$\displaystyle\textstyle{\leq(\mathtt{M}/\eta)\mathbf{W}_{1}(\mu_{n},\mu)}$
(73)
$\displaystyle\textstyle{+\int_{\mathbb{R}^{p}}\left|\log(\pi_{n}[\mathrm{k}(\cdot,y)]+\eta)-\log(\pi[\mathrm{k}(\cdot,y)]+\eta)\right|\mathrm{d}\mu(y),}$
(74)

where the second inequality follows using the dual representation of
$\mathbf{W}_{1}$.

Combining the result above, the dominated convergence theorem and the fact
that $\underset{n\to+\infty}{\lim}\mathbf{W}_{1}(\mu_{n},\mu)=0$, we get that
$\underset{n\to+\infty}{\lim}\mathscr{G}^{\eta}_{\alpha}(\pi_{n},\mu_{n})=\mathscr{G}^{\eta}_{\alpha}(\pi,\mu)$.
Thus, $\\{\mathscr{G}^{\eta}_{\alpha}(\cdot,\mu_{n})\,:n\in\mathbb{N}\\}$ is a
$\Gamma$-system. Since we have that for any
$\pi\in\mathscr{P}(\mathbb{R}^{d})$ and $n\in\mathbb{N}$,
$\mathscr{G}^{\eta}_{\alpha}(\pi,\mu_{n})\geq\alpha\text{KL}(\pi|\pi_{0})$, we
have that $\\{\mathscr{G}^{\eta}_{\alpha}(\cdot,\mu_{n})\,:n\in\mathbb{N}\\}$
is equicoercive. We conclude using Proposition 13.

###  A.5 Proof of Proposition 4

Take $\alpha>0$ and $\eta\geq 0$. Since by Proposition 2,
$\mathscr{G}_{\alpha}^{\eta}$ is proper,
$\mathscr{G}_{\alpha}^{\eta}(\pi_{\alpha,\eta}^{\star})<+\infty$ and
therefore, $\text{KL}(\pi_{\alpha,\eta}^{\star}|\pi_{0})<+\infty$. Hence,
$\pi_{\alpha,\eta}^{\star}$ admits a density w.r.t to $\pi_{0}$. Since
$\pi_{0}$ is equivalent to the Lebesgue measure, we get that
$\pi_{\alpha,\eta}^{\star}$ admits a density with respect to the Lebesgue
measure. In addition, let $\pi_{1}\in\mathscr{P}(\mathbb{R}^{d})$ be such that
for any $x\in\mathbb{R}^{d}$ we have for some $\tau>0$

$(\mathrm{d}\pi_{1}/\mathrm{d}\pi_{0})(x)=f(x)/\pi_{0}[f],\qquad
f(x)=\exp[(\tau/2)\left\|x\right\|^{2}].$ (75)

Since $\text{KL}(\pi_{\alpha,\eta}^{\star}|\pi_{0})<+\infty$ we have
$\int_{\mathbb{R}^{d}}\left|\log((\mathrm{d}\pi_{\alpha,\eta}^{\star}/\mathrm{d}\pi_{0})(x))\right|\mathrm{d}\pi_{\alpha,\eta}^{\star}(x)<+\infty$
and

$\displaystyle\textstyle{\int_{\mathbb{R}^{d}}\log((\mathrm{d}\pi_{\alpha,\eta}^{\star}/\mathrm{d}\pi_{1})(x))\mathbbm{1}_{\left[1,+\infty\right)}((\mathrm{d}\pi_{\alpha,\eta}^{\star}/\mathrm{d}\pi_{1})(x))\mathrm{d}\pi_{\alpha,\eta}^{\star}(x)}$
(76)
$\displaystyle\qquad\quad\textstyle{=\int_{\mathbb{R}^{d}}\left\\{\log((\mathrm{d}\pi_{\alpha,\eta}^{\star}/\mathrm{d}\pi_{0})(x))-(\tau/2)\left\|x\right\|^{2}+\log(\pi_{0}[f])\right\\}}$
(77)
$\displaystyle\qquad\qquad\times\textstyle{\mathbbm{1}_{\left[1,+\infty\right)}((\mathrm{d}\pi_{\alpha,\eta}^{\star}/\mathrm{d}\pi_{1})(x))\mathrm{d}\pi_{\alpha,\eta}^{\star}(x)}$
(78)
$\displaystyle\qquad\quad\textstyle{\leq\int_{\mathbb{R}^{d}}\left\\{\left|\log((\mathrm{d}\pi_{\alpha,\eta}^{\star}/\mathrm{d}\pi_{0})(x))\right|+\log(\pi_{0}[f])\right\\}}$
(79)
$\displaystyle\qquad\qquad\times\textstyle{\mathbbm{1}_{\left[1,+\infty\right)}((\mathrm{d}\pi_{\alpha,\eta}^{\star}/\mathrm{d}\pi_{1})(x))\mathrm{d}\pi_{\alpha,\eta}^{\star}(x)<+\infty.}$
(80)

Since $t\mapsto t\log(t)$ is bounded on $\left[0,1\right]$ we have that

$-\textstyle{\int_{\mathbb{R}^{d}}\log((\mathrm{d}\pi_{\alpha,\eta}^{\star}/\mathrm{d}\pi_{1})(x))\mathbbm{1}_{\left[0,1\right]}((\mathrm{d}\pi_{\alpha,\eta}^{\star}/\mathrm{d}\pi_{1})(x))\mathrm{d}\pi_{\alpha,\eta}^{\star}(x)<+\infty.}$
(81)

Combining (81) and (76) we get that
$\text{KL}(\pi_{\alpha,\eta}^{\star}|\pi_{1})<+\infty$. Therefore, using [28,
Equation 2.6], we get that

$\displaystyle\textstyle{(\tau/2)\int_{\mathbb{R}^{d}}\left\|x\right\|^{2}\mathrm{d}\pi_{\alpha,\eta}^{\star}(x)-\log(\pi_{0}[f])}$
(82)
$\displaystyle\qquad\qquad\qquad\textstyle{=\int_{\mathbb{R}^{d}}\log((\mathrm{d}\pi_{1}/\mathrm{d}\pi_{0})(x))\mathrm{d}\pi_{\alpha,\eta}^{\star}(x)=\text{KL}(\pi_{\alpha,\eta}^{\star}|\pi_{0})-\text{KL}(\pi_{\alpha,\eta}^{\star}|\pi_{1})<+\infty.}$
(83)

Therefore, $\pi_{\alpha,\eta}^{\star}\in\mathscr{P}_{2}(\mathbb{R}^{d})$.

Let
$(\alpha_{n})_{n\in\mathbb{N}}\left(0,+\infty\right)^{\mathbb{N}},(\eta_{n})_{n\in\mathbb{N}}\in\left[0,+\infty\right)^{\mathbb{N}}$
such that $\underset{n\to+\infty}{\lim}\alpha_{n}=0$,
$\underset{n\to+\infty}{\lim}\eta_{n}=0$ and
$\underset{n\to+\infty}{\lim}\pi_{\alpha_{n},\eta_{n}}^{\star}=\pi^{\star}$
with $\pi^{\star}\in\mathscr{P}_{2}(\mathbb{R}^{d})$. For any
$(\pi_{n})_{n\in\mathbb{N}}\in\mathscr{P}_{2}(\mathbb{R}^{d})^{\mathbb{N}}$
such that $\lim_{n\to+\infty}\pi_{n}=\pi$ we have using the monotonicity in
$\alpha$ of $\mathscr{G}_{\alpha}^{\eta}$ and Fatou’s lemma

$\liminf_{n\to+\infty}\mathscr{G}_{\alpha_{n}}^{\eta_{n}}(\pi_{n})\geq\liminf_{n\to+\infty}\mathscr{G}^{\eta_{n}}(\pi_{n})\geq\mathscr{G}(\pi).$
(84)

Let $\pi\in\mathscr{P}_{2}(\mathbb{R}^{d})$ and for any $n\in\mathbb{N}$
define $\pi_{n}\in\mathscr{P}_{2}(\mathbb{R}^{d})$ with density w.r.t. the
Lebesgue measure given for any $x\in\mathbb{R}^{d}$ by

$\textstyle{\pi_{n}(x)=\int_{\mathbb{R}^{d}}(2\uppi\alpha_{n})^{-d/2}\exp[-\left\|x-\tilde{x}\right\|^{2}/(2\alpha_{n})]\mathrm{d}\pi(\tilde{x}).}$
(85)

Let $X$ be a random variable with distribution $\pi$ and for any
$n\in\mathbb{N}$, let $X_{n}=X+\alpha_{n}^{1/2}Z$ where $Z$ is a Gaussian
random variable with zero mean and identity covariance matrix. For any
$n\in\mathbb{N}$, $X_{n}$ has distribution $\pi_{n}$. Therefore, we have
$\lim_{n\to+\infty}\mathbf{W}_{2}(\pi_{n},\pi)^{2}\leq\lim_{n\to+\infty}\mathbb{E}[\|X-X_{n}\|^{2}]=0$.

In what follows, we show that
$\lim_{n\to+\infty}\mathscr{G}^{\eta_{n}}(\pi_{n})=\mathscr{G}(\pi)$. Define
$h:\ \left[0,+\infty\right)\to\mathbb{R}$ as $h(t)=t\log(t)$. For any
$n\in\mathbb{N}$, using Jensen’s inequality and the Fubini-Tonelli theorem we
have

$\displaystyle\textstyle{\int_{\mathbb{R}^{d}}h(\pi_{n}(x))\mathrm{d}x}$
$\displaystyle\textstyle{\leq\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}h((2\uppi\alpha_{n})^{-d/2}\exp[-\left\|x-\tilde{x}\right\|^{2}/(2\alpha_{n})])\mathrm{d}\pi(\tilde{x})\mathrm{d}x}$
(86)
$\displaystyle\leq\textstyle{-(d/2)\log(2\uppi\alpha_{n})\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}(2\uppi\alpha_{n})^{-d/2}\exp[-\left\|x-\tilde{x}\right\|^{2}/(2\alpha_{n})]\mathrm{d}x\mathrm{d}\pi(\tilde{x})}$
(87) $\displaystyle\leq\textstyle{-(d/2)\log(2\uppi\alpha_{n}).}$ (88)

Using 2 there exists $C_{1}\geq 0$ and $\tau>0$ such that for any
$x\in\mathbb{R}^{d}$, $\left|U(x)\right|\leq C_{1}+\tau\left\|x\right\|^{2}$.
Using this result, the Fubini-Tonelli theorem, the fact that for any
$\tilde{x}\in\mathbb{R}^{d}$, $\left\|x\right\|^{2}\leq
2\left\|x-\tilde{x}\right\|^{2}+2\left\|\tilde{x}\right\|^{2}$ and that
$\int_{\mathbb{R}^{d}}\left\|\tilde{x}\right\|^{2}\mathrm{d}\pi(\tilde{x})<+\infty$
we have for any $n\in\mathbb{N}$

$\displaystyle\textstyle{\int_{\mathbb{R}^{d}}\left|U(x)\right|\mathrm{d}\pi_{n}(x)}$
$\displaystyle\textstyle{\leq
C_{1}+\tau\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\left\|x\right\|^{2}(2\uppi\alpha_{n})^{-d/2}\exp[-\left\|x-\tilde{x}\right\|^{2}/(2\alpha_{n})]\mathrm{d}\tilde{x}\mathrm{d}\pi(x)}$
(89) $\displaystyle\leq\textstyle{C_{1}+2\tau
d\sup_{n\in\mathbb{N}}\alpha_{n}+2\tau\int_{\mathbb{R}^{d}}\left\|\tilde{x}\right\|^{2}\mathrm{d}\pi(\tilde{x})<+\infty.}$
(90)

Combining (88) and (90) we have
$\lim_{n\to+\infty}\alpha_{n}\text{KL}(\pi_{n}|\pi_{0})=0$.

For any $n\in\mathbb{N}$ and $y\in\mathbb{R}^{p}$,
$\log(\pi_{n}[\mathrm{k}(\cdot,y)]+\eta_{n})\leq\log(\mathtt{M}+\sup_{n\in\mathbb{N}}\eta_{n})$.
Define $\Phi:\ \mathbb{R}^{d}\to\mathbb{R}$,
$\Phi:x\mapsto\left\|x\right\|^{2}$ and $\Psi:\ \mathbb{R}^{p}\to\mathbb{R}$,
$\Psi:y\mapsto\left\|y\right\|^{2}$. Using Jensen’s inequality and the fact
that under 2 there exists $C_{2}\geq 0$ such that for any
$(x,y)\in\mathbb{R}^{d}\times\mathbb{R}^{p}$,
$\exp[-\Phi(x)-\Psi(y)]/\mathrm{k}(x,y)\leq C_{2}\exp[C_{2}]$, we have for any
$y\in\mathbb{R}^{p}$ and $n\in\mathbb{N}$ that

$\log(\pi_{n}[\mathrm{k}(\cdot,y)]+\eta_{n})\geq\log(\pi_{n}[\mathrm{k}(\cdot,y)])\geq-\Psi(y)-\sup_{n\in\mathbb{N}}\pi_{n}[\Phi]-C_{2}-\log
C_{2}.$ (91)

Since, $(\pi_{n})_{n\in\mathbb{N}}$ is relatively compact in
$\mathscr{P}_{2}(\mathbb{R}^{d})$ and $\mu\in\mathscr{P}_{2}(\mathbb{R}^{p})$,
there exists $C\geq 0$ such that for any $n\in\mathbb{N}$, $\pi_{n}[\Phi]\leq
C$, see [96, Definition 6.8] and $\mu[\Psi]<+\infty$. Hence, there exists
$M\geq 0$ such that for any $n\in\mathbb{N}$ and $y\in\mathbb{R}^{p}$ such
that $|\log(\pi_{n}[\mathrm{k}(\cdot,y)]+\eta_{n})|\leq M$. Combining this
result and the dominated convergence theorem, we get that
$\lim_{n\to+\infty}\mathscr{G}^{\eta_{n}}(\pi_{n})=\mathscr{G}(\pi)$.

Therefore, we have that
$\mathscr{G}(\pi)=\lim_{n\to+\infty}\mathscr{G}_{\alpha_{n}}^{\eta_{n}}(\pi_{n})$,
and $\\{\mathscr{G}_{\alpha}^{\eta}\,:\alpha,\eta\geq 0\\}$ is a
$\Gamma$-system. Using Proposition 14, we get that $\pi^{\star}$ is a
minimizer of $\mathscr{G}$.

Assume that there exists
$\pi^{\dagger}\in\operatorname*{arg\,min}_{\mathscr{P}(\mathbb{R}^{d})}\mathscr{G}$
such that $\text{KL}(\pi^{\dagger}|\pi_{0})<\text{KL}(\pi^{\star}|\pi_{0})$.
In what follows we assume that $\text{KL}(\pi^{\star}|\pi_{0})<+\infty$, the
case where $\text{KL}(\pi^{\star}|\pi_{0})=+\infty$ is similar and omitted.
There exists $\varepsilon>0$ such that
$\text{KL}(\pi^{\dagger}|\pi_{0})=\text{KL}(\pi^{\star}|\pi_{0})-\varepsilon$.
Since
$\liminf_{n\to+\infty}\text{KL}(\pi_{\alpha_{n},\eta_{n}}^{\star}|\pi_{0})\geq\text{KL}(\pi^{\star}|\pi_{0})$,
there exists $n_{0}\in\mathbb{N}$ such that for any $n\in\mathbb{N}$ with
$n\geq n_{0}$ we have

$\text{KL}(\pi_{\alpha_{n},\eta_{n}}^{\star}|\pi_{0})\geq\text{KL}(\pi^{\star}|\pi_{0})-\varepsilon/2\geq\text{KL}(\pi^{\dagger}|\pi_{0})+\varepsilon/2.$
(92)

Therefore we have that for any $n\geq n_{0}$

$\displaystyle\mathscr{G}_{\alpha_{n}}^{\eta_{n}}(\pi_{\alpha_{n},\eta_{n}}^{\star})$
$\displaystyle=\mathscr{G}^{\eta_{n}}(\pi_{\alpha_{n},\eta_{n}}^{\star})+\alpha_{n}\text{KL}(\pi_{\alpha_{n},\eta_{n}}^{\star}|\pi_{0})$
(93)
$\displaystyle\geq\mathscr{G}^{\eta_{n}}(\pi^{\dagger})+\alpha_{n}\text{KL}(\pi^{\dagger}|\pi_{0})+\alpha_{n}\varepsilon/2>\mathscr{G}_{\alpha_{n}}^{\eta_{n}}(\pi^{\dagger}),$
(94)

which is a contradiction since for any $n\in\mathbb{N}$,
$\pi_{\alpha_{n},\eta_{n}}^{\star}$ is a minimizer of
$\mathscr{G}_{\alpha_{n}}^{\eta_{n}}$. Hence, for any
$\pi\in\operatorname*{arg\,min}_{\mathscr{P}(\mathbb{R}^{d})}\mathscr{G}$,
$\text{KL}(\pi^{\star}|\pi_{0})\leq\text{KL}(\pi|\pi_{0})$, which concludes
the proof.

##  Appendix B Proofs of Section 3

###  B.1 Basics on Wasserstein gradient flows and subdifferential of
$\mathscr{G}_{\alpha}^{\eta}$

In this section we recall basics results on gradient flows in Wasserstein
spaces and provide a slight extension of the classical variational formula. We
refer to [2] for further details. We denote by
$\mathrm{L}^{2}(\mathbb{R}^{d},\nu)$ the set of square integrable functions
w.r.t. a measure $\nu$, by $\mathrm{C}_{c}^{\infty}(\mathbb{R}^{d})$ the set
of infinitely many times differentiable functions with compact support and by
$\mathrm{W}^{1,1}_{\mathrm{loc}}(\mathbb{R}^{d})$ the set of functions which
are in $\mathrm{W}^{1,1}_{\mathrm{loc}}(\mathsf{X})$ for all compact sets
$\mathsf{X}\subset\mathbb{R}^{d}$, where
$\mathrm{W}^{1,1}_{\mathrm{loc}}(\mathsf{X})$ denotes the Sobolev space of
functions whose zero-th and first derivative are bounded in $\mathrm{L}^{1}$.
For any $\nu\in\mathscr{P}(\mathbb{R}^{d})$ and $t:\
\mathbb{R}^{d}\to\mathbb{R}^{d}$ measurable, we denote by $t_{\\#}\nu$ the
push-forward measure, $t_{\\#}\nu(A)=\nu(t^{-1}(A))$ for all
$A\in\mathcal{B}(\mathbb{R}^{d})$. We start by recalling the definition of the
subdifferentiability in Wasserstein spaces, see [2, Definition 10.1.1]. We
denote $\mathscr{P}_{2}^{r}(\mathbb{R}^{d})$ the space of probability
distributions in $\mathscr{P}_{2}(\mathbb{R}^{d})$ which are absolutely
continuous w.r.t the Lebesgue measure.

######  Definition 15 (Fréchet subdifferential).

Let $\Phi:\ \mathscr{P}_{2}(\mathbb{R}^{d})\to\left(-\infty,+\infty\right)$.
Let $\pi\in\mathscr{P}_{2}^{r}(\mathbb{R}^{d})$, then
$\xi\in\mathrm{L}^{2}(\mathbb{R}^{d},\pi)$ belongs to the strong Fréchet
subdifferential $\partial_{\mathrm{s}}\Phi(\pi)$ of $\Phi$ at $\pi$ if for any
$t\in\mathrm{L}^{2}(\mathbb{R}^{d},\pi)$

$\textstyle{\liminf_{t\to\operatorname{Id}}\left.\left\\{\Phi(t_{\\#}\pi)-\Phi(\pi)-\int_{\mathbb{R}^{d}}\langle\xi(x),t(x)-x\rangle\mathrm{d}\pi(x)\right\\}\middle/\left\|t-\operatorname{Id}\right\|_{\mathrm{L}^{2}(\mathbb{R}^{d},\pi)}\right.=0.}$
(95)

The following lemma is a specific case of the chain-rule formula.

######  Lemma 16.

Let $g\in\mathrm{C}^{1}(\mathbb{R},\mathbb{R})$ be Lipschitz, convex and
lower-bounded. For any $y\in\mathbb{R}^{p}$, let $\mathscr{L}:\
\mathscr{P}_{2}(\mathbb{R}^{d})\times\mathbb{R}^{p}\to\mathbb{R}$,
$\pi\in\mathscr{P}_{2}(\mathbb{R}^{d})$,
$\mu\in\mathscr{P}_{2}(\mathbb{R}^{p})$ and
$\xi\in\mathrm{L}^{2}(\mathbb{R}^{d}\times\mathbb{R}^{p},\pi\otimes\mu)$ such
that for any $y\in\mathbb{R}^{p}$ and
$\pi\in\mathscr{P}_{2}^{r}(\mathbb{R}^{d})$,
$\xi(\pi,y)\in\partial_{\mathrm{s}}\mathscr{L}(\pi,y)$ and
$-\xi(\pi,y)\in\partial_{\mathrm{s}}(-\mathscr{L})(\pi,y)$. In addition,
assume that there exists $h\in\mathrm{L}^{1}(\mathbb{R}^{p},\mu)$ such that
for any $\pi_{1},\pi_{2}\in\mathscr{P}_{2}(\mathbb{R}^{d})$ and
$y\in\mathbb{R}^{p}$ we have

$\left|\mathscr{L}(\pi_{1},y)-\mathscr{L}(\pi_{2},y)\right|\leq
h(y)\mathbf{W}_{2}(\pi_{1},\pi_{2}).$ (96)

Let $\mathscr{M}$ be given for any $\pi\in\mathscr{P}(\mathbb{R}^{d})$ by

$\textstyle{\mathscr{M}(\pi)=\int_{\mathbb{R}^{p}}g(\mathscr{L}(\pi,y))\mathrm{d}\mu(y).}$
(97)

Then, we have that $\partial_{\mathrm{s}}\mathscr{M}(\pi)\neq\emptyset$ and
$\\{x\mapsto\int_{\mathbb{R}^{p}}g^{\prime}(\mathscr{L}(\pi,y))\xi(\cdot,y)\mathrm{d}\mu(y)\\}\in\partial_{\mathrm{s}}\mathscr{M}(\pi)$.

###### Proof.

Let $\pi\in\mathscr{P}(\mathbb{R}^{d})$,
$t\in\mathrm{L}^{2}(\mathbb{R}^{d},\pi)$ and $y\in\mathbb{R}^{d}$. Since $g$
is convex we have

$g(\mathscr{L}(t_{\\#}\pi,y))-g(\mathscr{L}(\pi,y))\geq
g^{\prime}(\mathscr{L}(\pi,y))(\mathscr{L}(t_{\\#}\pi,y)-\mathscr{L}(\pi,y)).$
(98)

We have

$\displaystyle\textstyle{g(\mathscr{L}(t_{\\#}\pi,y))-g(\mathscr{L}(\pi,y))-g^{\prime}(\mathscr{L}(\pi,y))\int_{\mathbb{R}^{d}}\langle\xi(x,y),t(x)-x\rangle\mathrm{d}\pi(x)}$
(99) $\displaystyle\qquad\textstyle{\geq
g^{\prime}(\mathscr{L}(\pi,y))\left(\mathscr{L}(t_{\\#}\pi,y)-\mathscr{L}(\pi,y)-\int_{\mathbb{R}^{d}}\langle\xi(x,y),t(x)-x\rangle\mathrm{d}\pi(x)\right).}$
(100)

In addition, using that $\xi(\pi,y)\in\partial_{\mathrm{s}}\mathscr{L}(\pi,y)$
and $-\xi(\pi,y)\in\partial_{\mathrm{s}}(-\mathscr{L})(\pi,y)$ we have

$\displaystyle\textstyle{\liminf_{t\to\operatorname{Id}}\left(\mathscr{L}(t_{\\#}\pi,y)-\mathscr{L}(\pi,y)-\int_{\mathbb{R}^{d}}\langle\xi(x,y),t(x)-x\rangle\mathrm{d}\pi(x)\right)/\|t-\operatorname{Id}\|_{2}\geq
0,}$ (101)
$\displaystyle\textstyle{\limsup_{t\to\operatorname{Id}}\left(\mathscr{L}(t_{\\#}\pi,y)-\mathscr{L}(\pi,y)-\int_{\mathbb{R}^{d}}\langle\xi(x,y),t(x)-x\rangle\mathrm{d}\pi(x)\right)/\|t-\operatorname{Id}\|_{2}\leq
0.}$ (102)

Hence, we have

$\textstyle{\lim_{t\to\operatorname{Id}}\left(\mathscr{L}(t_{\\#}\pi,y)-\mathscr{L}(\pi,y)-\int_{\mathbb{R}^{d}}\langle\xi(x,y),t(x)-x\rangle\mathrm{d}\pi(x)\right)/\|t-\operatorname{Id}\|_{2}=0.}$
(103)

We also have that

$\displaystyle\textstyle{\left|\mathscr{L}(t_{\\#}\pi,y)-\mathscr{L}(\pi,y)-\int_{\mathbb{R}^{d}}\langle\xi(x,y),t(x)-x\rangle\mathrm{d}\pi(x)\right|/\|t-\operatorname{Id}\|_{2}}$
(104) $\displaystyle\qquad\qquad\qquad\textstyle{\leq
h(y)+\int_{\mathbb{R}^{d}}\left\|\xi(x,y)\right\|\mathrm{d}\pi(x).}$ (105)

We conclude upon using this result, the fact that $g$ is Lipschitz, (99) and
the dominated convergence theorem. ∎

We are now ready to derive the subdifferential of
$\mathscr{G}_{\alpha}^{\eta}$ for any $\eta>0$.

######  Lemma 17 (Strong subdifferential of $\mathscr{G}^{\eta}$).

Assume 1. Then for any $\pi\in\mathscr{P}_{2}^{r}(\mathbb{R}^{d})$ and
$\eta>0$, we have that
$\partial_{\mathrm{s}}\mathscr{G}^{\eta}(\pi)\neq\emptyset$ and

$\textstyle{\\{x\mapsto\int_{\mathbb{R}^{p}}(\eta+\pi[\mathrm{k}(x,y)])^{-1}\nabla_{1}\mathrm{k}(x,y)\mathrm{d}\mu(y)\\}\in\partial_{\mathrm{s}}\mathscr{G}^{\eta}(\pi).}$
(106)

###### Proof.

First, using that $\|\mathrm{d}^{2}\mathrm{k}\|\leq\mathtt{M}$ we get that for
any $x_{1},x_{2}\in\mathbb{R}^{d}$, $y\in\mathbb{R}^{p}$

$\mathrm{k}(x_{2},y)\geq\mathrm{k}(x_{1},y)+\langle\nabla_{1}\mathrm{k}(x_{1},y),x_{2}-x_{1}\rangle-\mathtt{M}\left\|x_{1}-x_{2}\right\|^{2}.$
(107)

Let $\mathscr{L}:\
\mathscr{P}_{2}(\mathbb{R}^{d})\times\mathbb{R}^{p}\to\left[0,+\infty\right)$
given for any $\pi\in\mathscr{P}_{2}(\mathbb{R}^{d})$ and $y\in\mathbb{R}^{p}$
by

$\textstyle{\mathscr{L}(\pi,y)=\int_{\mathbb{R}^{d}}\mathrm{k}(x,y)\mathrm{d}\pi(x).}$
(108)

Using (107) and [2, Proposition 10.4.2] we get that for any
$y\in\mathbb{R}^{p}$ and $\pi\in\mathscr{P}_{2}(\mathbb{R}^{d})$,
$\partial_{\mathrm{s}}\mathscr{L}(\pi,y)=\\{\partial_{1}\mathrm{k}(\cdot,y)\\}$.
Since there exists
$\left\|\partial_{1}\mathrm{k}\right\|_{\infty}\leq\mathtt{M}<+\infty$ we get
that for any $\pi_{1},\pi_{2}\in\mathscr{P}_{2}(\mathbb{R}^{d})$ and
$y\in\mathbb{R}^{p}$

$\left|\mathscr{L}(\pi_{1},y)-\mathscr{L}(\pi_{2},y)\right|\leq\mathtt{M}\mathbf{W}_{2}(\pi_{1},\pi_{2}).$
(109)

In addition, let $g:\ \left(0,+\infty\right)\to\mathbb{R}$ given for any
$t\geq 0$ by $g(t)=-\log(t+\eta)$. Using that $g$ is convex, (109) and that
$\left\|\partial_{1}\mathrm{k}\right\|_{\infty}\leq\mathtt{M}<+\infty$ in
Lemma 16, we get that for any $\pi\in\mathscr{P}_{2}(\mathbb{R}^{d})$,
$\partial_{\mathrm{s}}\mathscr{G}^{\eta}(\pi)\neq\emptyset$ and
$\\{x\mapsto\int_{\mathbb{R}^{d}}(\eta+\pi[\mathrm{k}(x,y)])^{-1}\nabla_{1}\mathrm{k}(x,y)\mathrm{d}\mu(y)\\}\in\partial_{\mathrm{s}}\mathscr{G}^{\eta}(\pi)$.
∎

Let $\mathscr{H}:\ \mathscr{P}(\mathbb{R}^{d})\to\left[0,+\infty\right]$ be
defined for any $\pi\in\mathscr{P}(\mathbb{R}^{d})$ by
$\mathscr{H}(\pi)=\textup{KL}\left(\pi|\pi_{0}\right)$. Lemma 18 is a
restatement of [2, Theorem 10.4.9].

######  Lemma 18 (Strong subdifferential of
$\textup{KL}\left(\pi|\pi_{0}\right)$).

Assume that $\pi_{0}$ admits a density, also denoted $\pi_{0}$, w.r.t. the
Lebesgue measure such that for any $x\in\mathbb{R}^{d}$,
$\pi_{0}(x)\propto\exp[-U(x)]$ for some convex
$U:\mathbb{R}^{d}\to\mathbb{R}$. Let
$\pi\in\mathscr{P}_{2}^{r}(\mathbb{R}^{d})$ such that
$\mathscr{H}(\pi)<+\infty$ and
$(\mathrm{d}\pi/\mathrm{d}\pi_{0})\in\mathrm{W}^{1,1}_{\mathrm{loc}}(\mathbb{R}^{d},\left(0,+\infty\right))$.
Then,

$\displaystyle\partial_{\mathrm{s}}\mathscr{H}(\pi)=\left\\{x\mapsto\nabla\log(\mathrm{d}\pi/\mathrm{d}\pi_{0})(x)\right\\}.$
(110)

Combining these two lemmas we are now ready to derive the subdifferential of
$\mathscr{G}_{\eta}^{\alpha}$ given by (10) as well as one associated
Wasserstein gradient flow.

######  Proposition 19 (Gradient flow).

Assume 1 and that $\pi_{0}$ admits a density, also denoted $\pi_{0}$, w.r.t.
the Lebesgue measure such that for any $x\in\mathbb{R}^{d}$,
$\pi_{0}(x)\propto\exp[-U(x)]$ for some convex
$U:\mathbb{R}^{d}\to\mathbb{R}$. Let
$\pi\in\mathscr{P}_{2}^{r}(\mathbb{R}^{d})$ be such that
$\textup{KL}\left(\pi|\pi_{0}\right)<+\infty$ and
$(\mathrm{d}\pi/\mathrm{d}\pi_{0})\in\mathrm{W}^{1,1}_{\mathrm{loc}}(\mathbb{R}^{d},\left(0,+\infty\right))$.
Then for any $\alpha,\eta>0$

$\textstyle{\left\\{x\mapsto\int_{\mathbb{R}^{p}}(\eta+\pi[\mathrm{k}(x,y)])^{-1}\nabla_{1}\mathrm{k}(x,y)\mathrm{d}\mu(y)+\alpha\nabla\log(\mathrm{d}\pi/\mathrm{d}\pi_{0})(x)\right\\}\in\partial_{\mathrm{s}}\mathscr{G}_{\alpha}^{\eta}(\pi).}$
(111)

In addition, if $U\in\mathrm{C}^{1}(\mathbb{R}^{d},\mathbb{R})$ and there
exists $(\pi_{t})_{t\geq 0}$ such that for any $t\in\left(0,+\infty\right)$,
$\pi_{t}$ admits a density w.r.t. the Lebesgue measure also denoted (with a
slight abuse of notation) by
$\pi_{t}\in\mathrm{C}^{1}(\mathbb{R}^{d},\mathbb{R})$, and for any
$\varphi\in\mathrm{C}_{c}^{\infty}(\mathbb{R}^{d})$

$\textstyle{\int_{\mathbb{R}}\varphi(x)\mathrm{d}\pi_{t}(x)-\int_{\mathbb{R}}\varphi(x)\mathrm{d}\pi_{0}(x)}\\\
=\textstyle{\int_{0}^{t}\int_{\mathbb{R}^{d}}\left[-\int_{\mathbb{R}^{d}}\left\\{\left.\langle\nabla_{1}\mathrm{k}(x,y),\nabla\varphi(x)\rangle\middle/\left(\int_{\mathbb{R}^{d}}\mathrm{k}(x,y)\mathrm{d}\pi_{s}(x)+\eta\right)\right\\}\mathrm{d}\mu(y)\right.\right.}\\\
\textstyle{\left.-\alpha\langle\nabla
U(x),\nabla\varphi(x)\rangle+\alpha\Delta\varphi(x)\vphantom{\int_{\mathbb{R}^{d}}}\right]\mathrm{d}\pi_{s}(x),}$
(112)

then $(\pi_{t})_{t\geq 0}$ is a Wasserstein gradient flow associated with
$\mathscr{G}_{\alpha}^{\eta}$.

###### Proof.

The first part of the proof is a straightforward application of Lemma 17 and
Lemma 18. The second part of the proof follows from the definition of a
Wasserstein gradient flow [2, Definition 11.1.1]. ∎

###  B.2 Proof of Proposition 5

Let $\alpha,\eta>0$. First, we show that $\tilde{b}_{\eta}$ is Lipschitz
continuous. Let $f(t)=(\eta+t)^{-1}$. Using that for any $t\geq 0$,
$\left|f^{\prime}(t)\right|\leq\eta^{-2}$ we have for any
$x_{1},x_{2}\in\mathbb{R}^{d}$, $y\in\mathbb{R}^{p}$ and
$\pi_{1},\pi_{2}\in\mathscr{P}_{1}(\mathbb{R}^{d})$

$\displaystyle\|\tilde{b}_{\eta}(x_{1},\pi_{1},y)-\tilde{b}_{\eta}(x_{2},\pi_{2},y)\|$
(113)
$\displaystyle\qquad\leq\|\nabla_{1}\mathrm{k}(x_{1},y)/(\pi_{1}[\mathrm{k}(\cdot,y)]+\eta)-\nabla_{1}\mathrm{k}(x_{2},y)/(\pi_{2}[\mathrm{k}(\cdot,y)]+\eta)\|$
(114)
$\displaystyle\qquad\leq\|\nabla_{1}\mathrm{k}(x_{1},y)/(\pi_{1}[\mathrm{k}(\cdot,y)]+\eta)-\nabla_{1}\mathrm{k}(x_{2},y)/(\pi_{1}[\mathrm{k}(\cdot,y)]+\eta)\|$
(115)
$\displaystyle\qquad\qquad+\|\nabla_{1}\mathrm{k}(x_{2},y)/(\pi_{1}[\mathrm{k}(\cdot,y)]+\eta)-\nabla_{1}\mathrm{k}(x_{2},y)/(\pi_{2}[\mathrm{k}(\cdot,y)]+\eta)\|$
(116)
$\displaystyle\qquad\leq(\mathtt{M}/\eta)\|x_{1}-x_{2}\|+\mathtt{M}\left|(\pi_{1}[\mathrm{k}(\cdot,y)]+\eta)^{-1}-(\pi_{2}[\mathrm{k}(\cdot,y)]+\eta)^{-1}\right|$
(117)
$\displaystyle\qquad\leq(\mathtt{M}/\eta)(1+(1/\eta))\left\\{\|x_{1}-x_{2}\|+\left|\pi_{1}[\mathrm{k}(\cdot,y)]-\pi_{2}[\mathrm{k}(\cdot,y)]\right|\right\\},$
(118)

where we have used the fact that, under 1, $\mathrm{k}$ is Lipschitz
continuous. The rest of the proof is classical, see for instance [91, Theorem
1.1], but is given for completeness. Denote
$b:\mathbb{R}^{d}\times\mathscr{P}_{1}(\mathbb{R}^{d})\to\mathbb{R}^{d}$ such
that for any $x\in\mathbb{R}^{d}$ and $\pi\in\mathscr{P}_{1}(\mathbb{R}^{d})$

$\textstyle{b(x,\pi)=\int_{\mathbb{R}^{p}}\tilde{b}_{\eta}(x,\pi,y)\mathrm{d}\mu(y)-\alpha\nabla
U(x).}$ (119)

Using 3-(a) and (113) we have that for any $x_{1},x_{2}\in\mathbb{R}^{d}$ and
$\pi_{1},\pi_{2}\in\mathscr{P}_{1}(\mathbb{R}^{d})$

$\displaystyle\textstyle{\left\|b(x_{1},\pi_{1})-b(x_{2},\pi_{2})\right\|\leq((\mathtt{M}/\eta)(1+(1/\eta))+\alpha\mathtt{L})}$
(120)
$\displaystyle\qquad\qquad\times\textstyle{\left(\left\|x_{1}-x_{2}\right\|+\int_{\mathbb{R}^{p}}\left|\pi_{1}[\mathrm{k}(\cdot,y)]-\pi_{2}[\mathrm{k}(\cdot,y)]\right|\mathrm{d}\mu(y)\right).}$
(121)

Combining this result and 1, we get that for any
$x_{1},x_{2}\in\mathbb{R}^{d}$ and
$\pi_{1},\pi_{2}\in\mathscr{P}_{1}(\mathbb{R}^{d})$

$\left\|b(x_{1},\pi_{1})-b(x_{2},\pi_{2})\right\|\leq((\mathtt{M}/\eta)(1+(1/\eta))+\alpha\mathtt{L})(1+\mathtt{M})\left(\left\|x_{1}-x_{2}\right\|+\mathbf{W}_{1}(\pi_{1},\pi_{2})\right),$
(122)

where the inequality follows using the dual representation of
$\mathbf{W}_{1}$. Let $T\geq 0$. For any
$\bm{\nu}\in\mathrm{C}(\left[0,T\right],\mathscr{P}_{1}(\mathbb{R}^{d}))$
denote by $(\mathbf{X}_{t}^{\bm{\nu}})_{t\in\left[0,T\right]}$ the unique
strong solution to (22) with initial condition
$\bm{\nu}\in\mathscr{P}_{1}(\mathbb{R}^{d})$ (see [57, Theorem 2.9]) given for
any $t\in\left[0,T\right]$ by

$\textstyle{\mathbf{X}_{t}^{\bm{\nu}}=\mathbf{X}_{0}^{\bm{\nu}}+\int_{0}^{t}b(\mathbf{X}_{s}^{\bm{\nu}},\bm{\nu}_{s})\mathrm{d}s+\sqrt{2\alpha}\mathbf{B}_{t},}$
(123)

where $(\mathbf{B}_{t})_{t\geq 0}$ is a $d$-dimensional Brownian motion with
$\mathbf{X}_{0}^{\bm{\nu}}=X_{0}\in\mathbb{R}^{d}$. Using 1 and (122), there
exists $C\geq 0$ such that for any $x\in\mathbb{R}^{d}$ and
$\pi\in\mathscr{P}_{1}(\mathbb{R}^{d})$

$\left\|b(x,\pi)\right\|\leq C\left(1+\left\|x\right\|\right).$ (124)

Denote $(\mathcal{F}_{t})_{t\geq 0}$ the filtration associated with
$(\mathbf{B}_{t})_{t\geq 0}$. Using Jensen’s inequality we have that
$(\|\mathbf{B}_{t}\|)_{t\geq 0}$ is a $(\mathcal{F}_{t})_{t\geq
0}$-supermartingale. Therefore, using [85, Chapter 2, Corollary 1.6], we get
that for any $t\geq 0$,

$\textstyle{\mathbb{E}[\sup_{s\in\left[0,t\right]}\|\mathbf{B}_{s}\|]\leq
2\mathbb{E}^{1/2}[\|\mathbf{B}_{t}\|^{2}]\leq 2(dt)^{1/2}.}$ (125)

Using this result and (124) we get that for any $t\geq 0$

$\displaystyle\textstyle{\mathbb{E}[\sup_{s\in\left[0,t\right]}\|\mathbf{X}_{s}^{\bm{\nu}}\|]}$
(126)
$\displaystyle\qquad\textstyle{\leq\mathbb{E}[\|\mathbf{X}_{0}^{\bm{\nu}}\|]+2(2\alpha
d)^{1/2}+(C+2(2\alpha
d)^{1/2})t+C\int_{0}^{t}\mathbb{E}[\sup_{u\in\left[0,s\right]}\|\mathbf{X}_{u}^{\bm{\nu}}\|]\mathrm{d}s.}$
(127)

Using Grönwall’s lemma we get that
$\mathbb{E}[\sup_{t\in\left[0,T\right]}\|\mathbf{X}_{t}^{\bm{\nu}}\|]<+\infty$.
In particular, for any $t\in\left[0,T\right]$,
$\bm{\lambda}^{\bm{\nu}}_{t}\in\mathscr{P}_{1}(\mathbb{R}^{d})$, where
$\bm{\lambda}^{\bm{\nu}}$ is the distribution of
$(\mathbf{X}_{t}^{\bm{\nu}})_{t\in\left[0,T\right]}$. Similarly there exists
$C\geq 0$ such that for any $t,s\in\left[0,T\right]$ we have

$\mathbf{W}_{1}(\bm{\lambda}^{\bm{\nu}}_{t},\bm{\lambda}^{\bm{\nu}}_{s})\leq\mathbb{E}\left[\left\|\mathbf{X}_{t}^{\bm{\nu}}-\mathbf{X}_{s}^{\bm{\nu}}\right\|\right]\leq
C(t-s).$ (128)

Therefore,
$\bm{\lambda}^{\bm{\nu}}\in\mathrm{C}(\left[0,T\right],\mathscr{P}_{1}(\mathbb{R}^{d}))$.
In addition, using (122) we get that for any
$\bm{\nu}_{1},\bm{\nu}_{2}\in\mathrm{C}(\left[0,T\right],\mathscr{P}_{1}(\mathbb{R}^{d}))$
and $t\in\left[0,T\right]$

$\displaystyle\textstyle{\mathbb{E}[\|\mathbf{X}_{t}^{\bm{\nu}_{1}}-\mathbf{X}_{t}^{\bm{\nu}_{2}}\|]\leq((\mathtt{M}/\eta)(1+(1/\eta))+\alpha\mathtt{L})(1+\mathtt{M})}$
(129)
$\displaystyle\qquad\qquad\times\textstyle{\int_{0}^{t}\mathbb{E}[\left\\{\|\mathbf{X}_{s}^{\bm{\nu}_{1}}-\mathbf{X}_{s}^{\bm{\nu}_{2}}\|+\mathbf{W}_{1}(\bm{\nu}_{1,s},\bm{\nu}_{2,s})\right\\}\mathrm{d}s].}$
(130)

By Grönwall’s lemma, there exists $C\geq 0$ such that for any
$\bm{\nu}_{1},\bm{\nu}_{2}\in\mathrm{C}(\left[0,T\right],\mathscr{P}_{1}(\mathbb{R}^{d}))$
and $t\geq 0$

$\mathbb{E}\left[\left\|\mathbf{X}_{t}^{\bm{\nu}_{1}}-\mathbf{X}_{t}^{\bm{\nu}_{2}}\right\|\right]\leq
T((\mathtt{M}/\eta)(1+(1/\eta))+\alpha\mathtt{L})(1+\mathtt{M})\\\
\times\exp[T((\mathtt{M}/\eta)(1+(1/\eta))+\alpha\mathtt{L})(1+\mathtt{M})]\sup_{s\in\left[0,T\right]}\mathbf{W}_{1}(\bm{\nu}_{1,s},\bm{\nu}_{2,s}).$
(131)

Let $F_{T}:\
\mathrm{C}(\left[0,T\right],\mathscr{P}_{1}(\mathbb{R}^{d}))\to\mathrm{C}(\left[0,T\right],\mathscr{P}_{1}(\mathbb{R}^{d}))$.
Then there exists $T_{0}\geq 0$ such that for any $T\in\left[0,T\right]$ and
$\bm{\nu}_{1},\bm{\nu}_{2}\in\mathrm{C}(\left[0,T\right],\mathscr{P}_{1}(\mathbb{R}^{d}))$

$\sup_{t\in\left[0,T_{0}\right]}\mathbf{W}_{1}(F_{T}(\bm{\nu}_{1})_{t},F_{T}(\bm{\nu}_{2})_{t})\leq(1/2)\sup_{t\in\left[0,T_{0}\right]}\mathbf{W}_{1}(\bm{\nu}_{1,t},\bm{\nu}_{2,t}).$
(132)

Since $\mathscr{P}_{1}(\mathbb{R}^{d})$ is complete, see [96, Theorem 6.18],
$F_{T}$ admits a unique fixed point using Picard’s theorem. Denote
$\bm{\lambda}^{\star}$ this fixed point and
$(\mathbf{X}_{t}^{\star})_{t\in\left[0,T\right]}$ the solution of (123) with
$\bm{\nu}\leftarrow\bm{\lambda}^{\star}$. Then
$(\mathbf{X}_{t}^{\star})_{t\in\left[0,T\right]}$ is a solution of (22) up to
time $T\geq 0$. By recursion, we obtain a solution
$(\mathbf{X}_{t}^{\star})_{t\geq 0}$.

Assume that there exist two solutions $(\mathbf{X}_{1,t}^{\star})_{t\geq 0}$
and $(\mathbf{X}_{2,t}^{\star})_{t\geq 0}$ and denote $\bm{\lambda}_{1}$ and
$\bm{\lambda}_{2}$ their associated distribution. Since there exists a unique
fixed point to $F_{T}$ we get that $\bm{\lambda}_{1}=\bm{\lambda}_{2}$. Since
there exists a unique strong solution to (123), we get that
$(\mathbf{X}_{1,t}^{\star})_{t\geq 0}=(\mathbf{X}_{2,t}^{\star})_{t\geq 0}$,
which concludes the proof.

###  B.3 Proof of Proposition 7

Let $N\in\mathbb{N}^{\star}$. The existence and strong uniqueness of a
solution to (24) is a straightforward consequence of (120), [57, Theorem 2.9]
and the fact that for any
$\\{x_{1}^{k,N}\\}_{k=1}^{N},\\{x_{2}^{k,N}\\}_{k=1}^{N}\in(\mathbb{R}^{d})^{N}$
and $\ell\in\\{1,\dots,N\\}$ we have

$\displaystyle\textstyle{\|\int_{\mathbb{R}^{p}}\\{\tilde{b}_{\eta}(x_{1}^{\ell,N},(1/N)\sum_{k=1}^{N}\updelta_{x_{1}^{k,N}},y)-\tilde{b}_{\eta}(x_{2}^{\ell,N},(1/N)\sum_{k=1}^{N}\updelta_{x_{2}^{k,N}},y)\\}\mathrm{d}\mu(y)\|}$
(133)
$\displaystyle\quad\textstyle{\leq((\mathtt{M}/\eta)(1+(1/\eta))+\alpha\mathtt{L})}$
(134)
$\displaystyle\quad\quad\textstyle{\times(\|x_{1}^{\ell,N}-x_{2}^{\ell,N}\|+(1/N)\sum_{k=1}^{N}\int_{\mathbb{R}^{p}}|\mathrm{k}(x_{1}^{k,N},y)-\mathrm{k}(x_{2}^{k,N},y)|\mathrm{d}\mu(y))}$
(135) $\displaystyle\quad\textstyle{\leq
2((\mathtt{M}/\eta)(1+(1/\eta))+\alpha\mathtt{L})\|x_{1}^{1:N}-x_{2}^{1:N}\|.}$
(136)

We now turn to the quantitative functional propagation of chaos result. For
any $x\in\mathbb{R}^{d}$ and $\pi\in\mathscr{P}_{1}(\mathbb{R}^{d})$ denote
$b(x,\pi)=\int_{\mathbb{R}^{p}}\tilde{b}_{\eta}(x,\pi,y)\mathrm{d}\mu(y)-\alpha\nabla
U(x)$. We recall that using (120) there exists $C\geq 0$ such that for any
$x_{1},x_{2}\in\mathbb{R}^{d}$ and
$\pi_{1},\pi_{2}\in\mathscr{P}_{1}(\mathbb{R}^{d})$ we have

$\textstyle{\left\|b(x_{1},\pi_{1})-b(x_{2},\pi_{2})\right\|\leq
C\left\\{\left\|x_{1}-x_{2}\right\|+\int_{\mathbb{R}^{p}}\left|\pi_{1}[\mathrm{k}(\cdot,y)]-\pi_{2}[\mathrm{k}(\cdot,y)]\right|\mathrm{d}\mu(y)\right\\}.}$
(137)

Using this result, we have for any $t\geq 0$

$\displaystyle\textstyle{\mathbb{E}[\sup_{s\in\left[0,t\right]}\|\mathbf{X}_{s}^{\star}-\mathbf{X}_{s}^{1,N}\|]\leq\int_{0}^{t}\mathbb{E}[\|b(\mathbf{X}_{s}^{\star},\bm{\lambda}_{s}^{\star})-b(\mathbf{X}_{s}^{1,N},\bm{\lambda}^{N}_{s})\|]\mathrm{d}s}$
(138) $\displaystyle\quad\textstyle{\leq
C\int_{0}^{t}\mathbb{E}[\sup_{u\in\left[0,s\right]}\|\mathbf{X}_{u}^{\star}-\mathbf{X}_{u}^{1,N}\|]\mathrm{d}s}$
(139)
$\displaystyle\quad\textstyle{+C\int_{0}^{t}\int_{\mathbb{R}^{p}}\mathbb{E}[|(1/N)\sum_{k=1}^{N}\mathrm{k}(\mathbf{X}_{s}^{k,N},y)-\bm{\lambda}_{s}^{\star}[\mathrm{k}(\cdot,y)]|]\mathrm{d}\mu(y)\mathrm{d}s}.$
(140)

Now, consider $N$ independent copies of the nonlinear process
$\mathbf{X}_{s}^{\star}$, $\\{(\mathbf{X}_{s}^{k,\star})_{t\geq
0}\\}_{k=1}^{N}$. We can bound the second term in the above with

$\displaystyle\mathbb{E}[|(1/N)\sum_{k=1}^{N}\mathrm{k}(\mathbf{X}_{s}^{k,N},y)-\bm{\lambda}_{s}^{\star}[\mathrm{k}(\cdot,y)]|]\leq(1/N)\mathbb{E}[|\sum_{k=1}^{N}\left\\{\mathrm{k}(\mathbf{X}_{s}^{k,N},y)-\mathrm{k}(\mathbf{X}_{s}^{k,\star},y)\right\\}|]$
(141)
$\displaystyle\qquad\qquad+(1/N)\mathbb{E}[|\sum_{k=1}^{N}\left\\{\mathrm{k}(\mathbf{X}_{s}^{k,\star},y)-\bm{\lambda}_{s}^{\star}[\mathrm{k}(\cdot,y)]\right\\}|]$
(142)
$\displaystyle\qquad\qquad\leq\mathtt{M}\mathbb{E}[\sup_{u\in\left[0,s\right]}\|\mathbf{X}_{u}^{\star}-\mathbf{X}_{u}^{1,N}\|]+(1/N)\mathbb{E}[|\sum_{k=1}^{N}\left\\{\mathrm{k}(\mathbf{X}_{s}^{k,\star},y)-\bm{\lambda}_{s}^{\star}[\mathrm{k}(\cdot,y)]\right\\}|],$
(143)

where we used the Lipschitz continuity of $\mathrm{k}$ and the fact that
$\\{(\mathbf{X}_{t}^{k,N})_{t\geq 0}\\}_{k=1}^{N}$ is exchangeable to obtain
the last inequality. Plugging the above into (138) and using Jensen’s
inequality we obtain

$\displaystyle\textstyle{\mathbb{E}[\sup_{s\in\left[0,t\right]}\|\mathbf{X}_{s}^{\star}-\mathbf{X}_{s}^{1,N}\|]\leq}$
(144) $\displaystyle\quad\textstyle{\leq
C(1+\mathtt{M})\int_{0}^{t}\mathbb{E}[\sup_{u\in\left[0,s\right]}\|\mathbf{X}_{u}^{\star}-\mathbf{X}_{u}^{1,N}\|]\mathrm{d}s}$
(145)
$\displaystyle\quad\textstyle{+C(1+\mathtt{M})(1/N)\int_{0}^{t}\int_{\mathbb{R}^{p}}\mathbb{E}^{1/2}\left[\sum_{k=1}^{N}\left(\mathrm{k}(\mathbf{X}_{s}^{k,\star},y)-\bm{\lambda}_{s}^{\star}[\mathrm{k}(\cdot,y)]\right)^{2}\right]\mathrm{d}\mu(y)\mathrm{d}s}$
(146) $\displaystyle\quad\textstyle{\leq
C(1+\mathtt{M})\\{\int_{0}^{t}\mathbb{E}[\sup_{u\in\left[0,s\right]}\|\mathbf{X}_{u}^{\star}-\mathbf{X}_{u}^{1,N}\|]\mathrm{d}s+\sqrt{2}\mathtt{M}tN^{-1/2}\\}.}$
(147)

Using Grönwall’s lemma we get that for any $T\geq 0$ there exists $C_{T}\geq
0$ such that for any $N\in\mathbb{N}$

$\textstyle{\mathbb{E}[\sup_{t\in\left[0,T\right]}\|\mathbf{X}_{t}^{\star}-\mathbf{X}_{t}^{1,N}\|]\leq
C_{T}N^{-1/2},}$ (148)

which concludes the proof.

###  B.4 Proof of Proposition 8

First, we recall that $b$ is given for any $x\in\mathbb{R}^{d}$,
$\nu\in\mathscr{P}(\mathbb{R}^{d})$ and $y\in\mathbb{R}^{p}$ by

$\textstyle{b(x,\nu)=\int_{\mathbb{R}^{p}}\tilde{b}_{\eta}(x,\nu,y)\mathrm{d}\mu(y)-\alpha\nabla
U(x),\quad\tilde{b}_{\eta}(x,\nu,y)=-\nabla_{1}\mathrm{k}(x,y)/(\nu[\mathrm{k}(\cdot,y)]+\eta).}$
(149)

Using 1 and (120), there exist $C_{0},C_{1}\geq 0$ such that for any
$x_{1},x_{2}\in\mathbb{R}^{d}$,
$\pi_{1},\pi_{2}\in\mathscr{P}(\mathbb{R}^{d})$ and $y\in\mathbb{R}^{p}$,

$\textstyle{\|\tilde{b}_{\eta}(x_{1},\pi_{1},y)\|\leq
C_{0},\qquad\left\|b(x_{1},\pi_{1})-b(x_{2},\pi_{2})\right\|\leq
C_{1}\left(\left\|x_{1}-x_{2}\right\|+\mathbf{W}_{1}(\pi_{1},\pi_{2})\right).}$
(150)

In addition, note that for any
$x_{1}^{1:N},x_{2}^{1:N}\in(\mathbb{R}^{d})^{N}$ we have for any
$i\in\\{1,2\\}$

$\textstyle{\mathbf{W}_{1}(\mu_{1}^{N},\mu_{2}^{N})\leq(1/N)\sum_{k=1}^{N}\|x_{1}^{k,N}-x_{2}^{k,N}\|,\qquad\mu_{i}^{N}=(1/N)\sum_{k=1}^{N}\updelta_{x^{k,N}}.}$
(151)

Let $N\in\mathbb{N}$ and denote
$B_{N}:(\mathbb{R}^{d})^{N}\to(\mathbb{R}^{d})^{N}$ given for any
$x^{1:N}\in(\mathbb{R}^{d})^{N}$ by

$\textstyle{B_{N}(x^{1:N})=\\{b(x^{k,n},\mu^{N})\\}_{k\in\\{1,\dots,N\\}},\qquad\mu^{N}=(1/N)\sum_{k=1}^{N}\updelta_{x^{k,N}}.}$
(152)

Therefore, using (150), (151) and the Cauchy-Schwarz inequality we have for
any $x^{1:N}_{1},x_{2}^{1:N}\in(\mathbb{R}^{d})^{N}$

$\displaystyle\textstyle{\|B_{N}(x^{1:N}_{1})-B_{N}(x^{1:N}_{2})\|\leq
C_{1}(\|x_{1}^{1:N}-x_{2}^{1:N}\|+\sum_{k=1}^{N}\|x_{1}^{k,N}-x_{2}^{k,N}\|)}$
(153) $\displaystyle\qquad\qquad\textstyle{\leq
C_{1}(1+N^{1/2})\|x_{1}^{1:N}-x_{2}^{1:N}\|.}$ (154)

Let
$R=\max(2C_{0}N^{1/2}/(\alpha\mathtt{m}),(2\mathtt{c}N/(\alpha\mathtt{m}))^{1/2})$
with $\mathtt{m}$ and $\mathtt{c}$ as in 3-(b). Using (150) and the Cauchy-
Schwarz inequality we have for any
$x^{1:N}_{1},x_{2}^{1:N}\in(\mathbb{R}^{d})^{N}$ with
$\|x_{1}^{1:N}-x_{2}^{1:N}\|\geq R$

$\displaystyle\langle
B_{N}(x^{1:N}_{1})-B_{N}(x^{1:N}_{2}),x_{1}^{1:N}-x_{2}^{1:N}\rangle$ (155)
$\displaystyle\qquad\textstyle{\leq-\alpha\mathtt{m}\|x_{1}^{1:N}-x_{2}^{1:N}\|^{2}+\alpha\mathtt{c}N+C_{0}\sum_{k=1}^{N}\|x_{1}^{k,N}-x_{2}^{k,N}\|}$
(156)
$\displaystyle\qquad\textstyle{\leq-\alpha\mathtt{m}\|x_{1}^{1:N}-x_{2}^{1:N}\|^{2}+\alpha\mathtt{c}N+C_{0}N^{1/2}\|x_{1}^{1:N}-x_{2}^{1:N}\|}$
(157)
$\displaystyle\qquad\textstyle{\leq-(\alpha\mathtt{m}/2)\|x_{1}^{1:N}-x_{2}^{1:N}\|^{2}.}$
(158)

We conclude upon combining (153), (155) and [31, Corollary 2].

###  B.5 Proof of Proposition 9

First, we start with the following lemma.

######  Lemma 20.

Assume 1, 3-(a) and 3-(b). Let
$\nu_{1},\nu_{2}\in\mathscr{P}_{1}(\mathbb{R}^{d})$. Denote
$(\mathbf{X}_{1,t}^{\star},\mathbf{X}_{2,t}^{\star})_{t\geq 0}$ the stochastic
process such that $(\mathbf{X}_{1,0}^{\star},\mathbf{X}_{2,0}^{\star})$ is the
optimal coupling between $\nu_{1}$ and $\nu_{2}$ w.r.t. to $\mathbf{W}_{1}$.
Then for any $T\geq 0$, there exists $C_{T}\geq 0$ such that for any
$t\in\left[0,T\right]$

$\mathbf{W}_{1}(\bm{\lambda}_{t}^{\star}(\nu_{1}),\bm{\lambda}_{t}^{\star}(\nu_{2}))\leq
C_{T}\mathbf{W}_{1}(\nu_{1},\nu_{2}),$ (159)

where for any $\nu\in\mathscr{P}_{1}(\mathbb{R}^{d})$,
$\bm{\lambda}^{\star}(\nu)$ is the distribution of
$(\mathbf{X}_{t}^{\star})_{t\geq 0}$ with initial distribution $\nu$.

###### Proof.

Let $T\geq 0$. Using (120), we have for any $t\in\left[0,T\right]$

$\textstyle{\mathbb{E}[\|\mathbf{X}_{1,t}^{\star}-\mathbf{X}_{2,t}^{\star}\|]\leq\mathbf{W}_{1}(\nu_{1},\nu_{2})+C\int_{0}^{t}\\{\mathbb{E}[\|\mathbf{X}_{1,s}^{\star}-\mathbf{X}_{2,s}^{\star}\|]+\mathbf{W}_{1}(\bm{\lambda}_{s}^{\star}(\nu_{1}),\bm{\lambda}_{s}^{\star}(\nu_{2}))\\}\mathrm{d}s,}$
(160)

where $C=(\mathtt{M}/\eta)(1+(1/\eta))(1+\alpha\mathtt{L})(1+\mathtt{M})$.
Using Grönwall’s lemma we have for any $t\in\left[0,T\right]$

$\textstyle{\mathbb{E}[\|\mathbf{X}_{1,t}^{\star}-\mathbf{X}_{2,t}^{\star}\|]\leq
C(1+T)e^{C(1+T)}\\{\mathbf{W}_{1}(\nu_{1},\nu_{2})+C\int_{0}^{t}\mathbf{W}_{1}(\bm{\lambda}_{s}^{\star}(\nu_{1}),\bm{\lambda}_{s}^{\star}(\nu_{2}))\mathrm{d}s\\}.}$
(161)

Using that for any $t\geq 0$,
$\mathbf{W}_{1}(\bm{\lambda}_{t}^{\star}(\nu_{1}),\bm{\lambda}_{t}^{\star}(\nu_{2}))\leq\mathbb{E}\left[\left\|\mathbf{X}_{1,t}^{\star}-\mathbf{X}_{2,t}^{\star}\right\|\right]$
and Grönwall’s lemma we get that there exists $C_{T}\geq 0$ such that for any
$t\in\left[0,T\right]$

$\mathbf{W}_{1}(\bm{\lambda}_{t}^{\star}(\nu_{1}),\bm{\lambda}_{t}^{\star}(\nu_{2}))\leq
C_{T}\mathbf{W}_{1}(\nu_{1},\nu_{2}),$ (162)

which concludes the proof. ∎

###### Proof of Proposition 9.

Let $\alpha,\eta>0$. First we show that $\\{\pi^{1,N}\,:N\in\mathbb{N}\\}$ is
relatively compact in $\mathscr{P}_{1}(\mathbb{R}^{d})$. Let $N\in\mathbb{N}$
and assume that $\mathbf{X}_{0}^{1:N}=0$. Using Itô’s formula we have that
$(\mathbf{M}_{t}^{N})_{t\geq 0}$ is a $\mathcal{F}_{t}^{N}$-martingale where
for any $t\geq 0$

$\textstyle{\mathbf{M}_{t}^{N}=(1/2)\|\mathbf{X}_{t}^{1,N}\|^{2}-\int_{0}^{t}\\{\langle\mathbf{X}_{s}^{1,N},b(\mathbf{X}_{s}^{1,N},\bm{\lambda}^{N}_{s})\rangle+\alpha\\}\mathrm{d}s.}$
(163)

and $(\mathcal{F}_{t}^{N})_{t\geq 0}$ is the filtration associated with
$\\{(\mathbf{B}_{t}^{k})_{t\geq 0}\\}_{k=1}^{N}$ and
$\\{(\mathbf{X}_{t}^{k,N})_{t\geq 0}\\}_{k=1}^{N}$ is given in (24). Note that
using 3-(b) with $x_{1}=x$ and $x_{2}=0$, we have for any $x\in\mathbb{R}^{d}$

$\langle\nabla U(x),x\rangle\geq\langle\nabla
U(0),x\rangle+\mathtt{m}\|x\|^{2}-\mathtt{c}.$ (164)

Therefore, using this result, (25) and the Cauchy-Schwarz inequality we obtain
that for any $t\geq 0$

$\displaystyle\textstyle{(1/2)\mathbb{E}[\|\mathbf{X}_{t}^{1,N}\|^{2}]-(1/2)\mathbb{E}[\|\mathbf{X}_{s}^{1,N}\|^{2}]}$
(165)
$\displaystyle\qquad\textstyle{\leq\int_{s}^{t}\\{\int_{\mathbb{R}^{d}}\mathbb{E}[\|\tilde{b}_{\eta}(\mathbf{X}_{u}^{1,N},\bm{\lambda}^{N}_{u},y)\|\|\mathbf{X}_{u}^{1,N}\|]\mathrm{d}\mu(y)\mathrm{d}u\\}}$
(166)
$\displaystyle\qquad\textstyle{+\int_{s}^{t}\\{-\alpha\mathtt{m}\mathbb{E}[\|\mathbf{X}_{u}^{1,N}\|^{2}]+\alpha\|\nabla
U(0)\|\mathbb{E}[\|\mathbf{X}_{u}^{1,N}\|]+\alpha\mathtt{c}+\alpha\\}\mathrm{d}u.}$
(167)

Let $\mathscr{V}_{t}^{N}=\mathbb{E}[\|\mathbf{X}_{t}^{1,N}\|^{2}]$. Using that
for any $x\in\mathbb{R}^{d}$ and $\pi\in\mathscr{P}(\mathbb{R}^{d})$ we have
$\|\tilde{b}_{\eta}(x,\pi)\|\leq\mathtt{M}/\eta$, we get that

$\displaystyle\mathrm{d}\mathscr{V}_{t}^{N}/\mathrm{d}t$
$\displaystyle\leq(2\mathtt{M}/\eta+\alpha\|\nabla
U(0)\|)(\mathscr{V}_{t}^{N})^{1/2}+\alpha+\alpha\mathtt{c}-\alpha\mathtt{m}\mathscr{V}_{t}^{N}$
(168) $\displaystyle\leq 2(2\mathtt{M}/\eta+\alpha\|\nabla
U(0)\|)^{2}/(\alpha\mathtt{m})+\alpha+\alpha\mathtt{c}-(\alpha\mathtt{m}/2)\mathscr{V}_{t}^{N}.$
(169)

Hence, for any $t\geq 0$ and any $N\in\mathbb{N}$ we get that
$\mathscr{V}_{t}^{N}\leq C$ with

$C=4\\{(2\mathtt{M}/\eta+\alpha\|\nabla
U(0)\|)^{2}/(\alpha\mathtt{m})+\alpha+\mathtt{c}\\}^{2}/(\alpha\mathtt{m}).$
(170)

Therefore, letting $t\to+\infty$ we get that for any $N\in\mathbb{N}$,
$\int_{\mathbb{R}^{d}}\left\|\tilde{x}\right\|^{2}\mathrm{d}\pi^{1,N}(\tilde{x})\leq
C$. Hence, $\\{\pi^{1,N}\,:N\in\mathbb{N}\\}$ is relatively compact in
$\mathscr{P}_{1}(\mathbb{R}^{d})$ using [2, Proposition 7.1.5].

Let $\pi^{\star}$ be a cluster point of $\\{\pi^{1,N}\,:N\in\mathbb{N}\\}$.
Let $(N_{k})_{k\in\mathbb{N}}$ be an increasing sequence such that
$\lim_{k\to+\infty}\mathbf{W}_{1}(\pi^{1,N_{k}},\pi^{\star})=0$. We have that
for any $t\geq 0$

$\displaystyle\mathbf{W}_{1}(\pi^{\star},\pi_{\alpha,\eta}^{\star})\leq\mathbf{W}_{1}(\pi^{\star},\pi^{1,N_{k}})$
(171)
$\displaystyle\qquad\qquad+\mathbf{W}_{1}(\pi^{1,N_{k}},\bm{\lambda}_{t}^{1,N_{k}}(\pi^{N_{k}}))+\mathbf{W}_{1}(\bm{\lambda}_{t}^{1,N_{k}}(\pi^{N_{k}}),\bm{\lambda}_{t}^{\star}(\pi^{1,N_{k}}))$
(172)
$\displaystyle\qquad\qquad+\mathbf{W}_{1}(\bm{\lambda}_{t}^{\star}(\pi^{1,N_{k}}),\bm{\lambda}_{t}^{\star}(\pi^{\star}))+\mathbf{W}_{1}(\bm{\lambda}_{t}^{\star}(\pi^{\star}),\pi_{\alpha,\eta}^{\star}).$
(173)

We now control each of these terms. Let $\varepsilon>0$ and set $t\geq 0$ such
that, using Proposition 6,
$\mathbf{W}_{1}(\bm{\lambda}_{t}^{\star}(\pi^{\star}),\pi_{\alpha,\eta}^{\star})\leq\varepsilon$.
Using Lemma 20, there exists $k_{0}\in\mathbb{N}$ such that for any $k\geq
k_{0}$ we have
$\mathbf{W}_{1}(\bm{\lambda}_{t}^{\star}(\pi^{1,N_{k}}),\bm{\lambda}_{t}^{\star}(\pi^{\star}))\leq\varepsilon$.
Using Proposition 7, there exists $k_{1}\in\mathbb{N}$ such that for any
$k\geq k_{1}$ we have that
$\mathbf{W}_{1}(\bm{\lambda}_{t}^{1,N_{k}}(\pi^{N_{k}}),\bm{\lambda}_{t}^{\star}(\pi^{1,N_{k}}))\leq\varepsilon$.
Since $\pi^{N_{k}}$ is invariant for $(\mathbf{X}_{t}^{1:N})_{t\geq 0}$ we get
that
$\mathbf{W}_{1}(\pi^{1,N_{k}},\bm{\lambda}_{t}^{1,N_{k}}(\pi^{N_{k}}))=0$.
Finally, there exists $k_{2}\in\mathbb{N}$ such that for any $k\geq k_{3}$ we
have $\mathbf{W}_{1}(\pi^{\star},\pi^{1,N_{k}})\leq\varepsilon$. Combining
these results in (171), we get that
$\mathbf{W}_{1}(\pi^{\star},\pi_{\alpha,\eta}^{\star})\leq\varepsilon$.
Therefore, since $\varepsilon>0$ is arbitrary, we have that
$\pi^{\star}=\pi_{\alpha,\eta}^{\star}$, which concludes the proof. ∎

##  Appendix C Proofs of Section 4

###  C.1 Basics on Lions derivatives

We briefly recall here the definition of Lions derivative of functionals
defined over $\mathscr{P}_{2}(\mathbb{R}^{d})$. In this section, we fix an
underlying probability space $(\Omega,\mathcal{F},\mathbb{P})$ and denote by
$\mathrm{L}^{2}(\Omega,\mathcal{F},\mathbb{P};\mathbb{R}^{d})$ the space of
$\mathbb{R}^{d}$-valued, $\mathcal{F}$-measurable random variables with finite
second moment. We refer to [20, 18, 48, 14, 8] for a thorough exposition of
Lions derivatives with applications to mean-field games. The following
definitions can be found in [8, 48]. We start by introducing the notion of
$L$-differentiability.

######  Definition 21.

Let $\mathscr{M}:\ \mathscr{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ and
$\bar{\mathscr{M}}:\mathrm{L}^{2}(\Omega,\mathcal{F},\mathbb{P};\mathbb{R}^{d})\to\mathbb{R}$
such that for any
$X\in\mathrm{L}^{2}(\Omega,\mathcal{F},\mathbb{P};\mathbb{R}^{d})$,
$\bar{\mathscr{M}}(X)=\mathscr{M}(\mathcal{L}(X))$, where $\mathcal{L}(X)$ is
the distribution of $X$. $\bar{\mathscr{M}}$ is called a lifted version of
$\mathscr{M}$. We say that $\mathscr{M}$ is $L$-differentiable at
$\nu_{0}\in\mathscr{P}_{2}(\mathbb{R}^{d})$ if there exists a random variable
$X_{0}$ such that $\mathcal{L}(X_{0})=\nu_{0}$ and $\bar{\mathscr{M}}$ is
Fréchet differentiable at $X_{0}$, i.e. there exists
$Y\in\mathrm{L}^{2}(\Omega,\mathcal{F},\mathbb{P},\mathbb{R}^{d})$ such that

$\lim_{\|X-X_{0}\|_{2}\to
0}\left\|\bar{\mathscr{M}}(X)-\bar{\mathscr{M}}(X_{0})-\langle
Y,X-X_{0}\rangle\right\|/\|X-X_{0}\|_{2}=0.$ (174)

If $\mathscr{M}$ is $L$-differentiable at $\nu_{0}$ for all
$\nu_{0}\in\mathscr{P}_{2}(\mathbb{R}^{d})$ then we say that $\mathscr{M}$ is
$L$-differentiable.

If $\bar{\mathscr{M}}$ is differentiable at $X_{0}$ then there exists $\xi:\
\mathbb{R}^{d}\to\mathbb{R}^{d}$ measurable and uniquely defined almost
everywhere such that $Y=\xi(X_{0})$. Note that $\xi$ does not depend on
$X_{0}$ except via $\mathcal{L}(X_{0})$. Hence $\xi$ does not depend on the
choice of $X_{0}$.

######  Definition 22.

Let $\mathscr{M}:\ \mathscr{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ be
$L$-differentiable at $\mu$. Then we write
$\mathrm{D}^{L}\mathscr{M}(\mu)=\xi$. Moreover we have
$\mathrm{D}^{L}\mathscr{M}:\
\mathscr{P}_{2}(\mathbb{R}^{d})\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ given by
$\mathrm{D}^{L}\mathscr{M}(\mu,z):=\mathrm{D}^{L}\mathscr{M}(\mu)(z)$.

We emphasize that $\mathrm{D}^{L}$ is well-defined since $\xi$ only depends on
$\mathcal{L}(X_{0})$. A mapping $\mathscr{M}:\
\mathscr{P}(\mathbb{R}^{d})\to\mathbb{R}^{p}$ is said to be $L$-differentiable
if for any $i\in\\{1,\dots,p\\}$, $\mathscr{M}_{i}$ is $L$-differentiable. We
present here a simple example of $L$-derivative which is relevant for the
proof of Lemma 26 below.

######  Example 23.

Consider $\mathscr{M}:\
\mathscr{P}(\mathbb{R}^{d})\times\mathbb{R}^{p}\to\mathbb{R}$ given by
$\mathscr{M}(\nu,y)=\frac{1}{\nu\left[\mathrm{k}(,y)\right]+\eta}$ with
$\mathrm{k}$ satisfying 1 and $\eta>0$. We have
$\bar{\mathscr{M}}(X,y)=\frac{1}{\mathbb{E}\left[\mathrm{k}(X,y)\right]+\eta}$
with $X\sim\nu$ and we have for any
$H\in\mathrm{L}^{2}(\Omega,\mathbb{R}^{d})$

$\displaystyle\bar{\mathscr{M}}(X+H)-\bar{\mathscr{M}}(X)-\left\langle\frac{\nabla_{1}\mathrm{k}(X,y)}{\mathbb{E}\left[\mathrm{k}(X,y)\right]+\eta},H\right\rangle$
(175)
$\displaystyle\qquad\qquad=\frac{1}{\mathbb{E}\left[\mathrm{k}(X+H,y)\right]+\eta}-\frac{1}{\mathbb{E}\left[\mathrm{k}(X,y)\right]+\eta}-\left\langle\frac{\nabla_{1}\mathrm{k}(X,y)}{\mathbb{E}\left[\mathrm{k}(X,y)\right]+\eta},H\right\rangle$
(176)
$\displaystyle\qquad\qquad=\frac{\mathbb{E}\left[\mathrm{k}(X,y)\right]-\mathbb{E}\left[\mathrm{k}(X+H,y)\right]-\langle\nabla_{1}\mathrm{k}(X,y),H\rangle}{\left(\mathbb{E}\left[\mathrm{k}(X,y)\right]+\eta\right)\left(\mathbb{E}\left[\mathrm{k}(X+H,y)\right]+\eta\right)}.$
(177)

Now, since $\mathbb{E}\left[\mathrm{k}(X+H,y)\right]+\eta\geq\eta>0$ and
$\mathbb{E}\left[\mathrm{k}(X,y)\right]+\eta\geq\eta>0$ and, as shown in [8,
Example 2] the $L$-derivative of
$\mathcal{Q}(\nu,y)=\nu\left[\mathrm{k}(,y)\right]$ is
$\mathrm{D}^{L}\mathcal{Q}(\nu,y)(x)=\nabla_{1}\mathrm{k}(\cdot,y)(x)$, we
have

$\mathrm{D}^{L}\mathscr{M}(\nu,y)(x)=-\frac{\nabla_{1}\mathrm{k}(x,y)}{\nu\left[\mathrm{k}(\cdot,y)\right]+\eta}.$
(178)

Similarly to Definition 22, we introduce the higher-order Lions derivatives
(see, e.g., [8, Appendix A] or [48, Appendix A.2]).

######  Definition 24.

Let $\mathscr{M}:\ \mathscr{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ be
$L$-differentiable. For any $x\in\mathbb{R}^{d}$ and $i\in\\{1,\dots,d\\}$,
define $g_{x}^{i}:\ \mathscr{P}(\mathbb{R}^{d})\to\mathbb{R}$ such that for
any $x\in\mathbb{R}^{d}$, $i\in\\{1,\dots,d\\}$ and
$\mu\in\mathscr{P}_{2}(\mathbb{R}^{d})$ we have
$g_{x}^{i}(\mu)=\mathrm{D}^{L}\mathscr{M}(\mu)(x)_{i}$. If for any
$x\in\mathbb{R}^{d}$ and $i\in\\{1,\dots,d\\}$, $g_{x}^{i}$ is
$L$-differentiable then we say that $\mathscr{M}$ is twice $L$-differentiable
and we define $(\mathrm{D}^{L})^{2}\mathscr{M}:\
\mathscr{P}_{2}(\mathbb{R}^{d})\times\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}^{d}\times\mathbb{R}^{d}$
given for any $\mu\in\mathscr{P}_{2}(\mathbb{R}^{d})$ and
$x_{0},x_{1}\in\mathbb{R}^{d}$ by

$(\mathrm{D}^{L})^{2}\mathscr{M}(\mu)(x_{0},x_{1})=(\mathrm{D}^{L}(g_{x_{0}}^{i})(\mu)(x_{1}))_{i\in\\{1,\dots,d\\}}.$
(179)

Finally, we define the class
$\mathrm{C}^{2,(2,1)}(\mathbb{R}^{d}\times\mathscr{P}_{2}(\mathbb{R}^{d}),\mathbb{R}^{p})$
used to establish the main strong approximation results in [8]. An equivalent
definition for $\mathbb{R}$-valued functionals $\mathscr{M}$ is given in [8,
Definition A.2].

######  Definition 25.

Let $\mathscr{M}:\
\mathbb{R}^{d}\times\mathscr{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}^{p}$. We say
that
$\mathscr{M}\in\mathrm{C}^{2,(2,1)}(\mathbb{R}^{d}\times\mathscr{P}_{2}(\mathbb{R}^{d}),\mathbb{R}^{p})$
if the following hold:

  1. (a)

For any $\mu\in\mathscr{P}_{2}(\mathbb{R}^{d})$, $\mathscr{M}(\cdot,\mu)$ is
twice differentiable.

  2. (b)

For any $x\in\mathbb{R}^{d}$, $\mathscr{M}(x,\cdot)$ is $L$-differentiable and
$(x,\mu,y)\mapsto\mathrm{D}^{L}\mathscr{M}(x,\mu)(y)\in\mathrm{C}(\mathbb{R}^{d}\times\mathscr{P}_{2}(\mathbb{R}^{d})\times\mathbb{R}^{d},\mathbb{R}^{d\times
p})$.

  3. (c)

For any $x\in\mathbb{R}^{d}$, $\mathscr{M}(x,\cdot)$ is twice
$L$-differentiable and we have that
$(x,\mu,y_{0},y_{1})\mapsto(\mathrm{D}^{L})^{2}\mathscr{M}(x,\mu)(y_{0},y_{1})\in\mathrm{C}(\mathbb{R}^{d}\times\mathscr{P}_{2}(\mathbb{R}^{d})\times\mathbb{R}^{d}\times\mathbb{R}^{d},\mathbb{R}^{d\times
d\times p})$.

  4. (d)

For any $\mu\in\mathscr{P}_{2}(\mathbb{R}^{d})$ and $y\in\mathbb{R}^{d}$,
$x\mapsto\mathrm{D}^{L}\mathscr{M}(x,\mu)(y)$ is differentiable and
$(x,\mu,y)\mapsto\mathrm{d}_{x}\mathrm{D}^{L}\mathscr{M}(x,\mu)(y)\in\mathrm{C}(\mathbb{R}^{d}\times\mathscr{P}_{2}(\mathbb{R}^{d})\times\mathbb{R}^{d},\mathbb{R}^{d\times
d\times p})$.

###  C.2 Proof of Proposition 10

In this section, we let $\alpha,\eta>0$ and verify that the drift in (22)
satisfies the conditions of [8, Theorem 2.2]. For completeness we report the
conditions of [8, Theorem 2.2] below. We denote by
$b(x,\nu):\mathbb{R}^{d}\times\mathscr{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}^{d}$
and
$\sigma(x,\nu):\mathbb{R}^{d}\times\mathscr{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}^{d\times
d}$ the drift and diffusion coefficient of the MKVSDE, respectively. In the
following, $\left\|s\right\|$ denotes the standard Euclidean norm if $s$ is a
vector and the Hilbert-Schmidt norm
$\left\|s\right\|=(\sum_{ij}s_{ij}^{2})^{1/2}$ if $s$ is a matrix.

_Conditions on the drift coefficient_

$(\mathbf{A}_{b}^{1})$

There exists constants $L_{b}^{1},\alpha_{1}>0$ such that

$\displaystyle\left\|b(x_{1},\nu)-b(x_{2},\nu)\right\|$ $\displaystyle\leq
L_{b}^{1}\left(1+\left\|x_{1}\right\|^{\alpha_{1}}+\left\|x_{2}\right\|^{\alpha_{1}}\right)\left\|x_{1}-x_{2}\right\|$
(180) $\displaystyle\langle x_{1}-x_{2},b(x_{1},\nu)-b(x_{2},\nu)\rangle$
$\displaystyle\leq L_{b}^{1}\left\|x_{1}-x_{2}\right\|^{2}$ (181)
$\displaystyle\left\|b(x,\nu_{1})-b(x,\nu_{2})\right\|$ $\displaystyle\leq
L_{b}^{1}\mathbf{W}_{2}(\nu_{1},\nu_{2})$ (182)

$(\mathbf{A}_{b}^{2})$

Let
$b_{i}\in\mathrm{C}^{2,(2,1)}(\mathbb{R}^{d}\times\mathscr{P}_{2}(\mathbb{R}^{d}),\mathbb{R}^{d})$,
where $b_{i}$ denotes the $i$-th component of $b$. There exists constants
$L_{b}^{2},\alpha_{2}>0$ such that for any $i=1,\dots,d$

$\displaystyle\left\|\nabla^{2}b_{i}(x,\nu)(z)\right\|\vee$
$\displaystyle\left\|\nabla\\{\mathrm{D}^{L}b_{i}(x,\nu)(\cdot)\\}(z)\right\|\vee\left\|(\mathrm{D}^{L})^{2}b_{i}(x,\nu)(z,z)\right\|$
(183) $\displaystyle\leq
L_{b}^{2}\left(1+\left\|x\right\|^{1+\alpha_{2}}+\left\|z\right\|^{1+\alpha_{2}}+\nu(|\cdot|^{2})^{(1+\alpha_{2})/2}\right)$
(184)

$(\mathbf{A}_{b}^{3})$

There exists constants $L_{b}^{3},\alpha_{3}>0$ such that

$\displaystyle\left\|\nabla b(x_{1},\nu_{1})-\nabla
b(x_{2},\nu_{2})\right\|\leq$ $\displaystyle
L_{b}^{3}\left(\left\|x_{1}-x_{2}\right\|+\mathbf{W}_{2}(\nu_{1},\nu_{2})\right)$
(185)
$\displaystyle\times\left(1+\left\|x_{1}\right\|^{\alpha_{3}}+\left\|x_{2}\right\|^{\alpha_{3}}+\right.$
$\displaystyle\left.\nu_{1}(|\cdot|^{2})^{\alpha_{3}/2}+\nu_{2}(|\cdot|^{2})^{\alpha_{3}/2}\right)$
(186)
$\displaystyle\left\|\mathrm{D}^{L}b(x_{1},\nu_{1})(z_{1})-\mathrm{D}^{L}b(x_{2},\nu_{2})(z_{2})\right\|\leq$
(187) $\displaystyle
L_{b}^{3}\left(\left\|x_{1}-z_{1}\right\|+\left\|x_{2}-z_{2}\right\|+\right.$
$\displaystyle\left.\mathbf{W}_{2}(\nu_{1},\nu_{2})\right)$ (188)
$\displaystyle\times\left(1+\left\|x_{1}\right\|^{\alpha_{3}}+\left\|x_{2}\right\|^{\alpha_{3}}+\left\|z_{1}\right\|^{\alpha_{3}}\right.$
$\displaystyle\left.+\left\|z_{2}\right\|^{\alpha_{3}}+\nu_{1}(|\cdot|^{2})^{\alpha_{3}/2}+\nu_{2}(|\cdot|^{2})^{\alpha_{3}/2}\right.\left.\right)$
(189)

$(\mathbf{A}_{b}^{4})$

There exists a constant $L_{b}^{4}>0$ such that

$\displaystyle\left\|b(0,\nu)\right\|\leq L_{b}^{4}.$ (190)

_Conditions on the diffusion coefficient_

$(\mathbf{A}_{\sigma}^{1})$

There exists a constant $L_{\sigma}^{1}>0$ such that

$\displaystyle\left\|\sigma(x_{1},\nu_{1})-\sigma(x_{2},\nu_{2})\right\|\leq
L_{\sigma}^{1}\left(\left\|x_{1}-x_{2}\right\|+\mathbf{W}_{2}(\nu_{1},\nu_{2})\right)$
(191)

$(\mathbf{A}_{\sigma}^{2})$

Let
$\sigma_{ij}\in\mathrm{C}^{2,(2,1)}(\mathbb{R}^{d}\times\mathscr{P}_{2}(\mathbb{R}^{d}),\mathbb{R})$,
where $\sigma_{ij}$ denotes the $(i,j)$-th component of $\sigma$. There exists
a constant $L_{\sigma}^{2}>0$ such that for any $i,j=1,\dots,d$

$\displaystyle\left\|\mathrm{D}^{L}\sigma_{ij}(x,\nu)(z)\right\|+\left\|\nabla\\{\mathrm{D}^{L}\sigma_{ij}(x,\nu)(\cdot)\\}(z)\right\|+\left\|(\mathrm{D}^{L})^{2}\sigma_{ij}(x,\nu)(z,z)\right\|\leq
L_{\sigma}^{2}$ (192)

$(\mathbf{A}_{\sigma}^{3})$

There exists a constant $L_{\sigma}^{3}>0$ such that

$\displaystyle\left\|\nabla\sigma(x_{1},\nu_{1})-\nabla\sigma(x_{2},\nu_{2})\right\|\leq
L_{\sigma}^{3}\left(\left\|x_{1}-x_{2}\right\|+\mathbf{W}_{2}(\nu_{1},\nu_{2})\right)$
(193)
$\displaystyle\left\|\mathrm{D}^{L}\sigma(x_{1},\nu_{1})(z_{1})-\mathrm{D}^{L}\sigma(x_{2},\nu_{2})(z_{2})\right\|$
(194) $\displaystyle\qquad\leq
L_{\sigma}^{3}\left(\left\|x_{1}-z_{1}\right\|+\left\|x_{2}-z_{2}\right\|+\mathbf{W}_{2}(\nu_{1},\nu_{2})\right)$
(195)

$(\mathbf{A}_{\sigma}^{4})$

There exists a constant $L_{\sigma}^{4}>0$ such that

$\displaystyle\left\|\sigma(0,\nu)\right\|+\left\|\mathrm{D}^{L}\sigma(x_{1},\nu)(x_{2})\sigma(x_{2},\nu)\right\|\leq
L_{\sigma}^{4}.$ (196)

We start by computing the $L$-derivatives of $b$. We recall that $b:\
\mathbb{R}^{d}\times\mathscr{P}(\mathbb{R}^{d})\to\mathbb{R}^{d}$ is given for
any $x\in\mathbb{R}^{d}$ and $\nu\in\mathscr{P}(\mathbb{R}^{d})$ by

$\textstyle{b(x,\nu)=\int_{\mathbb{R}^{p}}\tilde{b}_{\eta}(x,\nu,y)\mathrm{d}\mu(y)-\alpha\nabla
U(x),\quad\tilde{b}_{\eta}(x,\nu,y)=-\nabla_{1}\mathrm{k}(x,y)/(\nu[\mathrm{k}(\cdot,y)]+\eta).}$
(197)

In all of our results, $\mu$ can be replaced by $\mu^{M}$.

######  Lemma 26.

Let $\alpha,\eta>0$ and assume 1 and 3. Then
$b\in\mathrm{C}^{2,(2,1)}(\mathbb{R}^{d}\times\mathscr{P}_{2}(\mathbb{R}^{d}),\mathbb{R}^{d})$.
In addition, we have for any $\nu\in\mathscr{P}_{2}(\mathbb{R}^{d})$,
$x_{0},x_{1},x_{2}\in\mathbb{R}^{d}$

  1. 1.

$\textstyle{\mathrm{d}_{x}^{2}b(x_{0},\nu)=\int_{\mathbb{R}^{p}}\mathrm{d}^{3}_{x}\mathrm{k}(x_{0},y)/(\nu[\mathrm{k}(\cdot,y)]+\eta)\mathrm{d}\mu(y)-\alpha\mathrm{d}^{3}_{x}U(x_{0}),}$

  2. 2.

$\textstyle{\mathrm{D}^{L}b(x_{0},\nu)(x_{1})=\int_{\mathbb{R}^{p}}\nabla_{1}\mathrm{k}(x_{0},y)\otimes\nabla_{1}\mathrm{k}(x_{1},y)/(\nu[\mathrm{k}(\cdot,y)]+\eta)\mathrm{d}\mu(y),}$

  3. 3.

$\textstyle{\mathrm{d}_{x}\mathrm{D}^{L}b(x_{0},\nu)(x_{1})=\int_{\mathbb{R}^{p}}\mathrm{d}_{x}^{2}\mathrm{k}(x_{0},y)\otimes\nabla_{1}\mathrm{k}(x_{1},y)/(\nu[\mathrm{k}(\cdot,y)]+\eta)\mathrm{d}\mu(y),}$

  4. 4.

$\textstyle{(\mathrm{D}^{L})^{2}b(x_{0},\nu)(x_{1},x_{2})=\int_{\mathbb{R}^{p}}\nabla_{1}\mathrm{k}(x_{0},y)\otimes\nabla_{1}\mathrm{k}(x_{1},y)\otimes\nabla_{1}\mathrm{k}(x_{2},y)/(\nu[\mathrm{k}(\cdot,y)]+\eta)\mathrm{d}\mu(y),}$

where $\otimes$ denotes the outer product.

###### Proof.

We prove each result separately.

  1. 1.

First, we have that for any $x\in\mathbb{R}^{d}$,
$\nu\in\mathscr{P}_{2}(\mathbb{R}^{d})$ and $y\in\mathbb{R}^{p}$

$\displaystyle\mathrm{d}_{x}\tilde{b}_{\eta}(x,\eta,y)=-\nabla_{1}^{2}\mathrm{k}(x,y)/(\nu[\mathrm{k}(\cdot,y)]+\eta),$
(198)
$\displaystyle\mathrm{d}_{x}^{2}\tilde{b}_{\eta}(x,\eta,y)=-\mathrm{d}_{x}^{3}\mathrm{k}(x,y)/(\nu[\mathrm{k}(\cdot,y)]+\eta).$
(199)

We have that for any $x\in\mathbb{R}^{d}$,
$\nu\in\mathscr{P}_{2}(\mathbb{R}^{d})$ and $y\in\mathbb{R}^{p}$,
$\|\mathrm{d}_{x}\tilde{b}_{\eta}(x,\eta,y)\|\leq\mathtt{M}/\eta$ and
$\|\mathrm{d}_{x}^{2}\tilde{b}_{\eta}(x,\eta,y)\|\leq\mathtt{M}/\eta$. Hence,
using the dominated convergence theorem,
$b\in\mathrm{C}^{2}(\mathbb{R}^{d},\mathbb{R}^{d})$ and we have that for any
$x_{0}\in\mathbb{R}^{d}$ and $\nu\in\mathscr{P}_{2}(\mathbb{R}^{d})$

$\textstyle{\mathrm{d}_{x}^{2}b(x_{0},\nu)=\int_{\mathbb{R}^{p}}-\mathrm{d}^{3}_{x}\mathrm{k}(x_{0},y)/(\nu[\mathrm{k}(\cdot,y)]+\eta)\mathrm{d}\mu(y)-\alpha\mathrm{d}^{3}_{x}U(x_{0}).}$
(200)

  2. 2.

Define $\bar{b}:\
\mathbb{R}^{d}\times\mathrm{L}^{2}(\Omega,\mathbb{R}^{d})\to\mathbb{R}^{d}$,
for any $x\in\mathbb{R}^{d}$ and $X\in\mathrm{L}^{2}(\Omega,\mathbb{R}^{d})$,
via

$\textstyle{\bar{b}(x,X)=\int_{\mathbb{R}^{p}}-\nabla_{1}\mathrm{k}(x,y)/(\mathbb{E}[\mathrm{k}(X,y)]+\eta)\mathrm{d}\mu(y)-\alpha\nabla
U(x),}$ (201)

so that for any $X\in\mathrm{L}^{2}(\Omega,\mathbb{R}^{d})$,
$\bar{b}(\cdot,X)=b(\cdot,\mathcal{L}(X))$. In addition, using Example 23, the
fact that $\mathrm{D}^{L}\mathscr{M}$ there in is bounded under 1 and the
dominated convergence theorem we have that for any
$H\in\mathrm{L}^{2}(\Omega,\mathbb{R}^{d})$ and $x\in\mathbb{R}^{d}$

$\textstyle{\underset{\|H\|\to
0}{\lim}\|\bar{b}(x,X+H)-\bar{b}(x,X)-\int_{\mathbb{R}^{p}}\frac{\nabla_{1}\mathrm{k}(x,y)\langle\nabla_{1}\mathrm{k}(X,y),H\rangle}{\mathbb{E}[\mathrm{k}(X,y)]+\eta}\mathrm{d}\mu(y)\|/\|H\|=0.}$
(202)

Therefore, for any $x_{0},x_{1}\in\mathbb{R}^{d}$ and
$\nu\in\mathscr{P}_{2}(\mathbb{R}^{d})$

$\textstyle{\mathrm{D}^{L}b(x_{0},\nu)(x_{1})=\int_{\mathbb{R}^{p}}-\nabla_{1}\mathrm{k}(x_{0},y)\otimes\nabla_{1}\mathrm{k}(x_{1},y)/(\nu[\mathrm{k}(\cdot,y)]+\eta)\mathrm{d}\mu(y).}$
(203)

Using the boundedness of $\nabla_{1}\mathrm{k}$ ensured by 1 and the dominated
convergence theorem, we have
$(x_{0},\nu,x_{1})\mapsto\mathrm{D}^{L}b(x_{0},\nu)(x_{1})\in\mathrm{C}(\mathbb{R}^{d}\times\mathscr{P}_{2}(\mathbb{R}^{d})\times\mathbb{R}^{d},\mathbb{R}^{d\times
d})$.

  3. 3.

For any $x_{1}\in\mathbb{R}^{d}$ and $\nu\in\mathscr{P}_{2}(\mathbb{R}^{d})$,
$x_{0}\mapsto\mathrm{D}^{L}b(x_{0},\nu)(x_{1})\in\mathrm{C}^{1}(\mathbb{R}^{d},\mathbb{R}^{d\times
d})$ and we have that for any $x_{0}\in\mathbb{R}^{d}$

$\textstyle{\mathrm{d}_{x}\mathrm{D}^{L}b(x_{0},\nu)(x_{1})=\int_{\mathbb{R}^{p}}-\mathrm{d}_{x}^{2}\mathrm{k}(x_{0},y)\otimes\nabla_{1}\mathrm{k}(x_{1},y)/(\nu[\mathrm{k}(\cdot,y)]+\eta)\mathrm{d}\mu(y).}$
(204)

  4. 4.

Finally, we define $\bar{g}:\
\mathbb{R}^{d}\times\mathrm{L}^{2}(\Omega,\mathbb{R}^{d})\times\mathbb{R}^{d}\to\mathbb{R}^{d\times
d}$ such that for any $x_{0},x_{1}\in\mathbb{R}^{d}$ and
$X\in\mathrm{L}^{2}(\Omega,\mathbb{R}^{d})$

$\textstyle{\bar{g}(x_{0},X,x_{1})=\int_{\mathbb{R}^{p}}-\nabla_{1}\mathrm{k}(x_{0},y)\otimes\nabla_{1}\mathrm{k}(x_{1},y)/(\mathbb{E}[\mathrm{k}(X,y)]+\eta)\mathrm{d}\mu(y).}$
(205)

For any $X\in\mathrm{L}^{2}(\Omega,\mathbb{R}^{d})$,
$\bar{g}(\cdot,X)=\mathrm{D}^{L}(\cdot,\mathcal{L}(X),\cdot)$. In addition,
using the boundedness ensured by 1 and dominated convergence theorem we have
that for any $H\in\mathrm{L}^{2}(\Omega,\mathbb{R}^{d})$ and
$x_{0},x_{1}\in\mathbb{R}^{d}$

$\displaystyle\textstyle{\left.\lim_{\|H\|\to
0}\|\bar{g}(x_{0},X+H,x_{1})-\bar{g}(x_{0},X,x_{1})\right.}$ (206)
$\displaystyle\qquad\textstyle{\left.-\int_{\mathbb{R}^{p}}\nabla_{1}\mathrm{k}(x_{0},y)\otimes\nabla_{1}\mathrm{k}(x_{1},y)\langle\nabla_{1}\mathrm{k}(X,y),H\rangle/(\mathbb{E}[\mathrm{k}(X,y)]+\eta)\mathrm{d}\mu(y)\|/\|H\|=0\right..}$
(207)

Therefore, for any $x_{0},x_{1},x_{2}\in\mathbb{R}^{d}$ and
$\nu\in\mathscr{P}_{2}(\mathbb{R}^{d})$

$\displaystyle\textstyle{(\mathrm{D}^{L})^{2}b(x_{0},\nu)(x_{1},x_{2})}$ (208)
$\displaystyle\qquad\textstyle{=\int_{\mathbb{R}^{p}}-\nabla_{1}\mathrm{k}(x_{0},y)\otimes\nabla_{1}\mathrm{k}(x_{1},y)\otimes\nabla_{1}\mathrm{k}(x_{2},y)/(\nu[\mathrm{k}(\cdot,y)]+\eta)\mathrm{d}\mu(y).}$
(209)

We conclude using again the boundedness of $\nabla_{1}\mathrm{k}$ ensured by 1
and the dominated convergence theorem to show that
$(x_{0},\mu,x_{1},x_{2})\mapsto(\mathrm{D}^{L})^{2}b(x_{0},\nu)(x_{1},x_{2})\in\mathrm{C}(\mathbb{R}^{d}\times\mathscr{P}_{2}(\mathbb{R}^{d})\times\mathbb{R}^{d}\times\mathbb{R}^{d},\mathbb{R}^{d\times
d\times d})$.

∎

###### Proof of Proposition 10.

Using Lemma 26 there exists $\mathtt{K}\geq 0$ such that for any
$\nu\in\mathscr{P}_{2}(\mathbb{R}^{d})$ and
$x_{0},x_{1},x_{2}\in\mathbb{R}^{d}$ we have

$\displaystyle\textstyle{\|b(0,\nu)\|+\|\mathrm{d}_{x}^{2}b(x_{0},\nu)\|+\|\mathrm{D}^{L}b(x_{0},\nu)(x_{1})\|}$
(210)
$\displaystyle\qquad\qquad\textstyle{+\|\mathrm{d}_{x}\mathrm{D}^{L}b(x_{0},\nu)(x_{1})\|+\|(\mathrm{D}^{L})^{2}b(x_{0},\nu)(x_{1},x_{2})\|\leq\mathtt{K}.}$
(211)

In addition, using (120), for any $x_{1},x_{2}\in\mathbb{R}^{d}$ and
$\nu_{1},\nu_{2}\in\mathscr{P}_{2}(\mathbb{R}^{d})$ we have

$\left\|b(x_{1},\nu_{1})-b(x_{2},\nu_{2})\right\|\leq(\mathtt{M}/\eta)(1+(1/\eta))(1+\alpha\mathtt{L})(1+\mathtt{M})\left(\left\|x_{1}-x_{2}\right\|+\mathbf{W}_{2}(\nu_{1},\nu_{2})\right).$
(212)

Hence, $(\mathbf{A}_{b}^{1})$ in [8] is satisfied. Using (210) we have that
$(\mathbf{A}_{b}^{2})$ and $(\mathbf{A}_{b}^{4})$ in [8] are satisfied.
Similarly to (120), we have for any $x_{1},x_{2}\in\mathbb{R}^{d}$ and
$\nu_{1},\nu_{2}\in\mathscr{P}_{2}(\mathbb{R}^{d})$

$\displaystyle\|\mathrm{d}_{x}b(x_{1},\nu_{1})-\mathrm{d}_{x}b(x_{2},\nu_{2})\|$
(213)
$\displaystyle\quad\textstyle{\leq\|\int_{\mathbb{R}^{p}}\mathrm{d}^{2}_{x}\mathrm{k}(x_{1},y)/(\nu_{1}[\mathrm{k}(\cdot,y)]+\eta)-\int_{\mathbb{R}^{p}}\mathrm{d}^{2}_{x}\mathrm{k}(x_{2},y)/(\nu_{2}[\mathrm{k}(\cdot,y)]+\eta)\|}$
(214)
$\displaystyle\qquad+\alpha\textstyle{\|\mathrm{d}^{2}_{x}U(x_{1})-\mathrm{d}^{2}_{x}U(x_{2})\|}$
(215)
$\displaystyle\quad\leq(\mathtt{M}/\eta)(1+(1/\eta))(1+\alpha\mathtt{L}_{2})(1+\mathtt{M})\\{\|x_{1}-x_{2}\|+\mathbf{W}_{2}(\nu_{1},\nu_{2})\\}.$
(216)

Second, similarly to (120), we have for any
$x_{1},x_{2},x_{3},x_{4}\in\mathbb{R}^{d}$ and
$\nu_{1},\nu_{2}\in\mathscr{P}_{2}(\mathbb{R}^{d})$

$\displaystyle\textstyle{\|\mathrm{D}^{L}b(x_{1},\nu_{1})(x_{2})-\mathrm{D}^{L}b(x_{2},\nu_{2})(x_{3})\|}$
(217)
$\displaystyle\qquad\leq\textstyle{\|\int_{\mathbb{R}^{p}}\nabla_{1}\mathrm{k}(x_{1},y)\otimes\nabla_{1}\mathrm{k}(x_{2},y)/(\nu_{1}[\mathrm{k}(\cdot,y)]+\eta)\mathrm{d}\mu(y)}$
(218)
$\displaystyle\qquad\textstyle{\qquad-\int_{\mathbb{R}^{p}}\nabla_{1}\mathrm{k}(x_{3},y)\otimes\nabla_{1}\mathrm{k}(x_{4},y)/(\nu_{2}[\mathrm{k}(\cdot,y)]+\eta)\mathrm{d}\mu(y)\|}$
(219)
$\displaystyle\qquad\leq(\mathtt{M}^{2}/\eta)(2+(1/\eta))(1+\mathtt{M})\\{\|x_{1}-x_{3}\|+\|x_{2}-x_{4}\|+\mathbf{W}_{2}(\nu_{1},\nu_{2})\\}.$
(220)

Combining (213) and (217), we get that $(\mathbf{A}_{b}^{3})$ in [8] is
satisfied. In addition, note that $(\mathbf{A}_{\sigma}^{1})$,
$(\mathbf{A}_{\sigma}^{2})$, $(\mathbf{A}_{\sigma}^{3})$ and
$(\mathbf{A}_{\sigma}^{4})$ are immediately satisfied since
$\sigma=\sqrt{2\alpha}$ in our case. We conclude the proof upon using [8,
Theorem 2.5]. ∎

###  C.3 Proof of Proposition 11

Let $M,N\in\mathbb{N}^{\star}$ and $\alpha,\eta>0$. Using (120), we have for
any $x_{1},x_{2}\in\mathbb{R}^{d}$ and
$\nu_{1},\nu_{2}\in\mathscr{P}_{1}(\mathbb{R}^{d})$

$\displaystyle\textstyle{\|b(x_{1},\nu_{1})-b^{M}(x_{2},\nu_{2})\|}$ (221)
$\displaystyle\textstyle{\leq((\mathtt{M}/\eta)(1+(1/\eta))+\alpha\mathtt{L})(\left\|x_{1}-x_{2}\right\|+\int_{\mathbb{R}^{p}}\left|\nu_{1}[\mathrm{k}(\cdot,y)]-\nu_{2}[\mathrm{k}(\cdot,y)]\right|\mathrm{d}\mu(y))}$
(222)
$\displaystyle\textstyle{+\|(1/M)\sum_{j=1}^{M}\\{\tilde{b}_{\eta}(x_{2},\nu_{2},y^{j,M})-\int_{\mathbb{R}^{p}}\tilde{b}_{\eta}(x_{2},\nu_{2},y)\mathrm{d}\mu(y)\\}\|}.$
(223)

For any $i\in\\{1,2\\}$ and $x_{i}^{1:N}\in(\mathbb{R}^{d})^{N}$, let
$\nu_{i}^{N}=(1/N)\sum_{k=1}^{N}\updelta_{x_{i}^{k,N}}$. Using (221), we have
that for any $x_{1}^{1:N},x_{2}^{1:N}\in(\mathbb{R}^{d})^{N}$

$\displaystyle\textstyle{\|b(x_{1}^{1,N},\nu_{1}^{N})-b^{M}(x_{2}^{1,N},\nu_{2}^{N})\|}$
(224)
$\displaystyle\textstyle{\leq((\mathtt{M}/\eta)(1+(1/\eta))+\mathtt{L})(1+\mathtt{M})}$
(225)
$\displaystyle\qquad\times\textstyle{(\|x_{1}^{1,N}-x_{2}^{1,N}\|+(1/N)\sum_{k=1}^{N}\|x_{1}^{k,N}-x_{2}^{k,N}\|)}$
(226)
$\displaystyle\textstyle{+\|(1/M)\sum_{j=1}^{M}\\{\tilde{b}_{\eta}(x_{2}^{1,N},\nu_{2}^{N},y^{j,M})-\int_{\mathbb{R}^{p}}\tilde{b}_{\eta}(x_{2}^{1,N},\nu_{2}^{N},y)\mathrm{d}\mu(y)\\}\|}.$
(227)

Let $C\geq 0$ and $\\{A_{k}\\}_{k=1}^{M}$ be a family of $d$-dimensional
random variables such that for any $i,j\in\\{1,\dots,M\\}$ with $i\neq j$

$\textstyle{\mathbb{E}[\|A_{i}\|^{2}]\leq C,\qquad\mathbb{E}[\langle
A_{i},A_{j}\rangle]=0.}$ (228)

Then, we have that

$\textstyle{\mathbb{E}[\|(1/M)\sum_{k=1}^{N}A_{k}\|]\leq(C/M)^{1/2}.}$ (229)

Let
$A_{k}:=\tilde{b}_{\eta}(X_{N}^{1,N,M},\lambda_{n}^{N,M},y^{k,M})-\int_{\mathbb{R}^{p}}\tilde{b}_{\eta}(X_{N}^{1,N,M},\lambda_{n}^{N,M},y)\mathrm{d}\mu(y)$
for any $k\in\\{1,\dots,M\\}$. Then $\\{A_{k}\\}_{k=1}^{M}$ satisfies (228)
with $C=\mathtt{M}/\eta$. Combining this result, (224), (229) and that for any
$n\in\mathbb{N}$, $\\{X_{n}^{k,N}-X_{n}^{k,N,M}\\}_{k=1}^{N}$ is exchangeable,
we have

$\displaystyle\mathbb{E}[\textstyle{\|b(X_{n}^{1,N},\lambda_{n}^{N})-b^{M}(X_{n}^{1,N,M},\lambda_{n}^{N,M})\|}]$
(230)
$\displaystyle\qquad\textstyle{\leq((\mathtt{M}/\eta)(1+(1/\eta))+\mathtt{L})(1+\mathtt{M})}$
(231)
$\displaystyle\qquad\qquad\times\textstyle{(\mathbb{E}[\|X_{n}^{1,N}-X_{n}^{1,N,M}\|]+(1/N)\sum_{k=1}^{N}\mathbb{E}[\|X_{n}^{k,N}-X_{n}^{k,N,M}\|])}$
(232)
$\displaystyle\qquad\textstyle{+\mathbb{E}[\|(1/M)\sum_{k=1}^{M}A_{k}\|]}$
(233) $\displaystyle\qquad\textstyle{\leq
2((\mathtt{M}/\eta)(1+(1/\eta))+\mathtt{L})(1+\mathtt{M})\mathbb{E}[\|X_{n}^{1,N}-X_{n}^{1,N,M}\|]}+(\mathtt{M}/(M\eta))^{1/2}.$
(234)

Using this result, we have for any $n\in\mathbb{N}$

$\displaystyle\mathbb{E}[\|X_{n+1}^{1,N}-X_{n+1}^{1,N,M}\|]\leq\mathbb{E}[\|X_{n}^{1,N}-X_{n}^{1,N,M}\|]$
(235)
$\displaystyle\qquad+\gamma\mathbb{E}\left[\frac{\gamma\left\|b(X_{n}^{1,N},\lambda_{n}^{N})\right\|}{(1+\gamma\left\|b(X_{n}^{1,N},\lambda_{n}^{N})\right\|)}\frac{\left\|b(X_{n}^{1,N},\lambda_{n}^{N})-b^{M}(X_{n}^{1,N,M},\lambda_{n}^{N,M})\right\|}{(1+\gamma\left\|b^{M}(X_{n}^{1,N,M},\lambda_{n}^{N,M})\right\|)}\right]$
(236)
$\displaystyle\qquad+\gamma\mathbb{E}\left[\frac{\left\|b(X_{n}^{1,N},\lambda_{n}^{N})-b^{M}(X_{n}^{1,N,M},\lambda_{n}^{N,M})\right\|}{(1+\gamma\left\|b^{M}(X_{n}^{1,N,M},\lambda_{n}^{N,M})\right\|)}\right]$
(237)
$\displaystyle\qquad\leq\mathbb{E}[\|X_{n}^{1,N}-X_{n}^{1,N,M}\|]+2\gamma\mathbb{E}\left[\left\|b(X_{n}^{1,N},\lambda_{n}^{N})-b^{M}(X_{n}^{1,N,M},\lambda_{n}^{N,M})\right\|\right]$
(238) $\displaystyle\qquad\leq
C\left((1+2\gamma)\mathbb{E}[\|X_{n}^{1,N}-X_{n}^{1,N,M}\|]+2\gamma
M^{-1/2}\right).$ (239)

Proceeding recursively and recalling that $X_{0}^{1,N}=X_{0}^{1,N,M}$ the
result follows.

###  C.4 Proof of Proposition 12

The proof follows the same structure of that of Proposition 11. Let
$M,N,m\in\mathbb{N}^{\star}$ with $m\leq M$ and $\alpha,\eta>0$. Proceeding as
above we can write a result equivalent to (230)

$\displaystyle\mathbb{E}[\textstyle{\|b(X_{n}^{1,N},\lambda_{n}^{N})-b^{M}(X_{n}^{1,N,m},\lambda_{n}^{N,m})\|}]$
(240)
$\displaystyle\textstyle{\leq((\mathtt{M}/\eta)(1+(1/\eta))+\mathtt{L})(1+\mathtt{M})}$
(241)
$\displaystyle\quad\times\textstyle{(\mathbb{E}[\|X_{n}^{1,N}-X_{n}^{1,N,m}\|]+(1/N)\sum_{k=1}^{N}\mathbb{E}[\|X_{n}^{k,N}-X_{n}^{k,N,m}\|])}$
(242) $\displaystyle\textstyle{+\mathbb{E}[\|(1/m)\sum_{k=1}^{m}A_{k}\|]},$
(243)

where
$A_{k}:=\tilde{b}_{\eta}(X_{n}^{1,N,m},\lambda_{n}^{N,m},y^{j,m})-\int_{\mathbb{R}^{p}}\tilde{b}_{\eta}(X_{n}^{1,N,m},\lambda_{n}^{N,m},y)\mathrm{d}\mu(y)$
and $\\{y_{1},\dots,y_{m}\\}$ is a sample from $\mu^{M}$. Consider the last
expectation and decompose

$\displaystyle\mathbb{E}[\|(1/m)\sum_{j=1}^{m}\\{\tilde{b}_{\eta}(X_{n}^{1,N,m},\lambda_{n}^{N,m},y^{j,m})-\int_{\mathbb{R}^{p}}\tilde{b}_{\eta}(X_{n}^{1,N,m},\lambda_{n}^{N,m},y)\mathrm{d}\mu(y)\\}\|]$
(244)
$\displaystyle\qquad\leq\mathbb{E}[\|(1/m)\sum_{j=1}^{m}\\{\tilde{b}_{\eta}(X_{n}^{1,N,m},\lambda_{n}^{N,m},y^{j,m})-(1/M)\sum_{k=1}^{M}\tilde{b}_{\eta}(X_{n}^{1,N,m},\lambda_{n}^{N,m},y^{j,M})\\}\|]$
(245)
$\displaystyle\qquad+\mathbb{E}[\|(1/M)\sum_{k=1}^{M}\tilde{b}_{\eta}(X_{n}^{1,N,m},\lambda_{n}^{N,m},y^{j,M})-\int_{\mathbb{R}^{p}}\tilde{b}_{\eta}(X_{n}^{1,N,m},\lambda_{n}^{N,m},y)\mathrm{d}\mu(y)\\}\|].$
(246)

As shown in the proof of Proposition 11, the second term satisfies

$\displaystyle\mathbb{E}[\|(1/M)\sum_{k=1}^{M}\tilde{b}_{\eta}(X_{n}^{1,N,m},\lambda_{n}^{N,m},y^{j,M})-\int_{\mathbb{R}^{p}}\tilde{b}_{\eta}(X_{n}^{1,N,m},\lambda_{n}^{N,m},y)\mathrm{d}\mu(y)\\}\|]$
(247) $\displaystyle\qquad\qquad\leq(\mathtt{M}/(M\eta))^{1/2}.$ (248)

Let us denote the $\sigma$-field generated by $\mu^{M}$ by
$\mathcal{S}^{M}:=\sigma\left(y_{j}:j=1,\ldots,M\right)$. For any
$k\in\\{1,\dots,m\\}$, $x_{1}\in\mathbb{R}^{d}$ let
$A_{k}=\tilde{b}_{\eta}(x_{1},\nu_{1},y^{k,m})-(1/M)\sum_{k=1}^{M}\tilde{b}_{\eta}(X_{n}^{1,N,m},\lambda_{n}^{N,m},y^{j,M})$.
Conditionally on $\mathcal{S}^{M}$, $\\{A_{k}\\}_{k=1}^{m}$ satisfies (228)
with $C=\mathtt{M}/\eta$. Hence,

$\displaystyle\mathbb{E}[\|(1/m)\sum_{j=1}^{m}\\{\tilde{b}_{\eta}(X_{n}^{1,N,m},\lambda_{n}^{N,m},y^{j,m})-(1/M)\sum_{k=1}^{M}\tilde{b}_{\eta}(X_{n}^{1,N,m},\lambda_{n}^{N,m},y^{j,M})\\}\|]$
(249) $\displaystyle\qquad\qquad\leq(\mathtt{M}/(m\eta))^{1/2}.$ (250)

Combining the two expectations we obtain the following bound for (240)

$\displaystyle\mathbb{E}[\textstyle{\|b(X_{n}^{1,N},\lambda_{n}^{N})-b^{M}(X_{n}^{1,N,m},\lambda_{n}^{N,m})\|}]$
(251)
$\displaystyle\qquad\textstyle{\leq((\mathtt{M}/\eta)(1+(1/\eta))+\mathtt{L})(1+\mathtt{M})}$
(252)
$\displaystyle\qquad\qquad\times\textstyle{(\mathbb{E}[\|X_{n}^{1,N}-X_{n}^{1,N,m}\|]+(1/N)\sum_{k=1}^{N}\mathbb{E}[\|X_{n}^{k,N}-X_{n}^{k,N,m}\|])}$
(253)
$\displaystyle\qquad\textstyle{+(\mathtt{M}/(m\eta))^{1/2}+(\mathtt{M}/(M\eta))^{1/2}}$
(254)
$\displaystyle\qquad\textstyle{\leq((\mathtt{M}/\eta)(1+(1/\eta))+\mathtt{L})(1+\mathtt{M})}$
(255)
$\displaystyle\qquad\qquad\times\textstyle{(\mathbb{E}[\|X_{n}^{1,N}-X_{n}^{1,N,m}\|]+(1/N)\sum_{k=1}^{N}\mathbb{E}[\|X_{n}^{k,N}-X_{n}^{k,N,m}\|])}$
(256) $\displaystyle\qquad\textstyle{+2(\mathtt{M}/(m\eta))^{1/2}},$ (257)

where we used the fact that $m\leq M$. The above allows us to obtain an
equivalent result to (235). Proceeding recursively and recalling that
$X_{0}^{1,N}=X_{0}^{1,N,m}$ we have the result.

##  Appendix D Additional Experiments

###  D.1 Choice of initial distribution and reference measure

To study the influence of the reference measure $\pi_{0}$ on
$\mathscr{G}_{\alpha}^{\eta}$ and of the initial condition $X_{0}^{1:N}$ on
the speed of convergence we consider the following toy Fredholm integral
equation

$\textstyle{\mathcal{N}(y;0,\sigma_{\mathrm{k}}^{2}+\sigma_{\pi}^{2})=\int_{\mathbb{R}}\mathcal{N}(x;0,\sigma_{\pi}^{2})\mathcal{N}(y;x,\sigma_{\mathrm{k}}^{2})\mathrm{d}x,}$

where $\mathcal{N}(x;m,\sigma^{2})$ is a Gaussian distribution with mean $m$
and variance $\sigma^{2}$, with $\sigma_{\pi}^{2}=0.43^{2}$ and
$\sigma_{\mathrm{k}}^{2}=0.45^{2}$. We assume that $\mu$ is known through a
sample of size $M=10^{4}$, we consider four reference measures: a uniform
distribution, a zero-mean Gaussian distribution with variance given by the
empirical variance of the sample from $\mu$, a Student’s t-distribution with
100 degrees of freedom and a zero-mean Laplace distribution with variance
given by the empirical variance of the sample from $\mu$; and three initial
conditions: the solution $\pi$, a uniform over the interval $[-2,2]$ which
contains more than $99\%$ of the mass and a highly peaked zero-mean Gaussian
distribution. As an example we set $N=m=500$, $\gamma=10^{-2}$, $\alpha=0.02$
and iterate for 300 time steps. The behaviour for other values of $N,m$ is
similar.

$X_{0}^{1:N}\sim\pi$$\hat{\pi}$$\log\mathscr{G}_{\alpha}^{\eta}$$X_{0}^{1:N}\sim
U(-2,2)$$X_{0}^{1:N}\sim\mathcal{N}(0,10^{-8})$

Figure 7: Influence of reference measure and initial condition,
reconstructions (left panel) and approximate value of
$\mathscr{G}_{\alpha}^{\eta}$ (right panel).

Figure 7 shows that the reference measure does not influence the speed of
convergence but influences the shape of the reconstructions, which are more
peaked when the reference measure is Laplace and flatter when the reference
measure is a Gaussian distribution; this corresponds to slightly different
minima for $\mathscr{G}_{\alpha}^{\eta}$. In particular, the order of the
minima is not influenced by the initial condition, with the uniform reference
measure always giving the smallest minima. Indeed, if we compare the values of
$\mathscr{G}_{\alpha}^{\eta}$ at the solution $\pi$ we find that when the
reference measure is Uniform, $\textup{KL}\left(\pi|\pi_{0}\right)$ is not
defined and

$\displaystyle\mathscr{G}_{\alpha}^{\eta}(\textrm{t})\geq\mathscr{G}_{\alpha}^{\eta}(\textrm{Gaussian})\simeq\mathscr{G}_{\alpha}^{\eta}(\textrm{Laplace}).$
(258)

The initial condition $X_{0}^{1:N}$ affects the speed of convergence with
light tail distributions giving faster convergence as already observed in [12,
4], but does not affect the shape of the reconstructions. This is not
surprising as Proposition 7 guarantees the existence of a unique solution for
any initial condition $X_{0}^{1:N}$.

###  D.2 Choice of $\alpha$

####  D.2.1 Toy problem

Consider again the toy Fredholm integral equation in Section D.1 and consider
$\mathscr{G}_{\alpha}$ with the reference measure $\pi_{0}$ given by a
Gaussian distribution with mean 0 and variance $\sigma_{0}^{2}$. If we
restrict the class of distributions on which we look for a minimizer to that
of Gaussian distributions with mean 0 and variance $\beta$, i.e.
$\pi(x)=\mathcal{N}(x;0,\beta)$, the functional $\mathscr{G}_{\alpha}$ can be
computed exactly

$\displaystyle\mathscr{G}_{\alpha}(\pi)$
$\displaystyle\textstyle{=-\int_{\mathbb{R}^{p}}\log(\pi[\mathrm{k}(\cdot,y)])\mathrm{d}\mu(y)+\alpha\textup{KL}\left(\pi|\pi_{0}\right)}$
(259)
$\displaystyle\textstyle{=(1/2)\log(2\pi(\beta+\sigma_{\mathrm{k}}^{2}))+\sigma_{\mu}^{2}/(2(\beta+\sigma_{\mathrm{k}}^{2}))+\alpha/2\left(\log\sigma_{0}^{2}/\beta+\beta/\sigma_{0}^{2}-1\right).}$
(260)

Finding the optimal $\beta$ for given $\alpha$ amounts to solving the
following cubic equation

$\displaystyle\alpha\beta^{3}+\beta^{2}(2\alpha\sigma_{\mathrm{k}}^{2}+(1-\alpha)\sigma_{0}^{2})+\beta(\alpha\sigma_{\mathrm{k}}^{4}-\sigma_{\mu}^{2}\sigma_{0}^{2}+\sigma_{0}^{2}\sigma_{\mathrm{k}}^{2}-2\alpha\sigma_{\mathrm{k}}^{2})-\alpha\sigma_{0}^{2}\sigma_{\mathrm{k}}^{4}=0,$
(261)

clearly when $\alpha=0$ (no cross-entropy constraint),
$\beta=\sigma_{\pi}^{2}$, while for all $\alpha>0$ (at least) one optimal
$\beta$ exists. However, as $\alpha$ increases the value of $\beta$ stops
increasing and stabilizes around a fixed value which depends on
$\sigma_{0}^{2}$, Figure 8 shows the result for $\alpha\in[0,1]$ and
$\sigma_{0}^{2}=0.1^{2}$.

$\alpha$$\beta(\alpha)$$\alpha$$\mathscr{G}_{\alpha}$

Figure 8: Example of dependence of the variance of the minimizer
$\pi^{\star}_{\alpha}$ (left) and the functional $\mathscr{G}_{\alpha}$
(right) on the parameter $\alpha$. The red lines denote the values obtained
with $\alpha=0$, i.e. no regularization.

As discussed in Section 2.2, taking $\alpha=1$ amounts to Bayesian inference
for a model with prior $\pi_{0}$ and data given by the available sample from
$\mu$. Using the cubic equation above we obtain that the corresponding optimal
$\beta$ is $0.44$; comparing the minimizer obtained with $\alpha=1$ with the
solution $\pi$ and the minimizer corresponding to the value of $\alpha=4\cdot
10^{-5}$ obtained by cross-validation we find that $\alpha=1$ results in
significantly over-smoothed reconstructions (Figure 9) while the value chosen
by cross-validation provides reconstructions which are indistinguishable from
the solution $\pi$.

Figure 9: Effect of regularizing parameter $\alpha$ on reconstructions. We
compare $\alpha=1$ which corresponds to Bayesian inference for the incomplete
data model with prior $\pi_{0}$ and $\alpha=$ selected by cross-validation.