# The dynamical Schrödinger problem in abstract metric spaces

Léonard Monsaingeon GFM Universidade de Lisboa, Campo Grande, Edifício C6,
1749-016 Lisboa, Portugal and IECL Université de Lorraine, F-54506 Vandoeuvre-
lès-Nancy Cedex, France. email<EMAIL_ADDRESS>Luca
Tamanini CEREMADE (UMR CNRS 7534), Université Paris Dauphine PSL, Place du
Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France and INRIA-Paris,
MOKAPLAN, 2 Rue Simone Iff, 75012, Paris, France. email:
<EMAIL_ADDRESS>Dmitry Vorotnikov University of Coimbra, CMUC,
Department of Mathematics, 3001-501 Coimbra, Portugal. email:
<EMAIL_ADDRESS>
###### Abstract

In this paper we introduce the dynamical Schrödinger problem on abstract
metric spaces, defined for a wide class of entropy and Fisher information
functionals. Under very mild assumptions we prove a generic Gamma-convergence
result towards the geodesic problem as the noise parameter
$\varepsilon\downarrow 0$. We also investigate the connection with geodesic
convexity of the driving entropy, and study the dependence of the entropic
cost on the parameter $\varepsilon$. Some examples and applications are
discussed.

MSC [2020] 49J45, 49Q20, 58B20.
Keywords: Benamou-Brenier formulation, Schrödinger bridge, Fisher information,
Gamma-convergence, geodesic convexity, gradient flows, optimal transport

###### Contents

  1. 1 Introduction
  2. 2 Heuristics
    1. 2.1 Convergence of the Schrödinger problem
    2. 2.2 Quantifying the convexity
  3. 3 In metric spaces
    1. 3.1 Preliminaries and setting
    2. 3.2 A pseudo-Riemannian computation
  4. 4 Small-temperature limit and convexity
    1. 4.1 $\Gamma$-convergence of the Schrödinger problem
    2. 4.2 Displacement convexity
  5. 5 Derivative of the cost
  6. 6 Examples
    1. 6.1 The Boltzmann-Shannon relative entropy
    2. 6.2 Internal energies and the Rényi entropy
    3. 6.3 Mean-field Schrödinger problem
    4. 6.4 Non-linear mobilities
    5. 6.5 Hadamard spaces

##  1 Introduction

Gaspard Monge and Erwin Schrödinger came up with two a priori unrelated
problems that are concerned with finding a preeminent way of deforming a
prescribed probability distribution into another one. While Monge was
interested in optimizing the cost of transportation of goods [62, 63, 58],
Schrödinger’s original thought experiment [59, 60] aimed for finding the most
likely evolution between two subsequent observations of a cloud of independent
particles. So, even if in both cases we are facing an interpolation and
optimization problem, the former is deterministic in nature whereas the latter
is strongly related to large deviations theory, and is, at the first glance,
purely stochastic. We refer to a recent survey [22] for various formulations
and aspects of the Schrödinger problem, and to [64, 65] for a discussion of
its role in Euclidean Quantum Mechanics.

Anyway, several analogies and connections exist between the two problems. They
can be appreciated by looking carefully at the interpolation aspects of both
problems, namely at their dynamical formulations and the underlying equations
governing the respective evolutions. In the case of a quadratic transportation
cost over a Riemannian manifold $M$, the Monge-Kantorovich optimal transport
problem is solved (at least in a weak sense) by interpolating between the
source and the target distributions with a constant-speed, length-minimizing
geodesic in the Otto-Wasserstein space of probability measures
$\mathcal{P}_{2}(M)$. This gives a curve
$(\mu_{t})_{t\in[0,1]}\subset\mathcal{P}_{2}(M)$ which formally satisfies (in
the sense of the celebrated Otto calculus [55, 62, 63])

$\nabla_{\dot{\mu}_{t}}\dot{\mu}_{t}=0,$ (1.1)

where $\nabla_{\dot{\mu}_{t}}$ is the covariant derivative along the curve
$t\mapsto\mu_{t}$. The Schrödinger problem with parameter $\varepsilon$ (from
a physical viewpoint, $\varepsilon$ can be seen as a temperature or level of
noise) can also be translated into such a geometric language. By analogy, when
looking at the covariant derivative along the optimal evolution
$(\mu_{t}^{\varepsilon})_{t\in[0,1]}$, usually called Schrödinger bridge or
entropic interpolation, the resulting equation is surprising and can be viewed
[23] as Newton’s second law

$\nabla_{\dot{\mu}_{t}^{\varepsilon}}\dot{\mu}_{t}^{\varepsilon}=\frac{\varepsilon^{2}}{8}\nabla
I(\mu_{t}^{\varepsilon}),$ (1.2)

where in the right-hand side $\nabla$ denotes the gradient in the Otto-
Wasserstein pseudo-Riemannian sense and $I$ is the Fisher information

$I(\mu)=4\int_{M}|\nabla\sqrt{\rho}|^{2}\,{\mathrm{d}}\mathrm{vol}=\int_{M}|\nabla\log\rho|^{2}\rho\,{\mathrm{d}}\mathrm{vol}$

provided $\mu=\rho\cdot\mathrm{vol}$. A related observation is that the
(scaled) heat flow, coinciding with the (scaled) gradient flow of the
Boltzmann-Shannon entropy [40, 63]

$H(\mu)=\int_{M}\rho\log\rho\,{\mathrm{d}}\mathrm{vol}$

for $\mu=\rho\cdot\mathrm{vol}$, is also a solution to (1.2): a simple
differentiation in time of $\dot{\mu}_{t}=-\frac{\varepsilon}{2}\nabla
H(\mu_{t})$ and the fact that $I=|\nabla H|^{2}$ in the Otto-Wasserstein sense
automatically yield

$\nabla_{\dot{\mu}_{t}}\dot{\mu}_{t}=\frac{\varepsilon}{2}\nabla^{2}H(\mu_{t})\cdot\frac{\varepsilon}{2}\nabla
H(\mu_{t})=\frac{\varepsilon^{2}}{8}\nabla|\nabla
H(\mu_{t})|^{2}=\frac{\varepsilon^{2}}{8}\nabla I(\mu_{t}).$

This shows that the Schrödinger problem lies between optimal transport and
diffusion and is naturally intertwined with both deterministic behaviour and
Brownian motion. It shares the same Newton’s law as the gradient flow of the
entropy, but unlike the heat flow it has a prescribed final configuration to
match: it is up to the parameter $\varepsilon$ to tip the balance in favour of
deterministic transport or diffusion. With this heuristics in mind, we see
that as $\varepsilon\to 0$ the applied force $\varepsilon^{2}\nabla
I(\mu_{t}^{\varepsilon})$ in (1.2) vanishes, so that the Schrödinger problem
may be interpreted as a noisy (entropic) counterpart of the Monge-Kantorovich
optimal transport, corresponding to the unforced geodesic evolution (1.1)
discussed above. This informal relationship has a rigorous counterpart, which
dates back to the pioneering works on the asymptotic behavior of the
Schrödinger problem as $\varepsilon\to 0$ of T. Mikami, M. Thieullen [50, 51],
and C. Léonard [44, 45]. This was subsequently developed in [19, 8, 38]. Very
recently [47, 30], similar small-noise results were obtained for static Monge-
Kantorovich problems regularized with more general entropies.

This first connection can be investigated further and by doing so one can
remark that (1.2) is exactly the Euler-Lagrange optimality equation for the
dynamical Benamou-Brenier formulation of the Schrödinger problem [18, 22, 45,
39], which consists in minimizing the Lagrangian kinetic action perturbed by
the Fisher information: In more precise terms,

$\inf\bigg{\\{}\frac{1}{2}\int_{0}^{1}\int_{M}|v_{t}|^{2}\,{\mathrm{d}}\mu_{t}{\mathrm{d}}t+\frac{\varepsilon^{2}}{8}\int_{0}^{1}I(\mu_{t})\,{\mathrm{d}}t\bigg{\\}},$
(1.3)

where the infimum runs over all solutions of the continuity equation

$\partial\mu_{t}+\mathrm{div}(v_{t}\mu_{t})=0$

with prescribed initial and final densities. Also from this variational
standpoint the reader can see that as $\varepsilon\to 0$ the Schrödinger
problem formally reduces to

$\inf\frac{1}{2}\int_{0}^{1}\int_{M}|v_{t}|^{2}\,{\mathrm{d}}\mu_{t}{\mathrm{d}}t,$
(1.4)

namely to the dynamical Benamou-Brenier formulation of the (quadratic) optimal
transport problem [9]. This variational representation depicts in a way
clearer than (1.2) the double nature of the Schrödinger problem, the
competition between the determinism encoded in the kinetic energy and the
unpredictability coming from the Fisher information, and the role played by
$\varepsilon$ in balancing these two opposite behaviours.

The double bond of the Schrödinger problem with optimal transport on the one
hand and heat flow on the other hand results in fruitful and wide-ranging
applications of both theoretical and applied interest. Indeed, from the
connection with the heat flow the solutions to the Schrödinger problem gain
regularity properties which are not available in optimal transport, and thanks
to the asymptotic behaviour of the Schrödinger problem as $\varepsilon\to 0$
entropic interpolations represent an efficient way to approximate Wasserstein
geodesics with second-order accuracy [38, 25]. This approach has already
turned out to be successful in conjunction with functional inequalities [24,
34] and differential calculus along Wasserstein geodesics [38]. But the nice
behaviour of Schrödinger bridges is important also for computational purposes.
The impact of Schrödinger problem and Sinkhorn algorithms (deeply related to
the static formulation of the former) on the numerical methods used in optimal
transportation theory has been impressive, as witnessed by several recent
works (see [57] and references therein as well as [26, 10, 12, 13, 11, 29]).

As a matter of fact neither the particular structure of the Wasserstein space
nor the specific choice of the Boltzmann-Shannon functional are required to
define the two problems in question (cf. a related discussion in the heuristic
paper [43]): one can of course define length-minimizing geodesics in any
metric space $({\rm X},{\sf d})$, and the Schrödinger problem (or at least its
Benamou-Brenier formulation described above) merely involves an entropy
functional and a corresponding Fisher information. Given such a reasonable
entropy functional ${\sf E}$ on ${\rm X}$ that generates a gradient flow in a
suitable sense, the corresponding Fisher information is expected to be nothing
but the dissipation rate of ${\sf E}$ (along solutions of its own gradient
flow), just as $I$ coincides with the rate of dissipation of the entropy $H$
along the heat flow. This observation is the starting point of the present
paper, where we intend to study the abstract Schrödinger bridge problem or, in
other words, the entropic approximation of geodesics in metric spaces.

Under very mild assumptions on ${\rm X}$ and ${\sf E}$, we will prove the
solvability of the abstract $\varepsilon$-Schrödinger problem and the
$\Gamma$-convergence to the corresponding geodesic problem as $\varepsilon\to
0$. We will also rigorously justify, in the metric setting, that any
trajectory of a gradient flow solves an associated Schrödinger problem.
Leveraging a quantitative $AC^{2}$ estimate based on a straightforward chain-
rule in the smooth Riemannian setting, the cornerstone of our analysis will be
the systematic construction of an $\varepsilon$-regularized entropic copy
$(\gamma^{\varepsilon}_{t})_{t\in[0,1]}$ of any arbitrary curve
$(\gamma_{t})_{t\in[0,1]}$. These perturbed curves will provide recovery
sequences for the $\Gamma$-convergence. Our construction is completely
Eulerian and essentially consists in running the ${\sf E}$-gradient flow for a
short time $h_{\varepsilon}(t)$ starting at $\gamma_{t}$ for all $t$, for
well-chosen functions $h_{\varepsilon}\geq 0$. The challenge here will be to
reproduce the (formal, differential) Riemannian chain-rule in metric spaces.
Notably, an analogous pseudo-Riemannian idea has recently been used by A.
Baradat and some of the authors [8, 53] in order to prove the
$\Gamma$-convergence for the classical dynamical Schrödinger problem on the
Otto-Wasserstein space and for its counterpart on the non-commutative Fisher-
Rao space, respectively. However, in those papers the computations were ad hoc
and heavily exploited the underlying structures of the particular spaces as
well as the properties of the particular flows (namely, of the classical heat
flow and of its restriction to multivariate Gaussians), whereas here we derive
everything from the existence of an abstract gradient flow on ${\rm X}$ driven
by ${\sf E}$.

Another notion of paramount importance herein will be convexity. In the smooth
Riemannian setting, and given $\lambda\in\mathbb{R}$, elementary calculus
shows that the $\lambda$-convexity of ${\sf E}$ along geodesics is of course
equivalent to a uniform lower bound $\mathrm{Hess}\,{\sf
E}(x)\geq\lambda\operatorname{Id}$ as quadratic forms in the tangent space,
but also more importantly to the $\lambda$-contractivity of the ${\sf
E}$-gradient flow. In the metric setting no second order calculus is available
in general, and the very notion of gradient flow as well as its connection
with geodesic convexity and contractivity become much more subtle. The key
notion of gradient flow that we shall use throughout is that of _Evolution
Variational Inequality_ , or ${\rm EVI}_{\lambda}$ flow [2]. Under reasonable
assumptions it is well known that (a suitable variant of) convexity of ${\sf
E}$ generally provides existence of an ${\rm EVI}_{\lambda}$-flow starting at
any $x\in{\rm X}$, see [2]. A natural question to ask is whether the converse
also holds true, i.e. whether well-posedness of a reasonable gradient flow
implies some convexity. This was proved in [15] for the specific case of the
Euclidean Wasserstein space ${\rm X}=W_{2}(\Omega)$,
$\Omega\subset\mathbb{R}^{d}$, and at least for the so-called internal
energies, and it is shown therein that $0$-contractivity of the gradient flow
(or equivalently, of the associated nonlinear diffusion equation) implies
$0$-displacement convexity in the sense of McCann [49]. In the same spirit,
and building up on Otto and Westdickenberg [56], Daneri and Savaré proved in a
very general metric setting that the generation of an ${\rm
EVI}_{\lambda}$-flow indeed implies $\lambda$-geodesic convexity [27, Theorem
3.2]. A byproduct of our analysis for the $\Gamma$-convergence will give a new
independent proof of this latter fact by a completely different approach,
essentially by constructing an $\varepsilon$-entropic regularization of
geodesics and carefully examining the defect of optimality at order one in
$\varepsilon\to 0$.

As a main application of the $\Gamma$-convergence of the Schrödinger problem
to the geodesic problem as $\varepsilon\to 0$ (and more generally of the
$\varepsilon^{\prime}$-Schrödinger problem to the $\varepsilon$-one as
$\varepsilon^{\prime}\to\varepsilon$) we investigate the behaviour of the
optimal value of the dynamical Schrödinger problem, henceforth called
_entropic cost_ , as a function of the temperature parameter $\varepsilon$,
with particular emphasis on the regularity and the small-noise regime. For the
classical dynamical Schrödinger problem (1.3), it has recently been proved by
the second author with G. Conforti [25] that the entropic cost is of class
$C^{1}((0,\infty))\cap C([0,\infty))$ (actually $C^{1}([0,\infty))$ under
suitable assumptions) and twice a.e. differentiable; once this regularity
information is available, the formula for the first derivative is rather easy
to guess, as by the envelope theorem it coincides with the partial derivative
w.r.t. $\varepsilon$ of the functional in (1.3) evaluated at any critical
point. Denoting by $\mathscr{C}_{\varepsilon}(\mu,\nu)$ the value in (1.3)
with marginal constraints $\mu$ and $\nu$ and by
$(\mu_{t}^{\varepsilon})_{t\in[0,1]}$ the associated Schrödinger bridge, this
statement reads as

$\frac{{\mathrm{d}}}{{\mathrm{d}}\varepsilon}\mathscr{C}_{\varepsilon}(\mu,\nu)=\frac{\varepsilon}{4}\int_{0}^{1}I(\mu_{t}^{\varepsilon})\,{\mathrm{d}}t,\qquad\forall\varepsilon>0$

and in [25] this identity played an important role in the study of both the
large- and small-noise behaviour of the Schrödinger problem, obtaining in
particular a Taylor expansion around $\varepsilon=0$ with
$o(\varepsilon^{2})$-accuracy. Since the central object in the present paper
is an abstract and general formulation of (1.3), an analogous result is
expected to hold. However, from a technical viewpoint the proof is much more
subtle and challenging, because unlike (1.3) our metric version of the
dynamical Schrödinger problem may have multiple solutions. For this reason the
discussion about the regularity of the entropic cost in this paper is less
concise than in [25]. Nonetheless, we are still able to deduce the same kind
of Taylor expansion with the same accuracy. Given the previous interpretation
of the Schrödinger problem as a noisy Monge-Kantorovich problem and the
importance of quantitative estimates in approximating optimal transport by
means of the Schrödinger problem, it is reasonable to expect that such a
Taylor expansion (valid in a general framework for a wide choice of
functionals ${\sf E}$) will fit to a countless variety of examples, some of
which will be discussed here.

The paper is organized as follows: In Section 2 we give a short and formal
proof of our fundamental $AC^{2}$ estimate in the smooth Riemannian setting,
and show how it can be exploited to establish $\Gamma$-convergence and
convexity. Section 3 fixes the metric framework in which we work for the rest
of the paper, and extends the previous estimate to this metric setting. In
Section 4 we prove the $\Gamma$-convergence as $\varepsilon\downarrow 0$ and
establish the geodesic convexity. Section 5 studies the dependence of the
optimal entropic cost on the temperature parameter $\varepsilon>0$, and
provides a second order expansion. Finally, we list in Section 6 several
examples and applications covered by our abstract results.

##  2 Heuristics

Here we remain formal and the computations are carried in a Riemannian
setting, where classical calculus and chain-rules are available. (Significant
work will be required later on to adapt the computations in metric spaces.)
All the objects and functions in this section are therefore considered to be
smooth, and we deliberately ignore any regularity issue.

Let $M$ be a Riemannian manifold with scalar product $\langle.,.\rangle_{q}$
at a point $q\in M$ and induced Riemannian distance $d$, and let
$V:M\to\mathbb{R}$ be a given potential. For simplicity we assume here that
$V$ is globally bounded from below on $M$, and up to replacing $V$ by $V-\min
V$ we can assume that $V(q)\geq 0$. (In section 3 we will relax this
assumption and allow $V$ to be only _locally bounded_.) Given a small
temperature parameter $\varepsilon>0$, and following [43], the (dynamic)
_geometric Schrödinger problem_ consists in solving the optimization problem

$\frac{1}{2}\int_{0}^{1}\left|\frac{{\mathrm{d}}q_{t}}{{\mathrm{d}}t}\right|^{2}\,{{\mathrm{d}}t}+\frac{\varepsilon^{2}}{2}\int_{0}^{1}|\nabla
V|^{2}(q_{t})\,{{\mathrm{d}}t}\qquad\longrightarrow\qquad\min;\\\ \mbox{s.t.
}q\in C([0,1],M)\mbox{ with endpoints }q_{0},q_{1}.$ (2.1)

For $s\geq 0$ we denote by $\Phi(s,q_{0})$ the semi-flow corresponding to the
autonomous $V$-gradient flow started from $q_{0}\in M$,

$\left\\{\begin{array}[]{l}\displaystyle{\frac{{\mathrm{d}}}{{\mathrm{d}}s}\Phi(s,q_{0})=-\nabla
V(\Phi(s,q_{0}))},\\\ \Phi(0,q_{0})=q_{0}.\end{array}\right.$

The goal of this section is to give a straightforward proof of the following
two facts, assuming that the potential $V$ is well behaved:

  1. (i)

the $\varepsilon$-Schrödinger problem converges to the geodesic problem as
$\varepsilon\to 0$;

  2. (ii)

$\lambda$-contractivity of the generated flow $\Phi$ can be turned into
$\lambda$-convexity along geodesics.

With this goal in mind, fix any two endpoints $q_{0},q_{1}\in M$ and take an
arbitrary curve joining them

$q\in C([0,1],M),\hskip 42.67912ptq|_{t=0}=q_{0}\quad\mbox{and}\quad
q|_{t=1}=q_{1}.$

For any function $h(t)\geq 0$ with $h(0)=h(1)=0$, we perturb $q$ by defining

$\tilde{q}_{t}:=\Phi(h(t),q_{t}),\hskip 42.67912ptt\in[0,1]$

i.e. $\tilde{q}_{t}$ is the solution of the $V$-gradient flow at time
$s=h(t)\geq 0$ starting from $q_{t}$ at time $s=0$. We shall refer to
$t\in[0,1]$ as a “horizontal time” and to $s\in[0,h(t)]$ as a “vertical time”,
see Figure 1. Later on we will think of the curve $\tilde{q}$ as a
“regularized” version of $q$.

$q_{0}$$q_{1}$$\Phi(s,q_{t})$$t$

$s$

$\tilde{q}_{t}$$q_{t}$ Figure 1: The perturbed curve

Note that the endpoints remain invariant, $\tilde{q}_{0}=q_{0}$ and
$\tilde{q}_{1}=q_{1}$. Since by definition of the flow
$\partial_{s}\Phi(s,q_{t})=-\nabla V(\Phi(s,q_{t}))$, the speed of the
perturbed curve can be computed as

$\begin{split}\frac{{\mathrm{d}}\tilde{q}_{t}}{{\mathrm{d}}t}&={\frac{{\mathrm{d}}}{{{\mathrm{d}}t}}}\Big{(}\Phi(h(t),q_{t})\Big{)}=\partial_{s}\Phi(h(t),q_{t})h^{\prime}(t)+\partial_{q}\Phi(h(t),q_{t})\frac{{\mathrm{d}}q_{t}}{{\mathrm{d}}t}\\\
&=-h^{\prime}(t)\nabla
V(\tilde{q}_{t})+\partial_{q}\Phi(h(t),q_{t})\frac{{\mathrm{d}}q_{t}}{{\mathrm{d}}t}.\end{split}$

Bringing the $h^{\prime}(t)$ term to the left-hand side and taking the half
squared norm (in the tangent space $T_{\tilde{q}_{t}}M$) gives

$\frac{1}{2}\left|\frac{{\mathrm{d}}\tilde{q}_{t}}{{\mathrm{d}}t}\right|^{2}+\frac{1}{2}|h^{\prime}(t)|^{2}|\nabla
V(\tilde{q}_{t})|^{2}+h^{\prime}(t)\underbrace{\langle\nabla
V(\tilde{q}_{t}),\frac{{\mathrm{d}}\tilde{q}_{t}}{{\mathrm{d}}t}\rangle_{\tilde{q}_{t}}}_{={\frac{{\mathrm{d}}}{{{\mathrm{d}}t}}}V(\tilde{q}_{t})}=\frac{1}{2}\left|\partial_{q}\Phi(h(t),q_{t})\frac{{\mathrm{d}}q_{t}}{{\mathrm{d}}t}\right|^{2}.$
(2.2)

Assume now that, for whatever reason, the gradient flow satisfies the
following quantified contractivity estimate w.r.t. the Riemannian distance $d$

$d(\Phi(s,p_{0}),\Phi(s,p_{0}^{\prime}))\leq e^{-\lambda
s}d(p_{0},p_{0}^{\prime}),\hskip 28.45274pt\forall s\geq
0,\,p_{0},p_{0}^{\prime}\in M$ (2.3)

for some fixed $\lambda\in\mathbb{R}$. Then it is easy to check that the
linear map $v\mapsto\partial_{q}\Phi(s,p)\cdot v$ (from $T_{p}M$ to
$T_{\Phi(s,p)}M$) has norm less than $e^{-\lambda s}$, and therefore (2.2)
gives

$\frac{1}{2}\left|\frac{{\mathrm{d}}\tilde{q}_{t}}{{\mathrm{d}}t}\right|^{2}+\frac{1}{2}|h^{\prime}(t)|^{2}|\nabla
V(\tilde{q}_{t})|^{2}+h^{\prime}(t){\frac{{\mathrm{d}}}{{{\mathrm{d}}t}}}V(\tilde{q}_{t})\leq\frac{1}{2}e^{-2\lambda
h(t)}\left|\frac{{\mathrm{d}}q_{t}}{{\mathrm{d}}t}\right|^{2}.$ (2.4)

Integration by parts yields next

$\begin{split}\frac{1}{2}\int_{0}^{1}\left|\frac{{\mathrm{d}}\tilde{q}_{t}}{{\mathrm{d}}t}\right|^{2}\,{{\mathrm{d}}t}&+\frac{1}{2}\int_{0}^{1}|h^{\prime}(t)|^{2}|\nabla
V(\tilde{q}_{t})|^{2}\,{{\mathrm{d}}t}-\int_{0}^{1}h^{\prime\prime}(t)V(\tilde{q}_{t})\,{{\mathrm{d}}t}\\\
&\leq\frac{1}{2}\int_{0}^{1}e^{-2\lambda
h(t)}\left|\frac{{\mathrm{d}}q_{t}}{{\mathrm{d}}t}\right|^{2}{{\mathrm{d}}t}+\Big{(}h^{\prime}(0)V(q_{0})-h^{\prime}(1)V(q_{1})\Big{)},\end{split}$
(2.5)

where the invariance $\tilde{q}_{0}=q_{0}$, $\tilde{q}_{1}=q_{1}$ was used in
the last boundary terms. This fundamental estimate gives a quantified bound on
the kinetic energy (namely the $L^{2}$ speed) of $\tilde{q}$ in terms of that
of the original curve $q$, and will be the cornerstone of the whole analysis.

Both the convexity and the convergence of the Schrödinger problem will
actually follow by setting $h(t)=\varepsilon H(t)$ for suitable choices of
$H(t)\geq 0$, and then letting $\varepsilon\downarrow 0$. Note that in this
case we have $h(t)=\varepsilon H(t)\downarrow 0$ uniformly, hence the
perturbed curve

$q^{\varepsilon}_{t}:=\Phi(\varepsilon H(t),q_{t})$ (2.6)

will converge uniformly to $q$ as $\varepsilon\downarrow 0$ too.

###  2.1 Convergence of the Schrödinger problem

A first use of (2.5) will be crucial in proving the $\Gamma$-convergence of
the Schrödinger functional

$\mathcal{A}_{\varepsilon}(q):=\frac{1}{2}\int_{0}^{1}\left|\frac{{\mathrm{d}}q_{t}}{{\mathrm{d}}t}\right|^{2}\,{{\mathrm{d}}t}+\frac{\varepsilon^{2}}{2}\int_{0}^{1}|\nabla
V(q_{t})|^{2}\,{{\mathrm{d}}t}$

towards the kinetic action

$\mathcal{A}(q):=\frac{1}{2}\int_{0}^{1}\left|\frac{{\mathrm{d}}q_{t}}{{\mathrm{d}}t}\right|^{2}\,{{\mathrm{d}}t}$

as $\varepsilon\downarrow 0$.

######  Theorem 2.1 (formal $\Gamma$-limit).

For any $q_{0},q_{1}\in M$ it holds

$\mathcal{A}=\Gamma-\lim\limits_{\varepsilon\to 0}\mathcal{A}_{\varepsilon}$

for the uniform convergence on the space of curves with fixed endpoints
$q_{0},q_{1}$.

###### Proof.

We check separately the $\Gamma-\liminf$ and the $\Gamma-\limsup$ properties.
As for the former, given any curve $q$ joining $q_{0},q_{1}$ and any
$q^{\varepsilon}\to q$ uniformly, since the kinetic energy functional
$q\mapsto\mathcal{A}(q)$ is always lower semicontinuous for the uniform
convergence we get first

$\mathcal{A}(q)\leq\liminf_{\varepsilon\downarrow
0}\mathcal{A}(q^{\varepsilon})\leq\liminf_{\varepsilon\downarrow
0}\mathcal{A}_{\varepsilon}(q^{\varepsilon}).$

For the $\Gamma-\limsup$, let $H(t)=\min\\{t,1-t\\}$ be the hat function
centered at $t=1/2$ with height $1/2$ and vanishing at the boundaries, set
$h(t)=\varepsilon H(t)$, and let $q^{\varepsilon}$ be the regularized curve
constructed in (2.6). In this simple smooth setting it is not difficult to
check that $q^{\varepsilon}\to q$ uniformly. Moreover, our choice of $h(t)$
results in $|h^{\prime}(t)|^{2}=\varepsilon^{2}$ with
$h^{\prime}(0)=\varepsilon$, $h^{\prime}(1)=-\varepsilon$, and
$h^{\prime\prime}(t)=-2\varepsilon\delta_{1/2}(t)$ in the distributional
sense. Therefore (2.5) gives immediately

$\begin{split}\mathcal{A}_{\varepsilon}(q^{\varepsilon})+2\varepsilon
V(q^{\varepsilon}_{1/2})&=\frac{1}{2}\int_{0}^{1}\left|\frac{{\mathrm{d}}q^{\varepsilon}_{t}}{{\mathrm{d}}t}\right|^{2}\,{{\mathrm{d}}t}+\frac{\varepsilon^{2}}{2}\int_{0}^{1}|\nabla
V(q^{\varepsilon}_{t})|^{2}\,{{\mathrm{d}}t}+2\varepsilon
V(q^{\varepsilon}_{1/2})\\\
&\leq\frac{1}{2}\int_{0}^{1}e^{-2\varepsilon\lambda
H(t)}\left|\frac{{\mathrm{d}}q_{t}}{{\mathrm{d}}t}\right|^{2}\,{{\mathrm{d}}t}+\varepsilon\Big{(}V(q_{0})+V(q_{1})\Big{)}.\end{split}$

The singularity of $h^{\prime\prime}$ at $t=1/2$ can be easily and rigorously
worked around, simply integrating by parts (2.2) separately on each interval
$t\in[0,1/2]$ and $t\in[1/2,1]$ and keeping track of the boundary terms
resulting ultimately in the above $2\varepsilon V(q^{\varepsilon}_{1/2})\geq
0$. Discarding this latter non-negative term finally gives

$\limsup_{\varepsilon\downarrow
0}\mathcal{A}_{\varepsilon}(q^{\varepsilon})\leq\limsup_{\varepsilon\downarrow
0}\left\\{\frac{1}{2}\int_{0}^{1}e^{-2\varepsilon\lambda
H(t)}\left|\frac{{\mathrm{d}}q_{t}}{{\mathrm{d}}t}\right|^{2}\,{{\mathrm{d}}t}+\varepsilon\Big{(}V(q_{0})+V(q_{1})\Big{)}\right\\}=\mathcal{A}(q)$

and concludes the proof. ∎

###  2.2 Quantifying the convexity

The second consequence of our fundamental estimate (2.5) is the quantification
of the convexity of the potential $V$ in terms of the quantified contractivity
(2.3). The point here is that the result can be obtained directly from (2.2),
which can be established in a purely metric setting without relying on
differential calculus (see the next section for details).

######  Theorem 2.2.

Assume that $V$ satisfies 2.3. Then $V$ is $\lambda$-geodesically convex, i.e.

$V(q_{\theta})\leq(1-\theta)V(q_{0})+\theta
V(q_{1})-\frac{\lambda}{2}\theta(1-\theta)d^{2}(q_{0},q_{1}),\qquad\theta\in(0,1)$

for _any_ geodesic $(q_{\theta})_{\theta\in[0,1]}$ in $M$.

###### Proof.

Let $(q_{t})_{t\in[0,1]}$ be an arbitrary geodesic with endpoints
$q_{0},q_{1}$. For fixed $\theta\in(0,1)$ let

$H_{\theta}(t):=\begin{cases}\displaystyle{\frac{1}{\theta}}t&\mbox{if
}t\in[0,\theta],\\\ -\displaystyle{\frac{1}{1-\theta}(t-1)}&\mbox{if
}t\in[\theta,1],\end{cases}$

be the hat function centered at $t=\theta$ with height $1$ and vanishing at
$t=0,1$, and for any $\varepsilon>0$ let $q^{\varepsilon}$ be the regularized
curve constructed in (2.6) with $h(t)=\varepsilon H_{\theta}(t)$. Note
moreover that

$h^{\prime}(0)=\frac{\varepsilon}{\theta},\qquad
h^{\prime}(1)=-\frac{\varepsilon}{1-\theta},\qquad
h^{\prime\prime}(t)=-\varepsilon\left(\frac{1}{\theta}+\frac{1}{1-\theta}\right)\delta_{\theta}(t)$

in the distributional sense. Discarding the non-negative term
$|h^{\prime}(t)|^{2}|\nabla V(\tilde{q}_{t})|^{2}$ in (2.5), the optimality of
the geodesic $q$ from $q_{0}$ to $q_{1}$ gives

$\begin{split}0&\leq\frac{1}{2}\int_{0}^{1}\left|\frac{{\mathrm{d}}q^{\varepsilon}_{t}}{{\mathrm{d}}t}\right|^{2}\,{{\mathrm{d}}t}-\frac{1}{2}\int_{0}^{1}\left|\frac{{\mathrm{d}}q_{t}}{{\mathrm{d}}t}\right|^{2}\,{{\mathrm{d}}t}\\\
&\overset{\eqref{eq:fundamental}}{\leq}\int_{0}^{1}h^{\prime\prime}(t)V(q^{\varepsilon}_{t})\,{{\mathrm{d}}t}+\frac{1}{2}\int_{0}^{1}\left(e^{-2\lambda
h(t)}-1\right)\left|\frac{{\mathrm{d}}q_{t}}{{\mathrm{d}}t}\right|^{2}\,{{\mathrm{d}}t}+\Big{(}h^{\prime}(0)V(q_{0})-h^{\prime}(1)V(q_{1})\Big{)}\\\
&=-\varepsilon\left(\frac{1}{\theta}+\frac{1}{1-\theta}\right)V(q^{\varepsilon}_{\theta})+\frac{d^{2}(q_{0},q_{1})}{2}\int_{0}^{1}\left(e^{-2\lambda\varepsilon
H_{\theta}(t)}-1\right){{\mathrm{d}}t}\\\
&\qquad+\varepsilon\left(\frac{1}{\theta}V(q_{0})+\frac{1}{1-\theta}V(q_{1})\right),\end{split}$

where the last equality follows from the constant speed property
$|\frac{{\mathrm{d}}q_{t}}{{\mathrm{d}}t}|^{2}=d^{2}(q_{0},q_{1})$ of the
geodesic $(q_{t})_{t\in[0,1]}$ connecting $q_{0},q_{1}$ as well as from the
explicit properties of $h(t)=\varepsilon H_{\theta}(t)$ listed above.
Multiplying by $\frac{\theta(1-\theta)}{\varepsilon}>0$ and rearranging gives

$V(q^{\varepsilon}_{\theta})\leq(1-\theta)V(q_{0})+\theta
V(q_{1})+\theta(1-\theta)\frac{d^{2}(q_{0},q_{1})}{2}\underbrace{\int_{0}^{1}\frac{e^{-2\lambda\varepsilon
H_{\theta}(t)}-1}{\varepsilon}{{\mathrm{d}}t}}_{:=I_{\varepsilon}}.$

Since $\int_{0}^{1}H_{\theta}(t){\mathrm{d}}t=\frac{1}{2}$ for all $\theta$ we
see that
$I_{\varepsilon}\to-2\lambda\int_{0}^{1}H_{\theta}(t)\,{{\mathrm{d}}t}=-\lambda$
as $\varepsilon\downarrow 0$, and the result immediately follows since
$V(q^{\varepsilon}_{\theta})\to V(q_{\theta})$ as well in the left-hand side.
∎

##  3 In metric spaces

Before trying to adapt the previous computations to the metric context we need
to fix once and for all the framework to be used in the sequel.

###  3.1 Preliminaries and setting

  * •

By $C([0,1],({\rm X},{\sf d}))$, or simply $C([0,1],{\rm X})$, we denote the
space of continuous curves with values in the metric space $({\rm X},{\sf
d})$. The collection of absolutely continuous curves on $[0,1]$ is denoted by
$AC([0,1],({\rm X},{\sf d}))$, or simply by $AC([0,1],{\rm X})$. For any curve
$(\gamma_{t})\in AC([0,1],{\rm X})$, its length is well defined as

$\ell(\gamma):=\int_{0}^{1}|\dot{\gamma}_{t}|\,{\mathrm{d}}t,$

where $|\dot{\gamma}_{t}|$ denotes the metric speed of $\gamma$. If
$|\dot{\gamma}_{t}|\in L^{2}(0,1)$, then we shall say that $(\gamma_{t})\in
AC^{2}([0,1],{\rm X})$. For these notions of absolutely continuous curves and
metric speed in a metric space, see for instance [2, Section 1.1].

  * •

A curve $\gamma:[0,1]\to{\rm X}$ is called geodesic provided ${\sf
d}(\gamma_{t},\gamma_{s})=|t-s|{\sf d}(\gamma_{0},\gamma_{1})$ for all
$t,s\in[0,1]$.

  * •

The slope $|\partial{\sf E}|$ of a functional ${\sf E}:{\rm
X}\to\mathbb{R}\cup\\{+\infty\\}$ at a point $x\in{\rm X}$ is set as $+\infty$
if $x\notin D({\sf E})$, $0$ if $x$ is isolated, and defined as

$|\partial{\sf E}|(x):=\limsup_{y\to x}\frac{[{\sf E}(x)-{\sf E}(y)]^{+}}{{\sf
d}(x,y)}$

if $x\in D({\sf E})$.

  * •

A curve $(\gamma_{t})_{t>0}\subset{\rm X}$ is said to be a gradient flow of
${\sf E}$ in the ${\rm EVI}_{\lambda}$ sense (with $\lambda\in\mathbb{R}$)
provided $(\gamma_{t})\in AC_{loc}((0,\infty),{\rm X})$ and

$\frac{1}{2}{\frac{{\mathrm{d}}}{{{\mathrm{d}}t}}}{\sf
d}^{2}(\gamma_{t},y)+\frac{\lambda}{2}{\sf d}^{2}(\gamma_{t},y)+{\sf
E}(\gamma_{t})\leq{\sf E}(y),\qquad\forall y\in{\rm X},\,\textrm{a.e. }t>0.$
(${\rm EVI}_{\lambda}$)

If $\gamma_{t}\to x$ as $t\downarrow 0$ with $x\in\overline{D({\sf E})}$, then
we say that the gradient flow $(\gamma_{t})$ starts at $x$.

After this premise, let us fix the framework we shall work within.

######  Setting 3.1.

On the space $({\rm X},{\sf d})$ and on the functional ${\sf E}:{\rm
X}\to\mathbb{R}\cup\\{+\infty\\}$ we make the following assumptions:

  1. (A1)

$({\rm X},{\sf d})$ is a complete and separable metric space;

  2. (A2)

${\sf E}$ is lower semicontinuous with dense domain, i.e. $\overline{D({\sf
E})}={\rm X}$, and locally bounded from below in the following sense: for any
${\sf d}$-bounded set $B\subset{\rm X}$ there exists $c_{B}\in\mathbb{R}$ such
that ${\sf E}(x)\geq c_{B}$ for all $x\in B$;

  3. (A3)

there exists $\lambda\in\mathbb{R}$ such that for any $x\in{\rm X}$ there
exists an ${\rm EVI}_{\lambda}$-gradient flow of ${\sf E}$ starting from $x$.
In view of (3.3), the corresponding 1-parameter semigroup shall be denoted
$\mathsf{S}_{t}$.

Sometimes, and always explicitly indicated, we will also use the following
extra hypothesis.

######  Assumption 3.2.

There exists a Hausdorff topology $\sigma$ on ${\rm X}$ such that ${\sf
d}$-bounded sets are sequentially $\sigma$-compact. Moreover, the distance
${\sf d}$ and the slope $|\partial{\sf E}|$ are $\sigma$-sequentially lower
semicontinuous.

######  Remark 3.3.

Assumption 3.2 is in particular valid provided $({\rm X},{\sf d})$ is a
locally compact space. Indeed, in this case the metric topology of $({\rm
X},{\sf d})$ is an admissible candidate for $\sigma$, since bounded sets are
relatively compact (by [17, Proposition 2.5.22]) and the lower semicontinuity
of the slope $|\partial{\sf E}|$ w.r.t. the metric topology is a consequence
of the forthcoming identity (3.1).  $\blacksquare$

######  Remark 3.4.

Assumption 3.2 implies that ${\sf d}$-converging sequences are also
$\sigma$-converging. Indeed, given $(x_{n})\subset{\rm X}$ with ${\sf
d}(x,x_{n})\to 0$ as $n\to\infty$ for some limit $x\in{\rm X}$, by Assumption
3.2 and by the boundedness of $(x_{n})_{n}$ there exist a subsequence
$(x_{n_{k}})_{k}$ and $y\in{\rm X}$ such that

$x_{n_{k}}\stackrel{{\scriptstyle\sigma}}{{\to}}y\quad\textrm{as}\quad
k\to\infty.$

Since ${\sf d}$ is $\sigma$-sequentially lower semicontinuous (again by
Assumption 3.2) we deduce that

${\sf d}(x,y)\leq\liminf_{k\to\infty}{\sf
d}(x,x_{n_{k}})=\lim_{n\to\infty}{\sf d}(x,x_{n})=0,$

whence $x=y$. This classically implies that the whole sequence converges,
$x_{n}\stackrel{{\scriptstyle\sigma}}{{\to}}x$.  $\blacksquare$

We list now some useful properties of ${\rm EVI}$-gradient flows, which hold
true in Setting 3.1 and that we shall use extensively in the sequel. First of
all, whenever $x\in X$ is the starting point of an ${\rm EVI}_{\lambda}$ flow,
the slope there (a _local_ object, a priori) admits the _global_
representation

$|\partial{\sf E}|(x)=\sup_{y\neq x}\Big{(}\frac{{\sf E}(x)-{\sf E}(y)}{{\sf
d}(x,y)}+\frac{\lambda}{2}{\sf d}(x,y)\Big{)}^{+},$ (3.1)

see [54, Proposition 3.6]. Since we assume that any $x\in X$ is the starting
point of an ${\rm EVI}_{\lambda}$-gradient flow, this means in particular that
$|\partial{\sf E}|:X\to[0,\infty]$ is lower semicontinuous, since so is the
right-hand side above (as a supremum of lower semicontinuous functions). This
also implies by [2, Theorem 1.2.5] that $|\partial{\sf E}|$ is a _strong upper
gradient_ for ${\sf E}$ in the sense of [2, Definition 1.2.1], namely: for
every $(\gamma_{t})\in AC([0,1],{\rm X})$, the map $t\mapsto{\sf
E}(\gamma_{t})$ is Borel and

$|{\sf E}(\gamma_{t_{1}})-{\sf
E}(\gamma_{t_{0}})|\leq\int_{t_{0}}^{t_{1}}|\partial{\sf
E}|(\gamma_{t})|\dot{\gamma}_{t}|\,{\mathrm{d}}t,\qquad\forall 0\leq t_{0}\leq
t_{1}\leq 1,$ (3.2)

the right-hand side being possibly infinite. In addition, if $(\gamma_{t})$ is
an ${\rm EVI}_{\lambda}$-gradient flow of ${\sf E}$ then the following hold
[54, Theorem 3.5]:

  1. (i)

If $(\gamma_{t})$ starts from $x\in\overline{D({\sf E})}$ and
$(\tilde{\gamma}_{t})$ is a second ${\rm EVI}_{\lambda}$-gradient flow of
${\sf E}$ starting from $y\in\overline{D({\sf E})}$ respectively, then

${\sf d}^{2}(\gamma_{t},\tilde{\gamma}_{t})\leq e^{-2\lambda t}{\sf
d}^{2}(x,y),\qquad\forall t\geq 0.$ (3.3)

This means that EVI-gradient flows are unique (provided they exist) and thus
if there exists an EVI-gradient flow $(\gamma_{t})$ starting from $x$, then a
1-parameter semigroup $(\mathsf{S}_{t})_{t>0}$ is unambiguously associated to
it via $\mathsf{S}_{t}(x)=\gamma_{t}$.

  2. (ii)

The maps $t\mapsto\gamma_{t}$ and $t\mapsto{\sf E}(\gamma_{t})$ are locally
Lipschitz in $(0,\infty)$ with values in ${\rm X}$ and $\mathbb{R}$,
respectively, and satisfy the _Energy Dissipation Equality_

$-{\frac{{\mathrm{d}}}{{{\mathrm{d}}t}}}{\sf
E}(\gamma_{t})=\frac{1}{2}|\dot{\gamma}_{t}|^{2}+\frac{1}{2}|\partial{\sf
E}|^{2}(\gamma_{t})=|\dot{\gamma}_{t}|^{2}=|\partial{\sf
E}|^{2}(\gamma_{t}),\qquad\textrm{for a.e. }t>0.$ (3.4)

  3. (iii)

The map

$t\mapsto e^{\lambda t}|\partial{\sf E}|(\gamma_{t})\quad\textrm{is non-
increasing}.$ (3.5)

  4. (iv)

If $(\gamma_{t})$ starts from $x$ and $y\in D(|\partial{\sf E}|)$, then

$|\partial{\sf E}|^{2}(\gamma_{t})\leq\frac{1}{2e^{\lambda t}-1}|\partial{\sf
E}|^{2}(y)+\frac{1}{I_{\lambda}(t)^{2}}{\sf d}^{2}(x,y),\qquad\textrm{provided
}-\lambda t<\log 2,$ (3.6)

where $I_{\lambda}(t):=\int_{0}^{t}e^{\lambda s}\,{\mathrm{d}}s$.

We emphasize that these properties directly follow from the very definition
(${\rm EVI}_{\lambda}$) of gradient flows, and a priori do not require ${\sf
E}$ to be geodesically $\lambda$-convex. Although analogous statements can be
found in [2] and [1] under convexity assumptions on ${\sf E}$, the latter are
essentially needed to grant _existence_ of EVI-gradient flows. It is important
to stress this fact because in Setting 3.1 we _assume_ that for any $x\in{\rm
X}$ there exists an ${\rm EVI}_{\lambda}$-gradient flow of ${\sf E}$ starting
there, which from [27] is known to imply that ${\sf E}$ is geodesically
convex. In Section 3 we also provide an alternative proof of this latter fact,
whence the necessity for us to avoid all properties of ${\rm EVI}$-gradient
flows which are actually a consequence of geodesic convexity.

We conclude this preliminary part with a general integrability result about
${\rm EVI}_{\lambda}$-gradient flows, which we could not find explicitly
written in the literature and will be used later on in the proof of Lemma 3.6.

######  Lemma 3.5.

With the same assumptions and notations as in Setting 3.1, let $x\in{\rm X}$.
Then

$t\mapsto{\sf E}(\mathsf{S}_{t}x)$ is integrable in $[0,T]$, for all $T>0$,

regardless of whether ${\sf E}(x)$ is finite or not.

On intervals $[\varepsilon,T]$ this computation is easily justified by the
fact that $t\mapsto{\sf E}(\mathsf{S}_{t}x)$ is locally Lipschitz in
$(0,\infty)$, hence locally integrable therein. But this computation is
legitimate even if $\varepsilon=0$, as we are going to see.

###### Proof.

Let $x\in{\rm X}$ and $T>0$ be as in our statement. Since ${\sf E}$ is bounded
from below on ${\sf d}$-bounded sets by (A2), and because
$(\mathsf{S}_{t}x)_{t\in[0,T]}$ is bounded, there exists $c\in\mathbb{R}$ such
that ${\sf E}(\mathsf{S}_{t}x)\geq c$ for all $t\in[0,T]$. Combining with
(${\rm EVI}_{\lambda}$) this gives

$c\leq{\sf E}(\mathsf{S}_{t}x)\leq{\sf
E}(y)-\frac{1}{2}\frac{{\mathrm{d}}}{{\mathrm{d}}t}{\sf
d}^{2}(\mathsf{S}_{t}x,y)-\frac{\lambda}{2}{\sf d}^{2}(\mathsf{S}_{t}x,y)$

for any $y\in D({\sf E})$ and $t\in(0,T]$. Integrating from $t=\eta>0$ to
$t=T$ gives

$\begin{split}c(T-\eta)\leq\int_{\eta}^{T}{\sf
E}(\mathsf{S}_{t}x)\,{\mathrm{d}}t&\leq(T-\eta){\sf
E}(y)-\frac{1}{2}\Big{(}{\sf d}^{2}(\mathsf{S}_{T}x,y)-{\sf
d}^{2}(\mathsf{S}_{\eta}x,y)\Big{)}\\\
&\qquad-\frac{\lambda}{2}\int_{\eta}^{T}{\sf
d}^{2}(\mathsf{S}_{t}x,y)\,{\mathrm{d}}t.\end{split}$

As $t\mapsto{\sf E}(\mathsf{S}_{t}x)$ is bounded from below on $[0,T]$ and the
right-hand side has a finite limit as $\eta\downarrow 0$ (thanks to the fact
that $t\mapsto\mathsf{S}_{t}x$ is ${\sf d}$-continuous on $[0,\infty)$ by the
very definition of ${\rm EVI}_{\lambda}$-gradient flow), we deduce the desired
integrability. ∎

###  3.2 A pseudo-Riemannian computation

In this section the formal Riemannian computations carried out at the
beginning of Section 2, and more precisely (2.4), will be reproduced
rigorously in the abstract Setting 3.1. To this aim, a key role will be played
by the following purely metric estimate:

######  Lemma 3.6.

With the same assumptions and notations as in Setting 3.1, let
$(\gamma_{t})\in AC([0,1],{\rm X})$ with ${\sf E}(\gamma_{0}),{\sf
E}(\gamma_{1})<\infty$. For any fixed absolutely continuous function
$h:[0,1]\to\mathbb{R}$ with $h(t)>0$ for all $t\in(0,1)$ let

$\tilde{\gamma}_{t}:=\mathsf{S}_{h(t)}\gamma_{t},\qquad t\in[0,1],$

and for any $0\leq t_{0}<t_{1}\leq 1$ write

$t^{+}:=\left\\{\begin{array}[]{ll}t_{1}&\mbox{if }h(t_{1})\geq h(t_{0})\\\
t_{0}&\mbox{otherwise}\end{array}\right.\qquad\mbox{and}\qquad
t^{-}:=\left\\{\begin{array}[]{ll}t_{0}&\mbox{if }h(t_{1})\geq h(t_{0})\\\
t_{1}&\mbox{otherwise}\end{array}\right..$ (3.7)

Then we have the exact estimate

$\begin{split}\frac{1}{2}\left|\frac{{\sf
d}(\tilde{\gamma}_{t_{1}},\tilde{\gamma}_{t_{0}})}{t_{1}-t_{0}}\right|^{2}&+\frac{1}{2\lambda^{2}}|\partial{\sf
E}|^{2}(\tilde{\gamma}_{t^{+}})\frac{e^{\lambda(h(t_{1})-h(t_{0}))}+e^{\lambda(h(t_{0})-h(t_{1}))}-2}{(t_{1}-t_{0})^{2}}\\\
&+\frac{1-e^{-\lambda(h(t^{+})-h(t^{-}))}}{\lambda(t^{+}-t^{-})}\cdot\frac{{\sf
E}(\tilde{\gamma}_{t_{1}})-{\sf E}(\tilde{\gamma}_{t_{0}})}{t_{1}-t_{0}}\\\
&\qquad\leq\frac{1}{2}e^{-\lambda(h(t_{1})+h(t_{0}))}\left|\frac{{\sf
d}(\gamma_{t_{1}},\gamma_{t_{0}})}{t_{1}-t_{0}}\right|^{2}.\end{split}$ (3.8)

Here we use the convention that $(+\infty)\times 0=0$ whenever $|\partial{\sf
E}|(\tilde{\gamma}_{t^{+}})=+\infty$ and $h(t_{0})=h(t_{1})$ in the second
term on the left-hand side of (3.8). Since we assume that $h(t)>0$ for
$t\in(0,1)$, and because any ${\rm EVI}_{\lambda}$-gradient flow immediately
falls within $D(|\partial{\sf E}|)$ by standard regularizing effects, this
latter case is in fact only possible if $t_{0}=0$, $t_{1}=1$, and
$h(t_{0})=h(t_{1})=0$. In that case $\tilde{\gamma}_{0}=\gamma_{0}$ and
$\tilde{\gamma}_{1}=\gamma_{1}$, the third term in the left-hand side also
cancels owing to $e^{-\lambda(h(t^{+})-h(t^{-}))}=1$, and (3.8) then holds as
a trivial equality.

We shall rely on this lemma later on in two different ways: First, fixing
$t_{0}=0$ and letting $t_{1}\downarrow 0$ (resp. fixing $t_{1}=1$ and letting
$t_{0}\uparrow 1$) to control in Lemma 3.10 the continuity of $t\mapsto{\sf
E}(\tilde{\gamma}_{t})$ at the boundaries $t=0,1$, and second, fixing
$t_{0}\in(0,1)$ and letting $t_{1}\to t_{0}$ to obtain in Proposition 3.11 a
pointwise differential estimate similar to (2.4).

######  Remark 3.7.

The times $t^{\pm}$ are just a fancy notation, ordered as $h(t^{-})\leq
h(t^{+})$. Note that in our estimate (3.8) the Fisher information
$|\partial{\sf E}|^{2}(\tilde{\gamma}_{t^{+}})$ is evaluated at the time
$t=t^{+}$ for which the “smoothing time” $s=h(t_{0})$ or $s=h(t_{1})$ is the
largest, i.e. where the regularizing vertical flow has been run for the
longest time. This is somehow natural, as this specific point is “better” than
the other one in terms of regularity.  $\blacksquare$

###### Proof.

By symmetry we only discuss the case $h(t_{1})\geq h(t_{0})$, i.e.
$t^{+}=t_{1}$ and $t^{-}=t_{0}$. As already mentioned, if
$h(t_{0})=h(t_{1})=0$ our statement is actually vacuous, thus it is not
restrictive to further assume $h(t_{1})>0$. Let us write for simplicity

$\hat{\gamma}_{t_{1}}:=\mathsf{S}_{h(t_{0})}\gamma_{t_{1}}.$

From an intuitive point of view, this corresponds to freezing a “vertical
time” $s=h(t_{0})$ and “translating” $\tilde{\gamma}_{t_{0}}$ in the
“horizontal” $t$ direction parallel to the curve $\gamma$ until $t_{1}$. Here,
in the “vertical” direction above $t_{1}$ the smoothing semigroup
$\mathsf{S}_{s}$ associated with ${\sf E}$ has been run at least for a
strictly positive time $h(t_{1})>0$, so that by (3.6) the solution of the
“vertical” gradient flow at that time lies within the regular domain ${\rm
X}_{1}=D(|\partial{\sf E}|)\subset{\rm X}_{0}=D({\sf E})\subset{\rm X}$, see
Figure 2.

$\gamma_{t_{0}}$$\gamma_{t_{1}}$$\tilde{\gamma}_{t_{0}}$$\hat{\gamma}_{t_{1}}\,\,[s=h(t_{0})]$$\tilde{\gamma}_{t_{1}}\,\,[s=h(t_{1})]$$t$

$s$

Figure 2: The horizontal and vertical curves

The first step is to write (${\rm EVI}_{\lambda}$) for
$s\mapsto\mathsf{S}_{s}(\gamma_{t_{1}})$ with $\tilde{\gamma}_{t_{0}}$ as a
reference point, namely

$\frac{1}{2}\frac{{\mathrm{d}}}{{\mathrm{d}}s}{\sf
d}^{2}(\mathsf{S}_{s}\gamma_{t_{1}},\tilde{\gamma}_{t_{0}})+\frac{\lambda}{2}{\sf
d}^{2}(\mathsf{S}_{s}\gamma_{t_{1}},\tilde{\gamma}_{t_{0}})+{\sf
E}(\mathsf{S}_{s}\gamma_{t_{1}})\leq{\sf E}(\tilde{\gamma}_{t_{0}}),$

which holds true for a.e. $s\in[0,h(t_{1})]$ in the “vertical” direction. This
inequality can be equivalently rewritten as

$\frac{1}{2}\frac{{\mathrm{d}}}{{\mathrm{d}}s}\Big{(}e^{\lambda s}{\sf
d}^{2}(\mathsf{S}_{s}\gamma_{t_{1}},\tilde{\gamma}_{t_{0}})\Big{)}\leq
e^{\lambda s}\Big{(}{\sf E}(\tilde{\gamma}_{t_{0}})-{\sf
E}(\mathsf{S}_{s}\gamma_{t_{1}})\Big{)}.$ (3.9)

Note that this estimate carries significant information if and only if the
reference point has finite entropy, i.e. ${\sf
E}(\tilde{\gamma}_{t_{0}})<\infty$ in the right-hand side. This holds true for
$t_{0}\in(0,1)$ because $\tilde{\gamma}_{t_{0}}$ is the ${\rm
EVI}_{\lambda}$-gradient flow of ${\sf E}$ starting from $\gamma_{t_{0}}$ at a
strictly positive time $s=h(t_{0})>0$, but also for $h(t_{0})=0$ if $t_{0}=0$
since in this case $\tilde{\gamma}_{0}=\gamma_{0}$ is assumed to have finite
entropy.

Integrating (3.9) from $s=h(t_{0})$ to $s=h(t_{1})$ gives

$\begin{split}\frac{1}{2}e^{\lambda h(t_{1})}{\sf
d}^{2}(\tilde{\gamma}_{t_{1}},\tilde{\gamma}_{t_{0}})&-\frac{1}{2}e^{\lambda
h(t_{0})}{\sf d}^{2}(\hat{\gamma}_{t_{1}},\tilde{\gamma}_{t_{0}})\\\
&\leq\int_{h(t_{0})}^{h(t_{1})}e^{\lambda s}\Big{(}{\sf
E}(\tilde{\gamma}_{t_{0}})-{\sf
E}(\mathsf{S}_{s}\gamma_{t_{1}})\Big{)}\,{\mathrm{d}}s\\\
&=\int_{h(t_{0})}^{h(t_{1})}e^{\lambda s}\Big{(}\big{(}{\sf
E}(\tilde{\gamma}_{t_{0}})-{\sf E}(\tilde{\gamma}_{t_{1}})\big{)}+\big{(}{\sf
E}(\tilde{\gamma}_{t_{1}})-{\sf
E}(\mathsf{S}_{s}\gamma_{t_{1}})\big{)}\Big{)}\,{\mathrm{d}}s\\\
&=\int_{h(t_{0})}^{h(t_{1})}e^{\lambda s}\Big{(}{\sf
E}(\tilde{\gamma}_{t_{1}})-{\sf
E}(\mathsf{S}_{s}\gamma_{t_{1}})\Big{)}\,{\mathrm{d}}s\\\
&\qquad-\frac{e^{\lambda h(t_{1})}-e^{\lambda h(t_{0})}}{\lambda}\Big{(}{\sf
E}(\tilde{\gamma}_{t_{1}})-{\sf E}(\tilde{\gamma}_{t_{0}})\Big{)}.\end{split}$
(3.10)

If $h(t_{0})>0$ this computation is legitimate because
$s\mapsto\mathsf{S}_{s}\gamma_{t_{1}}$ and $s\mapsto{\sf
E}(\mathsf{S}_{s}\gamma_{t_{1}})$ are locally Lipschitz in $(0,\infty)$, hence
$s\mapsto{\sf d}^{2}(\mathsf{S}_{s}\gamma_{t_{1}},\tilde{\gamma}_{t_{0}})$ and
$s\mapsto{\sf E}(\mathsf{S}_{s}\gamma_{t_{1}})$ are locally integrable
therein. But this computation is also justified when $h(t_{0})=0$ by Lemma
3.5. More specifically $s\mapsto{\sf E}(\mathsf{S}_{s}\gamma_{t_{1}})$ is
absolutely integrable on $[0,T]$ for any $T>0$ and a fortiori so is $s\mapsto
e^{\lambda s}{\sf E}(\mathsf{S}_{s}\gamma_{t_{1}})$.

Now let us estimate the terms in (3.10) to get (3.8). First, since
$\hat{\gamma}_{t_{1}}=\mathsf{S}_{h(t_{0})}\gamma_{t_{1}}$,
$\tilde{\gamma}_{t_{0}}=\mathsf{S}_{h(t_{0})}\gamma_{t_{0}}$, and
$\mathsf{S}_{h(t_{0})}(\cdot)$ is $\lambda$-contractive by (3.3), we observe
that the second term in the left-hand side of (3.10) can be controlled as

${\sf d}^{2}(\hat{\gamma}_{t_{1}},\tilde{\gamma}_{t_{0}})={\sf
d}^{2}(\mathsf{S}_{h(t_{0})}\gamma_{t_{1}},\mathsf{S}_{h(t_{0})}\gamma_{t_{0}})\leq
e^{-2\lambda h(t_{0})}{\sf d}^{2}(\gamma_{t_{1}},\gamma_{t_{0}}).$ (3.11)

On the right-hand side, let us define

$I:=\int_{h(t_{0})}^{h(t_{1})}e^{\lambda s}\Big{(}{\sf
E}(\tilde{\gamma}_{t_{1}})-{\sf
E}(\mathsf{S}_{s}\gamma_{t_{1}})\Big{)}\,{\mathrm{d}}s.$

This integral is clearly non-positive by (3.4), but we need a finer analysis.
To this aim, for fixed $0<s<h(t_{1})$ let us write

$\begin{split}{\sf E}(\tilde{\gamma}_{t_{1}})-{\sf
E}(\mathsf{S}_{s}\gamma_{t_{1}})&={\sf
E}(\mathsf{S}_{h(t_{1})}\gamma_{t_{1}})-{\sf
E}(\mathsf{S}_{s}\gamma_{t_{1}})\\\
&=\int_{s}^{h(t_{1})}\frac{{\mathrm{d}}}{{\mathrm{d}}\tau}{\sf
E}(\mathsf{S}_{\tau}\gamma_{t_{1}})\,{\mathrm{d}}\tau=-\int_{s}^{h(t_{1})}|\partial{\sf
E}|^{2}(\mathsf{S}_{\tau}\gamma_{t_{1}})\,{\mathrm{d}}\tau,\end{split}$

where the second equality holds due to $\tau\mapsto{\sf
E}(\mathsf{S}_{\tau}\gamma_{t_{1}})$ being Lipschitz on $[s,h(t_{1})]$, and
the third one stems from (3.4) for the gradient flow
$\tau\mapsto\mathsf{S}_{\tau}\gamma_{t_{1}}$. By (3.5)

$\begin{split}-\int_{s}^{h(t_{1})}|\partial{\sf
E}|^{2}(\mathsf{S}_{\tau}\gamma_{t_{1}})\,{\mathrm{d}}\tau&\leq-\int_{s}^{h(t_{1})}|\partial{\sf
E}|^{2}(\mathsf{S}_{h(t_{1})}\gamma_{t_{1}})e^{2\lambda(h(t_{1})-\tau)}\,{\mathrm{d}}\tau\\\
&=-e^{2\lambda h(t_{1})}|\partial{\sf
E}|^{2}(\mathsf{S}_{h(t_{1})}\gamma_{t_{1}})\int_{s}^{h(t_{1})}e^{-2\lambda\tau}\,{\mathrm{d}}\tau\\\
&=\frac{1}{2\lambda}\Big{(}1-e^{2\lambda(h(t_{1})-s)}\Big{)}|\partial{\sf
E}|^{2}(\mathsf{S}_{h(t_{1})}\gamma_{t_{1}}),\end{split}$

so that, as a consequence,

$\begin{split}I&\leq\frac{1}{2\lambda}\int_{h(t_{0})}^{h(t_{1})}e^{\lambda
s}\Big{(}1-e^{2\lambda(h(t_{1})-s)}\Big{)}|\partial{\sf
E}|^{2}(\mathsf{S}_{h(t_{1})}\gamma_{t_{1}})\,{\mathrm{d}}s\\\
&=\frac{1}{2\lambda}|\partial{\sf
E}|^{2}(\mathsf{S}_{h(t_{1})}\gamma_{t_{1}})\int_{h(t_{0})}^{h(t_{1})}\Big{(}e^{\lambda
s}-e^{2\lambda h(t_{1})}\cdot e^{-\lambda s}\Big{)}\,{\mathrm{d}}s\\\
&=-\frac{1}{2\lambda^{2}}|\partial{\sf
E}|^{2}(\tilde{\gamma}_{t_{1}})e^{\lambda
h(t_{1})}\Big{(}e^{\lambda(h(t_{1})-h(t_{0}))}+e^{\lambda(h(t_{0})-h(t_{1}))}-2\Big{)}.\end{split}$

Plugging this estimate together with (3.11) into (3.10) and dividing by
$(t_{1}-t_{0})^{2}>0$ entails our claim. ∎

We also need to study the behaviour of the “regularized” curve
$\tilde{\gamma}_{t}:=\mathsf{S}_{h(t)}\gamma_{t}$ and of the entropy ${\sf E}$
along it: this is the content of the following two results.

######  Lemma 3.8.

With the same assumptions and notations as in Setting 3.1, if $(\gamma_{t})\in
AC([0,1],{\rm X})$ and $h:[0,1]\to\mathbb{R}$ is absolutely continuous with
$h(t)>0$ for all $t\in(0,1)$, then the curve
$\tilde{\gamma}_{t}:=\mathsf{S}_{h(t)}\gamma_{t}$ belongs to
$AC_{loc}((0,1),{\rm X})\cap C([0,1],{\rm X})$.

###### Proof.

Fix $\delta\in(0,1/2)$, $t_{0},t_{1}\in[\delta,1-\delta]$ with $t_{0}\leq
t_{1}$ and define

$m_{\delta}:=\min_{t\in[\delta,1-\delta]}h(t),\qquad
M_{\delta}:=\max_{t\in[\delta,1-\delta]}h(t),$ (3.12)

paying attention to the fact that $m_{\delta}>0$ by construction. Write as
before $\hat{\gamma}_{t_{1}}:=\mathsf{S}_{h(t_{0})}\gamma_{t_{1}}$ for the
“horizontal” translation of $\tilde{\gamma}_{t_{0}}$ (see Figure 2). By
triangular inequality and the contraction estimate (3.3) we get

${\sf d}(\tilde{\gamma}_{t_{0}},\tilde{\gamma}_{t_{1}})\leq{\sf
d}(\tilde{\gamma}_{t_{0}},\hat{\gamma}_{t_{1}})+{\sf
d}(\hat{\gamma}_{t_{1}},\tilde{\gamma}_{t_{1}})\leq
e^{\lambda^{-}M_{\delta}}{\sf d}(\gamma_{t_{0}},\gamma_{t_{1}})+{\sf
d}(\hat{\gamma}_{t_{1}},\tilde{\gamma}_{t_{1}}),$ (3.13)

where $\lambda^{-}:=\max\\{-\lambda,0\\}$. Since $(\gamma_{t})$ is absolutely
continuous the first term in the right-hand side can be controlled as ${\sf
d}(\gamma_{t_{0}},\gamma_{t_{1}})\leq\int_{t_{0}}^{t_{1}}|\dot{\gamma}_{t}|{\mathrm{d}}t$.
As regards the second one, by (3.4) and up to assuming $h(t_{0})\leq h(t_{1})$
(which is not restrictive, as otherwise it is sufficient to swap the boundary
values of integration below) it holds

${\sf d}(\hat{\gamma}_{t_{1}},\tilde{\gamma}_{t_{1}})={\sf
d}(\mathsf{S}_{h(t_{0})}\gamma_{t_{1}},\mathsf{S}_{h(t_{1})}\gamma_{t_{1}})\leq\int_{h(t_{0})}^{h(t_{1})}\Big{|}\frac{{\mathrm{d}}}{{\mathrm{d}}s}\mathsf{S}_{s}\gamma_{t_{1}}\Big{|}\,{\mathrm{d}}s=\int_{h(t_{0})}^{h(t_{1})}|\partial{\sf
E}|(\mathsf{S}_{s}\gamma_{t_{1}})\,{\mathrm{d}}s,$ (3.14)

where, to avoid possibly ambiguous notations,
$|\frac{{\mathrm{d}}}{{\mathrm{d}}s}\mathsf{S}_{s}\gamma_{t_{1}}|$ denotes the
metric speed of the “vertical” curve $s\mapsto\mathsf{S}_{s}\gamma_{t_{1}}$.
In order to control the slope in the right-most term uniformly both in
$s\in[h(t_{0}),h(t_{1})]\subset[m_{\delta},M_{\delta}]$ and in
$t_{1}\in[\delta,1-\delta]$, for fixed $\delta$, let $\varepsilon$ be such
that $-\lambda\varepsilon<\log 2$ (if $\lambda\geq 0$, choose
$\varepsilon=m_{\delta}$) and define
$\varepsilon^{\prime}:=\min\\{m_{\delta},\varepsilon\\}$. Then by (3.5) and
the fact that $s\geq h(t_{0})\geq m_{\delta}\geq\varepsilon^{\prime}$ we have

$|\partial{\sf E}|(\mathsf{S}_{s}\gamma_{t_{1}})\leq
e^{\lambda(\varepsilon^{\prime}-s)}|\partial{\sf
E}|(\mathsf{S}_{\varepsilon^{\prime}}\gamma_{t_{1}})\leq
e^{\lambda^{-}(M_{\delta}-\varepsilon^{\prime})}|\partial{\sf
E}|(\mathsf{S}_{\varepsilon^{\prime}}\gamma_{t_{1}}),\qquad\forall
s\in[m_{\delta},M_{\delta}]$

and by (3.6) for any reference point $x\in D(|\partial{\sf E}|)$ it holds

$|\partial{\sf
E}|^{2}(\mathsf{S}_{\varepsilon^{\prime}}\gamma_{t_{1}})\leq\frac{1}{2e^{\lambda\varepsilon^{\prime}}-1}|\partial{\sf
E}|^{2}(x)+\frac{1}{I_{\lambda}(\varepsilon^{\prime})^{2}}{\sf
d}^{2}(x,\gamma_{t_{1}}).$

The squared distance in the right-hand side above is bounded uniformly in
$t_{1}\in[\delta,1-\delta]$, since by triangular inequality

${\sf d}(x,\gamma_{t_{1}})\leq{\sf d}(x,\gamma_{0})+{\sf
d}(\gamma_{0},\gamma_{t_{1}})\leq{\sf d}(x,\gamma_{0})+\ell(\gamma).$

Therefore there exists $C_{\delta}>0$ such that

$|\partial{\sf E}|(\mathsf{S}_{s}\gamma_{t_{1}})\leq C_{\delta}\qquad\mbox{for
all}\qquad t_{1}\in[\delta,1-\delta]\mbox{ and }s\in[m_{\delta},M_{\delta}].$
(3.15)

and plugging this bound into (3.14) yields

${\sf d}(\hat{\gamma}_{t_{1}},\tilde{\gamma}_{t_{1}})\leq
C_{\delta}|h(t_{1})-h(t_{0})|\leq
C_{\delta}\int_{t_{0}}^{t_{1}}|h^{\prime}(t)|\,{\mathrm{d}}t.$

It is now sufficient to combine this inequality with ${\sf
d}(\gamma_{t_{0}},\gamma_{t_{1}})\leq\int_{t_{0}}^{t_{1}}|\dot{\gamma}_{t}|{\mathrm{d}}t$
and (3.13) to get

${\sf
d}(\tilde{\gamma}_{t_{0}},\tilde{\gamma}_{t_{1}})\leq\int_{t_{0}}^{t_{1}}\Big{(}e^{\lambda^{-}M_{\delta}}|\dot{\gamma}_{t}|+C_{\delta}|h^{\prime}(t)|\Big{)}\,{\mathrm{d}}t.$
(3.16)

As $e^{\lambda^{-}M_{\delta}}|\dot{\gamma}_{t}|+C_{\delta}|h^{\prime}(t)|\in
L^{1}(\delta,1-\delta)$ and $\delta$ is arbitrary, the fact that
$(\tilde{\gamma}_{t})\in AC_{loc}((0,1))$ is thus proved.

Turning now to the continuity of $(\tilde{\gamma}_{t})$ at the endpoints, let
$t_{0}=0$ and $t_{1}\in(0,1)$. Arguing as for (3.13) but with a crucial
difference in the choice of the third point in the triangular inequality, it
holds

$\begin{split}{\sf d}(\tilde{\gamma}_{0},\tilde{\gamma}_{t_{1}})&\leq{\sf
d}(\tilde{\gamma}_{0},\mathsf{S}_{h(t_{1})}\gamma_{0})+{\sf
d}(\mathsf{S}_{h(t_{1})}\gamma_{0},\tilde{\gamma}_{t_{1}})\\\ &\leq{\sf
d}(\mathsf{S}_{h(0)}\gamma_{0},\mathsf{S}_{h(t_{1})}\gamma_{0})+e^{-\lambda
h(t_{1})}{\sf d}(\gamma_{0},\gamma_{t_{1}}).\end{split}$

The second term on the right-hand side vanishes as $t_{1}\downarrow 0$ by
(absolute) continuity of $\gamma$ and so does the first one, since
$s\mapsto\mathsf{S}_{s}\gamma_{0}$ is continuous in $[0,\infty)$ with values
in ${\rm X}$ and $h(t_{1})\to h(0)$. The continuity at $t=1$ is obtained
similarly and the proof is complete. ∎

######  Remark 3.9.

If $h(t)>0$ also in $t=0,1$, then the previous argument can be extended to the
whole interval $[0,1]$ and therefore $(\tilde{\gamma}_{t})\in AC([0,1],{\rm
X})$.  $\blacksquare$

######  Lemma 3.10.

With the same assumptions and notations as in Lemma 3.8, the entropy is
locally absolutely continuous in $(0,1)$ along the regularized curve
$\tilde{\gamma}_{t}$, i.e.

$t\mapsto{\sf E}(\tilde{\gamma}_{t})\quad\in\quad AC_{loc}((0,1)).$

If in addition $(\gamma_{t})\in AC^{2}([0,1],{\rm X})$, ${\sf
E}(\tilde{\gamma}_{0}),{\sf E}(\tilde{\gamma}_{1})<\infty$ and $h$ is
differentiable at $t=0$ and $t=1$ with $h^{\prime}(0)>0$ and
$h^{\prime}(1)<0$, then

$t\mapsto{\sf E}(\tilde{\gamma}_{t})\quad\in\quad C([0,1]).$

Note that ${\sf E}(\tilde{\gamma}_{0}),{\sf E}(\tilde{\gamma}_{1})<\infty$ is
automatically satisfied if $h(t)>0$ also in $t=0,1$.

###### Proof.

Let us first prove that $t\mapsto{\sf E}(\tilde{\gamma}_{t})$ is locally
absolutely continuous. Since $|\partial{\sf E}|$ is a strong upper-gradient,
the chain rule (3.2) holds and it suffices to show that $|\partial{\sf
E}|(\tilde{\gamma}_{t})|\dot{\tilde{\gamma}}_{t}|\in L^{1}_{loc}(0,1)$, namely

$\int_{\delta}^{1-\delta}|\partial{\sf
E}|(\tilde{\gamma}_{t})|\dot{\tilde{\gamma}}_{t}|\,{\mathrm{d}}t<\infty,\qquad\forall\delta\in(0,1/2),$
(3.17)

as this would imply that ${\sf E}\circ\tilde{\gamma}\in AC_{loc}((0,1))$ with

$\left|{\frac{{\mathrm{d}}}{{{\mathrm{d}}t}}}({\sf
E}\circ\tilde{\gamma})(t)\right|\leq|\partial{\sf
E}|(\tilde{\gamma}_{t})\cdot|\dot{\tilde{\gamma}}_{t}|,\qquad\textrm{for a.e.
}t\in(0,1).$

To this aim, observe from (3.16) that $|\dot{\tilde{\gamma}}_{t}|\in
L^{1}_{loc}(0,1)$ with $|\dot{\tilde{\gamma}}_{t}|\leq
e^{\lambda^{-}M_{\delta}}|\dot{\gamma}_{t}|+C_{\delta}|h^{\prime}(t)|$ a.e. on
$[\delta,1-\delta]$, with $M_{\delta}$ defined in (3.12). Moreover from (3.15)
we also know that $|\partial{\sf E}|(\mathsf{S}_{s}\gamma_{t})\leq C_{\delta}$
uniformly in $t\in[\delta,1-\delta]$ and $s\in[m_{\delta},M_{\delta}]$, so
that by choosing $s=h(t)$ we get in particular $|\partial{\sf
E}|(\tilde{\gamma}_{t})\leq C_{\delta}$ for all $t\in[\delta,1-\delta]$. This
shows that $t\mapsto|\partial{\sf E}|(\tilde{\gamma}_{t})$ belongs to
$L^{\infty}_{loc}(0,1)$, whence (3.17).

Now assume that $(\gamma_{t})\in AC^{2}([0,1],{\rm X})$, ${\sf
E}(\tilde{\gamma}_{0})<\infty$, $h$ is differentiable at $t=0$ with
$h^{\prime}(0)>0$ and let us prove that $t\mapsto{\sf E}(\tilde{\gamma}_{t})$
is continuous at $t=0$. (The argument is identical for $t=1$.) On the one
hand, as $(\tilde{\gamma}_{t})$ is continuous at $t=0$ by Lemma 3.8 and ${\sf
E}$ is lower semicontinuous, we see that ${\sf
E}(\tilde{\gamma}_{0})\leq\liminf_{t\downarrow 0}{\sf E}(\tilde{\gamma}_{t})$.
On the other hand, choosing $t_{0}=0$ in (3.8), our assumption that
$h^{\prime}(0)>0$ gives $h(t_{1})>h(0)$ for $t_{1}>0$ small, hence $t^{-}=0$
and $t^{+}=t_{1}$. Discarding the first two (non-negative) terms on the left-
hand side, and multiplying by $(t_{1}-t_{0})=t_{1}$ yield

$\begin{split}\frac{1-e^{-\lambda(h(t_{1})-h(0))}}{\lambda(t_{1}-0)}\cdot\big{(}{\sf
E}(\tilde{\gamma}_{t_{1}})-{\sf
E}(\tilde{\gamma}_{0})\big{)}&\leq\frac{t_{1}}{2}e^{-\lambda(h(t_{1})+h(0))}\left|\frac{{\sf
d}(\gamma_{t_{1}},\gamma_{0})}{t_{1}}\right|^{2}\\\
&\leq\frac{t_{1}}{2}e^{-\lambda(h(t_{1})+h(0))}\Big{(}\frac{1}{t_{1}}\int_{0}^{t_{1}}|\dot{\gamma}_{t}|\,{\mathrm{d}}t\Big{)}^{2}\\\
&\leq\frac{1}{2}e^{-\lambda(h(t_{1})+h(0))}\int_{0}^{t_{1}}|\dot{\gamma}_{t}|^{2}\,{\mathrm{d}}t.\end{split}.$

Letting $t_{1}\downarrow 0$, the right-hand side vanishes owing to our
assumption that $(\gamma_{t})\in AC^{2}([0,1],{\rm X})$, and clearly the
exponential difference quotient in the left-hand side converges to
$h^{\prime}(0)$. Rearranging gives

$h^{\prime}(0)\limsup_{t_{1}\downarrow 0}{\sf E}(\tilde{\gamma}_{t_{1}})\leq
h^{\prime}(0){\sf E}(\tilde{\gamma}_{0}),$

since $h^{\prime}(0)>0$ the desired upper semicontinuity follows and the proof
is complete. ∎

Gathering the results proven so far, we deduce the following:

######  Proposition 3.11.

With the same assumptions and notations as in Lemma 3.8, for a.e. $t\in(0,1)$
it holds

$\frac{1}{2}\left|\dot{\tilde{\gamma}}_{t}\right|^{2}+\frac{1}{2}|h^{\prime}(t)|^{2}|\partial{\sf
E}|^{2}(\tilde{\gamma}_{t})+h^{\prime}(t){\frac{{\mathrm{d}}}{{{\mathrm{d}}t}}}{\sf
E}(\tilde{\gamma}_{t})\leq\frac{1}{2}e^{-2\lambda
h(t)}\left|\dot{\gamma}_{t}\right|^{2}.$ (3.18)

###### Proof.

The argument simply consists in taking the limit $t_{1}\to t_{0}$ in (3.8),
which should clearly lead (at least formally) to (3.18) by Taylor-expanding
the various exponential difference quotients. In order to make this rigorous,
note that the first and third terms in the left-hand side of (3.18) are well
defined for a.e. $t\in(0,1)$ by Lemma 3.8 and Lemma 3.10, respectively. The
second term is also unambiguously defined because $h(t)>0$, hence the
“vertical” ${\rm EVI}_{\lambda}$-gradient flow starting from $\gamma_{t}$ and
defining $\tilde{\gamma}_{t}=\mathsf{S}_{h(t)}\gamma_{t}$ falls immediately
within the domain ${\rm X}_{1}=D(|\partial{\sf E}|)$. The right-hand side is
well defined for a.e. $t$ since $\gamma\in AC([0,1],{\rm X})$.

After this premise, let $t\in(0,1)$ be any differentiation point for $h$,
$t\mapsto\gamma_{t}$, $t\mapsto\tilde{\gamma}_{t}$ and $t\mapsto{\sf
E}(\tilde{\gamma}_{t})$, choose $t_{0}=t$ in (3.8) and let us take the right
limit $t_{1}\downarrow t_{0}$ (since we are considering a differentiability
point the left and right limits exist and are equal, so there is no need to
address the left limit). From the very definition (3.7) of $t^{\pm}$ it
clearly holds $t^{\pm}\to t_{0}$ as $t_{1}\downarrow t$, hence the convergence
of the right-hand side of (3.8) to the right-hand side of (3.18) is clear and
so is the convergence of the two difference quotients of $h$. By Lemma 3.8 the
first term in the left-hand side also passes to the limit, as does the third
one according to Lemma 3.10. The only term left to handle is the Fisher
information $|\partial{\sf E}|^{2}(\tilde{\gamma}^{+})$. From the continuity
of $t\mapsto\tilde{\gamma}_{t}$ (cf. Lemma 3.8) we see that
$\tilde{\gamma}_{t^{+}}\to\tilde{\gamma}_{t}$ in $({\rm X},{\sf d})$, and the
lower semicontinuity of the slope (3.1) results in

$|\partial{\sf E}|(\tilde{\gamma}_{t})\leq\liminf_{t_{1}\downarrow
0}|\partial{\sf E}|(\tilde{\gamma}_{t^{+}}).$

Thus rigorously taking the liminf $t_{1}\downarrow t_{0}$ in (3.8) entails
(3.18) and achieves the proof. ∎

The interesting consequence for our purpose is then:

######  Theorem 3.12.

With the same assumptions and notations as in Setting 3.1, fix
$\varepsilon>0$, and set $h_{\varepsilon}(t):=\varepsilon\min\\{t,1-t\\}$. Let
$(\gamma_{t})\in AC^{2}([0,1],{\rm X})$ be such that ${\sf E}(\gamma_{0}),{\sf
E}(\gamma_{1})<\infty$ and define

$\gamma^{\varepsilon}_{t}:=\mathsf{S}_{h_{\varepsilon}(t)}\gamma_{t},\qquad
t\in[0,1].$

Then $(\gamma^{\varepsilon}_{t})\in AC^{2}([0,1],{\rm X})$, $t\mapsto{\sf
E}(\gamma^{\varepsilon}_{t})$ belongs to $AC([0,1])$ and it holds

$\begin{split}\frac{1}{2}\int_{0}^{1}|\dot{\gamma}^{\varepsilon}_{t}|^{2}\,{{\mathrm{d}}t}+\frac{\varepsilon^{2}}{2}\int_{0}^{1}|\partial{\sf
E}|^{2}(\gamma^{\varepsilon}_{t})\,{{\mathrm{d}}t}&\leq\frac{1}{2}e^{\lambda^{-}\varepsilon}\int_{0}^{1}|\dot{\gamma}_{t}|^{2}\,{{\mathrm{d}}t}-2\varepsilon{\sf
E}(\gamma^{\varepsilon}_{1/2})\\\ &\qquad+\varepsilon\Big{(}{\sf
E}(\gamma_{0})+{\sf E}(\gamma_{1})\Big{)}.\end{split}$ (3.19)

Note here that $h_{\varepsilon}(0)=h_{\varepsilon}(1)=0$, so that the
endpoints $\gamma^{\varepsilon}_{0}=\gamma_{0}$ and
$\gamma^{\varepsilon}_{1}=\gamma_{1}$ remain unchanged.

###### Proof.

The strategy of proof simply consists in integrating (3.18) between $0$ and
$1$ while integrating by parts of the term
$h_{\varepsilon}^{\prime}(t){\frac{{\mathrm{d}}}{{{\mathrm{d}}t}}}{\sf
E}(\gamma^{\varepsilon}_{t})$, separately on $[0,1/2]$ and $[1/2,1]$. Note
carefully that our specific choice gives
$h_{\varepsilon}^{\prime}=\varepsilon$ and
$h_{\varepsilon}^{\prime}=-\varepsilon$ on these two time intervals,
respectively. Taking into account $e^{-2\lambda h_{\varepsilon}(t)}\leq
e^{\lambda^{-}\varepsilon}$, where $\lambda^{-}:=\max\\{-\lambda,0\\}$, this
procedure yields

$\begin{split}\frac{1}{2}\int_{0}^{1}|\dot{\gamma}^{\varepsilon}_{t}|^{2}\,{{\mathrm{d}}t}+\frac{\varepsilon^{2}}{2}\int_{0}^{1}|\partial{\sf
E}|^{2}(\gamma^{\varepsilon}_{t})\,{{\mathrm{d}}t}&\leq\frac{1}{2}e^{\lambda^{-}\varepsilon}\int_{0}^{1}|\dot{\gamma}_{t}|^{2}\,{{\mathrm{d}}t}-2\varepsilon{\sf
E}(\gamma^{\varepsilon}_{1/2})\\\ &\quad+\varepsilon\Big{(}{\sf
E}(\gamma^{\varepsilon}_{0})+{\sf
E}(\gamma^{\varepsilon}_{1})\Big{)}.\end{split}$

The term $2\varepsilon{\sf E}(\gamma^{\varepsilon}_{1/2})$ simply arises from
the two boundary terms at $t=1/2$ in the two integrations by parts.
(Alternatively, it can be seen as the result of $-\int_{0}^{1}{\sf
E}(\gamma^{\varepsilon}_{t})h^{\prime\prime}(t)$ arising from the integration
by parts in the whole interval $[0,1]$, with the singularity
$h^{\prime\prime}(t)=-2\varepsilon\delta_{1/2}(t)$). However, this argument is
not fully rigorous because all the terms on the left-hand side of (3.18) are
only _locally_ integrable, hence we may not be allowed to integrate them all
the way to $t=0$ and $t=1$.

In order to circumvent this slight issue, choose $\delta\in(0,1/2)$ and carry
out the same argument on $[\delta,1/2]$ and $[1/2,1-\delta]$ rather than on
$[0,1/2]$ and $[1/2,1]$: Integration by parts is now justified by Lemma 3.10
and this provides us with

$\begin{split}\frac{1}{2}\int_{\delta}^{1-\delta}|\dot{\gamma}^{\varepsilon}_{t}|^{2}\,{{\mathrm{d}}t}+\frac{\varepsilon^{2}}{2}\int_{\delta}^{1-\delta}|\partial{\sf
E}|^{2}(\gamma^{\varepsilon}_{t})\,{{\mathrm{d}}t}&\leq\frac{1}{2}e^{\lambda^{-}\varepsilon}\int_{\delta}^{1-\delta}|\dot{\gamma}_{t}|^{2}\,{{\mathrm{d}}t}\\\
&\quad+\varepsilon\Big{(}{\sf E}(\gamma^{\varepsilon}_{\delta})-2{\sf
E}(\gamma^{\varepsilon}_{1/2})+{\sf
E}(\gamma^{\varepsilon}_{1-\delta})\Big{)}.\end{split}$ (3.20)

It is then sufficient to pass to the limit as $\delta\downarrow 0$. By
monotonicity the left-hand side above converges to the left-hand side in
(3.19) and for the same reason so does the first term on the right-hand side,
while by the current choice of $h$ and by Lemma 3.10 $t\mapsto{\sf
E}(\gamma_{t}^{\varepsilon})$ is continuous on the whole interval $[0,1]$, so
that

$\lim_{\delta\downarrow 0}\varepsilon\Big{(}{\sf
E}(\gamma^{\varepsilon}_{\delta})+{\sf
E}(\gamma^{\varepsilon}_{1-\delta})\Big{)}=\varepsilon\Big{(}{\sf
E}(\gamma^{\varepsilon}_{0})+{\sf
E}(\gamma^{\varepsilon}_{1})\Big{)}=\varepsilon\Big{(}{\sf E}(\gamma_{0})+{\sf
E}(\gamma_{1})\Big{)}$

and (3.19) follows.

Finally, since the right-hand side of (3.19) is finite we see that
$|\dot{\gamma}^{\varepsilon}|\in L^{2}(0,1)$ and $|\partial{\sf
E}|(\gamma^{\varepsilon})\in L^{2}(0,1)$. As a consequence
$|\dot{\gamma}^{\varepsilon}|\cdot|\partial{\sf E}(\gamma^{\varepsilon})|\in
L^{1}(0,1)$ in the strong upper-chain rule (3.2), and ${\sf
E}\circ\gamma^{\varepsilon}\in AC([0,1])$ as desired. ∎

##  4 Small-temperature limit and convexity

###  4.1 $\Gamma$-convergence of the Schrödinger problem

Relying on the results of the previous section, we can now turn to Theorem 2.1
and make it rigorous in the metric setting. To this end, let us first
introduce two action functionals: the kinetic energy $\mathcal{A}$ and the
(halved) Fisher information $\mathcal{I}$ along a curve, respectively defined
as

$\mathcal{A}(\gamma):=\frac{1}{2}\int_{0}^{1}|\dot{\gamma}_{t}|^{2}\,{{\mathrm{d}}t}\qquad\textrm{and}\qquad\mathcal{I}(\gamma):=\frac{1}{2}\int_{0}^{1}|\partial{\sf
E}|^{2}(\gamma_{t})\,{{\mathrm{d}}t}$

for all $(\gamma_{t})\in C([0,1],{\rm X})$, where it is understood that
$\mathcal{A}(\gamma)=+\infty$ whenever $\gamma$ is not absolutely continuous.
Given two points $x,y\in{\rm X}$ and a temperature/slowing-down parameter
$\varepsilon>0$, the (metric) Schrödinger problem reads as

$\inf_{(\gamma_{t})\,:\,x\leadsto
y}\Big{\\{}\mathcal{A}(\gamma)+\varepsilon^{2}\mathcal{I}(\gamma)\Big{\\}},$
($\mathrm{Sch}^{\varepsilon}$)

where $(\gamma_{t})\,:\,x\leadsto y$ is a short-hand notation meaning that the
infimum runs over all $(\gamma_{t})\in C([0,1],{\rm X})$ such that
$\gamma_{0}=x$ and $\gamma_{1}=y$. For sake of brevity we also introduce

$\mathcal{A}_{\varepsilon}(\gamma):=\mathcal{A}(\gamma)+\varepsilon^{2}\mathcal{I}(\gamma).$

From ($\mathrm{Sch}^{\varepsilon}$) it is thus clear that the Fisher
information $\mathcal{I}$ acts as a perturbation of $\mathcal{A}$ and this has
a regularizing effect, since minimizers of ($\mathrm{Sch}^{\varepsilon}$) live
within the regular domain ${\rm X}_{1}=D(|\partial{\sf E}|)$.

######  Remark 4.1.

The smoothing effect is well understood for the classic Schrödinger problem in
a regular setting, namely when ${\sf E}$ is the Boltzmann-Shannon relative
entropy and ${\rm X}$ is the Wasserstein space over a smooth Riemannian
manifold. In this case, under mild assumptions on the end-points, minimizers
of ($\mathrm{Sch}^{\varepsilon}$) are curves of absolutely continuous measures
whose densities are bounded, smooth, Lipschitz, with exponentially fast
decaying tails.

In the current metric framework the properties above are meaningless, but
still minimizers of ($\mathrm{Sch}^{\varepsilon}$) are “regular” from a metric
point of view, since as just said they live within $D(|\partial{\sf E}|)$.
Moreover, in Proposition 4.2 we are going to see that ${\sf E}$ is absolutely
continuous along optimal curves.  $\blacksquare$

Let us first deal with the solvability of ($\mathrm{Sch}^{\varepsilon}$).

######  Proposition 4.2.

With the same assumptions and notations as in Setting 3.1 and under Assumption
3.2, for any fixed $x,y\in{\rm X}$ and $\varepsilon>0$ the Schrödinger problem
($\mathrm{Sch}^{\varepsilon}$) is solvable if and only if ${\sf E}(x),{\sf
E}(y)<\infty$ and there exists $(\gamma_{t})\in AC([0,1],{\rm X})$ such that
$\gamma_{0}=x$ and $\gamma_{1}=y$.

As the condition characterizing the solvability of the Schrödinger problem
does not depend on $\varepsilon$, it is clear that if
($\mathrm{Sch}^{\varepsilon}$) is solvable for some $\varepsilon>0$, then it
is actually solvable for all $\varepsilon>0$.

###### Proof.

Assume that the endpoints have finite entropy and that there exists an
absolutely continuous curve $\gamma$ connecting $x$ to $y$. Up to
reparametrization, we can assume that $(\gamma_{t})\in AC^{2}([0,1],{\rm X})$.
Theorem 3.12 thus guarantees that $\mathcal{A}_{\varepsilon}$ is finite along
the regularization $(\gamma^{\varepsilon}_{t})_{t\in[0,1]}$ of this curve and
therefore the variational problem ($\mathrm{Sch}^{\varepsilon}$) is proper.
Let then $(\gamma^{n}_{t})$ be any minimizing sequence and observe that the
kinetic action $\mathcal{A}$ is bounded uniformly in $n$, say
$\mathcal{A}(\gamma^{n})\leq C$ for all $n$. We now observe that for any pair
$0\leq t_{0}<t_{1}\leq 1$ it holds

${\sf
d}(\gamma^{n}_{t_{0}},\gamma^{n}_{t_{1}})\leq\int_{t_{0}}^{t_{1}}|\dot{\gamma}^{n}_{t}|\,{{\mathrm{d}}t}\leq|t_{0}-t_{1}|^{1/2}\Big{(}\int_{t_{0}}^{t_{1}}|\dot{\gamma}^{n}_{t}|^{2}\,{{\mathrm{d}}t}\Big{)}^{1/2}\leq
C|t_{0}-t_{1}|^{1/2}.$ (4.1)

Since the endpoints are fixed, this implies that the set of points
$\gamma^{n}_{t}$ is bounded in $({\rm X},{\sf d})$ uniformly in $n,t$, thus it
is $\sigma$-relatively sequentially compact by Assumption 3.2. By the refined
Arzelà-Ascoli lemma [2, Proposition 3.3.1], there exists a limiting ${\sf
d}$-continuous (actually $1/2$-Hölder continuous) curve $\gamma$ such that

$\gamma^{n}_{t}\stackrel{{\scriptstyle\sigma}}{{\to}}\gamma_{t},\qquad\forall
t\in[0,1].$

We now observe that the kinetic action is lower semicontinuous for this
pointwise-in-time convergence w.r.t. $\sigma$, cf. [3, Section 2.2] (indeed,
${\sf d}$ is lower semicontinuous w.r.t. $\sigma$, hence the $2$-energies of
the finite partitions of $\gamma$ are lower semicontinuous w.r.t. $\sigma$
too, whence the lower semicontinuity of the $2$-energy of $\gamma$ itself).
Moreover $|\partial{\sf E}|^{2}$ is also lower semicontinuous w.r.t. $\sigma$
by hypothesis, and this fact together with Fatou’s lemma gives

$\int_{0}^{1}|\partial{\sf
E}|^{2}(\gamma_{t})\,{{\mathrm{d}}t}\leq\int_{0}^{1}\liminf_{n\to\infty}|\partial{\sf
E}|^{2}(\gamma^{n}_{t})\,{{\mathrm{d}}t}\leq\liminf_{n\to\infty}\int_{0}^{1}|\partial{\sf
E}|^{2}(\gamma^{n}_{t})\,{{\mathrm{d}}t}.$

Therefore $\gamma$ is a minimizer of ($\mathrm{Sch}^{\varepsilon}$).

Conversely, assume that there exists a minimizer, denoted by $\gamma$ (the
following argument actually works for any curve along which
$\mathcal{A}_{\varepsilon}$ is finite and without Assumption 3.2). Then in
particular $t\mapsto|\dot{\gamma}_{t}|$ and $t\mapsto|\partial{\sf
E}|(\gamma_{t})$ belong to $L^{2}(0,1)$ and by (3.2) we see that $t\mapsto{\sf
E}(\gamma_{t})$ is globally absolutely continuous with

$\left|{\frac{{\mathrm{d}}}{{{\mathrm{d}}t}}}({\sf
E}\circ\gamma)(t)\right|\leq|\partial{\sf
E}|(\gamma_{t})\cdot|\dot{\gamma}_{t}|\in L^{1}(0,1).$

The fact that $(|\dot{\gamma}_{t}|)\in L^{2}(0,1)\subset L^{1}(0,1)$ trivially
implies $(\gamma_{t})\in AC([0,1],{\rm X})$, whereas the fact that
$t\mapsto|\partial{\sf E}|(\gamma_{t})$ belongs to $L^{2}(0,1)$ also implies
that $|\partial{\sf E}|(\gamma_{t})$ is finite for a.e. $t\in[0,1]$ and a
fortiori so is ${\sf E}(\gamma_{t})$, since $D(|\partial{\sf E}|)\subset
D({\sf E})$. Hence let $t^{*}\in(0,1)$ be any point satifying ${\sf
E}(\gamma_{t^{*}})<\infty$ and note that together with (3.2) this gives the
following global upper bound valid for all $t<t^{*}$

${\sf E}(\gamma_{t})\leq{\sf
E}(\gamma_{t^{*}})+\int_{t}^{t^{*}}\left|{\frac{{\mathrm{d}}}{{{\mathrm{d}}t}}}({\sf
E}\circ\gamma)(t)\right|{{\mathrm{d}}t}\leq{\sf
E}(\gamma_{t^{*}})+\int_{0}^{1}\left|{\frac{{\mathrm{d}}}{{{\mathrm{d}}t}}}({\sf
E}\circ\gamma)(t)\right|{{\mathrm{d}}t}=:\overline{{\sf E}}<\infty.$

As a consequence, and taking also into account the facts that
$t\mapsto\gamma_{t}$ is ${\sf d}$-continuous and ${\sf E}$ is lower
semicontinuous, we get

${\sf E}(\gamma_{0})={\sf E}\big{(}\lim_{t\to
0}\gamma_{t}\big{)}\leq\liminf_{t\to 0}{\sf E}(\gamma_{t})\leq\overline{{\sf
E}}$

and the proof is thus complete, as the same argument applies _mutatis
mutandis_ for $t=1$ too. ∎

We now fix $x,y\in{\rm X}$ and let $C([0,1],{\rm
X})\ni\gamma\mapsto\iota_{01}(\gamma)$ denote the convex indicator of the
endpoint constraints, i.e.

$\iota_{01}(\gamma)=\begin{cases}0&\mbox{if }\gamma_{0}=x\mbox{ and
}\gamma_{1}=y,\\\ +\infty&\mbox{otherwise.}\end{cases}$

With this said, we can finally state our $\Gamma$-convergence result, where
the finite-entropy assumption on the endpoints is motivated by the previous
proposition.

######  Theorem 4.3.

With the same assumptions and notations as in Setting 3.1, if $x,y\in{\rm X}$
are such that ${\sf E}(x),{\sf E}(y)<\infty$, then

$\Gamma-\lim_{\varepsilon\to
0}\Big{\\{}\mathcal{A}_{\varepsilon}+\iota_{01}\Big{\\}}=\mathcal{A}+\iota_{01}$

for the uniform convergence on $C([0,1],{\rm X})$. If Assumption 3.2 holds,
then the $\Gamma$-convergence also takes place w.r.t. the pointwise-in-time
$\sigma$-topology.

###### Proof.

The $\Gamma-\liminf$ inequality is rather clear, since the kinetic energy
$\mathcal{A}$ is lower semicontinuous both w.r.t. uniform-in-time ${\sf
d}$-convergence and pointwise-in-time $\sigma$-convergence: for the former
topology the fact is well known, for the latter it has been discussed in the
proof of Proposition 4.2. An analogous claim is also true for the convex
indicator $\iota_{01}$. As a consequence, we have that for any
$\gamma^{\varepsilon}$ converging to $\gamma$ uniformly in time in the metric
topology or pointwise in time in the topology $\sigma$ (if applicable) it
holds

$\begin{split}\mathcal{A}(\gamma)+\iota_{01}(\gamma)&\leq\liminf_{\varepsilon\downarrow
0}\mathcal{A}(\gamma^{\varepsilon})+\liminf_{\varepsilon\downarrow
0}\iota_{01}(\gamma^{\varepsilon})\leq\liminf_{\varepsilon\downarrow
0}\Big{\\{}\mathcal{A}(\gamma^{\varepsilon})+\iota_{01}(\gamma^{\varepsilon})\Big{\\}}\\\
&\leq\liminf_{\varepsilon\downarrow
0}\Big{\\{}\mathcal{A}(\gamma^{\varepsilon})+\varepsilon^{2}\mathcal{I}(\gamma^{\varepsilon})+\iota_{01}(\gamma^{\varepsilon})\Big{\\}}=\liminf_{\varepsilon\downarrow
0}\Big{\\{}\mathcal{A}_{\varepsilon}(\gamma^{\varepsilon})+\iota_{01}(\gamma^{\varepsilon})\Big{\\}},\end{split}$

whence the desired $\Gamma-\liminf$ inequality.

For the $\Gamma-\limsup$, take any $(\gamma_{t})\in AC^{2}([0,1],{\rm X})$
connecting $x$ to $y$ (if it does not exist, then there is nothing to prove).
Then Theorem 3.12 precisely provides a recovery sequence
$\gamma^{\varepsilon}_{t}:=\mathsf{S}_{h_{\varepsilon}(t)}\gamma_{t}$ with
$h_{\varepsilon}$ defined as therein, both for the uniform-in-time ${\sf
d}$-convergence and the pointwise-in-time $\sigma$-convergence (the latter is
an easy consequence of the former by Remark 3.4). To prove this claim, note
that for any $n\in\mathbb{N}$ there exist $t_{1},...,t_{k}\in[0,1]$ such that,
for any $t\in[0,1]$, ${\sf d}(\gamma_{t},\gamma_{t_{i}})<1/n$ for at least one
$t_{i}$; in addition, since $\gamma_{t}^{\varepsilon}\to\gamma_{t}$ for all
$t\in[0,1]$ there exists $\varepsilon_{n}$ small enough such that ${\sf
d}(\gamma_{t_{i}},\gamma_{t_{i}}^{\varepsilon})<1/n$ for all
$\varepsilon<\varepsilon_{n}$ and $i=1,...,k$. As a consequence, taking (3.3)
into account,

$\begin{split}{\sf d}(\gamma_{t},\gamma_{t}^{\varepsilon})&\leq{\sf
d}(\gamma_{t},\gamma_{t_{i}})+{\sf
d}(\gamma_{t_{i}},\mathsf{S}_{h_{\varepsilon}(t)}\gamma_{t_{i}})+{\sf
d}(\mathsf{S}_{h_{\varepsilon}(t)}\gamma_{t_{i}},\gamma_{t}^{\varepsilon})\\\
&\leq{\sf d}(\gamma_{t},\gamma_{t_{i}})+{\sf
d}(\gamma_{t_{i}},\mathsf{S}_{h_{\varepsilon}(t)}\gamma_{t_{i}})+e^{-\lambda
h_{\varepsilon}(t)}{\sf d}(\gamma_{t},\gamma_{t_{i}})\\\
&\leq\frac{1}{n}(2+e^{\lambda^{-}\varepsilon/2})\end{split}$

for all $t\in[0,1]$ and $\varepsilon<\varepsilon_{n}$ and by the arbitrariness
of $n$ we conclude that $\gamma^{\varepsilon}\to\gamma$ uniformly.
Furthermore, the $\limsup$ inequality can be proved as follows:

$\begin{split}\limsup_{\varepsilon\downarrow
0}\Big{\\{}\mathcal{A}_{\varepsilon}(\gamma^{\varepsilon})&+\iota_{01}(\gamma^{\varepsilon})\Big{\\}}=\limsup_{\varepsilon\downarrow
0}\Big{\\{}\mathcal{A}(\gamma^{\varepsilon})+\varepsilon^{2}\mathcal{I}(\gamma^{\varepsilon})+0\Big{\\}}\\\
&\overset{\eqref{eq:upper_bound_recovery_metric}}{\leq}\limsup_{\varepsilon\downarrow
0}\Big{\\{}e^{\lambda^{-}\varepsilon}\mathcal{A}(\gamma)-2\varepsilon{\sf
E}(\gamma^{\varepsilon}_{1/2})+\varepsilon\big{(}{\sf E}(x)+{\sf
E}(y)\big{)}\Big{\\}}\\\ &\leq\limsup_{\varepsilon\downarrow
0}\Big{\\{}e^{\lambda^{-}\varepsilon}\mathcal{A}(\gamma)+\varepsilon\big{(}{\sf
E}(x)+{\sf E}(y)\big{)}\Big{\\}}-2\liminf_{\varepsilon\downarrow
0}\varepsilon{\sf E}(\gamma^{\varepsilon}_{1/2})\\\
&\leq\mathcal{A}(\gamma)=\mathcal{A}(\gamma)+\iota_{01}(\gamma),\end{split}$

where the third inequality comes from the fact that, for any
$\varepsilon\downarrow 0$, $(\gamma_{1/2}^{\varepsilon})$ is contained in a
bounded set and by assumption ${\sf E}$ is bounded from below on bounded sets,
whence ${\sf E}(\gamma^{\varepsilon}_{1/2})\geq c$ for some $c\in\mathbb{R}$.
The proof is thus complete. ∎

As an easy consequence of this result we obtain the following:

######  Corollary 4.4.

With the same assumptions and notations as in Setting 3.1 and under the
further requirements that Assumption 3.2 holds and the Schrödinger problem
($\mathrm{Sch}^{\varepsilon}$) relative to $x,y\in{\rm X}$ is solvable, let
$\varepsilon_{k}\downarrow 0$ and $\omega^{k}$ be a minimizer of the
corresponding Schrödinger problem ($\mathrm{Sch}^{\varepsilon}$) with
$\varepsilon=\varepsilon_{k}$.

Then

$\lim_{k\to\infty}\Big{\\{}\mathcal{A}(\omega^{k})+\varepsilon_{k}^{2}\mathcal{I}(\omega^{k})\Big{\\}}=\inf_{(\gamma_{t})\,:\,x\leadsto
y}\mathcal{A}(\gamma).$

Moreover, there exists $\omega^{0}\in C([0,1],{\rm X})$ such that, up to a
subsequence, $\omega^{k}\to\omega^{0}$ in the pointwise-in-time
$\sigma$-topology and

$\mathcal{A}(\omega^{0})=\inf_{(\gamma_{t})\,:\,x\leadsto
y}\mathcal{A}(\gamma).$

###### Proof.

Recall that, under a mild equi-coercivity condition, $\Gamma$-convergence
precisely guarantees that the limit of the optimal values of the approximating
problems is the optimal value of the limit problem and limits of minimizers
are minimizers, cf. [16, Theorem 1.21]. In view of Theorem 4.3 and [16,
Theorem 1.21], for the mild equi-coercivity condition to hold it suffices to
prove that the set of minimizers $\\{\omega^{k}\\}$ is relatively compact in
the pointwise-in-time $\sigma$-topology. To this aim, the kinetic energies of
the curves $\omega^{k}$ are uniformly bounded since

$\mathcal{A}(\omega^{k})\leq\mathcal{A}(\omega^{k})+\varepsilon_{k}^{2}\mathcal{I}(\omega^{k})\leq\mathcal{A}(\omega^{\overline{\varepsilon}})+\varepsilon_{k}^{2}\mathcal{I}(\omega^{\overline{\varepsilon}})\leq\mathcal{A}(\omega^{\overline{\varepsilon}})+\overline{\varepsilon}^{2}\mathcal{I}(\omega^{\overline{\varepsilon}})<+\infty,$

where $\omega^{\overline{\varepsilon}}$ is the minimizer for the problem with
$\varepsilon=\overline{\varepsilon}:=\sup_{k}{\varepsilon_{k}}$. Arguing as in
the proof of Proposition 4.2, we deduce that there exists a continuous curve
$(\omega^{0}_{t})_{t\in[0,1]}$ connecting $x$ and $y$ such that, up to
extracting a suitable subsequence, $\omega^{k}_{t}\to\omega^{0}_{t}$ w.r.t.
$\sigma$ as $k\to\infty$ for all $t\in[0,1]$. ∎

######  Remark 4.5.

Note that in Corollary 4.4 the curve $\omega^{0}$ is length-minimizing but not
necessarily distance-minimizing, namely it needs not be a geodesic between $x$
and $y$, since we only know that

$\inf_{(\gamma_{t})\,:\,x\leadsto y}\mathcal{A}(\gamma)\geq\frac{1}{2}{\sf
d}^{2}(x,y)$

and the inequality might be strict, e.g. if ${\rm X}$ is a non-convex subset
of $\mathbb{R}^{d}$. However, if $({\rm X},{\sf d})$ is a length metric space,
i.e. for all $x,y\in{\rm X}$ and $\varepsilon>0$ there exists $(\gamma_{t})\in
AC([0,1],{\rm X})$ such that $\gamma_{0}=x$, $\gamma_{1}=y$ and
$\ell(\gamma)\leq{\sf d}(x,y)+\varepsilon$, then the inequality above turns
out to be an identity and, as a consequence, $\omega^{0}$ is a geodesic. This
means that for any two points having finite energy there always exists a
geodesic connecting them.  $\blacksquare$

When the endpoints have infinite entropy, the following variant of Theorem 4.3
may be useful:

######  Theorem 4.6.

With the same assumptions and notations as in Setting 3.1, let $x,y\in{\rm X}$
with possibly ${\sf E}(x),{\sf E}(y)=+\infty$ and for any fixed
$(\varepsilon_{n})_{n\in\mathbb{N}}$, $\varepsilon_{n}\downarrow 0$, let
$(\eta_{n})_{n\in\mathbb{N}}$ be converging to 0 slowly enough so that

$\varepsilon_{n}\left({\sf E}(\gamma_{0}^{n})+{\sf
E}(\gamma_{1}^{n}\right))\to
0\qquad\mbox{with}\qquad\gamma_{0}^{n}:=\mathsf{S}_{\eta_{n}}x,\quad\gamma_{1}^{n}:=\mathsf{S}_{\eta_{n}}y.$

Then

$\Gamma-\lim\limits_{n\to\infty}\Big{\\{}\mathcal{A}_{\varepsilon_{n}}+\iota^{n}_{01}\Big{\\}}=\mathcal{A}+\iota_{01},$

for the uniform convergence on $C([0,1],{\rm X})$. If Assumption 3.2 holds,
then the $\Gamma$-convergence also takes place w.r.t. the pointwise-in-time
$\sigma$-topology. Here $\iota_{01}^{n}$ and $\iota_{01}$ are the convex
indicators of the endpoint constraints for $\gamma_{0}^{n},\gamma_{1}^{n}$ and
$x,y$, respectively.

###### Proof.

The proof of the $\Gamma-\liminf$ is almost identical to that in Theorem 4.3,
with the only extra observation that

$\iota_{01}(\gamma)\leq\liminf_{n\to\infty}\iota_{01}^{n}(\gamma^{n}).$

For the $\Gamma-\limsup$, observe that if there does not exist
$(\gamma_{t})\in AC^{2}([0,1],{\rm X})$ joining $x$ and $y$, then there is
nothing to prove. Hence let us suppose that at least one curve
$(\gamma_{t})\in AC^{2}([0,1],{\rm X})$ connecting $x$ and $y$ exists, fix it
and note that Theorem 3.12 applied to the curve
$\mathsf{S}_{\eta_{n}}\gamma_{t}$ still provides a recovery sequence
$\gamma^{\varepsilon_{n}}_{t}:=\mathsf{S}_{\eta_{n}+h_{\varepsilon_{n}}(t)}\gamma_{t}$
with the same choice $h_{\varepsilon_{n}}(t)=\varepsilon_{n}\min\\{t,1-t\\}$
as before. Indeed, on the one hand

$\begin{split}\limsup_{n\to\infty}\Big{\\{}&\mathcal{A}_{\varepsilon_{n}}(\gamma^{\varepsilon_{n}})+\iota_{01}^{n}(\gamma^{\varepsilon_{n}})\Big{\\}}=\limsup_{n\to\infty}\Big{\\{}\mathcal{A}(\gamma^{\varepsilon_{n}})+\varepsilon^{2}_{n}\mathcal{I}(\gamma^{\varepsilon_{n}})+0\Big{\\}}\\\
&\stackrel{{\scriptstyle\eqref{eq:upper_bound_recovery_metric}}}{{\leq}}\limsup_{n\to\infty}\Big{\\{}e^{\lambda^{-}\varepsilon_{n}}\mathcal{A}(\mathsf{S}_{\eta_{n}}\gamma)-2\varepsilon_{n}{\sf
E}(\gamma_{1/2}^{\varepsilon_{n}})+\varepsilon_{n}\big{(}{\sf
E}(\gamma_{0}^{n})+{\sf E}(\gamma_{1}^{n})\big{)}\Big{\\}}\\\
&\leq\limsup_{n\to\infty}\Big{\\{}e^{\lambda^{-}\varepsilon_{n}}\mathcal{A}(\mathsf{S}_{\eta_{n}}\gamma)+\varepsilon_{n}\big{(}{\sf
E}(\gamma_{0}^{n})+{\sf
E}(\gamma_{1}^{n})\big{)}\Big{\\}}-2\liminf_{\varepsilon\downarrow
0}\varepsilon{\sf E}(\gamma^{\varepsilon}_{1/2})\\\
&\leq\limsup_{n\to\infty}\mathcal{A}(\mathsf{S}_{\eta_{n}}\gamma)\leq\limsup_{n\to\infty}e^{-2\lambda\eta_{n}}\mathcal{A}(\gamma)=\mathcal{A}(\gamma),\end{split}$

where the third inequality follows by the same argument adopted in the proof
of the previous theorem and the last one is due to (3.18) with
$h(t)\equiv\eta_{n}$. On the other hand,
$\gamma^{\varepsilon_{n}}_{t}\to\gamma_{t}$ uniformly in $t\in[0,1]$ in the
${\sf d}$-topology and, if Assumption 3.2 holds, for all $t\in[0,1]$ w.r.t.
$\sigma$: the argument described in the previous proof applies also here
verbatim. ∎

As conclusion, in the next proposition we show that any ${\rm EVI}$-gradient
flow is a solution of the Schrödinger problem with suitable endpoints.
Intuitively this is clear, because up to a rescaling factor $\varepsilon^{2}$
both the trajectories of the gradient flow of ${\sf E}$ and the solutions to
($\mathrm{Sch}^{\varepsilon}$) must formally satisfy the same Newton equation,
namely $\ddot{\gamma}_{t}=-\nabla\Phi(\gamma_{t})$ where the potential $\Phi$
is given by (minus) the Fisher information $-|\partial{\sf E}|^{2}$, cf. [33,
Remark 6]. This is also in complete analogy with the standard Schrödinger
problem, which includes the heat flow as a particular entropic interpolation.

######  Proposition 4.7.

With the same assumptions and notations as in Setting 3.1, fix
$\varepsilon>0$. Then for all $x,y\in{\rm X}$ the following lower bound on the
optimal value of ($\mathrm{Sch}^{\varepsilon}$) holds

$\inf_{(\gamma_{t})\,:\,x\leadsto
y}\mathcal{A}_{\varepsilon}(\gamma)\geq\varepsilon\big{|}{\sf E}(x)-{\sf
E}(y)\big{|}.$ (4.2)

If either $y=\mathsf{S}_{\varepsilon}x$ or $x=\mathsf{S}_{\varepsilon}y$, then
equality is achieved. In the former case the curve $[0,1]\ni
t\mapsto\hat{\gamma}_{t}:=\mathsf{S}_{\varepsilon t}x$ is a minimizer in the
Schrödinger problem and the optimal value is

$\inf_{(\gamma_{t})\,:\,x\leadsto
y}\mathcal{A}_{\varepsilon}(\gamma)=\varepsilon\big{(}{\sf E}(x)-{\sf
E}(\mathsf{S}_{\varepsilon}x)\big{)}.$

An analogous statement holds when $x=\mathsf{S}_{\varepsilon}y$.

###### Proof.

By (3.2) and Young’s inequality it follows that for any
$(\tilde{\gamma}_{t})\in AC^{2}([0,\varepsilon],{\rm X})$ joining $x$ and $y$
(if it exists; if not, (4.2) is trivial) it holds

$\big{|}{\sf E}(\tilde{\gamma}_{0})-{\sf
E}(\tilde{\gamma}_{\varepsilon})\big{|}\leq\frac{1}{2}\int_{0}^{\varepsilon}|\dot{\tilde{\gamma}}_{t}|^{2}\,{{\mathrm{d}}t}+\frac{1}{2}\int_{0}^{\varepsilon}|\partial{\sf
E}|^{2}(\tilde{\gamma}_{t})\,{{\mathrm{d}}t}.$

By setting $\gamma_{t}:=\tilde{\gamma}_{\varepsilon t}$, $t\in[0,1]$, and by
the arbitrariness of $\tilde{\gamma}$ we thus see that for all
$(\gamma_{t})\in AC^{2}([0,1],{\rm X})$ joining $x$ and $y$ we have

$\varepsilon\big{|}{\sf E}(\gamma_{0})-{\sf
E}(\gamma_{1})\big{|}\leq\frac{1}{2}\int_{0}^{1}|\dot{\gamma}_{t}|^{2}\,{{\mathrm{d}}t}+\frac{\varepsilon^{2}}{2}\int_{0}^{1}|\partial{\sf
E}|^{2}(\gamma_{t})\,{{\mathrm{d}}t},$

so that

$\varepsilon\big{|}{\sf E}(\gamma_{0})-{\sf
E}(\gamma_{1})\big{|}\leq\inf_{(\gamma_{t})\,:\,x\leadsto
y}\mathcal{A}_{\varepsilon}(\gamma).$

Now assume that $y=\mathsf{S}_{\varepsilon}x$: integrating (3.4) for the ${\rm
EVI}_{\lambda}$-gradient flow $\hat{\gamma}$ (paying attention to the
rescaling factor $\varepsilon$) between 0 and 1 we get

$\mathcal{A}_{\varepsilon}(\hat{\gamma})=\frac{1}{2}\int_{0}^{1}|\dot{\hat{\gamma}}_{t}|^{2}\,{{\mathrm{d}}t}+\frac{\varepsilon^{2}}{2}\int_{0}^{1}|\partial{\sf
E}|^{2}(\hat{\gamma}_{t})\,{{\mathrm{d}}t}=\varepsilon\big{(}{\sf E}(x)-{\sf
E}(y)\big{)}=\varepsilon\big{|}{\sf E}(x)-{\sf E}(y)\big{|},$

where the last equality comes from the fact that $t\mapsto{\sf
E}(\mathsf{S}_{t}x)$ is non-increasing, as a consequence of (3.4). Combining
this identity with (4.2) yields the conclusion. ∎

###  4.2 Displacement convexity

In analogy with Section 2.2, in this short section we establish the geodesic
$\lambda$-convexity of ${\sf E}$. As already explained in the Introduction,
although the result is known (cf. [27, Theorem 3.2]), our proof is independent
and new and is a further evidence of the wide range of applications of the
Schrödinger problem. Let us stress once more that all the properties of ${\rm
EVI}_{\lambda}$-gradient flows stated in Section 3.1 and used so far do not
rely on geodesic $\lambda$-convexity, whence the genuine independence of our
approach.

######  Theorem 4.8.

With the same assumptions and notations as in Setting 3.1, the potential ${\sf
E}$ is $\lambda$-convex along _any_ geodesic.

###### Proof.

Let $(\gamma_{t})$ be any constant-speed geodesic. We want to prove that

${\sf E}(\gamma_{\theta})\leq(1-\theta){\sf E}(\gamma_{0})+\theta{\sf
E}(\gamma_{1})-\frac{\lambda}{2}\theta(1-\theta){\sf
d}^{2}(\gamma_{0},\gamma_{1}),\qquad\forall\theta\in[0,1].$

We will establish this inequality by carefully estimating at order one as
$\varepsilon\downarrow 0$ the defect of optimality, in the geodesic problem
from $\gamma_{0}$ to $\gamma_{1}$, of a suitably regularized version
$(\gamma^{\varepsilon}_{t})$ of the geodesic.

If ${\sf E}(\gamma_{0})=+\infty$ or ${\sf E}(\gamma_{1})=+\infty$ there is
nothing to prove, so we can assume without loss of generality that both
endpoints have finite entropy. If $\theta=0$ or $\theta=1$ the inequality is
trivial as well. Fix then an arbitrary parameter $\theta\in(0,1)$ and let

$H_{\theta}(t):=\left\\{\begin{array}[]{ll}\displaystyle{\frac{1}{\theta}}t&\mbox{if
}t\in[0,\theta],\\\ -\displaystyle{\frac{1}{1-\theta}(t-1)}&\mbox{if
}t\in[\theta,1].\end{array}\right.$

be the hat function centered at $t=\theta$ with height $1$ and vanishing at
$t=0,1$. Setting $h(t):=\varepsilon H_{\theta}(t)$ for small $\varepsilon>0$,
let $(\gamma^{\varepsilon}_{t})$ be the curve constructed as in Lemma 3.6,
i.e.

$\gamma^{\varepsilon}_{t}:=\mathsf{S}_{h(t)}\gamma_{t},\qquad\mbox{for all
}t\in[0,1].$

Arguing as in the proof of Theorem 3.12, it is easily verified that with the
current choice of $h$ it is still true that
$t\mapsto|\dot{\gamma}_{t}^{\varepsilon}|$ and $t\mapsto|\partial{\sf
E}|(\gamma_{t}^{\varepsilon})$ belong to $AC^{2}([0,1],{\rm X})$ and
$t\mapsto{\sf E}(\gamma_{t}^{\varepsilon})$ to $AC([0,1])$, so that we can
integrate (3.18) in time on the whole interval $[0,1]$. Discarding the non-
negative term $|h^{\prime}(t)|^{2}|\partial{\sf
E}|^{2}(\gamma^{\varepsilon}_{t})$ and using the optimality of the geodesic
$\gamma$ (namely its optimality between $\gamma_{0}$ and $\gamma_{1}$) give

$\begin{split}0&\leq\frac{1}{2}\int_{0}^{1}|\dot{\gamma}_{t}^{\varepsilon}|^{2}\,{{\mathrm{d}}t}-\frac{1}{2}\int_{0}^{1}|\dot{\gamma}_{t}|^{2}\,{{\mathrm{d}}t}\\\
&\overset{\eqref{eq:pseudo_Riemannian_estimate}}{\leq}-\int_{0}^{1}h^{\prime}(t){\frac{{\mathrm{d}}}{{{\mathrm{d}}t}}}{\sf
E}(\gamma^{\varepsilon}_{t})\,{{\mathrm{d}}t}+\frac{1}{2}\int_{0}^{1}\left(e^{-2\lambda
h(t)}-1\right)|\dot{\gamma}_{t}|^{2}\,{{\mathrm{d}}t}\\\
&=-\varepsilon\int_{0}^{1}H_{\theta}^{\prime}(t){\frac{{\mathrm{d}}}{{{\mathrm{d}}t}}}{\sf
E}(\gamma^{\varepsilon}_{t})\,{{\mathrm{d}}t}+\frac{{\sf
d}^{2}(\gamma_{0},\gamma_{1})}{2}\int_{0}^{1}\left(e^{-2\varepsilon\lambda
H_{\theta}(t)}-1\right)\,{{\mathrm{d}}t},\end{split}$

where the last equality follows from the constant speed property of the
geodesic $\gamma$, namely $|\dot{\gamma}_{t}|={\sf d}(\gamma_{0},\gamma_{1})$.
Dividing by $\varepsilon>0$ and leveraging the explicit piecewise constant
values of $H_{\theta}^{\prime}(t)$ on each interval $(0,\theta)$ and
$(\theta,1)$ gives

$\begin{split}0&\leq-\int_{0}^{1}H_{\theta}^{\prime}(t){\frac{{\mathrm{d}}}{{{\mathrm{d}}t}}}{\sf
E}(\gamma^{\varepsilon}_{t})\,{{\mathrm{d}}t}+\frac{{\sf
d}^{2}(\gamma_{0},\gamma_{1})}{2}\underbrace{\int_{0}^{1}\frac{e^{-2\varepsilon\lambda
H_{\theta}(t)}-1}{\varepsilon}\,{{\mathrm{d}}t}}_{:=I_{\varepsilon}}\\\
&=-\int_{0}^{\theta}\frac{1}{\theta}{\frac{{\mathrm{d}}}{{{\mathrm{d}}t}}}{\sf
E}(\gamma^{\varepsilon}_{t})\,{{\mathrm{d}}t}+\int_{\theta}^{1}\frac{1}{1-\theta}{\frac{{\mathrm{d}}}{{{\mathrm{d}}t}}}{\sf
E}(\gamma^{\varepsilon}_{t})\,{{\mathrm{d}}t}+\frac{{\sf
d}^{2}(\gamma_{0},\gamma_{1})}{2}I_{\varepsilon}\\\
&=\frac{1}{\theta}\Big{(}{\sf E}(\gamma_{0})-{\sf
E}(\gamma^{\varepsilon}_{\theta})\Big{)}+\frac{1}{1-\theta}\Big{(}{\sf
E}(\gamma_{1})-{\sf E}(\gamma^{\varepsilon}_{\theta})\Big{)}+\frac{{\sf
d}^{2}(\gamma_{0},\gamma_{1})}{2}I_{\epsilon}.\end{split}$

Now let us multiply by $\theta(1-\theta)>0$ and rearrange the terms in order
to get

${\sf E}(\gamma^{\varepsilon}_{\theta})\leq(1-\theta){\sf
E}(\gamma_{0})+\theta{\sf E}(\gamma_{1})+\theta(1-\theta)\frac{{\sf
d}^{2}(\gamma_{0},\gamma_{1})}{2}I_{\varepsilon}.$

It is easy to check that $\int_{0}^{1}H_{\theta}(t){\mathrm{d}}t=\frac{1}{2}$
for all $\theta$, so that

$\lim_{\varepsilon\downarrow
0}I_{\varepsilon}=-2\lambda\int_{0}^{1}H_{\theta}(t)\,{{\mathrm{d}}t}=-\lambda.$

On the other hand, by definition of $\gamma^{\varepsilon}$ and since
$h(\theta)=\varepsilon\to 0$ it is clear that
$\gamma^{\varepsilon}_{\theta}=\mathsf{S}_{h(\theta)}\gamma_{\theta}=\mathsf{S}_{\varepsilon}\gamma_{\theta}\to\gamma_{\theta}$
in ${\rm X}$ (an ${\rm EVI}_{\lambda}$-gradient flow is continuous up to
$t=0$). By lower semicontinuity of ${\sf E}$ this yields

${\sf E}(\gamma_{\theta})\leq\liminf_{\varepsilon\downarrow 0}{\sf
E}(\gamma^{\varepsilon}_{\theta})\leq(1-\theta){\sf E}(\gamma_{0})+\theta{\sf
E}(\gamma_{1})-\frac{\lambda}{2}\theta(1-\theta){\sf
d}^{2}(\gamma_{0},\gamma_{1}),$

whence the conclusion. ∎

##  5 Derivative of the cost

As a main application of the $\Gamma$-convergence results contained in Theorem
4.3 and Corollary 4.4 (and, in a wider sense, of their strategy of proof), in
this section we investigate the dependence of the optimal value of the
Schrödinger problem ($\mathrm{Sch}^{\varepsilon}$) on the regularization
parameter $\varepsilon$, focusing in particular on the regularity as a
function of $\varepsilon$ and on the behaviour in the small-time regime. More
precisely, and denoting

$\mathscr{C}_{\varepsilon}(x,y):=\inf_{(\gamma_{t})\,:\,x\leadsto
y}\Big{\\{}\mathcal{A}(\gamma)+\varepsilon^{2}\mathcal{I}(\gamma)\Big{\\}},\qquad\forall\varepsilon\geq
0$

the optimal entropic cost, we show that
$\varepsilon\mapsto\mathscr{C}_{\varepsilon}(x,y)$ is (locally) absolutely
continuous and admits explicit left and right derivatives in a pointwise
sense. Moreover, since $\mathscr{C}_{\varepsilon}(x,y)\to\mathscr{C}_{0}(x,y)$
as $\varepsilon\downarrow 0$ by Corollary 4.4, we aim at measuring the error
$\mathscr{C}_{\varepsilon}(x,y)-\mathscr{C}_{0}(x,y)$ and studying the
minimizers of the unperturbed problem $\mathscr{C}_{0}(x,y)$ selected by
$\Gamma$-convergence. Since we focus here on the dependence on $\varepsilon$
we will assume throughout the whole Section 5 and without further mention the
well-posedness of the $\varepsilon$-Schrödinger problem:

######  Assumption 5.1.

Fix $x,y\in{\rm X}$ and suppose that for some (hence for any, by Proposition
4.2) $\varepsilon>0$ the Schrödinger problem ($\mathrm{Sch}^{\varepsilon}$)
admits at least one minimizer, in other words the infimum is attained in the
definition of $\mathscr{C}_{\varepsilon}(x,y)$.

We accordingly denote the set of $\varepsilon$-minimizers as

$\Lambda_{\varepsilon}(x,y):=\Big{\\{}\omega\in AC^{2}([0,1],{\rm
X})\,:\,\omega_{0}=x,\,\omega_{1}=y\quad\mbox{and}\quad\mathcal{A}_{\varepsilon}(\omega)=\mathscr{C}_{\varepsilon}(x,y)\Big{\\}}.$

Let us start the analysis with a preliminary monotonicity statement for the
Fisher information and the entropic cost, which generalizes [25, Lemma 3.3].

######  Lemma 5.2.

With the same assumptions and notations as in Setting 3.1 and for any
$0\leq\varepsilon_{1}<\varepsilon_{2}<\infty$ there holds

$\inf_{\Lambda_{\varepsilon_{1}}(x,y)}\mathcal{I}\geq\sup_{\Lambda_{\varepsilon_{2}}(x,y)}\mathcal{I},$

with possibly $\inf_{\Lambda_{0}(x,y)}\mathcal{I}=+\infty$. Moreover,
$\varepsilon\mapsto\mathscr{C}_{\varepsilon}(x,y)$ is monotone non-decreasing
on $[0,\infty)$.

###### Proof.

Let $\varepsilon_{1},\varepsilon_{2}$ as in the statement and choose
$\omega^{i}\in\Lambda_{\varepsilon_{i}}(x,y)$ for $i=1,2$, so that by
optimality

$\begin{split}\mathcal{A}(\omega^{1})+\varepsilon_{1}^{2}\mathcal{I}(\omega^{1})&\leq\mathcal{A}(\omega^{2})+\varepsilon_{1}^{2}\mathcal{I}(\omega^{2}),\\\
\mathcal{A}(\omega^{2})+\varepsilon_{2}^{2}\mathcal{I}(\omega^{2})&\leq\mathcal{A}(\omega^{1})+\varepsilon_{2}^{2}\mathcal{I}(\omega^{1}).\end{split}$

Summing these inequalities and dividing by
$\varepsilon_{2}^{2}-\varepsilon_{1}^{2}$ we obtain
$\mathcal{I}(\omega^{1})\geq\mathcal{I}(\omega^{2})$, and since
$\omega^{1}\in\Lambda_{\varepsilon_{1}}$ and
$\omega^{2}\in\Lambda_{\varepsilon_{2}}$ are arbitrary the desired conclusion
follows. As regards the last part of the statement, it is sufficient to note
that since $\omega^{i}$ are minimizers of their respective problems and
$\varepsilon_{1}<\varepsilon_{2}$,

$\mathscr{C}_{\varepsilon_{1}}(x,y)=\mathcal{A}(\omega^{1})+\varepsilon_{1}^{2}\mathcal{I}(\omega^{1})\leq\mathcal{A}(\omega^{2})+\varepsilon_{1}^{2}\mathcal{I}(\omega^{2})\leq\mathcal{A}(\omega^{2})+\varepsilon_{2}^{2}\mathcal{I}(\omega^{2})=\mathscr{C}_{\varepsilon_{2}}(x,y).$

∎

Let us then extend Theorem 4.3 and Corollary 4.4 from $\varepsilon=0$ to any
$\varepsilon\geq 0$.

######  Proposition 5.3.

With the same assumptions and notations as in Setting 3.1 and under the
additional Assumption 3.2, for any $\varepsilon>0$ there holds

$\Gamma-\lim_{\varepsilon^{\prime}\to\varepsilon}\Big{\\{}\mathcal{A}_{\varepsilon^{\prime}}+\iota_{01}\Big{\\}}=\mathcal{A}_{\varepsilon}+\iota_{01}$
(5.1)

for the pointwise-in-time $\sigma$-topology and

$\lim_{\varepsilon^{\prime}\to\varepsilon}\mathscr{C}_{\varepsilon^{\prime}}(x,y)=\mathscr{C}_{\varepsilon}(x,y).$

Moreover, for any $\varepsilon_{k}\to\varepsilon$ and any minimizer
$\omega^{k}\in\Lambda_{\varepsilon_{k}}(x,y)$, there exists
$\omega\in\Lambda_{\varepsilon}(x,y)$ such that, up to a subsequence,

$\omega_{t}^{k}\stackrel{{\scriptstyle\sigma}}{{\to}}\omega_{t},\qquad\forall
t\in[0,1]$

as $k\to\infty$.

###### Proof.

It is sufficient to prove (5.1), as the other properties follow by a verbatim
application of the arguments in the proof of Corollary 4.4.

Fix $\varepsilon$ and take $\varepsilon^{\prime}\to\varepsilon$. The
$\Gamma-\limsup$ inequality is trivial: if $\gamma^{\varepsilon}$ is such that
the right-hand side of (5.1) is finite (otherwise there is nothing to prove),
then the constant sequence
$\gamma^{\varepsilon^{\prime}}\equiv\gamma^{\varepsilon}$ is an admissible
recovery sequence. For the $\Gamma-\liminf$ inequality, note that the kinetic
action $\mathcal{A}$ and the Fisher information $\mathcal{I}$ are lower
semicontinuous w.r.t. pointwise-in-time $\sigma$-convergence (see the proof of
Proposition 4.2), and clearly so is the convex indicator. Hence for any
$\gamma^{\varepsilon^{\prime}}$ converging to $\gamma^{\varepsilon}$ for the
pointwise-in-time $\sigma$-topology it holds

$\begin{split}\mathcal{A}_{\varepsilon}(\gamma^{\varepsilon})+\iota_{01}(\gamma^{\varepsilon})&\leq\liminf_{\varepsilon^{\prime}\to\varepsilon}\Big{\\{}\mathcal{A}_{\varepsilon}(\gamma^{\varepsilon^{\prime}})+\iota_{01}(\gamma^{\varepsilon^{\prime}})\Big{\\}}\\\
&=\liminf_{\varepsilon^{\prime}\to\varepsilon}\Big{\\{}\mathcal{A}(\gamma^{\varepsilon^{\prime}})+\varepsilon^{2}\mathcal{I}(\gamma^{\varepsilon^{\prime}})+\iota_{01}(\gamma^{\varepsilon^{\prime}})\Big{\\}}\\\
&=\liminf_{\varepsilon^{\prime}\to\varepsilon}\Big{\\{}\mathcal{A}(\gamma^{\varepsilon^{\prime}})+(\varepsilon^{\prime})^{2}\mathcal{I}(\gamma^{\varepsilon^{\prime}})+\iota_{01}(\gamma^{\varepsilon^{\prime}})\Big{\\}}\\\
&=\liminf_{\varepsilon^{\prime}\to\varepsilon}\Big{\\{}\mathcal{A}_{\varepsilon^{\prime}}(\gamma^{\varepsilon^{\prime}})+\iota_{01}(\gamma^{\varepsilon^{\prime}})\Big{\\}}.\end{split}$

∎

As an immediate consequence of this result we deduce the following

######  Lemma 5.4.

With the same assumptions and notations as in Setting 3.1 and under Assumption
3.2, the function $\varepsilon\mapsto\mathscr{C}_{\varepsilon}(x,y)$ is
continuous on $[0,\infty)$.

Moreover, if $\varepsilon\mapsto\omega^{\varepsilon}$ is a continuous (w.r.t.
the pointwise-in-time $\sigma$-topology) selection of minimizers, then
$\varepsilon\mapsto\mathcal{A}(\omega^{\varepsilon})$ and
$\varepsilon\mapsto\mathcal{I}(\omega^{\varepsilon})$ are also continuous, on
$[0,\infty)$ and $(0,\infty)$ respectively.

Note that if the minimizers are unique, then
$\varepsilon\mapsto\omega^{\varepsilon}$ is automatically continuous w.r.t.
the pointwise-in-time $\sigma$-topology, simply by Proposition 5.3, as any
sequence of minimizers admits a subsequence converging to a minimizer and the
limit is in fact unique. Also, the continuity of the Fisher information can be
strengthened up to $\varepsilon=0$, see later on Theorem 5.7.

###### Proof.

The continuity of $\mathscr{C}_{\varepsilon}(x,y)$ for $\varepsilon>0$ is
granted by Proposition 5.3, while continuity at $\varepsilon=0$ has already
been proved in Corollary 4.4.

As regards the kinetic energy $\mathcal{A}$ and the Fisher information
$\mathcal{I}$, recall that they are both lower semicontinuous in $[0,\infty)$
w.r.t. the pointwise-in-time $\sigma$-topology, as already discussed in the
proof of Proposition 4.2. Thus, if $\varepsilon\mapsto\omega^{\varepsilon}$ is
as in the statement, we are left to prove that
$\varepsilon\mapsto\mathcal{A}(\omega^{\varepsilon})$ and
$\varepsilon\mapsto\mathcal{I}(\omega^{\varepsilon})$ are upper
semicontinuous. To this aim, it is sufficient to observe that

$\begin{split}\limsup_{\varepsilon^{\prime}\to\varepsilon}\mathcal{A}(\omega^{\varepsilon^{\prime}})&=\limsup_{\varepsilon^{\prime}\to\varepsilon}\Big{\\{}\mathscr{C}_{\varepsilon^{\prime}}(x,y)-(\varepsilon^{\prime})^{2}\mathcal{I}(\omega^{\varepsilon^{\prime}})\Big{\\}}\\\
&\leq\limsup_{\varepsilon^{\prime}\to\varepsilon}\mathscr{C}_{\varepsilon^{\prime}}(x,y)-\liminf_{\varepsilon^{\prime}\to\varepsilon}(\varepsilon^{\prime})^{2}\mathcal{I}(\omega^{\varepsilon^{\prime}})\\\
&\leq\mathscr{C}_{\varepsilon}(x,y)-\varepsilon^{2}\mathcal{I}(\omega^{\varepsilon})=\mathcal{A}(\omega^{\varepsilon}),\end{split}$

where the last inequality holds by the continuity of
$\varepsilon\mapsto\mathscr{C}_{\varepsilon}(x,y)$ and the lower
semicontinuity of $\varepsilon\mapsto\mathcal{I}(\omega^{\varepsilon})$. Thus
$\varepsilon\mapsto\mathcal{A}(\omega^{\varepsilon})$ is upper semicontinuous
in $[0,\infty)$. Interchanging $\mathcal{A}$ and $\mathcal{I}$ and writing now
$\mathcal{I}=\frac{1}{\varepsilon^{2}}(\mathscr{C}_{\varepsilon}-\mathcal{A})$,
the same argument shows that
$\varepsilon\mapsto\mathcal{I}(\omega^{\varepsilon})$ is upper semicontinuous
in $(0,\infty)$ (continuity at $\varepsilon=0$ will require a special
treatment later). ∎

We have now all the ingredients to discuss the regularity of the cost
$\mathscr{C}_{\varepsilon}(x,y)$ as a function of the noise parameter
$\varepsilon$ and explicitly compute its left and right derivatives.

######  Proposition 5.5.

With the same assumptions and notations as in Setting 3.1 and if Assumption
3.2 holds, the map $\varepsilon\mapsto\mathscr{C}_{\varepsilon}(x,y)$ is
$AC_{loc}([0,\infty))$, left and right differentiable everywhere in
$(0,\infty)$ and, for any $\varepsilon>0$, the left and right derivatives are
given by

$\frac{{\mathrm{d}}^{-}}{{\mathrm{d}}\varepsilon}\mathscr{C}_{\varepsilon}(x,y)=2\varepsilon\max_{\Lambda_{\varepsilon}(x,y)}\mathcal{I},\qquad\frac{{\mathrm{d}}^{+}}{{\mathrm{d}}\varepsilon}\mathscr{C}_{\varepsilon}(x,y)=2\varepsilon\min_{\Lambda_{\varepsilon}(x,y)}\mathcal{I}$
(5.2)

respectively, and the former (resp. latter) is left (resp. right) continuous.
It is part of the statement the fact that the maximum and the minimum are
attained.

######  Remark 5.6.

Heuristically, (5.2) is nothing but the envelope theorem. Indeed, if
$\varepsilon\mapsto\mathscr{C}_{\varepsilon}(x,y)$ were differentiable, then
its derivative would be given by
$\partial_{\varepsilon}\mathcal{A}_{\varepsilon}=2\varepsilon\mathcal{I}$
evaluated at any critical point, i.e. at any
$\omega^{\varepsilon}\in\Lambda_{\varepsilon}(x,y)$. However, since we do not
know in our general metric framework that Schrödinger problem has a unique
solution, we are not able to prove pointwise differentiability as in [25] and
we have to face the possibility of a gap between the left and right
derivatives. In any case, for a.e. $\varepsilon>0$ this gap is zero, because
$\varepsilon\mapsto\mathscr{C}_{\varepsilon}(x,y)$ is locally absolutely
continuous in $(0,\infty)$ and therefore a.e. differentiable. This means that,
up to a negligible set of temperatures, the left and right derivatives match
and $\mathcal{I}$ is constant on $\Lambda_{\varepsilon}(x,y)$. If for whatever
reason the Schrödinger problem ($\mathrm{Sch}^{\varepsilon}$) were uniquely
solvable (which is in particular true for the classic Schrödinger problem, as
proved in [39, Theorem 4.2]), then the left and right derivatives would be
trivially equal and Lemma 5.4 would give that
$\varepsilon\mapsto\mathscr{C}_{\varepsilon}(x,y)$ is actually
$C^{1}((0,\infty))$. Furthermore, the cost would also be twice differentiable
a.e. since by Lemma 5.2 its first derivative
$2\varepsilon\mathcal{I}(\omega^{\varepsilon})$ would be the product of a
linear function and of a monotone one.  $\blacksquare$

###### Proof.

The continuity of $\varepsilon\mapsto\mathscr{C}_{\varepsilon}(x,y)$ follows
by Lemma 5.4, so let us focus on left and right differentiability/continuity
and local absolute continuity.

Right differentiability. Fix $\varepsilon>0$, let $\delta>0$, and choose
$\omega^{\varepsilon}\in\Lambda_{\varepsilon}(x,y)$,
$\omega^{\varepsilon+\delta}\in\Lambda_{\varepsilon+\delta}(x,y)$. Then write

$\begin{split}\frac{\mathscr{C}_{\varepsilon+\delta}(x,y)-\mathscr{C}_{\varepsilon}(x,y)}{\delta}&=\frac{\mathcal{A}_{\varepsilon+\delta}(\omega^{\varepsilon+\delta})-\mathcal{A}_{\varepsilon}(\omega^{\varepsilon})}{\delta}\\\
&=\frac{\mathcal{A}_{\varepsilon+\delta}(\omega^{\varepsilon+\delta})-\mathcal{A}_{\varepsilon+\delta}(\omega^{\varepsilon})}{\delta}+\frac{\mathcal{A}_{\varepsilon+\delta}(\omega^{\varepsilon})-\mathcal{A}_{\varepsilon}(\omega^{\varepsilon})}{\delta}\end{split}$
(5.3)

and note that the second term on the right-hand side can be rewritten as

$\mathcal{A}_{\varepsilon+\delta}(\omega^{\varepsilon})-\mathcal{A}_{\varepsilon}(\omega^{\varepsilon})=(2\varepsilon\delta+\delta^{2})\mathcal{I}(\omega^{\varepsilon}).$

The first one is non-positive by optimality of $\omega^{\varepsilon+\delta}$
for $\mathcal{A}_{\varepsilon+\delta}$, hence we obtain

$\limsup_{\delta\downarrow
0}\frac{\mathscr{C}_{\varepsilon+\delta}(x,y)-\mathscr{C}_{\varepsilon}(x,y)}{\delta}\leq\limsup_{\delta\downarrow
0}\,(2\varepsilon+\delta)\mathcal{I}(\omega^{\varepsilon})=2\varepsilon\mathcal{I}(\omega^{\varepsilon}).$

As this inequality holds for any
$\omega^{\varepsilon}\in\Lambda_{\varepsilon}(x,y)$, we infer that

$\limsup_{\delta\downarrow
0}\frac{\mathscr{C}_{\varepsilon+\delta}(x,y)-\mathscr{C}_{\varepsilon}(x,y)}{\delta}\leq
2\varepsilon\inf_{\Lambda_{\varepsilon}(x,y)}\mathcal{I}.$ (5.4)

On the other hand we can also write

$\begin{split}\frac{\mathscr{C}_{\varepsilon+\delta}(x,y)-\mathscr{C}_{\varepsilon}(x,y)}{\delta}&=\frac{\mathcal{A}_{\varepsilon+\delta}(\omega^{\varepsilon+\delta})-\mathcal{A}_{\varepsilon}(\omega^{\varepsilon})}{\delta}\\\
&=\frac{\mathcal{A}_{\varepsilon+\delta}(\omega^{\varepsilon+\delta})-\mathcal{A}_{\varepsilon}(\omega^{\varepsilon+\delta})}{\delta}+\frac{\mathcal{A}_{\varepsilon}(\omega^{\varepsilon+\delta})-\mathcal{A}_{\varepsilon}(\omega^{\varepsilon})}{\delta}.\end{split}$
(5.5)

Using now the optimality of $\omega^{\varepsilon}$ for
$\mathcal{A}_{\varepsilon}$, we observe that the second term on the right-hand
side is non-negative, whence

$\frac{\mathcal{A}_{\varepsilon+\delta}(\omega^{\varepsilon+\delta})-\mathcal{A}_{\varepsilon}(\omega^{\varepsilon})}{\delta}\geq\frac{\mathcal{A}_{\varepsilon+\delta}(\omega^{\varepsilon+\delta})-\mathcal{A}_{\varepsilon}(\omega^{\varepsilon+\delta})}{\delta}=(2\varepsilon+\delta)\mathcal{I}(\omega^{\varepsilon+\delta}).$

For any sequence $\delta_{n}\downarrow 0$, Proposition 5.3 guarantees (up to
extraction of a subsequence if needed) that
$\omega^{\varepsilon+\delta_{n}}\to\overline{\omega}^{\varepsilon}$ in the
pointwise-in-time $\sigma$-topology for some
$\overline{\omega}^{\varepsilon}\in\Lambda_{\varepsilon}(x,y)$. By lower
semicontinuity of $\mathcal{I}$ this implies

$\liminf_{n\to\infty}\frac{\mathscr{C}_{\varepsilon+\delta_{n}}(x,y)-\mathscr{C}_{\varepsilon}(x,y)}{\delta_{n}}\geq\liminf_{n\to\infty}(2\varepsilon+\delta_{n})\mathcal{I}(\omega^{\varepsilon+\delta_{n}})\geq
2\varepsilon\mathcal{I}(\overline{\omega}^{\varepsilon})\geq
2\varepsilon\inf_{\Lambda_{\varepsilon}(x,y)}\mathcal{I},$

and together with (5.4) this yields

$\exists\lim_{n\to\infty}\frac{\mathscr{C}_{\varepsilon+\delta_{n}}(x,y)-\mathscr{C}_{\varepsilon}(x,y)}{\delta_{n}}=2\varepsilon\mathcal{I}(\overline{\omega}^{\varepsilon})=2\varepsilon\inf_{\Lambda_{\varepsilon}(x,y)}\mathcal{I}.$

As the right-hand side does not depend on the particular sequence
$\delta_{n}\downarrow 0$ we conclude that

$\exists\lim_{\delta\downarrow
0}\frac{\mathscr{C}_{\varepsilon+\delta}(x,y)-\mathscr{C}_{\varepsilon}(x,y)}{\delta}=2\varepsilon\min_{\Lambda_{\varepsilon}(x,y)}\mathcal{I},$

in particular $\mathcal{I}$ is minimized by any accumulation point
$\overline{\omega}^{\varepsilon}$ of
$\\{\omega^{\varepsilon+\delta}\\}_{\delta>0}$.

Left differentiability. The argument is very similar. Indeed, if $\delta<0$,
then the first term on the right-hand side of (5.3) is non-negative and the
second one can be handled in the same way. Hence there holds

$\liminf_{\delta\uparrow
0}\frac{\mathscr{C}_{\varepsilon+\delta}(x,y)-\mathscr{C}_{\varepsilon}(x,y)}{\delta}\geq
2\varepsilon\mathcal{I}(\omega^{\varepsilon}),$

for any $\omega^{\varepsilon}\in\Lambda_{\varepsilon}(x,y)$, and therefore

$\liminf_{\delta\uparrow
0}\frac{\mathscr{C}_{\varepsilon+\delta}(x,y)-\mathscr{C}_{\varepsilon}(x,y)}{\delta}\geq
2\varepsilon\sup_{\Lambda_{\varepsilon}(x,y)}\mathcal{I}.$

Applying the same considerations to (5.5) and following the same argument as
above we retrieve the $\limsup$ inequality, first along _some_ subsequence
$\delta_{n}\uparrow 0$ and then along _any_ $\delta\uparrow 0$. Combining with
the inequality above gives

$\exists\lim_{\delta\uparrow
0}\frac{\mathscr{C}_{\varepsilon+\delta}(x,y)-\mathscr{C}_{\varepsilon}(x,y)}{\delta}=2\varepsilon\max_{\Lambda_{\varepsilon}(x,y)}\mathcal{I},\qquad\forall\varepsilon>0,$

whence the pointwise left differentiability of
$\varepsilon\mapsto\mathscr{C}_{\varepsilon}(x,y)$.

Left and right continuity. In order to prove the right continuity of the right
derivative of $\varepsilon\mapsto\mathscr{C}_{\varepsilon}(x,y)$, note that on
the one hand by Lemma 5.2 for any $\varepsilon_{n}\downarrow\varepsilon$ it
holds

$\inf_{\Lambda_{\varepsilon}(x,y)}\mathcal{I}\geq\limsup_{n\to\infty}\sup_{\Lambda_{\varepsilon_{n}}(x,y)}\mathcal{I}\geq\limsup_{n\to\infty}\inf_{\Lambda_{\varepsilon_{n}}(x,y)}\mathcal{I}.$

On the other hand, we can assume up to a subsequence if needed that

$\liminf_{n\to\infty}\inf_{\Lambda_{\varepsilon_{n}}(x,y)}\mathcal{I}=\lim_{n\to\infty}\inf_{\Lambda_{\varepsilon_{n}}(x,y)}\mathcal{I}.$

As shown in the proof of right differentiability,
$\inf_{\Lambda_{\varepsilon^{\prime}}(x,y)}\mathcal{I}$ is attained for any
$\varepsilon^{\prime}>0$, hence in particular
$\inf_{\Lambda_{\varepsilon_{n}}(x,y)}\mathcal{I}=\mathcal{I}(\omega^{n})$ for
some $\omega^{n}\in\Lambda_{\varepsilon_{n}}(x,y)$, for all $n$. Up to
extracting a further subsequence, by Proposition 5.3 we can assume that
$\omega^{n}\to\overline{\omega}^{\varepsilon}$ w.r.t. the pointwise-in-time
$\sigma$-topology for some
$\overline{\omega}^{\varepsilon}\in\Lambda_{\varepsilon}(x,y)$, and moreover
by Lemma 5.4

$\lim_{n\to\infty}\mathcal{I}(\omega^{n})=\mathcal{I}(\overline{\omega})\geq\inf_{\Lambda_{\varepsilon}(x,y)}\mathcal{I}.$

Putting all these inequalities together provides us with the right continuity
of $\varepsilon\mapsto\inf_{\Lambda_{\varepsilon}(x,y)}\mathcal{I}$ and, a
fortiori, of the right derivative. Left continuity for the left derivative
follows along an analogous reasoning.

Local absolute continuity. Let $0<\varepsilon_{1}<\varepsilon_{2}<\infty$ and,
for any $0<\delta<1$, define

$f_{\delta}(\varepsilon):=\frac{\mathscr{C}_{\varepsilon+\delta}(x,y)-\mathscr{C}_{\varepsilon}(x,y)}{\delta}.$

The monotonicity of $\varepsilon\mapsto\mathscr{C}_{\varepsilon}(x,y)$ from
Lemma 5.2 gives $f_{\delta}\geq 0$. Arguing as in the very beginning of the
proof of the right differentiability we see that
$f_{\delta}(\varepsilon)\leq(2\varepsilon+1)\mathcal{I}(\omega^{\varepsilon})$
for any $\omega^{\varepsilon}\in\Lambda_{\varepsilon}(x,y)$, and by Lemma 5.2

$f_{\delta}(\varepsilon)\leq(2\varepsilon_{2}+1)\sup_{\Lambda_{\varepsilon_{1}}(x,y)}\mathcal{I}<\infty,\qquad\forall\varepsilon\in(\varepsilon_{1},\varepsilon_{2}].$

Hence $|f_{\delta}|\leq M$ uniformly in $\delta$ and $f_{\delta}$ converges
pointwise to the right derivative of
$\varepsilon\mapsto\mathscr{C}_{\varepsilon}(x,y)$ as $\delta\downarrow 0$,
whence by the dominated convergence theorem

$\int_{\varepsilon_{1}}^{\varepsilon_{2}}\frac{{\mathrm{d}}^{+}}{{\mathrm{d}}\varepsilon}\mathscr{C}_{\varepsilon}(x,y)\,{\mathrm{d}}\varepsilon=\lim_{\delta\downarrow
0}\int_{\varepsilon_{1}}^{\varepsilon_{2}}f_{\delta}(\varepsilon)\,{\mathrm{d}}\varepsilon.$

The right-hand side can be rewritten as

$\begin{split}\lim_{\delta\downarrow
0}\int_{\varepsilon_{1}}^{\varepsilon_{2}}f_{\delta}(\varepsilon)\,{\mathrm{d}}\varepsilon&=\lim_{\delta\downarrow
0}\Big{(}\frac{1}{\delta}\int_{\varepsilon_{1}}^{\varepsilon_{2}}\mathscr{C}_{\varepsilon+\delta}(x,y)\,{\mathrm{d}}\varepsilon-\frac{1}{\delta}\int_{\varepsilon_{1}}^{\varepsilon_{2}}\mathscr{C}_{\varepsilon}(x,y)\,{\mathrm{d}}\varepsilon\Big{)}\\\
&=\lim_{\delta\downarrow
0}\Big{(}\frac{1}{\delta}\int_{\varepsilon_{2}}^{\varepsilon_{2}+\delta}\mathscr{C}_{\varepsilon}(x,y)\,{\mathrm{d}}\varepsilon-\frac{1}{\delta}\int_{\varepsilon_{1}}^{\varepsilon_{1}+\delta}\mathscr{C}_{\varepsilon}(x,y)\,{\mathrm{d}}\varepsilon\Big{)}\\\
&=\mathscr{C}_{\varepsilon_{2}}(x,y)-\mathscr{C}_{\varepsilon_{1}}(x,y),\end{split}$

where the last equality holds by the Lebesgue differentiation theorem for the
continuous function $\varepsilon\mapsto\mathscr{C}_{\varepsilon}(x,y)$ (cf.
Lemma 5.4). We have thus proved that the cost belongs to
$AC_{loc}((0,\infty))$, since

$\mathscr{C}_{\varepsilon_{2}}(x,y)-\mathscr{C}_{\varepsilon_{1}}(x,y)=\int_{\varepsilon_{1}}^{\varepsilon_{2}}\frac{{\mathrm{d}}^{+}}{{\mathrm{d}}\varepsilon}\mathscr{C}_{\varepsilon}(x,y)\,{\mathrm{d}}\varepsilon,\qquad\forall\,0<\varepsilon_{1}<\varepsilon_{2}.$

For the full $AC_{loc}([0,\infty))$ regularity it is then sufficient to let
$\varepsilon_{1}\downarrow 0$: the left-hand side converges to
$\mathscr{C}_{\varepsilon_{2}}(x,y)-\mathscr{C}_{0}(x,y)$ by Lemma 5.4, and by
the monotonicity
$\frac{{\mathrm{d}}^{+}}{{\mathrm{d}}\varepsilon}\mathscr{C}_{\varepsilon}\geq
0$ the right-hand side also converges by monotone convergence. ∎

Relying on our previous auxiliary results and on Proposition 5.5, we are
finally in position of estimating the error
$\mathscr{C}_{\varepsilon}(x,y)-\mathscr{C}_{0}(x,y)$ with
$o(\varepsilon^{2})$ precision. We will also significantly refine Corollary
4.4 by proving that any accumulation point of any sequence of minimizers is
not only optimal for the unperturbed problem $\mathscr{C}_{0}(x,y)$, but also
$\mathcal{I}$-minimizing among all competitors in $\Lambda_{0}(x,y)$.

######  Theorem 5.7.

With the same assumptions and notations as in Proposition 5.5, if there exists
$\omega^{0}\in\Lambda_{0}(x,y)$ such that $\mathcal{I}(\omega^{0})<\infty$,
then the map $\varepsilon\mapsto\mathscr{C}_{\varepsilon}(x,y)$ is right
differentiable also at $\varepsilon=0$ with

$\frac{{\mathrm{d}}^{+}}{{\mathrm{d}}\varepsilon}\mathscr{C}_{\varepsilon}(x,y)\Big{|}_{\varepsilon=0}=0,$

the right derivative is right continuous for any $\varepsilon\geq 0$, and

$\mathscr{C}_{\varepsilon}(x,y)-\mathscr{C}_{0}(x,y)=\varepsilon^{2}\inf_{\Lambda_{0}(x,y)}\mathcal{I}+o(\varepsilon^{2}).$
(5.6)

Moreover, for any $\varepsilon_{n}\downarrow 0$ and any minimizer
$\omega^{n}\in\Lambda_{\varepsilon_{n}}(x,y)$ there exists
$\omega^{*}\in\Lambda_{0}(x,y)$ such that (up to a subsequence)
$\omega^{n}\to\omega^{*}$ for the pointwise-in-time $\sigma$-topology, and
$\omega^{*}$ has minimal Fisher information in $\Lambda_{0}(x,y)$

$\mathcal{I}(\omega^{*})=\min_{\Lambda_{0}(x,y)}\mathcal{I}.$

###### Proof.

The right differentiability of
$\varepsilon\mapsto\mathscr{C}_{\varepsilon}(x,y)$ at $\varepsilon=0$ follows
by the same argument carried out in Proposition 5.5. Indeed, given
$\omega^{0}$ as in the statement, by (5.3) with $\varepsilon=0$ it holds

$\limsup_{\delta\downarrow
0}\frac{\mathscr{C}_{\delta}(x,y)-\mathscr{C}_{0}(x,y)}{\delta}\leq\limsup_{\delta\downarrow
0}\delta\mathcal{I}(\omega^{0})=0.$

The liminf inequality is straightforward, since $\mathcal{I}\geq 0$ and thus
by (5.5) with $\varepsilon=0$

$\liminf_{\delta\downarrow
0}\frac{\mathscr{C}_{\delta}(x,y)-\mathscr{C}_{0}(x,y)}{\delta}\geq\liminf_{\delta\downarrow
0}\delta\mathcal{I}(\omega^{\delta})\geq 0$

for any $\omega^{\delta}\in\Lambda_{\delta}(x,y)$. This also shows that the
right derivative vanishes at $\varepsilon=0$.

As regards the right continuity of the right derivative, the case
$\varepsilon>0$ has already been discussed in Proposition 5.5. For
$\varepsilon=0$ the same strategy still works, with the only minor difference
that we cannot rely on Lemma 5.4 anymore. Nonetheless, if
$\omega^{n}\in\Lambda_{\varepsilon_{n}}(x,y)$ is as in Proposition 5.5,
$\overline{\omega}\in\Lambda_{0}(x,y)$ and $\omega^{n}\to\overline{\omega}$
for the pointwise-in-time $\sigma$-topology (the existence of such
$\overline{\omega}$ is granted by Corollary 4.4) it is still true that

$\liminf_{n\to\infty}\mathcal{I}(\omega^{n})\geq\mathcal{I}(\overline{\omega}),$

simply by lower semicontinuity of $\mathcal{I}$. With this single change in
the proof we deduce that
$\varepsilon\mapsto\inf_{\Lambda_{\varepsilon}(x,y)}\mathcal{I}$ is right
continuous and finite also at $\varepsilon=0$, thanks to the present
assumptions, and so is the right derivative of the cost due to
$\varepsilon\inf_{\Lambda_{\varepsilon}(x,y)}\mathcal{I}\to 0$ as
$\varepsilon\downarrow 0$.

The last part of the statement is a slight modification of these lines of
thought. Indeed, given any sequence $\varepsilon_{n}\downarrow 0$ and
$\omega^{n}\in\Lambda_{\varepsilon_{n}}(x,y)$, the existence of
$\omega^{*}\in\Lambda_{0}(x,y)$ such that, up to subsequences,
$\omega^{n}\to\omega^{*}$ is ensured by Corollary 4.4. The fact that
$\omega^{*}$ has minimal Fisher information among all elements in
$\Lambda_{0}(x,y)$ follows from

$\inf_{\Lambda_{0}(x,y)}\mathcal{I}\geq\limsup_{n\to\infty}\sup_{\Lambda_{\varepsilon_{n}}(x,y)}\mathcal{I}\geq\limsup_{n\to\infty}\mathcal{I}(\omega^{n})\geq\liminf_{n\to\infty}\mathcal{I}(\omega^{n})\geq\mathcal{I}(\omega^{*})\geq\inf_{\Lambda_{0}(x,y)}\mathcal{I},$

where we used once again Lemma 5.2 and the lower semicontinuity of
$\mathcal{I}$.

Thus, it only remains to establish (5.6). As
$\varepsilon\mapsto\mathscr{C}_{\varepsilon}(x,y)$ belongs to
$AC_{loc}([0,\infty))$ and the right derivative coincides a.e. with the full
derivative, (5.2) and the fundamental theorem of calculus yield

$\mathscr{C}_{\varepsilon}(x,y)-\mathscr{C}_{0}(x,y)=2\int_{0}^{\varepsilon}s\inf_{\Lambda_{s}(x,y)}\mathcal{I}\,{\mathrm{d}}s\leq
2\int_{0}^{\varepsilon}s\inf_{\Lambda_{0}(x,y)}\mathcal{I}\,{\mathrm{d}}s=\varepsilon^{2}\inf_{\Lambda_{0}(x,y)}\mathcal{I}.$

Here we used the monotonicity of the Fisher information from Lemma 5.2 in the
middle inequality. By the same monotonicity and the right continuity at
$\varepsilon=0$ of
$\varepsilon\mapsto\inf_{\Lambda_{\varepsilon}(x,y)}\mathcal{I}$ we also
deduce that

$\begin{split}\mathscr{C}_{\varepsilon}(x,y)-\mathscr{C}_{0}(x,y)&\geq
2\int_{0}^{\varepsilon}s\inf_{\Lambda_{\varepsilon}(x,y)}\mathcal{I}\,{\mathrm{d}}s=\varepsilon^{2}\inf_{\Lambda_{\varepsilon}(x,y)}\mathcal{I}\\\
&=\varepsilon^{2}\inf_{\Lambda_{0}(x,y)}\mathcal{I}+\varepsilon^{2}\Big{(}\inf_{\Lambda_{\varepsilon}(x,y)}\mathcal{I}-\inf_{\Lambda_{0}(x,y)}\mathcal{I}\Big{)}\\\
&=\varepsilon^{2}\inf_{\Lambda_{0}(x,y)}\mathcal{I}+o(\varepsilon^{2})\end{split}.$

Combining this lower bound with the previous upper one entails (5.6). ∎

######  Remark 5.8.

It is worth stressing that the upper bound on
$\mathscr{C}_{\varepsilon}(x,y)-\mathscr{C}_{0}(x,y)$ is not asymptotic, but
pointwise. A possible way to improve (5.6) would rely on a refined analysis of
$\varepsilon\mapsto\inf_{\Lambda_{\varepsilon}(x,y)}\mathcal{I}$, its
derivative (which exists a.e. by monotonicity), and possibly absolute
continuity.  $\blacksquare$

######  Remark 5.9.

The $\mathcal{I}$-minimizing property of the accumulation point $\omega^{*}$
is not specific of the choice $\varepsilon=0$, but of the particular
“backward” direction of the sequence $\varepsilon_{n}\downarrow$. Repeating
the argument in the proof of Theorem 5.7 it is indeed not difficult to check
that, given any $\varepsilon>0$, a sequence
$\varepsilon_{n}\downarrow\varepsilon$, and
$\omega^{n}\in\Lambda_{\varepsilon_{n}}(x,y)$ there exists
$\omega^{\varepsilon}\in\Lambda_{\varepsilon}(x,y)$ such that, up to a
subsequence, $\omega^{n}\to\omega^{\varepsilon}$ for the pointwise-in-time
$\sigma$-topology and

$\mathcal{I}(\omega^{\varepsilon})=\inf_{\Lambda_{\varepsilon}(x,y)}\mathcal{I}.$

In a symmetric fashion, a closer look into the proof of Proposition 5.5
suggests that an opposite behaviour appears in the “forward” direction. More
precisely, if $\varepsilon_{n}\uparrow\varepsilon$ instead of
$\varepsilon_{n}\downarrow\varepsilon$, then any accumulation point
$\omega^{\varepsilon}$ of $(\omega^{n})$ is such that

$\mathcal{I}(\omega^{\varepsilon})=\sup_{\Lambda_{\varepsilon}(x,y)}\mathcal{I}.$

However, the “backward” direction and the case $\varepsilon=0$ are usually
more interesting, because of the connection with the unperturbed problem
$\mathscr{C}_{0}(x,y)$ for which there might be multiple solutions even if the
Schrödinger problem ($\mathrm{Sch}^{\varepsilon}$) has a unique minimizer for
all $\varepsilon>0$. It is therefore natural to look for the (properties of
the) solutions selected via Schrödinger regularization.  $\blacksquare$

##  6 Examples

In this section we collect several and heterogeneous situations where our
abstract approach (in particular Theorem 4.3, Corollary 4.4, and Theorem 5.7)
applies. We shall also comment the novelty of the results thus obtained in
comparison with the existing literature. In this perspective, it is worth
discussing in more detail the role played by Assumption 3.2 so far, singling
out when the sequential lower semicontinuity of $|\partial{\sf E}|$ w.r.t.
$\sigma$ is needed and when it is not:

  * •

to show the existence of a solution to the Schrödinger problem
($\mathrm{Sch}^{\varepsilon}$) (cf. Proposition 4.2) it is crucial, in order
to apply the direct method of the calculus of variations;

  * •

in Theorem 4.3 and Corollary 4.4 it is not used;

  * •

unlike Corollary 4.4, in Proposition 5.3 it is needed for the $\Gamma$-liminf
inequality and so is in Lemma 5.4;

  * •

Proposition 5.5 relies on Proposition 5.3 and Lemma 5.4, hence it is
implicitly used;

  * •

in Theorem 5.7 the continuity of
$\varepsilon\mapsto\inf_{\Lambda_{\varepsilon}(x,y)}\mathcal{I}$ at
$\varepsilon=0$ requires the lower semicontinuity of $|\partial{\sf E}|$ and
also Proposition 5.5 is used in the proof of (5.6); hence the lower
semicontinuity of $|\partial{\sf E}|$ is really needed.

This means that if one is able to show the solvability of the Schrödinger
problem ($\mathrm{Sch}^{\varepsilon}$) by means other than those used in
Proposition 4.2, then Theorem 4.3 and Corollary 4.4 are still valid under the
following weaker hypothesis.

######  Assumption 6.1.

There exists a Hausdorff topology $\sigma$ on ${\rm X}$ such that ${\sf
d}$-bounded sequences contain $\sigma$-converging subsequences. Moreover, the
distance ${\sf d}$ is sequentially lower semicontinuous w.r.t. $\sigma$.

Theorem 5.7, instead, requires the full validity of Assumption 3.2.

###  6.1 The Boltzmann-Shannon relative entropy

As a first example, let us consider the Boltzmann-Shannon relative entropy on
the Wasserstein space built over a (locally compact) ${\rm RCD}(K,\infty)$
space. To this end, let $(M,{\sf d},\mathfrak{m})$ be a complete and separable
locally compact length metric space endowed with a Radon measure and assume
that it is an ${\rm RCD}(K,\infty)$ space [4] for some $K\in\mathbb{R}$. Let
${\rm X}:=\mathcal{P}_{2}(M)$ be the 2-Wasserstein space over $M$, namely the
space of probability measures with finite second moments, and equip it with
the 2-Wasserstein distance $W_{2}$: it turns out to be a complete and
separable metric space [14] as well. The Boltzmann-Shannon relative entropy
${\sf E}$ on ${\rm X}$ is defined as

${\sf
E}(\mu):=\left\\{\begin{array}[]{ll}\displaystyle{\int_{M}\rho\log(\rho)\,{\mathrm{d}}\mathfrak{m}}&\qquad\text{
if }\mu=\rho\mathfrak{m},\\\ +\infty&\qquad\text{ if
}\mu\not\ll{\mathfrak{m}}.\end{array}\right.$

As by [61, Theorem 4.24] there exist $C>0$, $x\in M$ such that
$\int_{M}e^{-C{\sf d}^{2}(\cdot,x)}{\mathrm{d}}\mathfrak{m}<\infty$, ${\sf E}$
can be equivalently rewritten as

${\sf
E}(\mu)=\underbrace{\int_{M}\tilde{\rho}\log(\tilde{\rho})\,{\mathrm{d}}\tilde{\mathfrak{m}}}_{\geq
0}-C\int_{M}{\sf d}^{2}(\cdot,x)\,{\mathrm{d}}\mu-\log Z,$

where $\tilde{\rho}$ is the Radon-Nikodym derivative of $\mu$ w.r.t.
$\tilde{\mathfrak{m}}$, with the normalization

$Z:=\int_{M}e^{-C{\sf
d}^{2}(\cdot,x)}{\mathrm{d}}\mathfrak{m},\qquad\tilde{\mathfrak{m}}:=\frac{1}{Z}e^{-C{\sf
d}^{2}(\cdot,x)}\mathfrak{m}.$

From this very definition, it is easy to see that ${\sf E}$ is a proper lower
semicontinuous functional, bounded from below on $W_{2}$-bounded sets. In
addition, it has a dense domain, since by (one of the equivalent) definition
of ${\rm RCD}$ spaces, cf. [4, Theorem 5.1], for any $\mu\in{\rm X}$ there
exists an ${\rm EVI}_{K}$-gradient flow of ${\sf E}$ starting from it. Thus
Setting 3.1 holds.

As regards Assumption 6.1, note that ${\rm X}$ is not locally compact unless
$M$ is compact, so that in general the metric topology of $X$ is not an
admissible candidate for $\sigma$. Nonetheless there is a natural alternative:
the narrow convergence of probability measures. Indeed, $W_{2}$-bounded