^{n}\times\mathcal{M}([0,1]\times\Omega).$

Our goal is to calculate each term on the right-hand side inside the supremum
in (37). We start with the $G^{\ast}$ term. In the following, we always use
$\langle\cdot,\cdot\rangle$ to denote the canonical pairing of a vector space
with its dual. We have

$\displaystyle G^{*}(-A^{*}y^{*})$ $\displaystyle=\sup_{(\phi,\psi)=x\in
X}\\{\langle-A^{*}y^{*},x\rangle-G(x)\\}=\sup_{x\in X}\left\\{-\langle
y^{*},Ax\rangle-G(x)\right\\}$ (38) $\displaystyle=\sup_{x\in
X}\left\\{-\langle
y^{*},Ax\rangle+\sum_{i=1}^{d}\int_{0}^{1}\psi_{i}(t)F_{i}(t)dt+\int_{\Omega}\phi(1,\cdot)d\rho_{1}-\int_{\Omega}\phi(0,\cdot)d\rho_{0}\right\\}.$
(39)

Using the notation $y^{*}=(\rho,m,\zeta)$, we have, for $x=(\phi,\psi)\in X$,

$\displaystyle\langle y^{*},Ax\rangle$ $\displaystyle=\langle
y^{*},A(\phi,\psi)\rangle=\langle
y^{*},(\partial_{t}\phi+\textstyle\sum_{i=1}^{d}H_{i}\psi_{i},\nabla\phi,\phi)\rangle$
(40) $\displaystyle=\int_{[0,1]\times\Omega}\partial_{t}\phi
d\rho+\int_{[0,1]\times\Omega}\nabla\phi\cdot
d\omega+\int_{[0,1]\times\Omega}\phi
d\zeta+\sum_{i=1}^{d}\int_{[0,1]\times\Omega}\psi_{i}(t)H_{i}(t,x)d\rho$ (41)
$\displaystyle=\int_{[0,1]\times\Omega}\partial_{t}\phi
d\rho+\int_{[0,1]\times\Omega}\nabla\phi\cdot
d\omega+\int_{[0,1]\times\Omega}\phi
d\zeta+\sum_{i=1}^{d}\int_{0}^{1}\psi_{i}(t)\int_{\Omega}H_{i}(t,x)d\rho_{t}dt.$
(42)

The calculations so far show that $G^{*}(-A^{*}y)$ is equal to

$\displaystyle\sup_{\phi}\left\\{-\int_{[0,1]\times\Omega}\partial_{t}\phi
d\rho-\int_{[0,1]\times\Omega}\nabla\phi\cdot
d\omega-\int_{[0,1]\times\Omega}\phi
d\zeta+\int_{\Omega}\phi(1,\cdot)d\rho_{1}-\int_{\Omega}\phi(0,\cdot)d\rho_{0}\right\\}$
$\displaystyle\hskip
144.54pt+\sum_{i=1}^{d}\sup_{\psi_{i}}\left\\{\int_{0}^{1}\left(F_{i}(t)-\int_{\Omega}H_{i}(t,x)d\rho_{t}(x)\right)\psi_{i}(t)dt\right\\}$

and it follows that

$G^{*}(-A^{*}y)=\left\\{\begin{array}[]{rl}0&\mbox{if
}(\rho,m,\zeta)\in\mathcal{CE}_{H,F}(\rho_{0},\rho_{1})\\\
+\infty&\mbox{otherwise.}\end{array}\right.$

This implies that

$\sup_{y^{*}\in
Y^{*}}\\{-F^{*}(y^{*})-G^{*}(-A^{*}y^{*})\\}=-\inf\left\\{F^{*}(y^{*})\middle|\begin{aligned}
y^{*}=(\rho,\omega,\zeta)\in\mathcal{CE}_{H,F}(\rho_{0},\rho_{1})\end{aligned}\right\\},$
(43)

so the existence of a minimal energy path follows by showing that
$F^{*}(y^{*})$ matches the objective of (32). This step is done in the same
way as in [9], using [28, Theorem 5]. Following that, since the infimum is
finite, we have that there exists $y^{*}\in\mathcal{CE}(\rho_{0},\rho_{1})$
that attains the infimum by Fenchel-Rockafeller.

######  Remark 4.6.

It is worth pointing out that the statement and proof of Theorem 4.3 could be
easily adapted to constrained balanced optimal transport as well i.e. to the
setting in which $\rho_{0}$ and $\rho_{1}$ are in addition assumed to be
probability measures and only transport is considered (i.e. $\zeta=0$),
assuming again that a finite energy path of probability measures exist. Such a
model with a very specific constraint was analyzed in [8] .

###  4.2 Existence of finite energy paths for the WFR energy with time-varying
constraints.

According to Theorem 4.3, to guarantee the existence of a minimal energy path
between two measures, it suffices to find a finite energy path. As we shall
see, unlike the original unconstrained model, this turns out to be a non-
trivial question. Let us first evacuate a few straightforward cases:

  * •

If $H=0$, then there is no feasible solution unless $F=0$ for a.e.
$t\in[0,1]$.

  * •

If $F=H=0$, then the problem is unconstrained (i.e.,
$\mathcal{A}(0,0)=\mathcal{M}^{+}([0,1]\times\Omega)$), so there is a finite
energy path, as shown in [9].

Our next goal is to examine the existence of finite energy paths for various
relevant choices of $H$ and $F$. The main results are contained in the
following theorems; the first one considers general affine constraints but
restricts them to be scalar-valued, and the second one allows for vector-
valued constraints but assumes them to be linear (i.e. $F=0$).

######  Theorem 4.7 (Finite energy paths for affine scalar constraints).

Suppose $\Omega$ is convex. Let $H:[0,1]\times\Omega\to\mathbb{R}$ be a
differentiable function with bounded partial derivatives such that $H(t,x)>0$
for all $(t,x)\in[0,1]\times\Omega$. Let $F:[0,1]\to\mathbb{R}$ be an
absolutely continuous function with $F^{\prime}\in L^{2}([0,1])$, such that
either

  1. 1.

$F(t)>0$ for all $t\in[0,1]$, or

  2. 2.

$F(t)\geq 0$ for all $t\in[0,1]$ and $F(t)=0$ on finitely many subintervals of
$[0,1]$, with decay to zero controlled as described in the statement of
Corollary 4.17.

Let $\rho_{0},\rho_{1}$ be measures such that
$\int_{\Omega}H(0,x)d\rho_{0}=F(0)$ and $\int_{\Omega}H(1,x)d\rho_{1}=F(1)$.
Then, there exists a finite energy path in
$\mathcal{CE}_{H,F}(\rho_{0},\rho_{1})$.

######  Theorem 4.8 (Finite energy paths for linear vector valued
constraints).

Let $H:[0,1]\times\Omega\to\mathbb{R}^{d}$ be a continuous function and let
$\rho_{0},\rho_{1}\in\mathcal{M}^{+}(\Omega)$ satisfy
$\int_{\Omega}H(t,x)d\rho_{0}(x)=0$ and $\int_{\Omega}H(t,x)d\rho_{1}(x)=0$
for a.e. $t\in[0,1]$. Then there exists a finite energy path in
$\mathcal{CE}_{H,0}(\rho_{0},\rho_{1})$.

Observe that these results imply cases (1) and (2) of Theorem 3.5 as special
cases. As a direct consequence of the proof of Theorem 4.8, we will also
obtain the following upper bound for the value of the constrained WFR energy:

######  Corollary 4.9.

If $H,\rho_{0}$ and $\rho_{1}$ satisfy the conditions of Theorem 4.8, then we
have the upper bound

${\textrm{WFR}_{\delta}^{H,0}}(\rho_{0},\rho_{1})^{2}\leq
4\delta^{2}(\rho_{0}(\Omega)+\rho_{1}(\Omega)).$

######  Remark 4.10.

As already pointed out in Remark 3.6, we emphasize that the continuity
condition in Theorem 4.8 is not actually necessary, but we only consider
continuous functions here as this is the condition for $H$ in Theorem 4.3.

We will first consider a few special cases and then leverage these results to
prove Theorem 4.7. We start with the case where $H(t,x)=1$ for all
$(t,x)\in[0,1]\times\Omega$. We note that $\rho\in\mathcal{A}(1,F)$ is
equivalent to $\rho_{t}(\Omega)=F(t)$ for a.e. $t\in[0,1]$; that is, this
constraint prescribes the total mass of $\rho_{t}$ at a.e. time $t$. We first
consider the necessary condition for $F$. The following result ensures that
$F$ can be identified with an absolutely continuous function.

######  Lemma 4.11.

Suppose that $\mu=(\rho,\omega,\zeta)\in\mathcal{CE}_{1,F}(\rho_{0},\rho_{1})$
is a finite energy path. Then there exists an absolutely continuous function
$u:[0,1]\to\mathbb{R}$ such that $u(t)=F(t)$ for almost every $t\in[0,1]$.

######  Proof 4.12.

Since $\mu$ is a finite energy path, Proposition 2.4 ensures that $\rho$,
$\omega$ and $\zeta$ are disintegrable in time, i.e.,
$\rho=dt\otimes\rho_{t},\omega=dt\otimes\omega_{t},\zeta=dt\otimes\zeta_{t}$.
Consider a $C^{1}$ function which is independent of the space variables,
$\phi=\phi(t)\in C^{1}([0,1])$. Evaluating the continuity equation (9) with
such a $\phi$ gives

$\int_{0}^{1}\phi^{\prime}(t)\rho_{t}(\Omega)dt+\int_{0}^{1}\phi(t)\zeta_{t}(\Omega)dt=\phi(1)\rho_{1}(\Omega)-\phi(0)\rho_{0}(\Omega)$
(44)

Define $u:[0,1]\to\mathbb{R}$ by
$u(t)=\int_{0}^{t}\zeta_{s}(\Omega)ds+\rho_{0}(\Omega)$. Then $u$ is
absolutely continuous, and $u^{\prime}(t)=\zeta_{t}(\Omega)$ almost
everywhere. Applying (44) to the constant function $\psi(t)=1$ implies that
$\int_{0}^{1}\zeta_{t}(\Omega)dt=\rho_{1}(\Omega)-\rho_{0}(\Omega)$, so
$u(1)=\rho_{1}(\Omega),u(0)=\rho_{0}(\Omega)$. Integration by parts yields

$\displaystyle\int_{0}^{1}\phi(t)\zeta_{t}(\Omega)dt$
$\displaystyle=\int_{0}^{1}\phi(t)u^{\prime}(t)dt=\phi(1)u(1)-\phi(0)u(0)-\int_{0}^{1}\phi^{\prime}(t)u(t)dt$
(45)

Substituting this into (44) gives us

$\displaystyle\int_{0}^{1}\phi^{\prime}(t)\rho_{t}(\Omega)dt=\int_{0}^{1}\phi^{\prime}(t)u(t)dt.$
(46)

Since this equation is true for any $\phi(t)\in C^{1}([0,1])$, we obtain
$\rho_{t}(\Omega)=F(t)=u(t)$ almost everywhere.

Based on this lemma, we can restrict our attention to the case where $F$ is
absolutely continuous without loss of generality. The next result considers
the special case of nonvanishing functions.

######  Proposition 4.13.

Supose $\Omega$ is convex and $F:[0,1]\to\mathbb{R}$ be an absolutely
continuous function such that $F^{\prime}(t)\in L^{2}([0,1])$ and $F(t)>0$ for
all $t\in[0,1]$. Let $\rho_{0},\rho_{1}$ be measures such that
$\rho_{0}(\Omega)=F(0)$ and $\rho_{1}(\Omega)=F(1)$. Then a finite energy path
in $\mathcal{CE}_{1,F}(\rho_{0},\rho_{1})$ exists.

######  Proof 4.14.

We begin with a finite energy path $(\rho,\omega,0)$ between
$\frac{1}{\rho_{0}(\Omega)}\rho_{0}$ and $\frac{1}{\rho_{1}(\Omega)}\rho_{1}$,
which exists since $\Omega$ is a convex compact set and this is balanced,
unconstrained transport [30, Theorem 5.28]. Since it has finite energy, $\rho$
and $\omega$ are disintegrable in time,
$\rho=dt\otimes\rho_{t},\omega=dt\otimes\omega_{t}$. Now define a new path
between $\rho_{0}$ and $\rho_{1}$ by
$\mu=(\bar{\rho},\bar{\omega},\bar{\zeta})=(F(t)\rho,F(t)\omega,F^{\prime}(t)\rho)$.
We show that $\mu$ satisfies the continuity equation as follows: for any
$\phi\in C^{1}([0,1]\times\Omega)$,

$\displaystyle\int_{0}^{1}\int_{\Omega}\partial_{t}\phi(t,x)F(t)d\rho_{t}(x)dt$
$\displaystyle=\int_{0}^{1}\int_{\Omega}(\partial_{t}(\phi(t,x)F(t))-\phi(t,x)F^{\prime}(t))d\rho_{t}(x)dt$
(47) $\displaystyle=C-\int_{0}^{1}\int_{\Omega}\nabla\phi(t,x)F(t)\cdot
d\omega_{t}(x)dt-\int_{0}^{1}\int_{\Omega}\phi(t,x)F^{\prime}(t)d\rho_{t}(x)dt$
(48) $\displaystyle=C-\int_{0}^{1}\int_{\Omega}\nabla\phi(t,x)\cdot
d\bar{\omega}_{t}(x)dt-\int_{0}^{1}\int_{\Omega}\phi(t,x)d\zeta_{t}(x)dt,$
(49)

where
$C=\int_{\Omega}\phi(1,\cdot)d\rho_{1}-\int_{\Omega}\phi(0,\cdot)d\rho_{0}$.
Equation (48) follows from the fact that $(\rho,\omega,0)$ satisfies the
continuity equation and by substituting $\phi\mapsto\phi(t,x)F(t)$. While this
function is not of regularity $C^{1}$ in general, we can approximate it by
$C^{1}$ functions by a density argument. By definition,
$\bar{\omega},\bar{\zeta}\ll\bar{\rho}$ so taking $\lambda=\bar{\rho}$ allows
us to calculate the energy. We have

$\displaystyle\int_{[0,1]\times\Omega}f_{\delta}\left(\frac{d\mu}{d\lambda}\right)d\lambda$
$\displaystyle=\int_{[0,1]\times\Omega}f_{\delta}\left(\frac{d\bar{\rho}}{d\lambda},\frac{d\bar{\omega}}{d\lambda},0\right)d\lambda+\delta^{2}\int_{[0,1]\times\Omega}f_{\delta}\left(\frac{d\bar{\rho}}{d\lambda},0,\frac{d\bar{\zeta}}{d\lambda}\right)d\lambda$
(50)
$\displaystyle=\int_{[0,1]\times\Omega}F(t)f_{\delta}\left(\frac{d\rho}{d\lambda},\frac{d\omega}{d\lambda},0\right)d\lambda+\frac{\delta^{2}}{2}\int_{0}^{1}\frac{F^{\prime}(t)^{2}}{F(t)}dt$
(51) $\displaystyle\leq
M\int_{[0,1]\times\Omega}f_{\delta}\left(\frac{d\rho}{d\lambda},\frac{d\omega}{d\lambda},0\right)d\lambda+\frac{\delta^{2}}{2m}\|F^{\prime}(t)\|_{L^{2}}^{2}$
(52)

where $M=\textrm{max}_{t\in[0,1]}F(t)$ and $m=\textrm{min}_{t\in[0,1]}F(t)>0$
which exists because $F(t)$ is continuous on a compact domain $[0,1]$. The
last expression is finite because the first term is the energy of
$(\rho,\omega,0)$, and the second term is finite by the $L^{2}$ assumption on
the derivative.

The proof above yields the following corollary.

######  Corollary 4.15.

If $F,\rho_{0}$ and $\rho_{1}$ satisfy the conditions of Proposition 4.13,
then we have the upper bound

${{\textrm{WFR}_{\delta}^{1,F}}}(\rho_{0},\rho_{1})^{2}\leq
M\cdot\mathrm{W}_{2}\left(\frac{\rho_{0}}{\rho_{0}(\Omega)},\frac{\rho_{1}}{\rho_{1}(\Omega)}\right)^{2}+\frac{\delta^{2}}{2m}\|F^{\prime}(t)\|^{2}_{L^{2}},$
(53)

where $M=\max_{t\in[0,1]}F(t),m=\min_{t\in[0,1]}F(t)$ and $\mathrm{W}_{2}$ is
the Wasserstein 2-distance (i.e., the square root of (4)).

We now consider constraint functions $F$ that can possibly reach zero, but
assuming some additional control on their behavior near their zero level sets.

######  Proposition 4.16.

Let $F$ be an absolutely continuous function such that $F(t)>0$ for
$t\in(0,1]$, $F^{\prime}(t)\in L^{2}[0,1]$, and
$F^{\prime}(t)=\Theta(t^{\alpha})$ as $t\to 0^{+}$ for some $\alpha>0$—that
is, there exist $K_{1},K_{2},\epsilon>0$, such that $K_{1}t^{\alpha}\leq
F^{\prime}(t)\leq K_{2}t^{\alpha}$ for all $t\leq\epsilon$. Let
$\rho_{0},\rho_{1}$ be measures such that $\rho_{0}(\Omega)=F(0)=0$ and
$\rho_{1}(\Omega)=F(1)$. Then a finite energy path in
$\mathcal{CE}_{1,F}(\rho_{0},\rho_{1})$ exists.

Before proving the proposition, we observe that it easily extends to functions
that take zero values on finitely many intervals.

######  Corollary 4.17.

Let $F:[0,1]\to\mathbb{R}$ be an absolutely continuous function such that
$F(t)=0$ on finitely many subintervals $[a_{i},b_{i}]$, $i=1,\cdots,n$, with
$0\leq a_{1}<b_{1}<a_{2}<b_{2}<\cdots<a_{n}<b_{n}\leq 1$. Further, assume that
$F^{\prime}(t)\in L^{2}[0,1]$ and that
$F^{\prime}(t)=\Theta((t-a_{i})^{\alpha_{i}})$ as $t\to t_{0}^{-}$ and
$F^{\prime}(t)=\Theta((t-b_{i})^{\alpha_{i}})$ as $t\to t_{0}^{+}$ for some
choices of $\alpha_{i},\beta_{i}>0$. Let $\rho_{0},\rho_{1}$ be measures such
that $\rho_{0}(\Omega)=F(0)$ and $\rho_{1}(\Omega)=F(1)$. Then a finite energy
path in $\mathcal{CE}_{1,F}(\rho_{0},\rho_{1})$ exists.

######  Proof 4.18.

We construct a finite energy path by concatenating the following paths. First,
from $t=0$ to $t=a_{1}$, we consider a finite energy path between $\rho_{0}$
and the zero measure, which exists by running in reverse the path which is
guaranteed to exist by Proposition 4.16. Then, starting from $t=a_{1}$, we
consider a constant path with $\rho=\omega=\zeta=0$ on $[a_{1},b_{1}]$. On the
interval between $t=b_{1}$ to $t=a_{2}$, we consider the concatenation of
finite energy path from the zero measure to
$F(\frac{b_{1}+a_{2}}{2})\delta_{a}$ on $[b_{1},\frac{b_{1}+a_{2}}{2}]$, where
$a$ is any point on $\Omega$, and the path returning to zero measure on
$[\frac{b_{1}+a_{2}}{2},a_{2}]$. The rest of the paths to be concatenated are
constructed similarly. Since we are concatenating finitely many finite energy
paths, the result also has finite energy, so we have the desired finite energy
path.

######  Proof 4.19 (Proof of Proposition 4.16).

Define a path
$\mu=(\rho,\omega,\zeta)=(dt\otimes\rho_{t},dt\otimes\omega_{t},dt\otimes\zeta_{t})$
by

$\rho_{t}=\frac{F(t)}{F(1)}\rho_{1},\qquad\omega_{t}=0,\quad\mbox{and}\quad\zeta_{t}=\frac{F^{\prime}(t)}{F(1)}\rho_{1}.$

It is easy to check that this path satisfies the continuity equation and that
$\zeta\ll\rho$. Now, take $\lambda=\rho$, so that

$\frac{d\rho}{d\lambda}=1,\qquad\frac{d\omega}{d\lambda}=0,\quad\mbox{and}\quad\frac{d\zeta}{d\lambda}=\frac{F^{\prime}(t)}{F(t)}.$

Therefore,

$\int_{[0,1]\times\Omega}f_{\delta}\left(\frac{d\mu}{d\lambda}\right)d\lambda=\int_{[0,1]\times\Omega}\frac{\delta^{2}F^{\prime}(t)^{2}}{2F(t)^{2}}d\lambda=\int_{0}^{1}\int_{\Omega}\frac{\delta^{2}F^{\prime}(t)^{2}}{2F(t)^{2}}\frac{F(t)}{F(1)}d\rho_{1}dt=\frac{\delta^{2}}{2}\int_{0}^{1}\frac{F^{\prime}(t)^{2}}{F(t)}dt,$
(54)

where the last line follows from $F(1)=\rho_{1}(\Omega)$. We now use the
asymptotic assumption. For $t<\epsilon$, we have $F^{\prime}(t)^{2}\leq
K^{2}_{2}t^{2\alpha}$, and
$F(t)=\int_{0}^{t}F^{\prime}(s)ds\geq\int_{0}^{t}K_{1}t^{\alpha}ds=\frac{K_{1}}{\alpha+1}t^{\alpha+1}$.
Thus we have

$\int_{0}^{1}\frac{F^{\prime}(t)^{2}}{F(t)}dt\leq\int_{0}^{\epsilon}\frac{(\alpha+1)K_{2}^{2}t^{2\alpha}}{K_{1}t^{\alpha+1}}dt+\int_{\epsilon}^{1}\frac{F^{\prime}(t)^{2}}{F(t)}dt=\frac{(\alpha+1)K_{2}^{2}}{\alpha
K_{1}}\epsilon^{\alpha}+\int_{\epsilon}^{1}\frac{F^{\prime}(t)^{2}}{F(t)}dt$
(55)

Finally, we note that $1/F(t)$ is a continuous function on $[\epsilon,1]$ as
$F(t)\neq 0$ on the interval, so it is bounded. Using Hölder’s inequality, we
observe that the second integral is finite:

$\int_{\epsilon}^{1}\frac{F^{\prime}(t)^{2}}{F(t)}dt\leq\|F^{\prime}(t)^{2}\|_{L^{1}}\left\|\frac{1}{F(t)}\right\|_{L^{\infty}}=\|F^{\prime}(t)\|_{L^{2}}^{2}\left\|\frac{1}{F(t)}\right\|_{L^{\infty}}<\infty.$

Therefore, the energy is finite.

We are now prepared to proceed with the proof of our main existence theorem
for one-dimensional constraints.

######  Proof 4.20 (Proof of Theorem 4.7).

We first take a finite energy path $(\rho,\omega,\zeta)$ in
$\mathcal{CE}_{1,F}(\rho_{0},\rho_{1})$ that connects the measure
$H(0,\cdot)\rho_{0}$ to the measure $H(1,\cdot)\rho_{1}$—this exists because
we are assuming the sufficient conditions from either Proposition 4.16 or
Corollary 4.17. Then we scale the path by $g=1/H$; that is, we consider the
path
$\tilde{\mu}=(\tilde{\rho},\tilde{\omega},\tilde{\zeta})=(g\rho,g\omega,g\zeta+(\partial_{t}g)\rho+(\nabla_{x}g)\cdot\omega)$
where the source term is modified so that it satisfies the continuity
equation. We first note that $g$ is bounded. Indeed, since $H$ is continuous
and positive-valued, compactness of $[0,1]\times\Omega$ implies that there
exists $c>0$ such that $H(t,x)\geq c$ for all $(t,x)\in[0,1]\times\Omega$, and
we have $|g|=g=1/H\leq 1/c$. Furthermore, the derivatives of $g$ are bounded.
Indeed, $H$ is assumed to have bounded derivatives, so there exists a constant
$M$ such that $|\partial_{t}g|=|\partial_{t}1/H|=|\partial_{t}H/H^{2}|\leq
M/c^{2}$ and $\nabla_{x}H$ is also bounded by the same argument.

The measure $\tilde{\rho}$ satisfies the desired affine constraint because
$d\tilde{\rho}_{t}(x)=g(t,x)d\rho_{t}(x)$, so

$\int_{\Omega}H(t,x)d\tilde{\rho}_{t}(x)=\int_{\Omega}H(t,x)g(t,x)d\rho_{t}(x)=\int_{\Omega}d\rho_{t}(x)=\rho_{t}(\Omega)=F(t).$
(56)

We now show that the path $\tilde{\mu}$ satisfies the continuity equation. For
any $\phi\in C^{1}([0,1]\times\Omega)$, by the Product Rule,

$\displaystyle\int_{[0,1]\times\Omega}(\partial_{t}\phi)d\tilde{\rho}$
$\displaystyle=\int_{[0,1]\times\Omega}(\partial_{t}\phi)gd\rho=\int_{[0,1]\times\Omega}(\partial_{t}(\phi
g)-\phi\partial_{t}g)d\rho$ (57)
$\displaystyle\int_{[0,1]\times\Omega}(\nabla_{x}\phi)\cdot d\tilde{\omega}$
$\displaystyle=\int_{[0,1]\times\Omega}(\nabla_{x}\phi)\cdot
gd\omega=\int_{[0,1]\times\Omega}(\nabla_{x}(\phi g)-\phi\nabla_{x}g)\cdot
d\omega$ (58)

Moreover, by definition,

$\displaystyle\int_{[0,1]\times\Omega}\phi
d\tilde{\zeta}-\int_{[0,1]\times\Omega}\phi\partial_{t}gd\rho-\int_{[0,1]\times\Omega}\phi(\nabla_{x}g)\cdot
d\omega=\int_{[0,1]\times\Omega}\phi gd\zeta$ (59)

Therefore,

$\displaystyle\int_{[0,1]\times\Omega}\partial_{t}\phi
d\tilde{\rho}+\int_{[0,1]\times\Omega}(\nabla_{x}\phi)\cdot
d\tilde{\omega}+\int_{[0,1]\times\Omega}\phi d\tilde{\zeta}$ (60)
$\displaystyle=\int_{[0,1]\times\Omega}\partial_{t}(\phi
g)d\rho+\int_{[0,1]\times\Omega}\nabla_{x}(\phi g)\cdot
d\omega+\left(\int_{[0,1]\times\Omega}\phi
d\tilde{\zeta}-\int_{[0,1]\times\Omega}\phi\partial_{t}gd\rho-\int_{[0,1]\times\Omega}\phi(\nabla_{x}g)\cdot
d\omega\right)$ (61) $\displaystyle=\int_{[0,1]\times\Omega}\partial_{t}(\phi
g)d\rho+\int_{[0,1]\times\Omega}\nabla_{x}(\phi g)\cdot
d\omega+\int_{[0,1]\times\Omega}\phi gd\zeta$ (62)
$\displaystyle=\int_{\Omega}\phi(1,\cdot)g(1,\cdot)d\rho_{1}-\int_{\Omega}\phi(1,\cdot)g(0,\cdot)d\rho_{0}=\int_{\Omega}\phi(1,\cdot)d\tilde{\rho}_{1}-\int_{\Omega}\phi(0,\cdot)d\tilde{\rho}_{0},$
(63)

where we have used the assumption that $(\rho,\omega,\zeta)$ satisfies the
continuity equation. Although $\phi g$ is not $C^{1}$ in general, we can
approximate the function by a $C^{1}$ function by a density argument.

Next, we show that the path $\tilde{\mu}$ has a finite energy. Take any Borel
nonnegative measure $\lambda$ such that $\rho,\omega,\zeta\ll\lambda$. Then we
obtain that

$\displaystyle\frac{d\tilde{\rho}}{d\lambda}=g\frac{d\rho}{d\lambda},\ \
\frac{d\tilde{\omega}}{d\lambda}=g\frac{d\omega}{d\lambda}\ \ \text{and
}\frac{d\tilde{\zeta}}{d\lambda}=g\frac{d\zeta}{d\lambda}+\partial_{t}g\frac{d\rho}{d\lambda}+(\nabla_{x}g)\cdot\frac{d\omega}{d\lambda}$
(64)

For brevity, let us denote
$\tilde{\rho}_{\lambda}=\frac{d\tilde{\rho}}{d\lambda},\tilde{\omega}_{\lambda}=\frac{d\tilde{\omega}}{d\lambda},\tilde{\zeta}_{\lambda}=\frac{d\tilde{\zeta}}{d\lambda}$
and similarly for Radon-Nykodim derivatives of $(\rho,\omega,\zeta)$. Observe
that the _Wasserstein term_ in the WFR objective is bounded as

$\int_{[0,1]\times\Omega}\frac{\|\tilde{\omega}_{\lambda}\|^{2}}{\tilde{\rho}_{\lambda}}d\lambda=\int_{[0,1]\times\Omega}g\frac{\|\omega_{\lambda}\|^{2}}{\rho_{\lambda}}d\lambda\leq\frac{1}{c}\int_{[0,1]\times\Omega}\frac{\|\omega_{\lambda}\|^{2}}{\rho_{\lambda}}d\lambda,$
(65)

where $c>0$ is the lower bound of $h$ introduced earlier. Since the quantity
on the right is a constant multiple of the Wasserstein term in the WFR
objective evaluated on $(\rho,\omega,\zeta)$, the right hand side of (65) is
finite. Now, for the _Fisher-Rao term_ in the energy, we have

$\displaystyle\frac{\tilde{\zeta}_{\lambda}^{2}}{\tilde{\rho}_{\lambda}}=\frac{g^{2}\zeta_{\lambda}^{2}+(\partial_{t}g)^{2}\rho_{\lambda}^{2}+(\nabla_{x}g\cdot\omega_{\lambda})^{2}+2g\partial_{t}g\zeta_{\lambda}\rho_{\lambda}+2\partial_{t}g\rho_{\lambda}\nabla_{x}g\cdot\omega_{\lambda}+2g\zeta_{\lambda}(\nabla_{x}g\cdot\omega_{\lambda})}{g\rho_{\lambda}}$
(66)

We now investigate the integral with respect to $\lambda$ of each term of
$\frac{\tilde{\zeta}_{\lambda}^{2}}{\tilde{\rho}_{\lambda}}$.

  * •

$\frac{g^{2}\zeta_{\lambda}^{2}}{g\rho_{\lambda}}=g\frac{\zeta_{\lambda}^{2}}{\rho_{\lambda}}$
has a finite integral because $g$ is bounded and $(\rho,\omega,\zeta)$ has
finite energy.

  * •

$\frac{(\partial_{t}g)^{2}\rho_{\lambda}^{2}}{\rho_{\lambda}}=\frac{(\partial_{t}g)^{2}}{g}\rho_{\lambda}$
has a finite integral as we integrate a bounded function
$\frac{(\partial_{t}\phi)^{2}}{g}$ by a finite measure $\rho$.

  * •

By Cauchy-Schwartz, we have
$\frac{(\nabla_{x}g\cdot\omega_{\lambda})^{2}}{g\rho_{\lambda}}\leq\frac{\|\nabla_{x}g\|^{2}\|\omega_{\lambda}\|^{2}}{g\rho_{\lambda}}=\frac{g^{4}\|\nabla_{x}H\|^{2}\|\omega_{\lambda}\|^{2}}{g\rho_{\lambda}}=g^{3}\frac{\|\nabla_{x}H\|^{2}\|\omega_{\lambda}\|^{2}}{\rho_{\lambda}}$
which has a finite integral because of the boundedness and the finite energy
assumption. Here, we used
$\|\nabla_{x}g\|=\frac{\|\nabla_{x}H\|}{H^{2}}=g^{2}\|\nabla_{x}H\|$.

  * •

$\frac{2g(\partial_{t}g)\zeta_{\lambda}\rho_{\lambda}}{g\rho_{\lambda}}=2(\partial_{t}g)\zeta_{\lambda}$
has a finite integral because it is an integral of a bounded function
$2\partial_{t}g$ by a finite measure $\zeta$.

  * •

$\frac{2(\partial_{t}g)\rho_{\lambda}\nabla_{x}g\cdot\omega_{\lambda}}{g\rho_{\lambda}}=\frac{2(\partial_{t}g)\nabla_{x}g\cdot\omega_{\lambda}}{g}=\left(\frac{2(\partial_{t}g)\nabla_{x}g}{g}\right)\cdot\omega_{\lambda}$
has a finite integral because it is an integral of a bounded function
$\frac{2\partial_{t}g\nabla_{x}g}{g}$ by a bounded vector measure $\omega$.

  * •

For the last term, if $F(t)=0$ at some point, assuming $F(t)$ satisfies the
conditions of Corollary 4.17, we can take a finite energy path
$(\rho,\omega,\zeta)$ with $\omega=0$ so the last term is zero, and it has a
finite integral. If $F(t)>0$ with $F^{\prime}(t)\in L^{2}$ (i.e., $F$
satisfies the conditions of Proposition 4.13), we can take the path
$(\rho,\omega,\zeta)=(F(t)\rho^{B},F(t)\omega^{B},F^{\prime}(t)\rho^{B})$
where $(\rho^{B},\omega^{B})$ is a finite energy path in a balanced case as
discussed before. Therefore,

$\frac{2g\zeta_{\lambda}(\nabla_{x}g\cdot\omega_{\lambda})}{g\rho_{\lambda}}=\frac{2gF^{\prime}\rho_{\lambda}^{B}(F\nabla_{x}g\cdot\omega_{\lambda}^{B})}{gF\rho_{\lambda}^{B}}=2F^{\prime}\nabla_{x}g\cdot\omega_{\lambda}^{B}=2F^{\prime}\sqrt{\rho_{\lambda}^{B}}\cdot\frac{\nabla_{x}g\cdot\omega_{\lambda}^{B}}{\sqrt{\rho_{\lambda}^{B}}}$
(67)

Now, by Cauchy-Schwartz,

$\left\|F^{\prime}(t)\sqrt{\rho_{\lambda}^{B}}\cdot\frac{\nabla_{x}g\cdot\omega_{\lambda}^{B}}{\sqrt{\rho_{\lambda}^{B}}}\right\|_{L^{1}(\lambda)}\leq\left\|f^{\prime}(t)\sqrt{\rho_{\lambda}^{B}}\right\|_{L^{2}(\lambda)}\left\|\frac{\nabla_{x}g\cdot\omega_{\lambda}^{B}}{\sqrt{\rho_{\lambda}^{B}}}\right\|_{L^{2}(\lambda)}$
(68)

and each term on the right-hand side is finite because

$\displaystyle\left\|F^{\prime}\sqrt{\rho_{\lambda}^{B}}\right\|^{2}_{L^{2}(\lambda)}$
$\displaystyle=\int_{\Omega\times[0,1]}F^{\prime
2}\rho_{\lambda}^{B}d\lambda=\int_{\Omega\times[0,1]}F^{\prime
2}d\rho^{B}=\int_{0}^{1}\int_{\Omega}F^{\prime 2}d\rho_{t}^{B}dt$ (69)
$\displaystyle=\int_{0}^{1}F^{\prime
2}\rho_{t}^{B}(\Omega)dt=\int_{0}^{1}F^{\prime 2}\cdot
1dt=\|F^{\prime}\|_{L^{2}}^{2}<\infty$ (70)

Note that $\rho_{t}^{B}(\Omega)=1$ for all $t$ since $\rho^{B}$ is a path for
balanced transport. The other term is finite because

$\displaystyle\left\|\frac{\nabla_{x}g\cdot\omega_{\lambda}^{B}}{\sqrt{\rho_{\lambda}^{B}}}\right\|^{2}_{L^{2}(\lambda)}$
$\displaystyle=\int_{[0,1]\times\Omega}\frac{(\nabla_{x}g\cdot\omega_{\lambda}^{B})^{2}}{\rho_{\lambda}^{B}}d\lambda$
(71)

which is finite by using Cauchy-Schwartz again and applying the boundedness
and the finite energy assumption. Overall, the path has a finite energy.

Finally, we consider the existence of finite paths for general constraint
dimension $d$ but for the special case of $F=0$.

######  Proof 4.21 (Proof of Theorem 4.8).

Similar to the proof of [20, Theorem 1], define
$\mu=(\rho,m,\zeta)=(dt\otimes\rho_{t},dt\otimes\omega_{t},dt\otimes\zeta_{t})$
by

$\rho_{t}=\begin{cases}(1-2t)^{2}\rho_{0},&0\leq t\leq 1/2\\\
(2t-1)^{2}\rho_{1}&1/2<t\leq
1\end{cases},\quad\omega_{t}=0,\quad\mbox{and}\quad\zeta_{t}=\begin{cases}-4(1-2t)\rho_{0},&0\leq
t\leq 1/2\\\ 4(2t-1)\rho_{1}&1/2<t\leq 1.\end{cases}$ (72)

We note that $\rho$ is a concatenation of the Fisher-Rao geodesic ([9],
Theorem 3.1) from $\rho_{0}$ to $0$ with the one from $0$ to $\rho_{1}$. By
direct calculation, it is easy to show that $\mu$ satisfies the constraint and
the continuity equation. We now calculate the integral. Define $\lambda=\rho$.
Clearly, $\rho,\omega,\zeta\ll\lambda$, and we have

$d\rho/d\lambda=1,\quad d\omega/d\lambda=0,\quad\mbox{and }\quad
d\zeta/d\lambda=\begin{cases}-\frac{4}{1-2t}&0\leq t<1/2\\\
\frac{4}{2t-1}&1/2<t\leq 1.\end{cases}$ (73)

Therefore, we obtain

$\displaystyle\int_{[0,1]\times\Omega}f_{\delta}\left(\frac{d\mu}{d\lambda}\right)d\lambda=\int_{0}^{1}\int_{\Omega}\frac{0^{2}+\delta^{2}(d\zeta/d\rho)^{2}}{2\cdot
1}d\rho_{t}dt$ (74)
$\displaystyle=\frac{\delta^{2}}{2}\int_{0}^{1/2}\int_{\Omega}\left(\frac{-4}{1-2t}\right)^{2}(1-2t)^{2}d\rho_{0}dt+\frac{\delta^{2}}{2}\int_{1/2}^{1}\int_{\Omega}\left(\frac{4}{2t-1}\right)^{2}(2t-1)^{2}d\rho_{1}dt=4\delta^{2}(\rho_{0}(\Omega)+\rho_{1}(\Omega))<\infty.$

###  4.3 Optimality certification

In the following we will provide a sufficient condition for optimality, which
generalizes Theorem 2.3 of [9] to the constrained case studied in the present
article. We first recall an important result on which we will rely to prove
the optimality certification theorem:

######  Lemma 4.22 (Lemma 2.1, [9]).

Denote the objective integral in (11) as $D_{\delta}(\mu)$. The
subdifferential of $D_{\delta}$ at $\mu=(\rho,\omega,\mu)$ such that
$D_{\delta}(\mu)<+\infty$ contains the set

$\partial^{c}D_{\delta}(\mu)=\left\\{(\alpha,\beta,\gamma)\in
C([0,1]\times\Omega;B_{\delta})\middle|\alpha+\frac{1}{2}\left(|\beta|^{2}+\frac{\gamma^{2}}{\delta^{2}}\right)=0-\rho\textrm{
a.e.},\beta\rho=\omega\textrm{ and }\gamma\rho=\delta^{2}\zeta\right\\}$ (75)

To prove the main result of this section we will need one more technical
lemma. In the following we slightly abuse the notation $\mathcal{A}(H,F)$ and
use it for the convex set of all
$\mu=(\rho,\omega,\zeta)\in\mathcal{M}^{+}([0,1]\times\Omega)\times\mathcal{M}([0,1]\times\Omega)^{n}\times\mathcal{M}([0,1]\times\Omega)$
such that $\rho\in\mathcal{A}(H,F)$. We omit the proof as this is the normal
cone of an affine set, which is well-studied.

######  Lemma 4.23.

Let $H=(H_{1},\cdots H_{d})\in C([0,1]\times\Omega)^{d}$ and
$F:[0,1]\times\Omega\to\mathbb{R}^{d}$. The normal cone of $\mathcal{A}(H,F)$,
i.e., the subdifferential of $\iota_{\mathcal{A}(H,F)}$ at
$\mu=(\rho,\omega,\zeta)\in\mathcal{A}(H,F)$ is the set

$N(\mu)=\left\\{\left(\sum_{i=1}^{d}H_{i}(t,x)g_{i}(t),0,0\right)\in
C([0,1]\times\Omega;\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R})|g_{i}\in
C([0,1]),i=1,\cdots,d\right\\}$ (76)

We are now able to show our main result of this subsection:

######  Theorem 4.24 (Optimality certificate).

Let $\rho_{0},\rho_{1}\in\mathcal{M}_{+}(\Omega)$. For
$\mu=(\rho,\omega,\zeta)\in\mathcal{CE}_{H,F}(\rho_{0},\rho_{1})$, if there
exists $\phi\in C^{1}([0,1]\times\Omega)$ such that

$(\partial_{t}\phi,\nabla\phi,\phi)\in\partial^{c}D_{\delta}(\mu)+N(\mu)$ (77)

then $\mu$ is a minimizer for the infimum in $(\ref{eqn: constWFR_affine})$.
We refer to such $\phi$ as an optimality certificate for $\mu$.

######  Proof 4.25.

We note that $\partial^{c}D_{\delta}(\mu)+N(\mu)\subset\partial
D_{\delta}(\mu)+\partial\iota_{\mathcal{A}(H,F)}(\mu)\subset\partial(D_{\delta}+\iota_{\mathcal{A}(H,F)})(\mu)$,
so if
$(\partial_{t}\phi,\nabla\phi,\phi)\in\partial^{c}D_{\delta}(\mu)+N(\mu)$, the
triple belongs to the subgradient of
$D_{\delta}(\mu)+\iota_{\mathcal{A}(H,F)}$, i.e., for any
$\mu^{\prime}\in\mathcal{CE}(\rho_{0},\rho_{1})$,

$\displaystyle
D_{\delta}(\mu^{\prime})+\iota_{\mathcal{A}(H,F)}(\mu^{\prime})-D_{\delta}(\mu)\geq\langle(\partial_{t}\phi,\nabla\phi,\phi),\mu^{\prime}-\mu\rangle=0$
(78)

where the right-hand side equals zero by definition of the distributional
continuity equation. Therefore, we have that $D_{\delta}(\mu^{\prime})\geq
D_{\delta}(\mu)$ for any
$\mu^{\prime}\in\mathcal{CE}_{H,F}(\rho_{0},\rho_{1})$, which concludes the
proof.

######  Example 4.26 (Scaling with prescribed total mass).

As an application of the above optimality certification, we will find a
minimizer of a pure scaling with a prescribed total mass constraint, i.e., let
$F:[0,1]\to(0,\infty)$ be a $C^{2}$ function and consider the affine
constraint $\mathcal{A}(1,F)$. Furthermore let
$\rho_{0}\in\mathcal{M}^{+}(\Omega)$ such that $\rho_{0}(\Omega)=F(0)$ and let
$\rho_{1}=\frac{F(1)}{F(0)}\rho_{0}$ be a pure scaling of $\rho_{0}$. Then

$\rho=dt\otimes\frac{F(t)}{F(0)}\rho_{0}$ (79)

defines a minimizer for $(\ref{eqn: constWFR_affine})$ with $H=1$, and

${\textrm{WFR}_{\delta}^{1,F}}(\rho_{0},\rho_{1})=\frac{\delta}{\sqrt{2}}\left(\int_{0}^{1}\frac{F^{\prime}(t)^{2}}{F(t)}dt\right)^{1/2}.$
(80)

To see this we define the function $\phi\in C^{1}([0,1]\times\Omega)$ by

$\phi(t,x)=\delta^{2}\frac{F^{\prime}(t)}{F(t)}.$ (81)

We obtain

$(\partial_{t}\phi,\nabla\phi,\phi)=\left(\delta^{2}\frac{F^{\prime\prime}(t)F(t)-(F^{\prime}(t))^{2}}{F(t)^{2}},0,\delta^{2}\frac{F^{\prime}(t)}{F(t)}\right)$
(82)

We will see that this triple decomposes to the sum of elements of
$\partial^{c}D_{\delta}(\mu)$ and $N(\mu)$ where
$\mu=(\rho,0,dt\otimes\frac{f^{\prime}(t)}{f(0)}\rho_{0})$. Write down
$\partial^{c}D_{\delta}(\mu)$ in this special case to see that

$\partial^{c}D_{\delta}(\mu)=\left\\{(\alpha,\beta,\gamma)\middle|\alpha+\frac{1}{2}\left(|\beta|^{2}+\frac{\gamma^{2}}{\delta^{2}}\right)=0-\rho\textrm{
a.e.},\beta=0-\rho\textrm{ a.e.}\textrm{ and
}\gamma=\delta^{2}\frac{F^{\prime}(t)}{F(t)}-\rho\textrm{ a.e.}\right\\}$ (83)

Therefore, we observe that

$\left(-\frac{\delta^{2}}{2}\frac{F^{\prime}(t)^{2}}{F(t)^{2}},0,\delta^{2}\frac{F^{\prime}(t)}{F(t)}\right)\in\partial^{c}D_{\delta}(\mu)$
(84)

We will now consider $N(\mu)$. Since $H=1$, we have $N(\mu)=\\{(g,0,0)|g\in
C([0,1])\\}$, so

$\left(\delta^{2}\frac{F^{\prime\prime}(t)F(t)-(1/2)(F^{\prime}(t))^{2}}{F(t)^{2}},0,0\right)\in
N(\mu)$ (85)

Finally, we have

$\displaystyle\left(-\frac{\delta^{2}}{2}\frac{F^{\prime}(t)^{2}}{F(t)^{2}},0,\delta^{2}\frac{F^{\prime}(t)}{F(t)}\right)+\left(\delta^{2}\frac{F^{\prime\prime}(t)F(t)-(1/2)(F^{\prime}(t))^{2}}{F(t)^{2}},0,0\right)$
(86)
$\displaystyle=\left(\delta^{2}\frac{F^{\prime\prime}(t)F(t)-(F^{\prime}(t))^{2}}{F(t)^{2}},0,\delta^{2}\frac{F^{\prime}(t)}{F(t)}\right)=(\partial_{t}\phi,\nabla\phi,\phi)\in\partial^{c}D_{\delta}(\mu)+N(\mu)$
(87)

Thus, $\phi$ is an optimal certificate, and $\mu$ is a minimizer of the
problem. We now calculate the optimal energy. By definition, $\rho$ and the
source term $dt\otimes\frac{F^{\prime}(t)}{F(0)}\rho_{0}$ are absolutely
continuous with respect to $\lambda=dt\otimes\rho_{0}$, so

${\textrm{WFR}_{\delta}^{1,F}}(\rho_{0},\rho_{1})^{2}=\delta^{2}\int_{0}^{1}\int_{\Omega}\frac{(F^{\prime}(t)/F(0))^{2}}{2(F(t)/F(0))}d\rho_{0}dt=\frac{\delta^{2}}{2}\int_{0}^{1}\frac{F^{\prime}(t)^{2}}{F(t)}dt.$
(88)

##  5 Constrained unbalanced optimal transport on Riemannian manifolds

In this section we will briefly outline how one can generalize the developed
theory to the manifold domain case, albeit under certain restrictive
assumptions. Let us denote by $M$ a Riemannian manifold of dimension $m$.
There are several specific issues with dealing with measures on $M$. A first
one is the definition of the transport measure $\omega$. Following the setting
of [17], we define a momentum field on $M$ as follows:

######  Definition 5.1 (Momentum field).

A momentum field on a Riemannian manifold $M$ is an element of the dual to
$\Gamma_{0}([0,1]\times M)$, the space of continuous functions $v:[0,1]\times
M\to TM$ such that $\pi\circ v(t,x)=x$ for any $t\in[0,1]$ and $x\in M$ where
$\pi:TM\to M$ is the usual projection. We will denote the space of momentum
fields by $\mathcal{V}([0,1]\times M)$.

This definition generalizes the concept of $\mathbb{R}^{n}$-valued vector
measures by taking the Riesz-Markov-Kakutani theorem [15, Theorem 7.17] as
motivation. Just as an Euclidean momentum field
$\omega\in\mathcal{M}([0,1]\times\Omega)^{n}$ is the dual of
$\mathbb{R}^{n}$-valued continuous maps by the aforementioned theorem, we use
the dual of continuous vector fields for the momentum field. This allows us to
extend the continuity equation to the Riemannian case:

######  Definition 5.2 (Riemannian continuity equation).

Let $\rho_{0},\rho_{1}\in\mathcal{M}^{+}(M)$. We say that a triple
$(\rho,\omega,\zeta)\in\mathcal{M}^{+}([0,1]\times
M)\times\mathcal{V}([0,1]\times M)\times\mathcal{M}([0,1]\times M)$ satisfies
the continuity equation $\partial_{t}\rho+\nabla\cdot\omega=\zeta$ with
boundary conditons $\rho_{0}$ and $\rho_{1}$ in the distributional sense if,
for any $\phi\in C^{1}([0,1]\times M)$,

$\int_{[0,1]\times M}\partial_{t}\phi
d\rho+\omega(\nabla_{x}\phi)+\int_{[0,1]\times M}\phi
d\zeta=\int_{M}\phi(1,\cdot)d\rho_{1}-\int_{M}\phi(0,\cdot)d\rho_{0}$ (89)

We also define the set of triples $(\rho,\omega,\zeta)$ satisfying the
continuity equation above as $\mathcal{CE}(\rho_{0},\rho_{1})$. It can be
shown, in the same way as in the Euclidean situation (Proposition 2.4), that
if $\rho$ satisfies the continuity equation, then $\rho$ can be disintegrated
in time with respect to the Lebesgue measure i.e. $\rho=dt\otimes\rho_{t}$. We
can thus extend the previous notation and again write
$\mathcal{CE}_{H,F}(\rho_{0},\rho_{1})$ for the set of all triples
$(\rho,\omega,\zeta)\in\mathcal{CE}(\rho_{0},\rho_{1})$ that further satisfy
the constraint $\int_{M}Hd\rho_{t}=F(t)$ for a.a. $t\in[0,1]$.

Given two measures $\rho_{0},\rho_{1}\in\mathcal{M}^{+}(M)$, we may then adapt
the previous definition to the manifold case and introduce the following
constrained unbalanced OT problem:

${\textrm{WFR}_{\delta}^{H,F}}(\rho_{0},\rho_{1})^{2}=\inf\left\\{\int_{[0,1]\times
M}f_{\delta,x}\left(\frac{d\mu}{d\lambda}(t,x)\right)d\lambda(t,x)\ \middle|\
\mu=(\rho,\omega,\zeta)\in\mathcal{CE}_{H,F}(\rho_{0},\rho_{1})(\rho_{0},\rho_{1})\right\\}$
(90)

where $\lambda$ is any nonnegative measure such that
$\rho,\omega,\zeta\ll\lambda$ and $f_{\delta,x}:\mathbb{R}\times
T_{x}M^{*}\times\mathbb{R}\rightarrow[0,+\infty]$ is given by:

$f_{\delta,x}(a,b,c)=\begin{cases}\frac{g_{x}^{-1}(b,b)+\delta^{2}c^{2}}{2a},&a>0\\\
0,&(a,b,c)=(0,0,0)\\\ +\infty,&\textrm{otherwise}\end{cases}$ (91)

where $g_{x}$ denotes the Riemannian metric tensor at $x\in M$ while its
inverse $g_{x}^{-1}$ is the cometric.

As one can see, although the formulation remains quite similar to the
Euclidean case, the cost function integrand’s dependency on the point $x$
itself in particular requires adapting the convex duality argument used to
prove the existence of minimizing paths. First, let us state and prove the
following lemma:

######  Lemma 5.3.

Let $x\in M$ and denote
$B_{\delta,x}=\left\\{(\alpha,\beta,\gamma)\in\mathbb{R}\times
T_{x}M\times\mathbb{R}\ |\
\alpha+\frac{1}{2}\left(g_{x}(\beta,\beta)+\frac{\gamma^{2}}{\delta^{2}}\right)\leq
0\right\\}$. The Fenchel conjugate of the convex indicator
$\iota_{B_{\delta,x}}$ is the function $f_{\delta,x}$ defined in (91).

######  Proof 5.4.

By definition of the Fenchel conjugate, we have:

$f_{\delta,x}(a,b,c)=\sup_{(\alpha,\beta,\gamma)}a\alpha+\langle
b,\beta\rangle+c\gamma-\iota_{B_{\delta,x}}(\alpha,\beta,\gamma)=\sup_{(\alpha,\beta,\gamma)\in
B_{\delta,x}}a\alpha+\langle b,\beta\rangle+c\gamma$

First, note that if $(a,b,c)=(0,0,0)$ then obviously $f_{\delta,x}(a,b,c)=0$.
If $a<0$, taking $(\alpha,\beta,\gamma)=(ta,0,0)$, and letting
$t\rightarrow+\infty$ shows that $f_{\delta,x}(a,b,c)=+\infty$. In addition,
when $a=0$ but $(b,c)\neq(0,0)$, it suffices to set
$(\beta,\gamma)=t(\bar{b},c)$, where $\bar{b}\in T_{x}M$ is such that $\langle
b,v\rangle=g_{x}(\bar{b},v)$ for any $v\in T_{x}M$, and
$\alpha=-\frac{1}{2}\left(g_{x}(t\bar{b},t\bar{b})+\frac{(tc)^{2}}{\delta^{2}}\right)$.
Then letting $t\rightarrow+\infty$ leads again to
$f_{\delta,x}(a,b,c)=+\infty$. Let us thus consider the last remaining case in
which $a>0$. For any $(\alpha,\beta,\gamma)\in B_{\delta,x}$, one has
$\alpha\leq-\frac{1}{2}\left(g_{x}(\beta,\beta)+\frac{\gamma^{2}}{\delta^{2}}\right)$
so that:

$a\alpha+\langle
b,\beta\rangle+c\gamma\leq-\frac{a}{2}\left(g_{x}(\beta,\beta)+\frac{\gamma^{2}}{\delta^{2}}\right)+\langle
b,\beta\rangle+c\gamma$

The maximum of the right hand side with respect to $\gamma$ is achieved at
$\hat{\gamma}=\delta^{2}\frac{c}{a}$ for which we find
$-\frac{a\hat{\gamma}^{2}}{2\delta^{2}}+c\hat{\gamma}=\frac{\delta^{2}c^{2}}{2a}$.
Similarly, the maximum value with respect to $\beta$ is achieved for
$\hat{\beta}=\frac{1}{a}\bar{b}$ in which case:

$-\frac{a}{2}g_{x}(\hat{\beta},\hat{\beta})+\langle
b,\beta\rangle=\frac{g_{x}(\bar{b},\bar{b})}{2a}=\frac{g_{x}^{-1}(b,b)}{2a}$

Combined with the above, we therefore obtain
$f_{\delta,x}(a,b,c)=\frac{g_{x}^{-1}(b,b)+\delta^{2}c^{2}}{2a}$.

Note that $f_{\delta,x}$ is also a 1-homogeneous function. We can now state a
generalized version of Theorem 4.3. In the following, the manifold $M$ is said
to be parallelizable if it is possible to find a set of $m$ continuous tangent
vector fields on $M$ written $\\{v_{1},\ldots,v_{m}\\}$ such that for all
$x\in M$, $\\{v_{1}(x),\ldots,v_{m}(x)\\}$ is a basis of $T_{x}M$.

######  Theorem 5.5 (Existence of minimizers on parallelizable manifolds).

Assume that $M$ is a parallelizable manifold and let
$\rho_{0},\rho_{1}\in\mathcal{M}^{+}(M)$ and $H:[0,1]\times
M\to\mathbb{R}^{d}$ be a continuous function,
$F=(F_{1},\ldots,F_{d}):[0,1]\to\mathbb{R}^{d}$ be a function with integrable
component functions $F_{i}$. If there exists a finite energy path in
$\mathcal{CE}_{H,F}(\rho_{0},\rho_{1})$, then there exists a minimizing path
to problem (90).

######  Proof 5.6.

The proof in part follows a similar path as that of Theorem 4.3 above, so we
will mainly highlight the steps which differ. We shall apply again the
Fenchel-Rockafellar theorem to the primal problem:
$\inf_{\tau=(\phi,\psi)}F(A\tau)+G(\tau)$ in which $\tau\in
X=C^{1}([0,1]\times M)\times C([0,1])^{d}$, $Y=C([0,1]\times
M)\times\Gamma^{0}([0,1]\times M)\times C([0,1]\times M)$ and $A$ is the
bounded linear operator $A:X\rightarrow Y$ defined by
$A(\phi,\psi)=(\partial_{t}\phi+\sum_{i=1}^{d}H_{i}\psi_{i},\nabla\phi,\phi)$.
The functional $G:X\rightarrow\mathbb{R}\cup\\{+\infty\\}$ is defined in the
same way as in the proof of Theorem 4.3 while
$F:Y\rightarrow\mathbb{R}\cup\\{+\infty\\}$ is given by:

$F(\alpha,\beta,\gamma)=\int_{[0,1]\times
M}\iota_{B_{\delta,x}}(\alpha(t,x),\beta(t,x),\gamma(t,x))dt\,d\text{vol}_{M}(x)$

with $\text{vol}_{M}$ being the Riemannian volume measure on $M$. Note that
both $G$ and $F$ are convex functionals and that, by definition, $Y^{*}$ can
be identified with the product of measure spaces $\mathcal{M}([0,1]\times
M)\times\mathcal{V}([0,1]\times M)\times\mathcal{M}([0,1]\times M)$. Fenchel-
Rockafellar theorem once again applies so that:

$\inf_{\tau\in X}\\{F(A\tau)+G(\tau)\\}=\sup_{\mu\in
Y^{*}}\\{-F^{*}(\mu)-G^{*}(-A^{*}\mu)\\}$

The term involving the convex conjugate of $G$ can be obtained in the same way
as previously and one gets that $G^{*}(-A^{*}\mu)$ is again the convex
indicator of the constraint set $\mathcal{CE}_{H,F}(\rho_{0},\rho_{1})$. An
extra difficulty compared to the Euclidean case is to recover the expression
of $F^{*}(\mu)$. We here rely on the parallalizability assumption on $M$ and
the existence of continuous tangent vector fields $\\{v_{1},\ldots,v_{m}\\}$
introduced above. This allows us to identify $\Gamma_{0}([0,1]\times M)$ with
$C([0,1]\times M)^{m}$. Indeed, for any $\beta\in\Gamma_{0}([0,1]\times M)$,
we may decompose $\beta(t,x)$ for $(t,x)\in[0,1]\times M$ in the basis
$\\{v_{1}(x),\ldots,v_{m}(x)\\}$ as
$\beta(t,x)=\sum_{i=1}^{m}\beta_{i}(t,x)v_{i}(x)$ and thus identify $\beta$
with $(\beta_{1},\ldots,\beta_{m})\in C([0,1]\times M)^{m}$. Under this
convention, $(\alpha,\beta,\gamma)$ maps $(t,x)\in[0,1]\times M$ to the fixed
Euclidean space $\mathbb{R}^{m+2}$ and [28, Theorem 5] still applies to the
integral functional $F$ in this case as well. Using Lemma 5.3 above together
with the fact that $f_{\delta,x}$ is 1-homogeneous (which implies that it is
equal to its recession function), we then get:

$F^{*}(\mu)=\int_{[0,1]\times
M}f_{\delta,x}\left(\frac{d\mu}{d\lambda}(t,x)\right)d\lambda(t,x).$

From this, it follows once again that the dual problem $\sup_{\mu\in
Y^{*}}\\{-F^{*}(\mu)-G^{*}(-A^{*}\mu)\\}=-\inf_{\mu\in
Y^{*}}\\{F^{*}(\mu)+G^{*}(-A^{*}\mu)\\}$ is equivalent to the infimum of the
original problem (90) which allows to conclude the proof.

It is worth pointing out that the assumption made on $M$ in Theorem 5.5
includes manifolds such as $S^{1}$, $S^{3}$ and higher-dimensional tori as
well as manifolds that are the image of a single parametrization function on
an open domain of an Euclidean space. However, it does not cover several
important cases, including the 2-sphere or non-orientable manifolds. We
suspect however that this could be extended to general compact manifolds based
on a careful refinement of the results from [28] that we used in the proof.

Regarding the existence of a finite energy path between two given measures
$\rho_{0}$ and $\rho_{1}$ in $\mathcal{M}^{+}(M)$, adapting the proofs of
Theorems 3.5, 4.7 and 4.8 to the manifold case is rather straightforward as
most of the arguments do not rely on the particular Euclidean domain structure
of $\Omega$. The statements of these theorems extend mutatis mutandis by
replacing $\Omega$ with a compact Riemannian manifold $M$ and replacing the
convexity assumption on $\Omega$ (when needed) with $M$ being geodesically
complete.

######  Example 5.7 (Area measures of convex domains).

An example of measure space which naturally fits the manifold UOT framework of
this section is the case of the area measures of convex domains in
$\mathbb{R}^{d}$ mentioned earlier in the introduction, which was also
previously considered by some of the authors in [8] and [4]. Although we shall
leave aside the technical details related to the definition and properties of
area measures for the sake of concision, let us still emphasize the connection
with the present work. To any oriented convex domain $Q$ in $\mathbb{R}^{d}$,
one can associate its area measure $\rho_{Q}$ which is the positive Radon
measure on the unit sphere $S^{d-1}$ (viewed as a submanifold of
$\mathbb{R}^{d}$) such that for any Borel set $B$, $\rho_{Q}(B)$ measures the
$(d-1)$-area of the portion of the boundary $\partial Q$ where the unit
outward normal $\vec{n}(x)\in B$. A central result in convex geometry is that
the area measure characterizes a convex set up to translation [14]. In fact,
$Q\mapsto\rho_{Q}$ induces a bijective correpondence between convex sets
modulo translations and the space of all measures $\mathcal{M}^{+}(S^{d-1})$
that satisfy the “closure” contraint
$\int_{S^{d-1}}xd\rho(x)=0\in\mathbb{R}^{d}$. By defining $H:x\in
S^{d-1}\mapsto x\in\mathbb{R}^{d}$, the latter space is precisely
$\mathcal{M}^{+}_{H,0}(S^{d-1})$ in the setting of this paper. Thus, by
pulling back the metric ${\textrm{WFR}_{\delta}^{H,0}}$ on
$\mathcal{M}^{+}_{H,0}(S^{d-1})$, one can in turn view the space of all convex
shapes as a geodesic space with some potentially interesting characteristics
of the associated metric and the geodesics, which were hinted in [8] in the
case $d=2$. While our present results mainly address the question of existence
of geodesics, we leave a more in depth study of the properties of such metrics
to future work.

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