As mentioned in the previous proof we already know, thanks to Proposition 2.2,
that

$\log P_{T}g\in L^{1}(\mu),\,\log P_{T}f\in L^{1}(\nu),\,\log P_{T}\bar{g}\in
L^{1}(\bar{\mu}),\text{ and }\log P_{T}\bar{f}\in L^{1}(\bar{\nu})\,.$

Now, given our assumptions, the left-hand side of (4.4) is finite (thanks to
Theorem 1.2) and since we have $\log P_{T}g\in L^{1}(\mu),\,\log P_{T}f\in
L^{1}(\nu)$ and $\int\log P_{T}\bar{g}\mathrm{d}\mu,\,\int\log
P_{T}\bar{f}\mathrm{d}\nu\in(-{\infty},+{\infty}]$ (from the proof of Theorem
1.2), we deduce that

$\log P_{T}\bar{g}\in L^{1}(\mu)\quad\text{ and }\quad\log P_{T}\bar{f}\in
L^{1}(\nu)\,.$

Similarly, working with (4.5) we get $\log P_{T}g\in L^{1}(\bar{\mu})$ and
$\log P_{T}f\in L^{1}(\bar{\nu})$.

Finally, from what we have just proven above and thanks to
$\mathcal{H}^{\mathrm{sym}}(\mu,\bar{\mu}),\,\mathcal{H}^{\mathrm{sym}}(\nu,\bar{\nu})$
being finite we deduce

$\log\frac{f}{\bar{f}}\in L^{1}(\mu)\cap L^{1}(\bar{\mu})\quad\text{ and
}\quad\log\frac{g}{\bar{g}}\in L^{1}(\nu)\cap L^{1}(\bar{\nu})\,,$

which combined with $\log f\in L^{1}(\mu)$, $\log g\in L^{1}(\nu)$,
$\log\bar{f}\in L^{1}(\bar{\mu})$ and $\log\bar{g}\in L^{1}(\bar{\nu})$ gives
our final assertion. ∎

###### Resuming the proof of Theorem 1.2.

Let us now prove (1.11). Without loss of generality we may assume that the
Fisher information of the marginals and the negative Sobolev norms are all
finite, otherwise the claimed estimate is trivial. Let us start by showing
that this guarantees the finiteness of the symmetric relative entropies
$\mathcal{H}^{\mathrm{sym}}(\mu,\bar{\mu}),\,\mathcal{H}^{\mathrm{sym}}(\nu,\bar{\nu})$.
Indeed, since $\mathcal{I}(\mu)<+{\infty}$ we have
$\nabla\log\frac{\mathrm{d}\mu}{\mathrm{d}\mathfrak{m}}\in L^{2}(\mu)$ and
therefore by the definition of
$\left\lVert\mu-\bar{\mu}\right\rVert_{\dot{H}^{-1}(\mu)}$ we deduce

$\left\lvert\int\log\frac{\mathrm{d}\mu}{\mathrm{d}\mathfrak{m}}\,\mathrm{d}(\mu-\bar{\mu})\right\rvert\leq\sqrt{\mathcal{I}(\mu)}\,\left\lVert\mu-\bar{\mu}\right\rVert_{\dot{H}^{-1}(\mu)}<+{\infty}\,.$

Thanks to the above bound and to the finiteness of
$\mathcal{H}(\mu|\mathfrak{m})$ and $\mathcal{H}(\bar{\mu}|\mathfrak{m})$ we
are allowed to write

$\displaystyle\int\log\frac{\mathrm{d}\bar{\mu}}{\mathrm{d}\mu}\,\mathrm{d}\bar{\mu}=\int\left[\frac{\mathrm{d}\bar{\mu}}{\mathrm{d}\mathfrak{m}}\log\frac{\mathrm{d}\bar{\mu}}{\mathrm{d}\mathfrak{m}}+\left(\frac{\mathrm{d}\mu}{\mathrm{d}\mathfrak{m}}-\frac{\mathrm{d}\bar{\mu}}{\mathrm{d}\mathfrak{m}}\right)\log\frac{\mathrm{d}\mu}{\mathrm{d}\mathfrak{m}}-\frac{\mathrm{d}\mu}{\mathrm{d}\mathfrak{m}}\log\frac{\mathrm{d}\mu}{\mathrm{d}\mathfrak{m}}\right]\mathrm{d}\mathfrak{m}$
$\displaystyle=\mathcal{H}(\bar{\mu}|\mathfrak{m})-\mathcal{H}(\mu|\mathfrak{m})+\int\log\frac{\mathrm{d}\mu}{\mathrm{d}\mathfrak{m}}\,\mathrm{d}(\mu-\bar{\mu})<+{\infty}\,,$

which reads as $\mathcal{H}(\bar{\mu}|\mu)<+{\infty}$. Let us just mention
that above the Radon-Nikodym derivative is well defined since, under our
assumption, $\bar{\mu}\sim\mu$ are equivalent. Indeed, if there were a Borel
subset $A\subset M$ such that $\mu(A)=0$ and $\bar{\mu}(A)>0$, then by
choosing as test function $h$ (a mollified version of) $\mathbbm{1}_{A}$ we
would get $\left\lVert\mu-\bar{\mu}\right\rVert_{\dot{H}^{-1}(\mu)}=+{\infty}$
which we are assuming to be finite; therefore $\bar{\mu}\ll\mu$. Similarly one
can prove $\mu\ll\bar{\mu}$, and hence $\bar{\mu}\sim\mu$. We have thus proven
that $\mathcal{H}(\bar{\mu}|\mu)$ is finite. By reasoning in the same fashion
one can prove that $\mathcal{H}(\mu|\bar{\mu})$, $\mathcal{H}(\bar{\nu}|\nu)$
and $\mathcal{H}(\nu|\bar{\nu})$ are also finite, whence the finiteness of the
symmetric entropies.

At this stage, the proof is similar to the one already presented above. It is
indeed enough to notice that

$\displaystyle\mathcal{H}^{\mathrm{sym}}$
$\displaystyle({\pi}^{{\mu}\to{\nu},{T}},{\pi}^{{\bar{\mu}}\to{\bar{\nu}},{T}})$
$\displaystyle=\int\log
f\,\mathrm{d}(\mu-\bar{\mu})+\int\log\bar{f}\,\mathrm{d}(\bar{\mu}-\mu)+\int\log
g\,\mathrm{d}(\nu-\bar{\nu})+\int\log\bar{g}\,\mathrm{d}(\bar{\nu}-\nu)$
$\displaystyle\leq\left\lVert\nabla\log
f\right\rVert_{L^{2}(\mu)}\left\lVert\mu-\bar{\mu}\right\rVert_{\dot{H}^{-1}(\mu)}+\left\lVert\nabla\log\bar{f}\right\rVert_{L^{2}(\bar{\mu})}\left\lVert\bar{\mu}-\mu\right\rVert_{\dot{H}^{-1}(\bar{\mu})}$
$\displaystyle+\left\lVert\nabla\log
g\right\rVert_{L^{2}(\nu)}\left\lVert\nu-\bar{\nu}\right\rVert_{\dot{H}^{-1}(\nu)}+\left\lVert\nabla\log\bar{g}\right\rVert_{L^{2}(\bar{\nu})}\left\lVert\bar{\nu}-\nu\right\rVert_{\dot{H}^{-1}(\bar{\nu})}\,,$

and that we can bound the $L^{2}$-norm of the gradient of the potentials in
terms of the Fisher information, e.g.

$\left\lVert\nabla\log
f\right\rVert_{L^{2}(\mu)}\leq\left\lVert\nabla\log\frac{\mathrm{d}\mu}{\mathrm{d}\mathfrak{m}}\right\rVert_{L^{2}(\mu)}+\left\lVert\nabla\log
P_{T}g\right\rVert_{L^{2}(\mu)}=\sqrt{\mathcal{I}(\mu)}+\left\lVert\nabla\log
P_{T}g\right\rVert_{L^{2}(\mu)}$

and analogously for the other three summands. ∎

Before moving to the proof of the stability estimates presented in (1.12) and
(1.13), we need another technical result given by the following

######  Lemma 4.3.

Let $(M,{\sf d},\mathfrak{m})$ satisfy (CD) and (I). For any $(\mu,\nu)$ and
$(\bar{\mu},\bar{\nu})$ satisfying (H1), (H2) and such that
$\left\lVert\mu-\bar{\mu}\right\rVert_{\dot{H}^{-1}(\mu)},\,\left\lVert\bar{\mu}-\mu\right\rVert_{\dot{H}^{-1}(\bar{\mu})},\,\left\lVert\nu-\bar{\nu}\right\rVert_{\dot{H}^{-1}(\nu)}$,
and $\left\lVert\bar{\nu}-\nu\right\rVert_{\dot{H}^{-1}(\bar{\nu})}$ are all
finite and with
$\mathcal{H}^{\mathrm{sym}}(\mu,\bar{\mu}),\,\mathcal{H}^{\mathrm{sym}}(\nu,\bar{\nu})<{\infty}$,
it holds

$\displaystyle\int\log\frac{P_{T}f}{P_{T}\bar{f}}\,\mathrm{d}\nu\leq\int\log\frac{f}{\bar{f}}\,\mathrm{d}\mu\qquad\text{and}\qquad\int\log\frac{P_{T}g}{P_{T}\bar{g}}\,\mathrm{d}\mu\leq\int\log\frac{g}{\bar{g}}\,\mathrm{d}\nu\,.$

Analogous bounds hold when we exchange the roles between $(\mu,\nu,f,g)$ and
$(\bar{\mu},\bar{\nu},\bar{f},\bar{g})$.

###### Proof.

Since
$\mathcal{H}(\nu|\bar{\nu})\leq\mathcal{H}({\pi}^{{\mu}\to{\nu},{T}}|{\pi}^{{\mu}\to{\bar{\nu}},{T}})$,
by means of the $fg$-decomposition the inequality reads as

$\int\log\frac{P_{T}f}{P_{T}\bar{f}}\mathrm{d}\nu+\int\log\frac{g}{\bar{g}}\mathrm{d}\nu\leq\int\log\frac{f}{\bar{f}}\mathrm{d}\mu+\int\log\frac{g}{\bar{g}}\mathrm{d}\nu\,,$

which yields the first inequality. The other bound can be proven in the same
way. ∎

By following the same line of reasoning applied for the stability of the
optimal plans, we can now deduce the following

######  Proposition 4.4.

Let $(M,{\sf d},\mathfrak{m})$ satisfy (CD) and (I). For any $(\mu,\nu)$ and
$(\bar{\mu},\bar{\nu})$ satisfying (H1), (H2), and such that
$\left\lVert\mu-\bar{\mu}\right\rVert_{\dot{H}^{-1}(\mu)},\,\left\lVert\bar{\mu}-\mu\right\rVert_{\dot{H}^{-1}(\bar{\mu})},\,\left\lVert\nu-\bar{\nu}\right\rVert_{\dot{H}^{-1}(\nu)}$,
and $\left\lVert\bar{\nu}-\nu\right\rVert_{\dot{H}^{-1}(\bar{\nu})}$ are all
finite and with
$\mathcal{H}^{\mathrm{sym}}(\mu,\bar{\mu}),\,\mathcal{H}^{\mathrm{sym}}(\nu,\bar{\nu})<{\infty}$,
it holds

$\displaystyle\mathcal{S}_{T}(\mu,\nu)-\mathcal{S}_{T}(\bar{\mu},\bar{\nu})\leq
T\biggl{[}\mathcal{H}(\bar{\mu}|\mu)\wedge\mathcal{H}(\bar{\nu}|\nu)\biggr{]}$
$\displaystyle+\frac{T}{\sqrt{E_{2\kappa}(T)}}\,\sqrt{\mathcal{C}_{T}(\mu,\nu)-\mathcal{H}(\mu|\mathfrak{m})}\left\lVert\mu-\bar{\mu}\right\rVert_{\dot{H}^{-1}(\mu)}$
$\displaystyle+\frac{T}{\sqrt{E_{2\kappa}(T)}}\,\sqrt{\mathcal{C}_{T}(\mu,\nu)-\mathcal{H}(\nu|\mathfrak{m})}\left\lVert\nu-\bar{\nu}\right\rVert_{\dot{H}^{-1}(\nu)},$

where $E_{2\kappa}$ is defined as in (1.8), from which it follows

$\displaystyle\left\lvert\mathcal{S}_{T}(\bar{\mu},\bar{\nu})-\mathcal{S}_{T}(\mu,\nu)\right\rvert\leq
T\biggl{[}\mathcal{H}^{\mathrm{sym}}(\mu,\bar{\mu})\wedge\mathcal{H}^{\mathrm{sym}}(\nu,\bar{\nu})\biggr{]}$
$\displaystyle\qquad+\frac{T}{\sqrt{E_{2\kappa}(T)}}\biggl{[}\sqrt{\mathcal{C}_{T}(\mu,\nu)-\mathcal{H}(\mu|\mathfrak{m})}\left\lVert\mu-\bar{\mu}\right\rVert_{\dot{H}^{-1}(\mu)}+\sqrt{\mathcal{C}_{T}(\mu,\nu)-\mathcal{H}(\nu|\mathfrak{m})}\left\lVert\nu-\bar{\nu}\right\rVert_{\dot{H}^{-1}(\nu)}\biggr{]}$
$\displaystyle\qquad+\frac{T}{\sqrt{E_{2\kappa}(T)}}\biggl{[}\sqrt{\mathcal{C}_{T}(\bar{\mu},\bar{\nu})-\mathcal{H}(\bar{\mu}|\mathfrak{m})}\left\lVert\bar{\mu}-\mu\right\rVert_{\dot{H}^{-1}(\bar{\mu})}+\sqrt{\mathcal{C}_{T}(\bar{\mu},\bar{\nu})-\mathcal{H}(\bar{\nu}|\mathfrak{m})}\left\lVert\bar{\nu}-\nu\right\rVert_{\dot{H}^{-1}(\bar{\nu})}\biggr{]}\,.$

###### Proof.

First of all, since the assumptions of Corollary 4.2 are met, the following
computations are all well defined. With this said, let us start by noticing
that

$\mathcal{S}_{T}(\mu,\nu)=T\,\mathcal{C}_{T}(\mu,\nu)-T\,\mathcal{H}(\mu|\mathfrak{m})-T\,\mathcal{H}(\nu|\mathfrak{m})=-T\int\log
P_{T}g\,\mathrm{d}\mu-T\int\log P_{T}f\,\mathrm{d}\nu\,,$

and therefore

$\displaystyle\mathcal{S}_{T}(\bar{\mu},\bar{\nu})-\mathcal{S}_{T}(\mu,\nu)=T\int\log
P_{T}g\,\mathrm{d}\mu+T\int\log P_{T}f\,\mathrm{d}\nu-T\int\log
P_{T}\bar{g}\,\mathrm{d}\bar{\mu}-T\int\log P_{T}\bar{f}\,\mathrm{d}\bar{\nu}$
$\displaystyle=T\int\log\frac{P_{T}g}{P_{T}\bar{g}}\mathrm{d}\mu+T\int\log\frac{P_{T}f}{P_{T}\bar{f}}\mathrm{d}\nu-T\int\log
P_{T}\bar{g}\,\mathrm{d}(\bar{\mu}-\mu)-T\int\log
P_{T}\bar{f}\,\mathrm{d}(\bar{\nu}-\nu)\,.$

Applying Lemma 4.3 we deduce that

(4.6)
$\displaystyle\mathcal{S}_{T}(\bar{\mu},\bar{\nu})-\mathcal{S}_{T}(\mu,\nu)$
$\displaystyle\leq
T\int\log\frac{P_{T}g}{P_{T}\bar{g}}\mathrm{d}\mu+T\int\log\frac{f}{\bar{f}}\mathrm{d}\mu$
$\displaystyle\qquad\qquad\qquad-T\int\log
P_{T}\bar{g}\,\mathrm{d}(\bar{\mu}-\mu)-T\int\log
P_{T}\bar{f}\,\mathrm{d}(\bar{\nu}-\nu)$
$\displaystyle=T\int\log\frac{\mathrm{d}\mu}{\mathrm{d}\bar{\mu}}\mathrm{d}\mu-T\int\log
P_{T}\bar{g}\,\mathrm{d}(\bar{\mu}-\mu)-T\int\log
P_{T}\bar{f}\,\mathrm{d}(\bar{\nu}-\nu)$
$\displaystyle=T\mathcal{H}(\mu|\bar{\mu})-T\int\log
P_{T}\bar{g}\,\mathrm{d}(\bar{\mu}-\mu)-T\int\log
P_{T}\bar{f}\,\mathrm{d}(\bar{\nu}-\nu)\,.$

Notice that above we could have used Lemma 4.3 on the other integral
$\int\log\frac{P_{T}g}{P_{T}\bar{g}}\mathrm{d}\mu$ and therefore we would have
got $\mathcal{H}(\nu|\bar{\nu})$ instead of $\mathcal{H}(\mu|\bar{\mu})$.
Therefore we can state that

$\displaystyle\mathcal{S}_{T}(\bar{\mu},\bar{\nu})-\mathcal{S}_{T}(\mu,\nu)$
$\displaystyle\leq
T\Bigl{[}\mathcal{H}(\mu|\bar{\mu})\wedge\mathcal{H}(\nu|\bar{\nu})\Bigr{]}-T\int\log
P_{T}\bar{g}\,\mathrm{d}(\bar{\mu}-\mu)-T\int\log
P_{T}\bar{f}\,\mathrm{d}(\bar{\nu}-\nu)$ $\displaystyle\leq
T\Bigl{[}\mathcal{H}(\mu|\bar{\mu})\wedge\mathcal{H}(\nu|\bar{\nu})\Bigr{]}+T\left\lVert\nabla\log
P_{T}\bar{g}\right\rVert_{L^{2}(\bar{\mu})}\left\lVert\bar{\mu}-\mu\right\rVert_{\dot{H}^{-1}(\bar{\mu})}$
$\displaystyle\qquad\qquad+T\left\lVert\nabla\log
P_{T}\bar{f}\right\rVert_{L^{2}(\bar{\nu})}\left\lVert\bar{\nu}-\nu\right\rVert_{\dot{H}^{-1}(\bar{\nu})}.$

By exchanging the roles between $(\mu,\nu)$ and $(\bar{\mu},\bar{\nu})$ and by
arguing in the same way we also get

$\displaystyle\mathcal{S}_{T}(\mu,\nu)-\mathcal{S}_{T}(\bar{\mu},\bar{\nu})\leq
T\Bigl{[}\mathcal{H}(\bar{\mu}|\mu)\wedge\mathcal{H}(\bar{\nu}|\nu)\Bigr{]}$
$\displaystyle+T\left\lVert\nabla\log
P_{T}g\right\rVert_{L^{2}(\mu)}\left\lVert\mu-\bar{\mu}\right\rVert_{\dot{H}^{-1}(\mu)}$
$\displaystyle+T\left\lVert\nabla\log
P_{T}f\right\rVert_{L^{2}(\nu)}\left\lVert\nu-\bar{\nu}\right\rVert_{\dot{H}^{-1}(\nu)},$

and therefore we deduce

$\displaystyle\left\lvert\mathcal{S}_{T}(\bar{\mu},\bar{\nu})-\mathcal{S}_{T}(\mu,\nu)\right\rvert\leq$
$\displaystyle\,T\Bigl{[}\mathcal{H}^{\mathrm{sym}}(\mu,\bar{\mu})\wedge\mathcal{H}^{\mathrm{sym}}(\nu,\bar{\nu})\Bigr{]}$
$\displaystyle+$ $\displaystyle\,T\left\lVert\nabla\log
P_{T}g\right\rVert_{L^{2}(\mu)}\left\lVert\mu-\bar{\mu}\right\rVert_{\dot{H}^{-1}(\mu)}+T\left\lVert\nabla\log
P_{T}f\right\rVert_{L^{2}(\nu)}\left\lVert\nu-\bar{\nu}\right\rVert_{\dot{H}^{-1}(\nu)}$
$\displaystyle+$ $\displaystyle\,T\left\lVert\nabla\log
P_{T}\bar{g}\right\rVert_{L^{2}(\bar{\mu})}\left\lVert\bar{\mu}-\mu\right\rVert_{\dot{H}^{-1}(\bar{\mu})}+T\left\lVert\nabla\log
P_{T}\bar{f}\right\rVert_{L^{2}(\bar{\nu})}\left\lVert\bar{\nu}-\nu\right\rVert_{\dot{H}^{-1}(\bar{\nu})}.$

Given the above, the thesis follows from the corrector estimates (cf.
Proposition 2.4). ∎

######  Remark 4.5.

Notice that in the cases where we change just one marginal (e.g.
$\mu=\bar{\mu}$), we can get rid of the relative entropy between the marginals
in the above result. Therefore in order to get a bound without the symmetric
relative entropy of the marginals it is enough to consider the case where one
of the marginals is frozen by means of the trivial inequality

(4.7)
$\left\lvert\mathcal{S}_{T}(\bar{\mu},\bar{\nu})-\mathcal{S}_{T}(\mu,\nu)\right\rvert\leq\left\lvert\mathcal{S}_{T}(\bar{\mu},\bar{\nu})-\mathcal{S}_{T}(\mu,\bar{\nu})\right\rvert+\left\lvert\mathcal{S}_{T}(\mu,\bar{\nu})-\mathcal{S}_{T}(\mu,\nu)\right\rvert\,.$

Let us conclude by showing that, under the finite Fisher information
assumption, we may write a stability estimate for the Schrödinger costs
without involving the symmetric relative entropies on the right-hand side.

######  Proposition 4.6.

Let $(M,{\sf d},\mathfrak{m})$ satisfy (CD) and (I). For any $(\mu,\nu)$ and
$(\bar{\mu},\bar{\nu})$ satisfying (H1) and (H2) we have

$\displaystyle\left\lvert\mathcal{C}_{T}(\bar{\mu},\bar{\nu})-\mathcal{C}_{T}(\mu,\nu)\right\rvert\leq$
$\displaystyle\frac{1}{\sqrt{E_{2\kappa}(T)}}\biggl{[}\sqrt{\mathcal{I}(\mu)}+\sqrt{\mathcal{C}_{T}(\mu,\nu)-\mathcal{H}(\mu|\mathfrak{m})}\biggr{]}\left\lVert\mu-\bar{\mu}\right\rVert_{\dot{H}^{-1}(\mu)}$
$\displaystyle+$
$\displaystyle\frac{1}{\sqrt{E_{2\kappa}(T)}}\biggl{[}\sqrt{\mathcal{I}(\bar{\mu})}+\sqrt{\mathcal{C}_{T}(\bar{\mu},\bar{\nu})-\mathcal{H}(\bar{\mu}|\mathfrak{m})}\biggr{]}\left\lVert\bar{\mu}-\mu\right\rVert_{\dot{H}^{-1}(\bar{\mu})}$
$\displaystyle+$
$\displaystyle\frac{1}{\sqrt{E_{2\kappa}(T)}}\biggl{[}\sqrt{\mathcal{I}(\nu)}+\sqrt{\mathcal{C}_{T}(\mu,\nu)-\mathcal{H}(\nu|\mathfrak{m})}\biggr{]}\left\lVert\nu-\bar{\nu}\right\rVert_{\dot{H}^{-1}(\nu)}$
$\displaystyle+$
$\displaystyle\frac{1}{\sqrt{E_{2\kappa}(T)}}\biggl{[}\sqrt{\mathcal{I}(\bar{\nu})}+\sqrt{\mathcal{C}_{T}(\bar{\mu},\bar{\nu})-\mathcal{H}(\bar{\nu}|\mathfrak{m})}\biggr{]}\left\lVert\bar{\nu}-\nu\right\rVert_{\dot{H}^{-1}(\bar{\nu})}\,,$

where $E_{2\kappa}$ is defined as in (1.8).

###### Proof.

The proof runs like the one given in Proposition 4.4 observing that, similarly
to (4.6), we can write

$\mathcal{C}_{T}(\bar{\mu},\bar{\nu})-\mathcal{C}_{T}(\mu,\nu)\leq\int\log\bar{f}\,\mathrm{d}(\bar{\mu}-\mu)+\int\log\bar{g}\,\mathrm{d}(\bar{\nu}-\nu)-\biggl{[}\mathcal{H}(\mu|\bar{\mu})\vee\mathcal{H}(\nu|\bar{\nu})\biggr{]}$

and therefore it holds

$\begin{split}\left\lvert\mathcal{C}_{T}(\bar{\mu},\bar{\nu})-\mathcal{C}_{T}(\mu,\nu)\right\rvert\leq\left\lVert\nabla\log
f\right\rVert_{L^{2}(\mu)}\left\lVert\mu-\bar{\mu}\right\rVert_{\dot{H}^{-1}(\mu)}+\left\lVert\nabla\log\bar{f}\right\rVert_{L^{2}(\bar{\mu})}\left\lVert\bar{\mu}-\mu\right\rVert_{\dot{H}^{-1}(\bar{\mu})}\\\
+\left\lVert\nabla\log
g\right\rVert_{L^{2}(\nu)}\left\lVert\nu-\bar{\nu}\right\rVert_{\dot{H}^{-1}(\nu)}+\left\lVert\nabla\log\bar{g}\right\rVert_{L^{2}(\bar{\nu})}\left\lVert\bar{\nu}-\nu\right\rVert_{\dot{H}^{-1}(\bar{\nu})}\,.\end{split}$

∎

Notice that in the above result we are able to consider the stability directly
between the Schrödinger costs $\mathcal{C}_{T}(\mu,\nu)$ and
$\mathcal{C}_{T}(\bar{\mu},\bar{\nu})$. This is indeed due to the fact that we
are working in a finite Fisher information setting.

###  4.1. Application to EOT with quadratic cost

In this section we translate the stability results stated in Theorem 1.2 and
Proposition 4.4 to the Euclidean EOT setting with quadratic cost

(4.8)
$\mathcal{S}^{\varepsilon}(\mu,\nu)\coloneqq\inf_{\pi\in\Pi(\mu,\nu)}\int|x-y|^{2}\,\mathrm{d}\pi+\varepsilon\,\mathcal{H}(\pi|\mu\otimes\nu)\,,\qquad\varepsilon>0\,.$

We will manage to do that under sufficiently general conditions, so that our
results will apply to any couple of marginals
$\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ with finite relative entropy
w.r.t. the Lebesgue measure $\mathcal{L}_{d}$ on $\mathbb{R}^{d}$ and with
densities w.r.t. $\mathcal{L}_{d}$ locally bounded away from $0$ on their
support and such that
$\mu(\partial\mathrm{supp}(\mu))=\nu(\partial\mathrm{supp}(\nu))=0$; or with
bounded and compactly supported densities. For later reference, let
${\pi}^{{\mu},{\nu}}_{\varepsilon}$ denote the minimizer (which exists and is
unique since $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, cf. [40, Theorem
4.2]).

In what follows we are going to consider a Schrödinger problem equivalent to
the above (4.8), where the underlying stochastic dynamics is given by the law
of the Ornstein–Uhlenbeck process

(4.9) $\mathrm{d}X_{t}=-\kappa X_{t}\mathrm{d}t+\sqrt{2}\,\mathrm{d}B_{t}\,,$

$(B_{t})_{t}$ being a $d$-dimensional Brownian motion and $\kappa>0$ a
curvature parameter (whose value will be specified later). The above SDE
admits as unique invariant measure the Gaussian distribution
$\mathfrak{m}\sim\mathcal{N}\bigl{(}0,\kappa^{-1}\operatorname{Id}\bigr{)}$,
i.e. $\mathrm{d}\mathfrak{m}(x)\sim e^{-\frac{\kappa}{2}\left\lvert
x\right\rvert^{2}}\mathrm{d}x$, which satisfies the curvature condition ${\sf
CD}(\kappa,{\infty})$ and hence a logarithmic Sobolev inequality with
parameter $\kappa^{-1}$ [5, Corollary 5.7.1]. Then, Herbst’s argument [5,
Proposition 5.4.1] implies that $\mathfrak{m}$ satisfies the integrability
condition (I) for any $r<\kappa/2$. Notice that for any choice of $\kappa>0$
we have

(4.10)
$\mathcal{H}(\mu|\mathfrak{m})=\mathcal{H}(\mu|\mathcal{L}_{d})+\frac{\kappa}{2}\,M_{2}(\mu)+\frac{d}{2}\log\frac{2\pi}{\kappa}<+{\infty}$

and similarly $\mathcal{H}(\nu|\mathfrak{m})<+{\infty}$.

Now, let us consider a final time in SP such that
$\frac{\varepsilon\kappa}{4}=\sinh(\kappa T)$, i.e.

(4.11)
$T\coloneqq\frac{1}{\kappa}\log\left(\frac{\varepsilon\kappa}{4}+\sqrt{\frac{\varepsilon^{2}\kappa^{2}}{16}+1}\right)\,$

and let $\mathrm{R}_{0,T}=\mathcal{L}(X_{0},X_{T})$ denote the joint law at
times $0$ and $T$ of the Ornstein–Uhlenbeck process solving (4.9) started at
the invariant measure $\mathfrak{m}$, whose density is given by the transition
kernel

${\sf
p}_{T}(x,y)=\frac{\mathrm{d}\mathrm{R}_{0,T}}{\mathrm{d}(\mathfrak{m}\otimes\mathfrak{m})}(x,y)=\frac{1}{(1-e^{-2\kappa
T})^{\frac{d}{2}}}\,\exp\left\\{-\frac{\left\lvert
x\right\rvert^{2}-2e^{\kappa T}\,x\cdot y+\left\lvert
y\right\rvert^{2}}{\frac{2}{\kappa}(e^{2\kappa T}-1)}\right\\}\,.$

Then, thanks to our choice of $T$, for any coupling $\pi\in\Pi(\mu,\nu)$ it
holds

$\displaystyle\int_{\mathbb{R}^{2d}}\left\lvert
x-y\right\rvert^{2}\mathrm{d}\pi+\varepsilon\,\mathcal{H}(\pi|\mu\otimes\nu)=M_{2}(\mu)+M_{2}(\nu)-2\int_{\mathbb{R}^{2d}}x\cdot
y\,\mathrm{d}\pi+\varepsilon\,\mathcal{H}(\pi|\mu\otimes\nu)$
$\displaystyle=-\frac{d\varepsilon}{2}\log(1-e^{-2\kappa T})+(1-e^{-\kappa
T})\bigl{(}M_{2}(\mu)+M_{2}(\nu)\bigr{)}-\varepsilon\int_{\mathbb{R}^{2d}}\log{\sf
p}_{T}(x,y)\,\mathrm{d}\pi+\varepsilon\,\mathcal{H}(\pi|\mu\otimes\nu)$
$\displaystyle=-\frac{d\varepsilon}{2}\log(1-e^{-2\kappa T})+(1-e^{-\kappa
T})\bigl{(}M_{2}(\mu)+M_{2}(\nu)\bigr{)}+\varepsilon\left[\mathcal{H}(\pi|\mu\otimes\nu)-\int_{\mathbb{R}^{2d}}\log\frac{\mathrm{d}\mathrm{R}}{\mathrm{d}(\mathfrak{m}\otimes\mathfrak{m})}\,\mathrm{d}\pi\right]$
$\displaystyle=-\frac{d\varepsilon}{2}\log(1-e^{-2\kappa T})+(1-e^{-\kappa
T})\bigl{(}M_{2}(\mu)+M_{2}(\nu)\bigr{)}+\varepsilon\,\mathcal{H}(\pi|\mathrm{R}_{0,T})-\varepsilon\,\mathcal{H}(\mu|\mathfrak{m})-\varepsilon\,\mathcal{H}(\nu|\mathfrak{m})\,.$

Therefore the above EOT problem (4.8) has the same minimizer of SP with
reference given by the above-chosen $\mathrm{R}_{0,T}$ and their values are
linked according to the following identity

(4.12) $\displaystyle\mathcal{S}^{\varepsilon}(\mu,\nu)=$
$\displaystyle\,\varepsilon\,\mathcal{C}_{T}(\mu,\nu)-\varepsilon\,\mathcal{H}(\mu|\mathfrak{m})-\varepsilon\,\mathcal{H}(\nu|\mathfrak{m})-\frac{d\varepsilon}{2}\log(1-e^{-2\kappa
T})+(1-e^{-\kappa T})\bigl{(}M_{2}(\mu)+M_{2}(\nu)\bigr{)}$ $\displaystyle=$
$\displaystyle\,\frac{\varepsilon}{T}\,\mathcal{S}_{T}(\mu,\nu)-\frac{d\varepsilon}{2}\log(1-e^{-2\kappa
T})+(1-e^{-\kappa T})\bigl{(}M_{2}(\mu)+M_{2}(\nu)\bigr{)}\,.$

Since we are interested in stability results, take also
$\bar{\mu},\,\bar{\nu}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ such that

$\mathcal{H}(\bar{\mu}|\mathcal{L}_{d}),\,\mathcal{H}(\bar{\nu}|\mathcal{L}_{d})<+{\infty}\quad\text{
and
}\quad\mathcal{H}^{\mathrm{sym}}(\mu,\bar{\mu}),\,\mathcal{H}^{\mathrm{sym}}(\nu,\bar{\nu})<+{\infty}\,.$

We are now ready to apply Proposition 4.4 and get

(4.13) $\displaystyle|\mathcal{S}^{\varepsilon}(\bar{\mu},\bar{\nu})$
$\displaystyle-\mathcal{S}^{\varepsilon}(\mu,\nu)|$ $\displaystyle\leq$
$\displaystyle\frac{\varepsilon}{T}\left\lvert\mathcal{S}_{T}(\bar{\mu},\bar{\nu})-\mathcal{S}_{T}(\mu,\nu)\right\rvert+\left(1-e^{-\kappa
T}\right)\Bigl{(}\left\lvert
M_{2}(\bar{\mu})-M_{2}(\mu)\right\rvert+\left\lvert
M_{2}(\bar{\nu})-M_{2}(\nu)\right\rvert\Bigr{)}$ $\displaystyle\leq$
$\displaystyle\,\varepsilon\Bigl{[}\mathcal{H}^{\mathrm{sym}}(\mu,\bar{\mu})\wedge\mathcal{H}^{\mathrm{sym}}(\nu,\bar{\nu})\Bigr{]}+\left(1-e^{-\kappa
T}\right)\Bigl{(}\left\lvert
M_{2}(\bar{\mu})-M_{2}(\mu)\right\rvert+\left\lvert
M_{2}(\bar{\nu})-M_{2}(\nu)\right\rvert\Bigr{)}$ $\displaystyle+$
$\displaystyle\,\frac{\varepsilon}{\sqrt{E_{2\kappa}(T)}}\biggl{[}\sqrt{\mathcal{C}_{T}(\mu,\nu)-\mathcal{H}(\mu|\mathfrak{m})}\left\lVert\mu-\bar{\mu}\right\rVert_{\dot{H}^{-1}(\mu)}+\sqrt{\mathcal{C}_{T}(\mu,\nu)-\mathcal{H}(\nu|\mathfrak{m})}\left\lVert\nu-\bar{\nu}\right\rVert_{\dot{H}^{-1}(\nu)}\biggr{]}$
$\displaystyle+$
$\displaystyle\,\frac{\varepsilon}{\sqrt{E_{2\kappa}(T)}}\biggl{[}\sqrt{\mathcal{C}_{T}(\bar{\mu},\bar{\nu})-\mathcal{H}(\bar{\mu}|\mathfrak{m})}\left\lVert\bar{\mu}-\mu\right\rVert_{\dot{H}^{-1}(\bar{\mu})}+\sqrt{\mathcal{C}_{T}(\bar{\mu},\bar{\nu})-\mathcal{H}(\bar{\nu}|\mathfrak{m})}\left\lVert\bar{\nu}-\nu\right\rVert_{\dot{H}^{-1}(\bar{\nu})}\biggr{]}\,.$

Since for any coupling $\pi\in\Pi(\bar{\mu},\mu)$ we can write

$\displaystyle\left\lvert M_{2}(\bar{\mu})-M_{2}(\mu)\right\rvert=$
$\displaystyle\,\left\lvert\int\left\lvert x\right\rvert^{2}-\left\lvert
y\right\rvert^{2}\,\mathrm{d}\pi\right\rvert\leq\int\left\lvert
x(x-y)\right\rvert\mathrm{d}\pi+\int\left\lvert
y(x-y)\right\rvert\mathrm{d}\pi$ $\displaystyle\leq$
$\displaystyle\,\left(\sqrt{M_{2}(\bar{\mu})}+\sqrt{M_{2}(\mu)}\right)\left(\int\left\lvert
x-y\right\rvert^{2}\mathrm{d}\pi\right)^{\frac{1}{2}}\,,$

by minimizing the right-hand side over $\pi\in\Pi(\mu,\nu)$ we end up with

$\displaystyle\left\lvert
M_{2}(\bar{\mu})-M_{2}(\mu)\right\rvert\leq\left(\sqrt{M_{2}(\bar{\mu})}+\sqrt{M_{2}(\mu)}\right)\mathcal{W}_{2}(\bar{\mu},\mu)$

and (4.13) reads as

(4.14) $\displaystyle|\mathcal{S}^{\varepsilon}$
$\displaystyle(\bar{\mu},\bar{\nu})-\mathcal{S}^{\varepsilon}(\mu,\nu)|\leq\varepsilon\Bigl{[}\mathcal{H}^{\mathrm{sym}}(\mu,\bar{\mu})\wedge\mathcal{H}^{\mathrm{sym}}(\nu,\bar{\nu})\Bigr{]}$
$\displaystyle+\left(1-e^{-\kappa
T}\right)\left(\sqrt{M_{2}(\bar{\mu})}+\sqrt{M_{2}(\mu)}\right)\mathcal{W}_{2}(\bar{\mu},\mu)+\left(1-e^{-\kappa
T}\right)\left(\sqrt{M_{2}(\bar{\nu})}+\sqrt{M_{2}(\nu)}\right)\mathcal{W}_{2}(\bar{\nu},\nu)$
$\displaystyle+\frac{\varepsilon}{\sqrt{E_{2\kappa}(T)}}\biggl{[}\sqrt{\mathcal{C}_{T}(\mu,\nu)-\mathcal{H}(\mu|\mathfrak{m})}\left\lVert\mu-\bar{\mu}\right\rVert_{\dot{H}^{-1}(\mu)}+\sqrt{\mathcal{C}_{T}(\mu,\nu)-\mathcal{H}(\nu|\mathfrak{m})}\left\lVert\nu-\bar{\nu}\right\rVert_{\dot{H}^{-1}(\nu)}\biggr{]}$
$\displaystyle+\frac{\varepsilon}{\sqrt{E_{2\kappa}(T)}}\biggl{[}\sqrt{\mathcal{C}_{T}(\bar{\mu},\bar{\nu})-\mathcal{H}(\bar{\mu}|\mathfrak{m})}\left\lVert\bar{\mu}-\mu\right\rVert_{\dot{H}^{-1}(\bar{\mu})}+\sqrt{\mathcal{C}_{T}(\bar{\mu},\bar{\nu})-\mathcal{H}(\bar{\nu}|\mathfrak{m})}\left\lVert\bar{\nu}-\nu\right\rVert_{\dot{H}^{-1}(\bar{\nu})}\biggr{]}\,.$

Similarly, since ${\pi}^{{\mu}\to{\nu},{T}}$ is also the optimizer of the EOT
problem (4.8), we can translate Theorem 1.2 into a stability result between
the optimal plans for (4.8), that is, between
${\pi}^{{\mu},{\nu}}_{\varepsilon}$ and
${\pi}^{{\bar{\mu}},{\bar{\nu}}}_{\varepsilon}$:

(4.15) $\displaystyle\varepsilon$
$\displaystyle\mathcal{H}^{\mathrm{sym}}({\pi}^{{\mu},{\nu}}_{\varepsilon},{\pi}^{{\bar{\mu}},{\bar{\nu}}}_{\varepsilon})=\frac{\varepsilon}{T}\,T\,\mathcal{H}^{\mathrm{sym}}({\pi}^{{\mu}\to{\nu},{T}},{\pi}^{{\bar{\mu}}\to{\bar{\nu}},{T}})\leq\,\varepsilon\,\mathcal{H}^{\mathrm{sym}}(\mu,\bar{\mu})+\varepsilon\,\mathcal{H}^{\mathrm{sym}}(\nu,\bar{\nu})$
$\displaystyle+$
$\displaystyle\,\frac{\varepsilon}{\sqrt{E_{2\kappa}(T)}}\biggl{[}\sqrt{\mathcal{C}_{T}(\mu,\nu)-\mathcal{H}(\mu|\mathfrak{m})}\left\lVert\mu-\bar{\mu}\right\rVert_{\dot{H}^{-1}(\mu)}+\sqrt{\mathcal{C}_{T}(\mu,\nu)-\mathcal{H}(\nu|\mathfrak{m})}\left\lVert\nu-\bar{\nu}\right\rVert_{\dot{H}^{-1}(\nu)}\biggr{]}$
$\displaystyle+$
$\displaystyle\,\frac{\varepsilon}{\sqrt{E_{2\kappa}(T)}}\biggl{[}\sqrt{\mathcal{C}_{T}(\bar{\mu},\bar{\nu})-\mathcal{H}(\bar{\mu}|\mathfrak{m})}\left\lVert\bar{\mu}-\mu\right\rVert_{\dot{H}^{-1}(\bar{\mu})}+\sqrt{\mathcal{C}_{T}(\bar{\mu},\bar{\nu})-\mathcal{H}(\bar{\nu}|\mathfrak{m})}\left\lVert\bar{\nu}-\nu\right\rVert_{\dot{H}^{-1}(\bar{\nu})}\biggr{]}\,.$

Given the above bounds one can now get explicit estimates by choosing the
curvature parameter $\kappa>0$.

######  Remark 4.7 (Small-noise limit).

Let us point out that the above bounds are stable when $\varepsilon\to 0$.
Indeed (4.11) guarantees us that the small-noise limit $\varepsilon\to 0$ is
equivalent to the small-time limit $T\to 0$ and moreover the ratio
$\frac{\varepsilon}{T}$ stays finite since

$\displaystyle\frac{\varepsilon}{T}=\frac{\varepsilon\kappa}{\kappa
T}=\frac{4\sinh(\kappa T)}{\kappa T}=2\,\frac{e^{\kappa T}-e^{-\kappa
T}}{\kappa T}\,\to\,4\,,\quad\text{as $T\to 0$.}$

Therefore in the small-noise limit, from (1.14) we deduce for (4.8) the
following stability results

(4.16) $\displaystyle\limsup_{\varepsilon\to
0}\left\lvert\mathcal{S}^{\varepsilon}(\bar{\mu},\bar{\nu})-\mathcal{S}^{\varepsilon}(\mu,\nu)\right\rvert\leq$
$\displaystyle
2\,\mathcal{W}_{2}(\mu,\nu)\,\biggl{[}\left\lVert\mu-\bar{\mu}\right\rVert_{\dot{H}^{-1}(\mu)}+\left\lVert\nu-\bar{\nu}\right\rVert_{\dot{H}^{-1}(\nu)}\biggr{]}$
$\displaystyle+$ $\displaystyle
2\,\mathcal{W}_{2}(\bar{\mu},\bar{\nu})\,\biggl{[}\left\lVert\bar{\mu}-\mu\right\rVert_{\dot{H}^{-1}(\bar{\mu})}+\left\lVert\bar{\nu}-\nu\right\rVert_{\dot{H}^{-1}(\bar{\nu})}\biggr{]}\,,$
$\displaystyle\limsup_{\varepsilon\to
0}\,\varepsilon\,\mathcal{H}^{\mathrm{sym}}({\pi}^{{\mu},{\nu}}_{\varepsilon},{\pi}^{{\bar{\mu}},{\bar{\nu}}}_{\varepsilon})\leq$
$\displaystyle
2\,\mathcal{W}_{2}(\mu,\nu)\,\biggl{[}\left\lVert\mu-\bar{\mu}\right\rVert_{\dot{H}^{-1}(\mu)}+\left\lVert\nu-\bar{\nu}\right\rVert_{\dot{H}^{-1}(\nu)}\biggr{]}$
$\displaystyle+$ $\displaystyle
2\,\mathcal{W}_{2}(\bar{\mu},\bar{\nu})\,\biggl{[}\left\lVert\bar{\mu}-\mu\right\rVert_{\dot{H}^{-1}(\bar{\mu})}+\left\lVert\bar{\nu}-\nu\right\rVert_{\dot{H}^{-1}(\bar{\nu})}\biggr{]}\,,$

which agree, up to a scaling constant, with Remark 1.7 in the Schrödinger
setting.

Notice that the above small-noise limit is independent from the choice of
$\kappa>0$. This suggests us to take the limit $\kappa\downarrow 0$ directly
in (4.14) and (4.15). Indeed from (4.11) it follows $T\to\,\varepsilon/4$ as
$\kappa\downarrow 0$. Furthermore, by rearranging (4.12) we notice that

$\begin{split}T\,\mathcal{C}_{T}(\mu,\nu)-T\,\mathcal{H}(\mu|\mathfrak{m})&=\frac{T}{\varepsilon}\,\mathcal{S}^{\varepsilon}(\mu,\nu)-\frac{T}{\varepsilon}(1-e^{-\kappa
T})\biggl{(}M_{2}(\mu)+M_{2}(\nu)\biggr{)}+T\,\mathcal{H}(\nu|\mathfrak{m})\\\
&\qquad+\frac{d\,T}{2}\log\left(1-e^{-2\kappa T}\right)\\\
&=\frac{T}{\varepsilon}\,\mathcal{S}^{\varepsilon}(\mu,\nu)-\frac{T}{\varepsilon}(1-e^{-\kappa
T})\biggl{(}M_{2}(\mu)+M_{2}(\nu)\biggr{)}+T\,\mathcal{H}(\nu|\mathcal{L}_{d})\\\
&\qquad+\frac{\kappa T}{2}M_{2}(\nu)+\frac{d\,T}{2}\log\left(4\pi
T\,\frac{1-e^{-2\kappa T}}{2\kappa T}\right)\end{split}$

and therefore we have

$\lim_{\kappa\to
0}\,T\,\mathcal{C}_{T}(\mu,\nu)-T\,\mathcal{H}(\mu|\mathfrak{m})=\frac{1}{4}\,\mathcal{S}^{\varepsilon}(\mu,\nu)+\frac{\varepsilon}{4}\,\mathcal{H}(\mu|\mathcal{L}_{d})+\frac{1}{4}\,C_{\varepsilon}\,,$

where $C_{\varepsilon}=\frac{d\varepsilon}{2}\log(4\pi\varepsilon)$. As a
consequence, by taking the limit in the right-hand sides of (4.14) and (4.15),
we deduce that

$\displaystyle\left\lvert\mathcal{S}^{\varepsilon}(\bar{\mu},\bar{\nu})-\mathcal{S}^{\varepsilon}(\mu,\nu)\right\rvert\leq\varepsilon\Bigl{[}\mathcal{H}^{\mathrm{sym}}(\mu,\bar{\mu})\wedge\mathcal{H}^{\mathrm{sym}}(\nu,\bar{\nu})$
$\displaystyle\Bigr{]}$
$\displaystyle+2\biggl{[}\sqrt{\mathcal{S}^{\varepsilon}(\mu,\nu)+\varepsilon\,\mathcal{H}(\nu|\mathcal{L}_{d})+C_{\varepsilon}}\left\lVert\mu-\bar{\mu}\right\rVert_{\dot{H}^{-1}(\mu)}$
$\displaystyle+\sqrt{\mathcal{S}^{\varepsilon}(\mu,\nu)+\varepsilon\,\mathcal{H}(\mu|\mathcal{L}_{d})+C_{\varepsilon}}\left\lVert\nu-\bar{\nu}\right\rVert_{\dot{H}^{-1}(\nu)}\biggr{]}$
$\displaystyle+2\biggl{[}\sqrt{\mathcal{S}^{\varepsilon}(\bar{\mu},\bar{\nu})+\varepsilon\,\mathcal{H}(\bar{\nu}|\mathcal{L}_{d})+C_{\varepsilon}}\left\lVert\bar{\mu}-\mu\right\rVert_{\dot{H}^{-1}(\bar{\mu})}$
$\displaystyle+\sqrt{\mathcal{S}^{\varepsilon}(\bar{\mu},\bar{\nu})+\varepsilon\,\mathcal{H}(\bar{\mu}|\mathcal{L}_{d})+C_{\varepsilon}}\left\lVert\bar{\nu}-\nu\right\rVert_{\dot{H}^{-1}(\bar{\nu})}\biggr{]}$

and

$\displaystyle\varepsilon\,\mathcal{H}^{\mathrm{sym}}({\pi}^{{\mu},{\nu}}_{\varepsilon},{\pi}^{{\bar{\mu}},{\bar{\nu}}}_{\varepsilon})\leq\,\varepsilon\,\mathcal{H}^{\mathrm{sym}}(\mu,\bar{\mu})+\varepsilon\,\mathcal{H}^{\mathrm{sym}}(\nu,\bar{\nu})$
$\displaystyle+2\biggl{[}\sqrt{\mathcal{S}^{\varepsilon}(\mu,\nu)+\varepsilon\,\mathcal{H}(\nu|\mathcal{L}_{d})+C_{\varepsilon}}\left\lVert\mu-\bar{\mu}\right\rVert_{\dot{H}^{-1}(\mu)}$
$\displaystyle+\sqrt{\mathcal{S}^{\varepsilon}(\mu,\nu)+\varepsilon\,\mathcal{H}(\mu|\mathcal{L}_{d})+C_{\varepsilon}}\left\lVert\nu-\bar{\nu}\right\rVert_{\dot{H}^{-1}(\nu)}\biggr{]}$
$\displaystyle+2\biggl{[}\sqrt{\mathcal{S}^{\varepsilon}(\bar{\mu},\bar{\nu})+\varepsilon\,\mathcal{H}(\bar{\nu}|\mathcal{L}_{d})+C_{\varepsilon}}\left\lVert\bar{\mu}-\mu\right\rVert_{\dot{H}^{-1}(\bar{\mu})}$
$\displaystyle+\sqrt{\mathcal{S}^{\varepsilon}(\bar{\mu},\bar{\nu})+\varepsilon\,\mathcal{H}(\bar{\mu}|\mathcal{L}_{d})+C_{\varepsilon}}\left\lVert\bar{\nu}-\nu\right\rVert_{\dot{H}^{-1}(\bar{\nu})}\biggr{]}\,.$

Notice that from the above bounds, we get the validity of (4.16) in the small-
noise limit $\varepsilon\downarrow 0$,.

##  Appendix A A log-integrability Lemma

Let us start by recalling a few interesting facts about Orlicz spaces. Let
$\mathfrak{q}$ be a probability measure on a measurable space
$(\Omega,\Sigma)$ and consider the Young functions

$\displaystyle\theta(t)\coloneqq
e^{t}-1\,,\qquad\theta^{*}(s)=\begin{cases}s\,\log s-s+1\quad&\text{if
}s>0,\\\ 1\quad&\text{if }s=0\,.\end{cases}$

where
$\theta^{*}(s)\coloneqq\sup_{t\in\mathbb{R}}\bigl{\\{}st-\theta(t)\bigr{\\}}$.
The Orlicz space associated to the Young function $\theta$, denoted by
$L_{\theta}(\mathfrak{q})$, is defined as the space of measurable functions
$f:\Omega\to\mathbb{R}$ with finite Luxemburg norm

$\left\lVert
f\right\rVert_{\theta}\coloneqq\inf\left\\{b>0\colon\int\theta\left(\frac{\left\lvert
f\right\rvert}{b}\right)\mathrm{d}\mathfrak{q}\leq 1\right\\}\,.$

To be precise we should make explicit reference to the underlying probability
$\mathfrak{q}$, when talking about Luxemburg norms; however we will omit such
reference when it is clear from the context.

When dealing with finite relative entropies, the Orlicz space associated to
$\theta^{*}$ (defined analogously as for $\theta$) plays a natural role.
Indeed, for any probability $\mathfrak{p}$ with
$\mathcal{H}(\mathfrak{p}|\mathfrak{q})<+\infty$ we have

$\displaystyle\left\lVert\frac{\mathrm{d}\mathfrak{p}}{\mathrm{d}\mathfrak{q}}\right\rVert_{\theta^{*}}=$
$\displaystyle\inf\left\\{b>0\colon\int\theta^{*}\left(\frac{1}{b}\,\frac{\mathrm{d}\mathfrak{p}}{\mathrm{d}\mathfrak{q}}\right)\mathrm{d}\mathfrak{q}\leq
1\right\\}=\inf\left\\{b>0\colon\frac{1}{b}\mathcal{H}(\mathfrak{p}|\mathfrak{q})-\frac{1}{b}\left(1+\log
b\right)+1\leq 1\right\\}$ $\displaystyle=$
$\displaystyle\inf\left\\{b>0\colon\mathcal{H}(\mathfrak{p}|\mathfrak{q})-1\leq\log
b\right\\}=e^{\mathcal{H}(\mathfrak{p}|\mathfrak{q})-1}\,.$

Finally, let us recall that for any $f\in L_{\theta}(\mathfrak{q})$ and $g\in
L_{\theta^{*}}(\mathfrak{q})$ we have

(A.1) $\int\left\lvert fg\right\rvert\mathrm{d}\mathfrak{q}\leq 2\left\lVert
f\right\rVert_{\theta}\,\left\lVert g\right\rVert_{\theta^{*}}\,,$

which is a consequence of the trivial inequality

$st\leq\theta(t)+\theta^{*}(s)\quad\forall s,t>0\,.$

After this digression, let us prove the following lemma, which is pivotal in
our reasoning since it allows to deduce the log-integrability of any positive
measurable function under the sole hypotheses of finite relative entropy and
some positive and negative integrability of the same function.

######  Lemma A.1.

Let $h$ be a positive measurable function and
$\mathfrak{p},\mathfrak{q}\in\mathcal{P}(\Omega)$ with
$\mathcal{H}(\mathfrak{p}|\mathfrak{q})<{\infty}$. If there exist $p,\,q>0$
such that $h\in L^{q}(\mathfrak{q})$ and $h^{-1}\in L^{p}(\mathfrak{q})$, then
$\log h\in L^{1}(\mathfrak{p})$. More precisely, it holds

(A.2) $\int\left\lvert\log h\right\rvert\mathrm{d}\mathfrak{p}\leq
2\,e^{\mathcal{H}(\mathfrak{p}|\mathfrak{q})-1}\,\left[\frac{1}{1\wedge
p\wedge q}\vee\log_{2}\left(\mathfrak{q}\\{h\geq
1\\}^{1-\frac{1}{q}}\left\lVert
h\right\rVert_{L^{q}(\mathfrak{q})}+\mathfrak{q}\\{h<1\\}^{1-\frac{1}{p}}\left\lVert
h^{-1}\right\rVert_{L^{p}(\mathfrak{q})}\right)\right]\,,$

and also

(A.3) $\int\left\lvert\log
h\right\rvert\mathrm{d}\mathfrak{p}\leq\frac{2\,e^{\mathcal{H}(\mathfrak{p}|\mathfrak{q})-1}}{p\wedge
q}\,\left[1\vee\log_{2}\left(\left\lVert
h\right\rVert_{L^{q}(\mathfrak{q})}^{p\wedge q}+\left\lVert
h^{-1}\right\rVert_{L^{p}(\mathfrak{q})}^{p\wedge q}\right)\right]\,.$

###### Proof.

By means of the Orlicz-Young inequality (A.1) we have

(A.4) $\int\left\lvert\log
h\right\rvert\mathrm{d}\mathfrak{p}=\int\left\lvert\frac{\mathrm{d}\mathfrak{p}}{\mathrm{d}\mathfrak{q}}\,\log
h\right\rvert\mathrm{d}\mathfrak{q}\leq 2\left\lVert\log
h\right\rVert_{\theta}\,\left\lVert\frac{\mathrm{d}\mathfrak{p}}{\mathrm{d}\mathfrak{q}}\right\rVert_{\theta^{*}}=2\left\lVert\log
h\right\rVert_{\theta}\,e^{\mathcal{H}(\mathfrak{p}|\mathfrak{q})-1}\,.$

Therefore it suffices to show that the above Luxemburg norm $\left\lVert\log
h\right\rVert_{\theta}$ is finite. For any $b>0$ we have

(A.5) $\displaystyle\int\theta\left(\frac{\left\lvert\log
h\right\rvert}{b}\right)\mathrm{d}\mathfrak{q}=\int\theta\left(\left\lvert\log
h^{\frac{1}{b}}\right\rvert\right)\mathrm{d}\mathfrak{q}=\int_{\\{h\geq
1\\}}h^{\frac{1}{b}}\mathrm{d}\mathfrak{q}+\int_{\\{h<1\\}}h^{-\frac{1}{b}}\mathrm{d}\mathfrak{q}-1\,.$

Before proceeding, let us point out that the above right-hand side is always
non-negative since it is greater or equal to $\mathfrak{q}\\{h\geq
1\\}+\mathfrak{q}\\{h<1\\}-1=0$.

Now consider any parameter $\delta\in(0,{\infty})$ (to be fixed later) and fix
$b\geq\frac{1}{\delta\wedge p\wedge q}$. For the sake of notation, let
$P\coloneqq\\{h\geq 1\\}$ and its complementary $N\coloneqq\\{h<1\\}$. In what
follows we are going to assume that both $\mathfrak{q}(P)>0$ and
$\mathfrak{q}(N)>0$; the case $\mathfrak{q}(P)\in\\{0,1\\}$ can be treated in
the same fashion. Now, introduce the probability measures on these sets
induced by $\mathfrak{q}$, i.e.

$\mathrm{d}\mathfrak{q}_{|_{P}}(x)\coloneqq\frac{\mathrm{d}\mathfrak{q}(x)}{\mathfrak{q}(P)}\qquad\text{and}\qquad\mathrm{d}\mathfrak{q}_{|_{N}}(x)\coloneqq\frac{\mathrm{d}\mathfrak{q}(x)}{\mathfrak{q}(N)}\,.$

Since $qb\geq 1$, from Jensen’s inequality we deduce that

$\displaystyle\int_{\\{h\geq
1\\}}h^{\frac{1}{b}}\mathrm{d}\mathfrak{q}=\mathfrak{q}(P)\int_{P}h^{\frac{1}{b}}\mathrm{d}\mathfrak{q}_{|_{P}}=\mathfrak{q}(P)\int_{P}h^{q\,\frac{1}{qb}}\mathrm{d}\mathfrak{q}_{|_{P}}\leq\mathfrak{q}(P)\left(\int_{P}h^{q}\mathrm{d}\mathfrak{q}_{|_{P}}\right)^{\frac{1}{qb}}$
$\displaystyle=\mathfrak{q}(P)^{1-\frac{1}{qb}}\left(\int_{P}h^{q}\mathrm{d}\mathfrak{q}\right)^{\frac{1}{qb}}\leq\mathfrak{q}(P)^{1-\frac{1}{qb}}\left\lVert
h\right\rVert_{L^{q}(\mathfrak{q})}^{\frac{1}{b}}\,.$

Similarly, from $pb\geq 1$ we deduce that

$\displaystyle\int_{\\{h<1\\}}\left(\frac{1}{h}\right)^{\frac{1}{b}}\mathrm{d}\mathfrak{q}=\mathfrak{q}(N)\int_{N}h^{-\frac{1}{b}}\mathrm{d}\mathfrak{q}_{|_{N}}=\mathfrak{q}(N)\int_{N}h^{-p\,\frac{1}{pb}}\mathrm{d}\mathfrak{q}_{|_{N}}\leq\mathfrak{q}(N)\left(\int_{N}h^{-p}\mathrm{d}\mathfrak{q}_{|_{N}}\right)^{\frac{1}{pb}}$
$\displaystyle=\mathfrak{q}(N)^{1-\frac{1}{pb}}\left(\int_{N}h^{-p}\mathrm{d}\mathfrak{q}\right)^{\frac{1}{pb}}\leq\mathfrak{q}(N)^{1-\frac{1}{pb}}\left\lVert
h^{-1}\right\rVert_{L^{p}(\mathfrak{q})}^{\frac{1}{b}}\,.$

For the sake of brevity let $\lambda\coloneqq\mathfrak{q}(P)$. Since also
$\delta b\geq 1$, from (A.5) and the concavity of $x^{\frac{1}{\delta b}}$ we
deduce

$\begin{split}\int\theta\left(\frac{\left\lvert\log
h\right\rvert}{b}\right)\mathrm{d}\mathfrak{q}&\leq\lambda^{1-\frac{1}{qb}}\left\lVert
h\right\rVert_{L^{q}(\mathfrak{q})}^{\frac{1}{b}}+(1-\lambda)^{1-\frac{1}{pb}}\left\lVert
h^{-1}\right\rVert_{L^{p}(\mathfrak{q})}^{\frac{1}{b}}-1\\\
&=\lambda\left(\frac{\left\lVert
h\right\rVert_{L^{q}(\mathfrak{q})}^{\delta}}{\lambda^{\frac{\delta}{q}}}\right)^{\frac{1}{\delta
b}}+(1-\lambda)\left(\frac{\left\lVert
h^{-1}\right\rVert_{L^{p}(\mathfrak{q})}^{\delta}}{(1-\lambda)^{\frac{\delta}{p}}}\right)^{\frac{1}{\delta
b}}-1\\\ &\leq\left(\lambda^{1-\frac{\delta}{q}}\left\lVert
h\right\rVert_{L^{q}(\mathfrak{q})}^{\delta}+(1-\lambda)^{1-\frac{\delta}{p}}\left\lVert
h^{-1}\right\rVert_{L^{p}(\mathfrak{q})}^{\delta}\right)^{\frac{1}{\delta
b}}-1\leq 1\,,\end{split}$

where the last inequality holds as soon as
$b\geq\frac{1}{\delta}\,\log_{2}\left(\lambda^{1-\frac{\delta}{q}}\left\lVert
h\right\rVert_{L^{q}(\mathfrak{q})}^{\delta}+(1-\lambda)^{1-\frac{\delta}{p}}\left\lVert
h^{-1}\right\rVert_{L^{p}(\mathfrak{q})}^{\delta}\right)$. Therefore, if we
take

$\bar{b}\coloneqq\frac{1}{\delta\wedge p\wedge
q}\vee\frac{1}{\delta}\,\log_{2}\left(\lambda^{1-\frac{\delta}{q}}\left\lVert
h\right\rVert_{L^{q}(\mathfrak{q})}^{\delta}+(1-\lambda)^{1-\frac{\delta}{p}}\left\lVert
h^{-1}\right\rVert_{L^{p}(\mathfrak{q})}^{\delta}\right)\,,$

we deduce that $\int\theta\left(\frac{\left\lvert\log
h\right\rvert}{\bar{b}}\right)\mathrm{d}\mathfrak{q}\leq 1$, and hence

(A.6) $\left\lVert\log h\right\rVert_{\theta}\leq\bar{b}=\frac{1}{\delta\wedge
p\wedge
q}\vee\frac{1}{\delta}\,\log_{2}\left(\lambda^{1-\frac{\delta}{q}}\left\lVert
h\right\rVert_{L^{q}(\mathfrak{q})}^{\delta}+(1-\lambda)^{1-\frac{\delta}{p}}\left\lVert
h^{-1}\right\rVert_{L^{p}(\mathfrak{q})}^{\delta}\right)\,.$

From this last result and (A.4) it follows that $\log h\in
L^{1}(\mathfrak{q})$ and that

(A.7) $\int\left\lvert\log h\right\rvert\mathrm{d}\mathfrak{p}\leq
2\,e^{\mathcal{H}(\mathfrak{p}|\mathfrak{q})-1}\,\left[\frac{1}{\delta\wedge
p\wedge
q}\vee\frac{1}{\delta}\,\log_{2}\left(\lambda^{1-\frac{\delta}{q}}\left\lVert
h\right\rVert_{L^{q}(\mathfrak{q})}^{\delta}+(1-\lambda)^{1-\frac{\delta}{p}}\left\lVert
h^{-1}\right\rVert_{L^{p}(\mathfrak{q})}^{\delta}\right)\right]\,.$

By taking $\delta=1$ we immediately get (A.2). In order to get the second
bound, take $\delta\coloneqq p\wedge q$. Then (A.6) reads as

(A.8) $\displaystyle\left\lVert\log h\right\rVert_{\theta}\leq\frac{1}{p\wedge
q}\left[1\vee\log_{2}\left(\lambda^{1-\frac{p\wedge q}{q}}\left\lVert
h\right\rVert_{L^{q}(\mathfrak{q})}^{p\wedge q}+(1-\lambda)^{1-\frac{p\wedge
q}{p}}\left\lVert h^{-1}\right\rVert_{L^{p}(\mathfrak{q})}^{p\wedge
q}\right)\right]\,.$

Since $\lambda=\mathfrak{q}(P)\in[0,1]$ we deduce that

$\displaystyle\log_{2}\left(\mathfrak{q}(P)^{1-\frac{p\wedge q}{q}}\left\lVert
h\right\rVert_{L^{q}(\mathfrak{q})}^{p\wedge
q}+(1-\mathfrak{q}(P))^{1-\frac{p\wedge q}{p}}\left\lVert
h^{-1}\right\rVert_{L^{p}(\mathfrak{q})}^{p\wedge q}\right)$
$\displaystyle\leq\log_{2}\sup_{\lambda\in[0,1]}\left(\lambda^{1-\frac{p\wedge
q}{q}}\left\lVert h\right\rVert_{L^{q}(\mathfrak{q})}^{p\wedge
q}+(1-\lambda)^{1-\frac{p\wedge q}{p}}\left\lVert
h^{-1}\right\rVert_{L^{p}(\mathfrak{q})}^{p\wedge q}\right)$
$\displaystyle=\begin{cases}\log_{2}\left(\left\lVert
h\right\rVert_{L^{q}(\mathfrak{q})}^{p}+\left\lVert
h^{-1}\right\rVert_{L^{p}(\mathfrak{q})}^{p}\right)\quad\text{if }p\leq q\\\
\log_{2}\left(\left\lVert h\right\rVert_{L^{q}(\mathfrak{q})}^{q}+\left\lVert
h^{-1}\right\rVert_{L^{p}(\mathfrak{q})}^{q}\right)\quad\text{if
}q<p\end{cases}$ $\displaystyle=\log_{2}\left(\left\lVert
h\right\rVert_{L^{q}(\mathfrak{q})}^{p\wedge q}+\left\lVert
h^{-1}\right\rVert_{L^{p}(\mathfrak{q})}^{p\wedge q}\right)\,,$

and hence

(A.9) $\left\lVert\log h\right\rVert_{\theta}\leq\frac{1}{p\wedge
q}\left[1\vee\log_{2}\left(\left\lVert
h\right\rVert_{L^{q}(\mathfrak{q})}^{p\wedge q}+\left\lVert
h^{-1}\right\rVert_{L^{p}(\mathfrak{q})}^{p\wedge q}\right)\right]\,.$

As a byproduct of the above bound and (A.4) we get (A.3). ∎

######  Remark A.2.

When $\mathfrak{q}\\{h\geq 1\\}\in\\{0,1\\}$, the bound (A.3) can be improved.
Indeed from (A.8) we straightforwardly deduce that

(A.10) $\displaystyle\int\left\lvert\log
h\right\rvert\mathrm{d}\mathfrak{p}\leq\,\begin{cases}2\,e^{\mathcal{H}(\mathfrak{p}|\mathfrak{q})-1}\,\displaystyle{\bigg{(}\frac{1}{p\wedge
q}\vee\log_{2}\left\lVert
h\right\rVert_{L^{q}(\mathfrak{q})}\bigg{)}}\qquad&\text{if
}\mathfrak{q}\\{h\geq 1\\}=1\,,\\\ \\\
2\,e^{\mathcal{H}(\mathfrak{p}|\mathfrak{q})-1}\,\displaystyle{\bigg{(}\frac{1}{p\wedge
q}\vee\log_{2}\left\lVert
h^{-1}\right\rVert_{L^{p}(\mathfrak{q})}\bigg{)}}\qquad&\text{if
}\mathfrak{q}\\{h\geq 1\\}=0\,.\end{cases}$

######  Corollary A.3.

If $p\wedge q\leq 1$ in the previous Lemma, then it holds

(A.11) $\int\left\lvert\log h\right\rvert\mathrm{d}\mathfrak{p}\leq
2\,e^{\mathcal{H}(\mathfrak{p}|\mathfrak{q})-1}\,\left[\frac{1}{p\wedge
q}+\left(\log_{2}\,\frac{\left\lVert
h\right\rVert_{L^{q}(\mathfrak{q})}+\left\lVert
h^{-1}\right\rVert_{L^{p}(\mathfrak{q})}}{2}\right)^{+}\,\right]\,.$

## References

  * [1] Stefan Adams, Nicolas Dirr, Mark A. Peletier, and Johannes Zimmer.  From a Large-Deviations Principle to the Wasserstein Gradient Flow: A New Micro-Macro Passage.  Communications in Mathematical Physics, 307(3):791–815, 2011.
  * [2] Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré.  Calculus and heat flow in metric measure spaces and applications to spaces with ricci bounds from below.  Inventiones mathematicae, 195(2):289–391, 2014.
  * [3] Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré.  Metric measure spaces with Riemannian Ricci curvature bounded from below.  Duke Mathematical Journal, 163(7):1405–1490, 2014.
  * [4] Julio Backhoff, Giovanni Conforti, Ivan Gentil, and Christian Léonard.  The mean field Schrödinger problem: ergodic behavior, entropy estimates and functional inequalities.  Probability Theory and Related Fields, 178(1):475–530, 2020.
  * [5] Dominique Bakry, Ivan Gentil, and Michel Ledoux.  Analysis and geometry of Markov diffusion operators, volume 348\.  Springer Science & Business Media, 2013.
  * [6] Franck Barthe and Alexander V. Kolesnikov.  Mass Transport and Variants of the Logarithmic Sobolev Inequality.  Journal of Geometric Analysis, 18:921–979, 2008.
  * [7] Jean-David Benamou.  Optimal transportation, modelling and numerical simulation.  Acta Numerica, 30:249–325, 2021.
  * [8] Espen Bernton, Promit Ghosal, and Marcel Nutz.  Entropic Optimal Transport: Geometry and Large Deviations.  Duke Mathematical Journal, forthcoming.
  * [9] Yann Brenier.  Polar factorization and monotone rearrangement of vector-valued functions.  Communications on pure and applied mathematics, 44(4):375–417, 1991\.
  * [10] Guillaume Carlier and Maxime Laborde.  A Differential Approach to the Multi-Marginal Schrödinger System.  SIAM Journal on Mathematical Analysis, 52(1):709–717, 2020.
  * [11] Guillaume Carlier, Paul Pegon, and Luca Tamanini.  Convergence rate of general entropic optimal transport costs.  Preprint, arXiv:2206.03347, 2022.
  * [12] Yongxin Chen, Tryphon T Georgiou, and Michele Pavon.  Stochastic Control Liaisons: Richard Sinkhorn Meets Gaspard Monge on a Schrodinger Bridge.  SIAM Review, 63(2):249–313, 2021.
  * [13] Alberto Chiarini, Giovanni Conforti, Giacomo Greco, and Zhenjie Ren.  Entropic turnpike estimates for the kinetic schrödinger problem.  Preprint, arXiv:2108.09161, 2021.
  * [14] Lenaic Chizat, Pierre Roussillon, Flavien Léger, François-Xavier Vialard, and Gabriel Peyré.  Faster Wasserstein distance estimation with the Sinkhorn divergence.  Advances in Neural Information Processing Systems, 33:2257–2269, 2020.
  * [15] Gauthier Clerc, Giovanni Conforti, and Ivan Gentil.  On the variational interpretation of local logarithmic Sobolev inequalities.  forthcoming in Annales de la faculté des Sciences de Tolouse, 2020.
  * [16] Giovanni Conforti.  A second order equation for Schrödinger bridges with applications to the hot gas experiment and entropic transportation cost.  Probability Theory and Related Fields, 174(1):1–47, 2019.
  * [17] Giovanni Conforti and Luca Tamanini.  A formula for the time derivative of the entropic cost and applications.  Journal of Functional Analysis, 280(11):964–1008, 2021.
  * [18] Guido De Philippis and Alessio Figalli.  The Monge–Ampère equation and its link to optimal transportation.  Bulletin of the American Mathematical Society, 51(4):527–580, 2014\.
  * [19] George Deligiannidis, Valentin De Bortoli, and Arnaud Doucet.  Quantitative Uniform Stability of the Iterative Proportional Fitting Procedure.  Preprint, arXiv:2108.08129, 2021.
  * [20] Stephan Eckstein and Marcel Nutz.  Quantitative Stability of Regularized Optimal Transport and Convergence of Sinkhorn’s Algorithm.  Preprint, arXiv:2110.06798, 2021.
  * [21] Matthias Erbar, Jan Maas, and Michiel Renger.  From large deviations to Wasserstein gradient flows in multiple dimensions.  Electronic Communications in Probability, 20:1–12, 2015.
  * [22] Alessio Figalli.  Existence, uniqueness, and regularity of optimal transport maps.  SIAM Journal on Mathematical Analysis, 39(1):126–137, 2007.
  * [23] Alessio Figalli and Nicola Gigli.  Local semiconvexity of Kantorovich potentials on non-compact manifolds.  ESAIM: Control, Optimisation and Calculus of Variations, 17(3):648–653, 2011.
  * [24] Ivan Gentil, Christian Léonard, and Luigia Ripani.  Dynamical aspects of the generalized Schrödinger problem via Otto calculus–A heuristic point of view.  Revista Matemática Iberoamericana, 36(4):1071–1112, 2020.
  * [25] Promit Ghosal, Marcel Nutz, and Espen Bernton.  Entropic Optimal Transport: Geometry and Large Deviations.  Preprint, arXiv:2102.04397, 2021.
  * [26] Promit Ghosal, Marcel Nutz, and Espen Bernton.  Stability of Entropic Optimal Transport and Schrödinger Bridges.  Preprint, arXiv:2106.03670, 2021.
  * [27] Nicola Gigli and Luca Tamanini.  Benamou-Brenier and duality formulas for the entropic cost on ${RCD}^{*}({K},{N})$ spaces.  Probability Theory and Related Fields, pages 1–34, 2019.
  * [28] Nicola Gigli and Luca Tamanini.  Second order differentiation formula on ${RCD}^{*}({K},{N})$ spaces.  Journal of the European Mathematical Society, 23(5):1727–1795, 2021\.
  * [29] Robert E Greene.  S. gallot, d. hulin and j. lafontaine, riemannian geometry.  Bulletin (New Series) of the American Mathematical Society, 21(1):157–162, 1989.
  * [30] Alexander Grigoryan.  Heat kernel and analysis on manifolds, volume 47.  American Mathematical Soc., 2009.
  * [31] Richard S. Hamilton.  A matrix Harnack estimate for the heat equation.  Comm. Anal. Geom., 1(1):113–126, 1993.
  * [32] Renjin Jiang, Huaiqian Li, and Huichun Zhang.  Heat Kernel Bounds on Metric Measure Spaces and Some Applications.  Potential Analysis, 44:601–627, 2016.
  * [33] Renjin Jiang and Huichun Zhang.  Hamilton’s gradient estimates and a monotonicity formula for heat flows on metric measure spaces.  Nonlinear Analysis, 131:32–47, 2016.
  * [34] Brett L. Kotschwar.  Hamilton’s gradient estimate for the heat kernel on complete manifolds.  Proc. Amer. Math. Soc., 135(9):3013–3019, 2007.
  * [35] Christian Léonard.  From the Schrödinger problem to the Monge–Kantorovich problem.  Journal of Functional Analysis, 262(4):1879–1920, 2012.
  * [36] Christian Léonard.  Girsanov Theory Under a Finite Entropy Condition, pages 429–465.  Springer Berlin Heidelberg, Berlin, Heidelberg, 2012.
  * [37] Christian Léonard.  A survey of the Schrödinger problem and some of its connections with optimal transport.  Discrete and Continuous Dynamical Systems, 34(4):1533–1574, 2014\.
  * [38] Toshio Mikami.  Monge’s problem with a quadratic cost by the zero-noise limit of $h$-path processes.  Probability Theory and Related Fields, 129:245–260, 2004.
  * [39] James R Norris.  Heat kernel asymptotics and the distance function in Lipschitz Riemannian manifolds.  Acta Mathematica, 179(1):79–103, 1997.
  * [40] Marcel Nutz.  Introduction to Entropic Optimal Transport.  http://www.math.columbia.edu/~mnutz/docs/EOT_lecture_notes.pdf, 2021.
  * [41] Marcel Nutz and Johannes Wiesel.  Entropic optimal transport: Convergence of potentials.  Probability Theory and Related Fields, pages 1–24, 2021.
  * [42] Marcel Nutz and Johannes Wiesel.  Stability of Schrödinger Potentials and Convergence of Sinkhorn’s Algorithm.  Preprint, arXiv:2201.10059, 2022.
  * [43] Soumik Pal.  On the difference between entropic cost and the optimal transport cost.  Preprint, arXiv:1905.12206, 2019.
  * [44] Gabriel Peyré and Marco Cuturi.  Computational optimal transport.  Foundations and Trends in Machine Learning, 11(5-6):355–607, 2019\.
  * [45] Rémi Peyre.  Comparison between $\mathcal{W}_{2}$ distance and $\dot{H}^{-1}$ norm, and Localisation of Wasserstein dinstance.  ESAIM: Control, Optimisation and Calculus of Variations, 24(4):1489–1501, 2018.
  * [46] Aram-Alexandre Pooladian and Jonathan Niles-Weed.  Entropic estimation of optimal transport maps.  Preprint, arXiv:2109.12004, 2021.
  * [47] Erwin Schrödinger.  La théorie relativiste de l’électron et l’ interprétation de la mécanique quantique.  Ann. Inst Henri Poincaré, 2:269 – 310, 1932.
  * [48] Karl-Theodor Sturm.  On the geometry of metric measure spaces. I.  Acta mathematica, 196(1):65–131, 2006.
  * [49] Karl-Theodor Sturm.  On the geometry of metric measure spaces. II.  Acta Math., 196(1):133–177, 2006.
  * [50] Luca Tamanini.  A generalization of Costa’s entropy power inequality.  IEEE Transactions on Information Theory, 68(7):4224–4229, 2022\.
  * [51] Cédric Villani.  Topics in Optimal Transportation, volume 58.  American Mathematical Society, 2003.
  * [52] Feng-Yu Wang.  Equivalent semigroup properties for the curvature-dimension condition.  Bulletin des sciences mathematiques, 135(6-7):803–815, 2011.


# Early Acceptance Matching Game for User-Centric Clustering in Scalable Cell-
free MIMO Networks††thanks: This work has been accepted for publication in
2024 European Conference on Networks and Communications (EuCNC) & 6G Summit

Ala Eddine Nouali1, Mohamed Sana1, Jean-Paul Jamont2
1CEA-Leti, Université Grenoble Alpes, F-38000 Grenoble, France,
2LCIS, Université Grenoble Alpes, Valence, France
Email : {ala-eddine.nouali<EMAIL_ADDRESS>jean-paul.jamont@univ-
grenoble-alpes.fr

###### Abstract

The canonical setup is the primary approach adopted in cell-free multiple-
input multiple-output (MIMO) networks, in which all access points (APs)
jointly serve every user equipment (UE). This approach is not scalable in
terms of computational complexity and fronthaul signaling becoming impractical
in large networks. This work adopts a user-centric approach, a scalable
alternative in which only a set of preferred APs jointly serve a UE. Forming
the optimal cluster of APs for each UE is a challenging task, especially, when
it needs to be dynamically adjusted to meet the quality of service (QoS)
requirements of the UE. This complexity is even exacerbated when considering
the constrained fronthaul capacity of the UE and the AP. We solve this problem
with a novel many-to-many matching game. More specifically, we devise an early
acceptance matching algorithm, which immediately admits or rejects UEs based
on their requests and available radio resources. The proposed solution
significantly reduces the fronthaul signaling while satisfying the maximum of
UEs in terms of requested QoS compared to state-of-the-art approaches.

##  I Introduction

Commercialized networks use cellular architecture, where terminals suffer from
large variation of quality of service (QoS) [1]. User equipment (UE) close to
the access point (AP), which references the cell center can achieve a high
QoS. However, the QoS fluctuates significantly as user moves further away from
the center, preventing cellular networks from providing high, consistent and
uniform QoS to every UE regardless of their location in the network. Cell-free
aims to solve this problem. It is a proposed new paradigm that eliminates cell
boundaries and manages interference by joint processing at APs to serve UEs
[2]. The canonical setup is the trivial approach adopted in cell-free
networks, in which all APs jointly serve each UE. This approach does not scale
well in terms of fronthaul requirements and computational complexity, becoming
impractical in large networks [3]. User-centric clustering is the proposed
scalable alternative, in which each UE is served by the nearest preferred APs
[4, 5].

Figure 1: A user-centric clustering operation in a cell-free MIMO network.
Here, $M$ geographically distributed APs jointly serve $K$ UEs. Each UE is
served by a cluster of nearby preferred APs depending on its QoS requirements.

Different techniques are adopted to form clusters of APs for UEs. The works
[3, 6] study centralized and distributed association AP-UE based on pilot
assignment. Authors in [7, 8] propose to minimize fronthaul signaling while
guaranteeing minimum QoS for UEs. Due to limited network resource, which need
to be efficiently allocated according to demand and network conditions,
various studies investigate user-centric clustering to manage resource
allocation in cell-free networks. For instance, [9] maximizes network energy
efficiency constrained to UEs QoS and APs transmission power capabilities.
Similarly, [10, 11] study cluster formation to maximize the network
throughput. Although there are works that restrict the total number of APs
serving a UE, these works consider that the capacity of the APs is unlimited:
an AP can accept any number of UEs. This assumption is unrealistic as in
practice, UEs and APs have limited fronthaul capacity and computational
capability. Considering these constraints, various methods have been employed
to allocate radio resources efficiently, with many-to-many (M2M) matching game
emerging as a particularly effective solution [12]. This approach is based on
the principles of matching theory, which has attracted the interest of
researchers because of its ability to enable distributed user association with
low computational complexity and fast convergence time [12]. Deferred
acceptance (DA) is the common strategy used in M2M matching game. In this
method, APs buffer user association requests at each round, finalizing the
association procedure only at the last iteration [13]. This results in a
prohibitive delays and slow convergence time.

Different from previous work, we propose to optimize radio resource allocation
by forming clusters of APs for UEs depending on their specific QoS
requirements (as shown in Fig. 1). We present a novel user association scheme
based on M2M matching game between the set of UEs and APs, aiming to satisfy
the maximum UEs in the network in terms of requested QoS. In contrast to the
DA strategy proposed in [13], we adopt an early acceptance (EA) strategy to
speed up the AP-UE association process. In this approach, UEs apply to a
specific set of APs, which immediately accept or reject the association
requests based on their preferences and available radio resources. The
proposed solution significantly reduces the computational complexity of the
cluster formation process, while providing an acceptable satisfaction level
for the maximum number of UEs in the network, depending on their requested
QoS.

##  II System model & problem formulation

###  II-A System Model

We consider a downlink cell-free MIMO network consists of $M$ geographically
distributed APs that cooperate to serve $K$ geographically distributed UEs.
Let $\mathcal{M}=\\{1,\dots,M\\}$ and $\mathcal{K}=\\{1,\dots,K\\}$ denote the
set of APs and UEs, respectively. Each AP $m$ is equipped with $N$ antennas,
while each UE has a single antenna. We consider the Rayleigh fading channel
model adopted in [7], in which the complex random channel
$\mathbf{h}_{k,m}\in\mathcal{C}^{N\times 1}$, independent and identically
distributed (iid), between UE $k$ and AP $m$ at time $t$ reads as:

$\mathbf{h}_{k,m}(t)=\mathbf{\alpha}_{k,m}(t)\sqrt{g_{k,m}(t)},$ (1)

where $\mathbf{\alpha}_{k,m}(t)\sim\mathcal{CN}(0,I_{N})$ denotes the small-
scale fading and $g_{k,m}(t)$ is the channel gain, which captures the
distance-dependent path loss and the shadowing effect as follows:

$g_{k,m}(t)=\left(\frac{\lambda}{4\pi}\right)^{2}\left(\frac{1}{d_{k,m}(t)}\right)^{\eta}\chi_{k,m}(t).$
(2)

Here, $\lambda$ is the wavelength of the operated carrier frequency $f$,
$d_{k,m}$ is the distance between UE $k$ and AP $m$, $\eta$ denotes the path
loss exponent and $\chi_{k,m}\sim\mathcal{LN}(0,\sigma_{s}^{2})$ denotes the
random log-normally distributed shadowing.

In the downlink scenario, we assume a full knowledge of channels between APs
and UEs. We denote with $\mathbf{v}_{k,m}(t)\in\mathcal{C}^{N\times 1}$ the
downlink transmit beamformer at AP $m$ for UE $k$ at time $t$, which we obtain
via the linear minimum mean squared error (LMMSE) precoding algorithm [14]:

$\mathbf{v}_{k,m}(t)=\mathbf{h}_{k,m}(t)(\mathbf{h}_{k,m}^{H}(t)\mathbf{h}_{k,m}(t)+\sigma_{n}^{2}(t))^{-1}.$
(3)

Here, $\mathbf{h}_{k,m}^{H}$ is the conjugate transpose of $\mathbf{h}_{k,m}$
and $\sigma_{n}^{2}(t)$ denotes the receiver noise power at time $t$, which is
a scalar in our case. Accordingly, the signal-to-interference-plus-noise ratio
(SINR) perceived by UE $k$ at time $t$ reads as:

$\mathrm{SINR}_{k}(t)=\frac{S_{k}(t)}{I_{k}(t)+\sigma_{n}^{2}(t)},$ (4)

where $S_{k}(t)$ is the received signal power given as:

$S_{k}(t)=\left|\displaystyle\sum_{m=1}^{M}\sqrt{P_{k,m}(t)}\mathbf{h}_{k,m}^{H}(t)\mathbf{v}_{k,m}(t)\delta_{k,m}(t)\right|^{2},$
(5)

and $I_{k}(t)$ is the perceived interference power given as:

$I_{k}(t)=\displaystyle\sum_{\begin{subarray}{c}j=1\\\ j\neq
k\end{subarray}}^{K}\left|\displaystyle\sum_{m=1}^{M}\sqrt{P_{j,m}(t)}\mathbf{h}_{k,m}^{H}(t)\mathbf{v}_{j,m}(t)\delta_{j,m}(t)\right|^{2}.$
(6)

In Eq. (5) and (6), $P_{k,m}(t)$ denotes the power allocated by AP $m$ to UE
$k$ at time $t$ and $\delta_{k,m}(t)$ is a binary association variable, which
indicates whether UE $k$ is associated with AP $m$ at time $t$, in which case
$\delta_{k,m}(t)=1$ and $\delta_{k,m}(t)=0$ otherwise. Hence, the data rate
perceived by UE $k$ at time $t$ is given by:

$R_{k}(t)=B\cdot\mathrm{SE}_{k}(t),$ (7)

where $\mathrm{SE}_{k}(t)=\log_{2}(1+\mathrm{SINR}_{k}(t))$ is the spectral
efficiency (SE) of UE $k$ at time $t$ and $B$ is the total system bandwidth.

###  II-B Problem Formulation

Let $R_{k}^{\rm req}(t)$ denote the time-varying data rate demand of UE $k$
and $\kappa_{k}(t)$ indicates its QoS satisfaction at time $t$, which we
define as in [15] as follows:

$\kappa_{k}(t)=\min\left(1,\frac{R_{k}(t)}{R_{k}^{\mathrm{req}}(t)}\right).$
(8)

Let $\kappa_{0}\in[0,1]$ define the minimum QoS satisfaction level of UEs. We
say a UE $k$ is $\kappa_{0}$-satisfied if $\kappa_{k}(t)\geq\kappa_{0}$. In
this case, when $\kappa_{0}=1$, we say the UE is fully satisfied. The QoS of a
UE varies according to multiple factors, including power allocation and user-
clustering strategy, which affect cell interference and hence network
performance.

In this context, we are interested in maximizing the total number of
$\kappa_{0}$-satisfied UEs, while minimizing the number of fronthaul
associations. To this end, we propose the following optimization problem:

$\displaystyle\underset{\Psi(t)}{\max}$
$\displaystyle\quad\mathbb{E}\left\\{\displaystyle\sum_{\begin{subarray}{c}k\in\mathcal{K}\end{subarray}}\mathbb{1}_{\\{\kappa_{k}(t)\geq\kappa_{0}\\}}\right\\},$
($\mathcal{P}$) $\displaystyle\mathrm{s.t.~{}}~{}$
$\displaystyle{}\delta_{k,m}(t)\in\\{0,1\\},\qquad\forall
k\in\mathcal{K},m\in\mathcal{M},$ ($\mathcal{C}_{1}$) $\displaystyle
P_{k,m}(t)\geq 0,\qquad\forall k\in\mathcal{K},m\in\mathcal{M},$
($\mathcal{C}_{2}$)
$\displaystyle\displaystyle\sum_{\begin{subarray}{c}k\in\mathcal{K}\end{subarray}}\delta_{k,m}(t)P_{k,m}(t)\leq
P_{\max},\qquad\forall m\in\mathcal{M},$ ($\mathcal{C}_{3}$)
$\displaystyle\sum_{\begin{subarray}{c}k\in\mathcal{K}\end{subarray}}\delta_{k,m}(t)\leq
K_{\max},\qquad\forall m\in\mathcal{M},$ ($\mathcal{C}_{4}$)
$\displaystyle\sum_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}\delta_{k,m}(t)\leq
M_{\max},\qquad\forall k\in\mathcal{K},$ ($\mathcal{C}_{5}$)

where $\Psi(t)=\\{\delta_{k,m}(t),P_{k,m}(t),\;\forall
k,m\in\mathcal{K}\times\mathcal{M}\\}$ is the set of parameters to adjust to
maximize our objective function. The expectation in ($\mathcal{P}$) is taken
w.r.t. the random traffic requests and channel realizations. The constraint
($\mathcal{C}_{1}$) defines the variable of association $\delta_{k,m}$ as a
binary variable. Constraints ($\mathcal{C}_{2}$) and ($\mathcal{C}_{3}$)
restricts the transmission power of an AP $m$ to do not exceed its maximum of
transmission power $P_{\max}$. Constraint ($\mathcal{C}_{4}$) indicates that
each AP $m$ can serve $K_{\max}$ UEs at most. Finally, constraint
($\mathcal{C}_{5}$) ensures that each UE $k$ is associated to at most
$M_{\max}$ APs.

In this work, we focus on the user-clustering problem. We assume that each AP
equally shares its transmit power with its served UEs. Thus, the power
allocation strategy is fully determined by the user-clustering operation.

##  III User-centric clustering as a many-to-many matching game

As shown in Fig. 2, user association can be seen as a many-to-many matching
game between two sets of players: APs and UEs. During this game, each player,
from a set of local observations, constructs a preference list based on its
own objective. UEs request association to APs according to their preference
lists and APs, in turn, decide individually the UEs to serve based on their
own preference lists.

2cm(1.34cm,1.775cm)

2cm(1.34cm,3.7cm)

AP setAP $1$AP $2$$\dots$AP $m$$\dots$AP $M$UE setUE $1$UE $2$$\dots$UE
$k$$\dots$UE $K$

Figure 2: Many-to-many matching between APs and UEs.

###  III-A Background on many-to-many matching concepts

Before formulating the user-centric clustering as a many-to-many matching
game, we first introduce some basic concepts based on two-sided matching
theory [12].

In matching game, each UE $k$ starts by building a preference list of ordered
APs $\mathcal{P}_{k}^{\rm UE}$, from most to least preferred. To do so, it
relies on a preference metric $\vartheta_{k,m}^{\rm UE}$ (e.g. the perceived
SINR w.r.t. AP $m$): UE $k$ prefers AP $m$ to AP $m^{\prime}$ ($m\neq
m^{\prime}$) if $\vartheta_{k,m}^{\rm UE}\geq\vartheta_{k,m^{\prime}}^{\rm
UE}$. Similarly, each AP $m$ uses a preference metric $\vartheta_{k,m}^{\rm
AP}$ (e.g. the channel gain $g_{k,m}$ w.r.t. UE $k$) to build its preference
list of ordered UEs $\mathcal{P}_{m}^{\rm AP}$.

######  Definition 1.

A many-to-many matching $\mu$ is a mapping function that assigns a matching
vector of APs $\mathcal{C}_{k}^{\rm UE}$ to each UE $k$ and a matching vector
of UEs $\mathcal{C}_{m}^{\rm AP}$ to each AP $m$. The matching process is
constrained to:

  1. 1.

$\mu(k)=\mathcal{C}_{k}^{\rm UE}\subseteq\mathcal{M}$ and $|\mu(k)|\leq
M_{\max}$ $\forall k\in\mathcal{K}$,

  2. 2.

$\mu(m)=\mathcal{C}_{m}^{\rm AP}\subseteq\mathcal{K}$ and $|\mu(m)|\leq
K_{\max}$ $\forall m\in\mathcal{M}$,

  3. 3.

$k\in\mu(m)\Leftrightarrow m\in\mu(k)$.

Conditions (1) and (2) represent the matching process between APs and UEs
whereas condition (3) guarantees a mutually accepted match between the set of
UEs and APs.

The initial step in a matching game involves constructing the preference lists
of the players. In the case of wireless networks, we can build these
preference lists through various metrics including channels gains, UEs SINR or
data rates. In our case, we build the preference lists for both UEs and APs
based on the channel gains.

###  III-B Proposed algorithm

_Deferred acceptance_ (DA) is a two-sided matching game in which, at each
iteration, each AP retains in its waiting list only the $K_{\max}$ UEs
preferred among the first new UE applicants and those previously in its
waiting list, and reject the others [13]. UEs in APs waiting list will be
associated after the final iteration of the game i.e. the association
procedure is deferred until the end of the game. As a result, the association
between APs and UEs can be a time-consuming process.

To overcome this problem, we adopt a new matching game called _early
acceptance_ (EA) with preference list updating (EA-PLU) [16]. In this game,
APs immediately decide to accept or reject UEs at each iteration. This
procedure allows to accelerate the association process and control the number
of associations between APs and UEs. Consequently, EA game reduces the total
number of fronthaul connections unlike DA game with known number of
associations equal to $\min(MK_{\max},KM_{\max}$). With lower AP-UE
connections, few messages are exchanged between UEs and APs via fronthaul
links, thus, limiting the communication overhead.

START Initialization - Initialize $\kappa_{0}$. - Set $m_{k}=1,\forall
k\in\mathcal{K}$. - Put all UEs in a rejection set $\mathcal{R}$. - Initialize
empty sets: - $\mathcal{U}~{}$: unassociated UEs - $\mathcal{A}~{}$:
associated UEs - $\mathcal{S}^{+}$: satisfied UEs - $\mathcal{S}^{-}$:
unsatisfied UEs Is there any UE $k\in\mathcal{R}$ in the updated preference
list quota of an unsaturated AP ? Request each UE $k\in\mathcal{R}$
association with its $m_{k}–$th preferred unsaturated AP. - If success: -
Associate UE $k$ with AP $m$ using Alg. 1. - Remove UE $k$ from $\mathcal{R}$
and add it to $\mathcal{A}$. - Else: - If $\mathcal{P}_{k}^{\rm
UE}=\varnothing$: - Remove UE $k$ from $\mathcal{R}$ and add it to
$\mathcal{U}$. - Else: - $m_{k}\leftarrow m_{k}+1$. - keep UE $k$ in
$\mathcal{R}$. Force each UE $k\in\mathcal{R}$ association with its first
preferred unsaturated AP. - If success: - Associate UE $k$ with its first
preferred unsaturated AP using Alg. 1. - Remove UE $k$ from $\mathcal{R}$ and
add it to $\mathcal{A}$. - Else: - Remove UE $k$ from $\mathcal{R}$ and add it
to $\mathcal{U}$. Is $\mathcal{R}$ empty ? Update clusters based on the
evolution strategy of Alg. 2. STOP YesNoYesNo

Figure 3: Flowchart representing our EA user-centric clustering procedure.

Our algorithm is summarized in the flowchart of Fig. 3. The proposed EA game
takes as an input the preference lists of APs ($\mathcal{P}_{m}^{\rm
AP},\forall m\in\mathcal{M}$) and UEs ($\mathcal{P}_{k}^{\rm UE},\forall
k\in\mathcal{K}$) and the quotas of APs ($q_{m}^{\rm AP},\forall
m\in\mathcal{M}$) and UEs ($q_{k}^{\rm UE},\forall k\in\mathcal{K}$). The
quota of UE $k$ (resp. AP $m$) is the number of remaining possible
associations it can establish. The output of the EA game is the matching
vector for UEs $\mathcal{C}^{\rm UE}=\left[\mathcal{C}^{\rm
UE}_{1},\mathcal{C}^{\rm UE}_{2},\ldots,\mathcal{C}^{\rm UE}_{K}\right]$.

1 Add AP $m$ to $\mathcal{C}^{\rm UE}_{k}$;

2 Remove UE $k$ from $\mathcal{P}_{m}^{\rm AP}$;

3 Remove AP $m$ from $\mathcal{P}_{k}^{\rm UE}$;

4 $q_{k}^{\rm UE}\leftarrow q_{k}^{\rm UE}-1$;

5 $q_{m}^{\rm AP}\leftarrow q_{m}^{\rm AP}-1$;

6 if  _$q_{m}^{\rm AP}=0$_ then

7 Remove AP $m$ from $\mathcal{P}_{k}^{\rm UE},\forall k\in\mathcal{K}$;

8if  _$q_{k}^{\rm UE}=0$_ then

9 Remove UE $k$ from $\mathcal{P}_{m}^{\rm AP},\forall m\in\mathcal{M}$;

Algorithm 1 User association between ${\rm AP}_{m}$ and ${\rm UE}_{k}$

We initialize our algorithm by setting the preference index of all UEs to one
($m_{k}=1,\forall k\in\mathcal{K}$), forming a rejection set $\mathcal{R}$
containing all UEs, and creating empty sets of associated UEs
($\mathcal{A}=\varnothing$), unassociated UEs ($\mathcal{U}=\varnothing$),
satisfied UEs ($\mathcal{S}^{+}=\varnothing$) and unsatisfied UEs
($\mathcal{S}^{-}=\varnothing$). The first step of our algorithm consists in
maximizing the number of associated UEs. We start by associating each UE to
one AP. At each iteration of the game, each UE $k$ applies to each $m_{k}$-th
preferred AP $m$ and it will be immediately accepted if it is among the
top-$q_{m}^{\rm AP}$ UEs in the preference list of the AP $m$. Algorithm 1
details the association procedure between UE $k$ and AP $m$.

When the updated preference list of a UE $k$ becomes empty, it will be added
to the set of unassociated UEs $\mathcal{U}$. UEs that are rejected by all APs
remain in $\mathcal{R}$. For each UE $k$ remaining in $\mathcal{R}$ (not
preferred by APs), we force the association with its first AP $m$ in its
updated preference list even though it is not among the top-$q_{m}^{\rm AP}$
UEs in its updated preference list. When all APs run out of quota, UEs
remaining in $\mathcal{R}$ will be added to the set of unassociated UEs
$\mathcal{U}$. Through those forced associations, we provide best link quality
for UEs remaining in $\mathcal{R}$ after prioritizing UEs favored by APs in
the previous step. In this way, we guarantee that either all UEs are served by
at least one AP or the quotas of the APs are fully exploited.

######  Definition 2 (favorable-association pair).

(${\rm AP}_{m}$, ${\rm UE}_{k}$) is a favorable-association pair if and only
if it satisfies

  1. 1.

$\mathrm{UE}$ $k\in\mathcal{P}_{m}^{\rm AP}\left(1:q_{m}^{\rm AP}\right)$,

  2. 2.

Associate ${\rm UE}_{k}$ to ${\rm AP}_{m}$ $\Rightarrow$ $\kappa_{k}(\mu_{\rm
evolve})>\kappa_{k}(\mu)$
and
$\displaystyle\sum_{\begin{subarray}{c}i\in\mathcal{K}\backslash\mathcal{U}\end{subarray}}\kappa_{i}(\mu_{\rm
evolve})\geq\displaystyle\sum_{\begin{subarray}{c}i\in\mathcal{K}\backslash\mathcal{U}\end{subarray}}\kappa_{i}(\mu)$,

where $\mu_{\rm evolve}$ is the matching strategy obtained after letting $\mu$
evolve with the new association between AP $m$ and UE $k$.

Data: $\mathcal{P}_{m}^{\rm AP},\mathcal{P}_{k}^{\rm UE},q_{m}^{\rm
AP},q_{k}^{\rm UE},\forall k\in\mathcal{K},m\in\mathcal{M}$,

and sets $\mathcal{A},\mathcal{S}^{+}$ and $\mathcal{S}^{-}$

Result: Matching vector $\mathcal{C}^{\rm UE}$

1 while  _$\mathcal{A}\neq\varnothing$ and there exists a favorable-
association pair_ do

2 Each UE $k\in\mathcal{A}$ tests its satisfaction;

3 if  _$\kappa_{k}\geq\kappa_{0}$_ then

4 Remove UE $k$ from $\mathcal{A}$ and add it to $\mathcal{S}^{+}$;

5

6 else

7 if  _$\mathcal{P}_{k}^{\rm UE}\neq\varnothing$_ then

8 Keep UE $k$ in $\mathcal{A}$ ;

9

10 else

11 Remove UE $k$ from $\mathcal{A}$ and add it to $\mathcal{S}^{-}$;

12

13

14 Each UE $k\in\mathcal{A}$ tries to improve its $\kappa_{k}$;

15 if  _$\mathcal{P}_{k}^{\rm UE}\neq\varnothing$_ then

16 Set $m_{k}=1$;

17 while  _$m_{k}\leq\min(q_{k}^{\rm UE},|\mathcal{P}_{k}^{\rm UE}|)$ and
there exists no favorable-association pair_ do

18 UE $k$ applies to its $m_{k}$-th preferred AP (namely AP $m$ with
$q_{m}^{\rm AP}\neq 0$);

19 if  _$({\rm AP}_{m},{\rm UE}_{k})$ is a favorable-association pair_ then

20 Associate ${\rm UE}_{k}$ with ${\rm AP}_{m}$ using Alg. 1;

21 Keep UE $k$ in $\mathcal{A}$ ;

22

23 else

24 $m_{k}\leftarrow m_{k}+1$ ;

25

26

27

28 else

29 Remove UE $k$ from $\mathcal{A}$ and add it to $\mathcal{S}^{-}$;

30

31

32if  _$\mathcal{A}\neq\varnothing$_ then

33 Remove each UE $k\in\mathcal{A}$ from $\mathcal{A}$ and add it to
$\mathcal{S}^{-}$;

34

Algorithm 2 Cluster evolution process

Condition (1) indicates that UE $k$ must be among the top-$q_{m}^{\rm AP}$ UEs
in the updated preference list of AP $m$. Condition (2) implies that the
association ${\rm AP}_{m}-{\rm UE}_{k}$ should improve the satisfaction level
of UE $k$ and should not decrease the sum of satisfaction level of all
associated UEs.

Based on definition 2, we propose the cluster evolution process described in
Algorithm 2. It starts by testing the satisfaction of each UE $k$ in
$\mathcal{A}$. When UE $k$ is $\kappa_{0}$-satisfied (i.e.
$\kappa_{k}\geq\kappa_{0}$) , we add it to the set of satisfied UEs
$\mathcal{S}^{+}$. Otherwise, it remains in $\mathcal{A}$ when its preference
list is not empty or we add it to $\mathcal{S}^{-}$.

The next steps in the Algorithm try to improve the satisfaction level of each
UE $k$ remaining in $\mathcal{A}$. We start by setting the reference index of
all UEs to one ($m_{k}=1,\forall k\in\mathcal{A}$), and each UE $k$ will apply
to its $m_{k}$-th preferred AP $m$ among the top-$q_{k}^{\rm UE}$ APs in its
updated preference list. When the pair (${\rm AP}_{m}$, ${\rm UE}_{k}$) is a
favorable-association pair, the association ${\rm AP}_{m}-{\rm UE}_{k}$ is set
(see Algorithm 1). Otherwise, $m_{k}$ is updated as $m_{k}\leftarrow m_{k}+1$
and UE $k$ will apply to the next $m_{k}$-th preferred AP. The cluster
evolution process is repeated until there exists no favorable-association pair
or $\mathcal{A}$ becomes empty. Hence, we obtain the association vector
$\mathcal{C}^{\rm UE}$.

##  IV Numerical results

In this section, we evaluate the performance of our proposed algorithm,
focusing specifically on its effectiveness in enhancing the satisfaction level
of UEs while limiting the number of AP-UE associations. In our simulations, we
consider a cell-free MIMO network consists of fixed number of $M=50$ APs, each
equipped with $N=16$ antennas that jointly serve a fixed number of $K=20$
single-antenna UEs. The APs and UEs are randomly distributed in a 200 m by 200
m area. During $T=100$ realizations, the APs are static and UEs follow a
random way-point mobility where the UE chooses a random direction and moves
with 1 m/s. The UEs request random traffic data from the set $\\{5,30,100\\}$
Mb/s. Table I summarizes the simulation parameters.

Benchmarks. We compare our solution with six classic benchmarks from the
literature:

  * •

Best channel (BC) associates each UE to the AP with the highest channel gain.

  * •

Min distance (MD) associates each UE to the closest AP based on distance.

  * •

Canonical setup (CS) associates each UE to all APs.

  * •

Greedy combining algorithm (GCA) forms clusters of APs for each UE by
improving the minimum of QoS per UE with deactivation of certain APs [9].

  * •

Many-to-many without swap-matching process (M2M w/o SMP) associates UEs via
many-to-many matching game with DA [13].

  * •

Many-to-many with swap-matching process (M2M w/ SMP) is an extension of
previous benchmark. After obtaining clusters, we will try to switch, between
two UEs, one of their associated APs while all other associations remain
unchanged. We will allow swap between two different associated APs of each
pair of UEs when the substitution should not reduce the total satisfaction
level of the network and should strictly increase at least $\kappa$ of one UE
while $\kappa$ of the other should not decrease.

TABLE I: Simulation parameters

Notations | Parameters | Values [17]
---|---|---
$f$ | Carrier frequency | 3.5 GHz
$B$ | System bandwidth | 20 MHz
$T$ | Simulation duration | 100
$M$ | Number of APs | 50
$K$ | Number of UEs | 20
$K_{\max}$ | Maximum number of served UEs per AP | 12
$M_{\max}$ | Maximum number of associated APs per UE | 8
$N$ | Number of antennas per AP | 16
$R_{k}^{\rm req}$ |  Requested throughput by UE $k$ |  $\\{$5, 30, 100$\\}$ Mb/s
$P_{\rm max}$ | Maximum power of each AP | 200 mW
$\eta$ | Path loss exponent | 2
$\sigma_{s}^{2}$ | Shadowing variance | 6
$\sigma_{n}^{2}$ | Noise variance | $10^{-5}$
$D_{\mathrm{th}}$ | Power difference threshold for APs selection |  30 dB1
$\lambda_{\rm GCA}$ | SE-EE ratio for GCA |  02
$\kappa_{0}$ | UE satisfaction level threshold |  $\\{$0.8, 0.9, 1$\\}$

  * 1

We use the same threshold as in [9] to form clusters of APs for each UE.

  * 2

We choose $\lambda_{\rm GCA}=0$ to consider only minimum of SE maximization
problem for the improvement process [9].

Complexity analysis. Table II details the computational complexity of the
different benchmarks and our proposed algorithm, which we refer to as PA
(EA-M2M). In particular, in the matching process of M2M w/ SMP, each AP sorts
the list of UEs requesting association and chooses the top $K_{\rm max}$
preferred UEs. This procedure costs $\mathcal{O}(KM\log(K))$ at each
iteration. In addition, the swap-matching process (SMP) in M2M w/ SMP
requires, in the worse case, $\mathcal{O}(M_{\max}^{2}K^{3})$ prohibitive
permutations further complexifying this approach. In contrast, our solution
does not require a sorting procedure, and the cluster evolution process we
propose as an alternative to SMP, limits the computational cost to
$\mathcal{O}(N^{\rm EA}K)$. Therefore, PA (EA-M2M) significantly reduces the
total execution time compared to M2M w/ SMP.

TABLE II: Computational complexity of compared algorithms.

User Clustering Alg. | Computational Complexity
---|---
BC | $\mathcal{O}(KM\log(M))$
MD | $\mathcal{O}(KM\log(M))$
CS | $\mathcal{O}(KM)$
GCA | $\mathcal{O}(KM^{3}\log(M))$
M2M w/o SMP | $\mathcal{O}(KM\log(M)+KM^{2}\log(K))$
M2M w/ SMP1 | $\mathcal{O}(KM\log(M)+KM^{2}\log(K)+N^{\rm SMP}M_{\rm max}K^{3})$
PA (EA-M2M)2 | $\mathcal{O}(KM\log(KM)+N^{\rm EA}\min(K_{\rm max},M_{\rm max})K)$

  * 1

Our simulations show that, in general, $N^{\rm SMP}$ does not exceed $M_{\rm
max}$.

  * 2

$N^{\rm EA}$ denotes the number of potential tests to form favorable-
association pairs given as $N^{\rm EA}=\sum_{k\in\mathcal{K}}\min(q_{k}^{\rm
UE},|\mathcal{P}_{k}^{\rm UE}|)$. It is upper-bounded by $M_{\rm max}K$.

Figure 4: Percentage of $\kappa_{0}$-satisfied UEs in the network.

Percentage of $\kappa_{0}$-satisfied UEs. This metric is exactly our objective
function that we aim to maximize. Here, we measure how many UEs reached a
$\kappa_{0}$-satisfaction level using our algorithm versus benchmarks. As
depicted in Fig. 4, our approach surpasses the benchmarks significantly,
irrespective of the $\kappa_{0}$ value. Our proposed algorithm shows higher
$\kappa_{0}$-satisfied UEs percentage that exceeds 10$\%$ in the case of fully
satisfaction (i.e. $\kappa_{0}=1$). Thus, we were able to improve cluster
formation process with less computational cost compared to the most
competitive benchmark M2M w/ SMP.

Figure 5: Percentage of satisfaction level per UE in the network. Figure 6:
Number of AP-UE associations in the network.

Level of satisfaction per UE. With this metric, we evaluate the QoS perceived
by UE (i.e. $\kappa$ in Eq. (8)). As shown in Fig. 5, the results demonstrate
that our proposed method ensures the highest satisfaction level, exceeding
$98\%$ regardless of $\kappa_{0}$ value. Specifically, in scenarios where
fully satisfaction is desired, our method reaches a satisfaction rate of
$99.1\%$. Consequently, we guarantee that even for $5.9\%$ of UEs that do not
achieve complete satisfaction (see Fig. 4), they still experience a
satisfactory QoS.

Number of AP-UE associations. Here, we assess the number of associations
established between APs and UEs. The results presented in Fig. 6 show that our
solution significantly improves the QoS for each UE while substantially
reducing the number of necessary associations between APs and UEs. Compared to
M2M w/ SMP, which complies with the association limits of UEs and APs, our
algorithm reduces by $84\%$ the number of associations required by M2M w/ SMP.
Consequently, we significantly limit the fronthaul signaling, as each AP now
interacts with a smaller number of UEs. Furthermore, it also limit the
computational complexity needed for e.g. channel precoding schemes.

##  V Conclusion

In this paper, we propose a matching game and devise scalable algorithms for
user-centric clustering in cell-free MIMO networks. Our solution jointly takes
into account the QoS requirements of UEs and limited radio resources to
maximize the number of satisfied UEs in terms of requested QoS. To do so, we
propose an EA-based many-to-many matching algorithm to speed the convergence
time and limit the signaling overhead. Numerical results show that our
proposed solution outperforms state-of-the-art benchmarks. For instance,
compared to DA-based schemes, it reduces the number of associations by $84\%$,
while providing up to $10\%$ improvement in the number of fully satisfied UEs.

Future work will investigate advanced ways for defining UEs and APs preference
lists to strengthen user-centric clustering process. Moreover, our solution
will include sleep mode mechanisms to improve energy efficiency of the
network.

## Acknowledgements

This work was funded by the French government under the France 2030 ANR
program “PEPR Networks of the Future” (ref. 22-PEFT-0003).

## References

  * [1] Ö. Tuğfe Demir, E. Björnson, and L. Sanguinetti, “Foundations of user-centric cell-free massive mimo,” arXiv e-prints, pp. arXiv–2108, 2021.
  * [2] H. Q. Ngo, A. Ashikhmin, H. Yang, E. G. Larsson, and T. L. Marzetta, “Cell-free massive MIMO versus small cells,” IEEE Transactions on Wireless Communications, vol. 16, no. 3, pp. 1834–1850, 2017.
  * [3] E. Björnson and L. Sanguinetti, “Scalable cell-free massive MIMO systems,” IEEE Transactions on Communications, vol. 68, no. 7, pp. 4247–4261, 2020.
  * [4] S. Buzzi et al., “Cell-free massive MIMO: User-centric approach,” IEEE Wireless Com. Letters, vol. 6, no. 6, pp. 706–709, 2017.
  * [5] S. Buzzi et al., “User-centric communications versus cell-free massive MIMO for 5G cellular networks,” in 21th International ITG Workshop on Smart Antennas, pp. 1–6, 2017.
  * [6] Z. Ren, A. Doufexi, and M. A. Beach, “Scalable pilot assignment for user-centric cell-free massive MIMO networks,” in IEEE International Conference on Communications (ICC), pp. 2555–2560, 2022.
  * [7] C. F. Mendoza, S. Schwarz, and M. Rupp, “Cluster formation in scalable cell-free massive MIMO networks,” in IEEE International Conference on Wireless and Mobile Computing, Networking and Communications (WiMob), pp. 62–67, 2020.
  * [8] C. F. Mendoza et al., “User-Centric Clustering in Cell-Free MIMO Networks using Deep Reinforcement Learning,” in IEEE International Conference on Communications (ICC), pp. 1036–1041, 2023.
  * [9] M. Ito, I. Kanno, Y. Amano, Y. Kishi, W.-Y. Chen, T. Choi, and A. F. Molisch, “Joint AP On/Off and User-Centric Clustering for Energy-Efficient Cell-Free Massive MIMO Systems,” in IEEE Vehicular Technology Conference (VTC2022-Fall), pp. 1–5, 2022.
  * [10] H. A. Ammar, R. Adve, S. Shahbazpanahi, G. Boudreau, and K. V. Srinivas, “Downlink resource allocation in multiuser cell-free MIMO networks with user-centric clustering,” IEEE Transactions on Wireless Communications, vol. 21, no. 3, pp. 1482–1497, 2021.
  * [11] J. Jiang, J. Wang, H. Chu, Q. Gao, and J. Zhang, “Whale Swarm Reinforcement Learning Based Dynamic Cooperation Clustering Method for Cell-Free Massive MIMO Systems,” IEEE Transactions on Vehicular Technology, vol. 72, no. 3, pp. 4114–4118, 2022.
  * [12] Y. Gu, W. Saad, M. Bennis, M. Debbah, and Z. Han, “Matching theory for future wireless networks: Fundamentals and applications,” IEEE Communications Magazine, vol. 53, no. 5, pp. 52–59, 2015.
  * [13] Z. Si, G. Chuai, W. Gao, and K. Zhang, “Many-to-Many matching user association scheme in ultra-dense millimeter-wave networks,” in IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), pp. 739–744, 2022.
  * [14] B. Bandemer, M. Haardt, and S. Visuri, “Linear MMSE multi-user MIMO downlink precoding for users with multiple antennas,” in IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), pp. 1–5, 2006.
  * [15] M. Sana, N. di Pietro, and E. Calvanese Strinati, “Transferable and Distributed User Association Policies for 5G and Beyond Networks,” in Proc. IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), pp. 966–971, 2021.
  * [16] A. Alizadeh and M. Vu, “Distributed user association in B5G networks using early acceptance matching game,” IEEE Transactions on Wireless Communications, vol. 20, no. 4, pp. 2428–2441, 2020.
  * [17] G. Interdonato, M. Karlsson, E. Björnson, and E. G. Larsson, “Local partial zero-forcing precoding for cell-free massive MIMO,” IEEE Trans. on Wireless Communications, vol. 19, no. 7, pp. 4758–4774, 2020.