from $\varepsilon,z^{1:n}_{i}$ for $i\in\\{0,1\\}$ and $n\in\mathbb{N}$) such
that for any $i\in\\{0,1\\}$

$|\updelta_{z^{1:n}_{i}}\mathrm{S}_{0}u(\cdot,z)-\updelta_{z^{1:n}_{i}}\mathrm{S}_{\varepsilon}u(\cdot,z)|\leq
C_{0}\mathbf{W}_{2}(\updelta_{z^{1:n}_{i}}\mathrm{S}_{0},\updelta_{z^{1:n}_{i}}\mathrm{S}_{\varepsilon})\;.$
(141)

Using this result, Section B.3 and Section B.3, there exists $C_{1}\geq 0$
(independent from $\varepsilon,z^{1:n}_{i}$ for $i\in\\{0,1\\}$ and
$n\in\mathbb{N}$) such that for any $i\in\\{0,1\\}$

$|\updelta_{z^{1:n}_{i}}\mathrm{S}_{0}u(\cdot,z)-\updelta_{z^{1:n}_{i}}\mathrm{S}_{\varepsilon}u(\cdot,z)|\leq
C_{1}\mathbf{W}_{1}^{\eta/2}(\updelta_{z^{1:n}_{i}}\mathrm{S}_{0},\updelta_{z^{1:n}_{i}}\mathrm{S}_{\varepsilon})\;.$
(142)

In addition, using Section 4.3.1, there exists $C_{2}\geq 0$ (independent from
$\varepsilon,z^{1:n}_{i}$ for $i\in\\{0,1\\}$ and $n\in\mathbb{N}$) such that
for any $i\in\\{0,1\\}$

$\textstyle{\int_{\mathsf{Z}^{n}}\mathbf{W}_{1}(\updelta_{z^{1:n}_{i}}\mathrm{S}_{0},\updelta_{z^{1:n}_{i}}\mathrm{S}_{\varepsilon})\mathrm{d}\mu^{\otimes
n}(z_{i}^{1:n})\leq C_{2}\max(\varepsilon^{\alpha},\varepsilon^{1/2})\;.}$
(143)

Combining this result, (142) and the fact that $t\mapsto t^{\eta/2}$ is
concave, we get that there exists $C_{3}\geq 0$ (independent from
$\varepsilon,z^{1:n}_{i}$ for $i\in\\{0,1\\}$ and $n\in\mathbb{N}$) such that
for any $i\in\\{0,1\\}$

$\textstyle{\int_{\mathsf{Z}^{n}}|\updelta_{z^{1:n}_{i}}\mathrm{S}_{0}u(\cdot,z)-\updelta_{z^{1:n}_{i}}\mathrm{S}_{\varepsilon}u(\cdot,z)|\mathrm{d}\mu^{\otimes
n}(z_{i}^{1:n})\leq
C_{3}\max(\varepsilon^{\eta\alpha/2},\varepsilon^{\eta/4})\;.}$ (144)

In addition, using Section 4.3.2 we have

$\textstyle{\int_{\mathsf{Z}^{n+1}}|\updelta_{z^{1:n}_{0}}\mathrm{S}_{\varepsilon}u(\cdot,z)-\updelta_{z^{1:n}_{1}}\mathrm{S}_{\varepsilon}u(\cdot,z)|\mathrm{d}\mu^{\otimes}(z^{1:n}_{0})\mathrm{d}\mu(z_{1,j})\leq
4(\mathtt{M}^{2}(\mathtt{c}+d\varepsilon)/\mathtt{m}+B^{2})\mathtt{c_{LS}}/(n\varepsilon)\;,}$
(145)

Combining this result, (139) and (144) we get

$\displaystyle\textstyle{\int_{\mathsf{Z}^{n+1}}|\updelta_{z^{1:n}_{0}}\mathrm{S}_{0}u(\cdot,z)-\updelta_{z^{1:n}_{1}}\mathrm{S}_{0}u(\cdot,z)|\mathrm{d}\mu^{\otimes}(z^{1:n}_{0})\mathrm{d}\mu(z_{1,j})}$
(146) $\displaystyle\qquad\qquad\leq
4(\mathtt{M}^{2}(\mathtt{c}+d\bar{\varepsilon})/\mathtt{m}+B^{2})\mathtt{c_{LS}}/(n\varepsilon)+2C_{3}\max(\varepsilon^{\eta\alpha/2},\varepsilon^{\eta/4})$
(147) $\displaystyle\qquad\qquad\leq
4(\mathtt{M}^{2}(\mathtt{c}+d\bar{\varepsilon})/\mathtt{m}+B^{2})\mathtt{c_{LS}}/(n\varepsilon)+2C_{3}\varepsilon^{\eta
s}\;.$ (148)

with $s=\min(\alpha/2,1/4)$. We conclude the proof upon letting
$\varepsilon=n^{-1/(1+\eta s)}$ ∎

The stability of the limiting measures allows us to establish Section 3.2.2
which provides quantitative bounds on $\mu^{\otimes
n}\mathrm{S}_{0}[U]-U^{\star}$ for large values of $n\in\mathbb{N}$. Indeed,
once Section 4.3.2 is established the proof of Section 3.2.2 is classical and
follows the lines of [60, Section 3.7].

###### Proof.

Let $n\in\mathbb{N}$ and $n\geq n_{0}$ with $n_{0}$ given by Section 3.2.2.
For any $z^{1:n}\in\mathsf{Z}^{n}$ we define
$U_{n}^{\star}(z^{1:n})=\inf\\{U_{n}(x,z^{1:n})\,:\;x\in\mathbb{R}^{d}\\}$.
Using that
$U^{\star}\geq\int_{\mathsf{Z}^{n}}U_{n}^{\star}(z^{1:n})\mathrm{d}\mu^{\otimes
n}(z^{1:n})$ we have

$\displaystyle\mu^{\otimes
n}\mathrm{S}_{0}[U]-U^{\star}\leq\textstyle{\mu^{\otimes
n}\mathrm{S}_{0}[U]-\int_{\mathsf{Z}^{n}}U_{n}^{\star}(z^{1:n})\mathrm{d}\mu^{\otimes
n}(z^{1:n})}$ (149) $\displaystyle\leq\textstyle{\mu^{\otimes
n}\mathrm{S}_{0}[U]-\int_{\mathsf{Z}^{n}}\int_{\mathbb{R}^{d}}U_{n}(x,z^{1:n})\mathrm{S}_{0}(z^{1:n},\mathrm{d}x)\mathrm{d}\mu^{\otimes
n}(z^{1:n})}$ (150)
$\displaystyle\leq\textstyle{\int_{\mathsf{Z}^{n}}\int_{\mathsf{Z}^{n}}\int_{\mathbb{R}^{d}}U_{n}(x,\tilde{z}^{1:n})\mathrm{S}_{0}(z^{1:n},\mathrm{d}x)\mathrm{d}\mu^{\otimes
n}(z^{1:n})\mathrm{d}\mu^{\otimes n}(\tilde{z}^{1:n})}$ (151)
$\displaystyle\qquad\textstyle{-\int_{\mathsf{Z}^{n}}\int_{\mathbb{R}^{d}}U_{n}(x,z^{1:n})\mathrm{S}_{0}(z^{1:n},\mathrm{d}x)\mathrm{d}\mu^{\otimes
n}(z^{1:n})}$ (152)
$\displaystyle\leq\textstyle{(1/n)\sum_{i=1}^{n}\int_{\mathsf{Z}}\int_{\mathsf{Z}^{n}}\int_{\mathbb{R}^{d}}\\{u(x,\tilde{z}_{i})-u(x,z_{i})\\}\mathrm{S}_{0}(z^{1:n},\mathrm{d}x)\mathrm{d}\mu^{\otimes
n}(z^{1:n})\mathrm{d}\mu(\tilde{z}_{i})}$ (153)
$\displaystyle\leq\textstyle{(1/n)\sum_{i=1}^{n}\int_{\mathsf{Z}}\int_{\mathsf{Z}^{n}}\\{\int_{\mathbb{R}^{d}}u(x,z_{i})\mathrm{S}_{0}(z^{1:n},\mathrm{d}x)-\int_{\mathbb{R}^{d}}u(x,z_{i})\mathrm{S}_{0}(\tilde{z}_{i}^{1:n},\mathrm{d}x)\\}\\}\mathrm{d}\mu^{\otimes
n}(z^{1:n})\mathrm{d}\mu(\tilde{z}_{i})\;,}$ (154)

where for any $i\in\\{1,\dots,n\\}$, we have that for any
$j\in\\{1,\dots,n\\}$, $\tilde{z}_{i,j}=z_{i}$ and
$\tilde{z}_{i,i}=\tilde{z}_{i}$. Using Section 3.2.2 we have $\mu^{\otimes
n}\mathrm{S}_{0}U-U^{\star}\leq C_{\eta}n^{-s\eta/(1+\eta s)}$ with
$C_{\eta}\geq 0$, which concludes the proof.

####  4.3.3 Proof of Section 3.2.2

We recall that $u$ is given in (59). We divide the proof into two parts.

  1. (a)

First, we prove that $\textstyle{\mu^{\otimes
n}\mathrm{S}_{0}(\left(-\uppi/2,\uppi/2\right))-\pi_{0}(\left(-\uppi/2,\uppi/2\right))=-1/2}$
for all $n\in\mathbb{N}$. Let $n\in\mathbb{N}$. Assume that $z<0$. Then the
minimum of $x\mapsto u(x,z)$ is attained on $\left[\uppi,+\infty\right)$.
Denote $\bar{u}:\ \mathbb{R}\times\left[-1/2,0\right)$ such that for any
$x\in\mathbb{R}$ and $z\in\left[-1/2,0\right)$, $\bar{u}(x,z)=h(x)+xz+\uppi
z+1-\cos(3x)$ with $h(x)=x^{4}/(1+x^{2})$. Note that for any $x\geq 0$ and
$z\in\left[-1/2,0\right)$, $\bar{u}(x,z)=u(x+\uppi,z)$. There exists
$a\in\left[0,\uppi/3\right]$ such that for any $x\geq 0$, $h^{\prime}(x)-1\leq
0$ if $x\leq a$ and $h^{\prime}(x)-1>0$ otherwise. Hence, we get that for any
$x\in\mathbb{R}$ with $x\geq\uppi/3$ and $z\in\left[-1/2,0\right)$

$\textstyle{\bar{u}(x,z)-\bar{u}(0,z)=\bar{u}(x,z)-\uppi z\geq
h(\uppi/3)-\uppi/6>0\;.}$ (155)

Therefore, for any $z\in\left[-1/2,0\right)$, the global minimum of $x\mapsto
u(x,z)$ is attained on $\left(\uppi,4\uppi/3\right)$. We have that for any
$x\in\left[\uppi/6,\uppi/3\right]$ and $z\in\left[-1/2,0\right)$

$\textstyle{\partial_{1}\bar{u}(x,z)\geq
h^{\prime}(x)-1/2+3(2-(6/\uppi)x)>0\;.}$ (156)

In addition, we have that for any $z\in\left[-1/2,0\right)$,
$\partial_{1}\bar{u}(0,z)=-z$. Hence, there exists
$\bar{x}(z)\in\left[0,\uppi/6\right]$ such that
$\partial_{1}\bar{u}(\bar{x}(z),z)=0$. In addition we have that for any
$z\in\left[-1/2,0\right)$, $x\mapsto\partial_{1}\bar{u}(x,z)$ is increasing on
$\left[0,\uppi/6\right]$. Therefore, for any $z\in\left[-1/2,0\right)$ there
exists a unique minimizer of $x\mapsto u(x,z)$ on
$\left[\uppi,5\uppi/6\right]$ given by $x^{\star}(z)=\uppi+\bar{x}(z)$. The
same conclusion holds with $x^{\star}(z)\in\left[-5\uppi/6,-\uppi\right]$ if
$z\in\left(0,1/2\right]$. We have that $\lim_{z\to
0}\sup\\{\|u(x,z)-u(x,0)\|\,:\;x\in\left[-5\uppi/6,5\uppi/6\right]\\}=0$.
Therefore we have that every limit point of $\\{x^{\star}(z)\\}_{z<0}$ when
$z\to 0$ is a global minimizer of $u(\cdot,0)$. But recall that
$\\{x^{\star}(z)\\}_{z<0}\subset\left[\uppi,5\uppi/6\right]$. Therefore, every
limit point of $\\{x^{\star}(z)\\}_{z<0}$ is equal to $\uppi$ and we have that
$\lim_{z\to 0,z>0}x^{\star}(z)=\uppi$. For any $n\in\mathbb{N}$, denote
$g_{n}$ the density of $T_{\\#}\mu^{\otimes n}$ where $T:\
\mathsf{Z}^{n}\to\mathsf{Z}$ is given by
$T(z^{1:n})=(1/n)\sum_{i=1}^{n}z_{i}$. For any $r,\varepsilon>0$ there exists
$n_{0}\in\mathbb{N}$ such that for any $n\in\mathbb{N}$ with $n\geq n_{0}$,
$\int_{\operatorname{B}(0,r)^{\mathrm{c}}}g_{n}(z)\mathrm{d}z\leq\varepsilon$.
Let $\varphi\in\mathrm{C}(\mathbb{R},\mathbb{R})$ bounded and $\varepsilon>0$.
Let $r>0$ such that for any $z\in[0,r]$,
$|\varphi(x^{\star}(z))-\varphi(-\uppi)|\leq\varepsilon$ and for any
$z\in[-r,0]$, $|\varphi(x^{\star}(z))-\varphi(\uppi)|\leq\varepsilon$. Using
this result, we have for any $n\in\mathbb{N}$ with $n\geq n_{0}$

$\displaystyle\textstyle{|\mu^{\otimes
n}\mathrm{S}_{0}[\varphi]-(\varphi(-\uppi)+\varphi(\uppi))/2|}$
$\displaystyle\leq\textstyle{\int_{0}^{+\infty}|\varphi(x^{\star}(z))-\varphi(-\uppi)|g_{n}(z)\mathrm{d}z+\int_{-\infty}^{0}|\varphi(x^{\star}(z))-\varphi(\uppi)|g_{n}(z)\mathrm{d}z}$
(157) $\displaystyle\leq(1+2\|\varphi\|_{\infty})\varepsilon\;.$ (158)

Therefore, we get that $\lim_{n\to+\infty}\mu^{\otimes
n}\mathrm{S}_{0}[\varphi]=(\varphi(-\uppi)+\varphi(\uppi))/2$, which concludes
the first part of the proof.

  2. (b)

Second, we prove that for any $n\in\mathbb{N}$, $\mu^{\otimes
n}\mathrm{S}_{0}[U]-U^{\star}\leq(\pi/(6\sqrt{3}))n^{-1/2}$. Note that for any
$n\in\mathbb{N}$ and $z^{1:n}\in\left[-1/2,1/2\right]^{n}$, with
$\sum_{i=1}^{n}z_{i}\neq 0$,
$\updelta_{z^{1:n}}\mathrm{S}_{0}[U]=U(x^{\star}(\bar{z}^{1:n}))$ with
$x^{\star}(\bar{z}^{1:n})\in\left[-5\uppi/6,5\uppi/6\right]$. We also have
that for any $x\in\mathbb{R}$ and $z_{1},z_{2}\in\left[-1/2,1/2\right]$,
$\textstyle{|u(x,z_{1})-u(x,z_{2})|\leq|x||z_{1}-z_{2}|}$. In particular, we
have that for any $z^{1:n}\in\left[-1/2,1/2\right]^{n}$ and
$x\in\left[-\uppi/3,\uppi/3\right]$

$\textstyle{|U(x)-U_{n}(x,z^{1:n})|\leq(\uppi/3)|(1/n)\sum_{i=1}^{n}z_{i}|\;.}$
(159)

Hence, using this result and that $U^{\star}=U(\uppi/3)=0$, we have that for
any $z^{1:n}\in\left[-1/2,1/2\right]$

$\textstyle{U_{n}(x^{\star}(\bar{z}^{1:n}),z^{1:n})\leq
U_{n}(\uppi/3,z^{1:n})\leq U^{\star}+(\pi/3)|(1/n)\sum_{i=1}^{n}z_{i}|\;.}$
(160)

Combining this result and that $\int_{\mathbb{R}}z^{2}\mathrm{d}\mu(z)=1/12$
we have $\mu^{\otimes
n}\mathrm{S}_{0}[U]-U^{\star}\leq(\pi/(6\sqrt{3}))n^{-1/2}$, which concludes
the proof.

∎

## Acknowledgement

This work is part of the research program MISTIC, supported by the French
Agence Nationale pour la Recherche (ANR-19-CE40-0005). V. De Bortoli was also
supported by EPSRC grant EP/R034710/1. We thank Francesca Crucinio for
pointing to us the use of _thermodynamic barriers_ and _kinetic barriers_ in
chemistry.

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## Organization of the appendix

In this supplementary material we derive technical lemmas and additional
results. In particular, we gather the technical lemmas of Section 2 in
Appendix A and the ones of Section 4.3.1 in Appendix B. In Appendix C, we
recall basic results from differential geometry and geometric measure theory.

##  Appendix A Technical results for Section 2

In this section, we derive some technical lemmas used in Section 4.1 in order
to prove Theorem 2 and other results from Section 2. We recall that for any
$\varphi:\ \mathbb{R}^{d}\to\mathbb{R}_{+}$ and $\varepsilon>0$, when this is
well-defined, we set

$\displaystyle\textstyle{\mathcal{I}_{\varepsilon}(\varphi)=C_{\varepsilon}^{-1}\int_{\mathbb{R}^{d}}\varphi(x)\Psi(x)\exp[-\left\|F(x)\right\|^{k}/\varepsilon]\mathrm{d}x\;,\quad\mathcal{J}_{\varepsilon}=\mathcal{I}_{\varepsilon}(1)\;,}$
(161)
$\displaystyle\textstyle{C_{\varepsilon}=\int_{\mathbb{R}^{d}}\exp[-\left\|x\right\|^{k}/\varepsilon]\mathrm{d}x=\varepsilon^{d/k}\int_{\mathbb{R}^{d}}\exp[-\left\|x\right\|^{k}]\mathrm{d}x=\varepsilon^{d/k}C_{1}\;.}$
(162)

In addition, we define

$\textstyle{\mathcal{I}_{0}(\varphi)=\int_{F^{-1}(0)}\varphi(x)\Psi(x)\mathrm{J}F(x)^{-1}\mathrm{d}\mathcal{H}^{d-\hat{d}}(x)\;,\quad\mathcal{J}_{0}=\mathcal{I}_{0}(1)\;,}$
(163)

with $\hat{d}=\min(d,p)$.

In Section A.1 we establish a link between a Hessian computed on the normal
bundle of a manifold and the generalized Jacobian. In Section A.2 we derive
technical truncation bounds for the proof of Theorem 2. Explicit controls of
some derivative are presented in Section A.3 in order to derive Section 4.1.1.
Finally, we present a smoothing lemma in Section A.4 which is key to weaken
the regularity assumptions of Section 4.1.3.

###  A.1 From normal Hessian to generalized Jacobian

Let $f\in\mathrm{C}^{2}(\mathbb{R}^{d})$ and $\mathrm{M}$ a manifold in
$\mathbb{R}^{d}$. For any $x\in\mathrm{M}$ we define $\nabla_{\perp}^{2}f(x)$
to be the projection of the Hessian on the orthogonal of the tangent space of
$\mathrm{M}$ at $x$, see [44].

######  Lemma A.0.

Let $U:\ \mathbb{R}^{d}\to\mathbb{R}$ and
$F\in\mathrm{C}^{\infty}(\mathbb{R}^{d},\mathbb{R}^{p})$ such that for any
$x\in\mathbb{R}^{d}$, $U(x)=\|F(x)\|^{2}$. In addition, assume that
$F^{-1}(0)\neq\emptyset$ and that for any $x\in F^{-1}(0)$,
$\mathrm{J}F(x)>0$. Then,
$\operatorname*{arg\,min}\\{U(x)\,:\;x\in\mathbb{R}^{d}\\}$ is a smooth
manifold and for any $x\in F^{-1}(0)$ we have that
$\det(\nabla^{2}_{\perp}U(x))=\mathrm{J}F(x)^{2}$.

###### Proof.

First, we have that
$\operatorname*{arg\,min}\\{U(x)\,:\;x\in\mathbb{R}^{d}\\}=F^{-1}(0)$. Hence,
$\operatorname*{arg\,min}\\{U(x)\,:\;x\in\mathbb{R}^{d}\\}$ is a smooth
manifold since $F\in\mathrm{C}^{\infty}(\mathbb{R}^{d},\mathbb{R}^{p})$. Let
$x\in F^{-1}(0)$. We have that
$\nabla^{2}U(x)=\mathrm{D}F(x)^{\top}\mathrm{D}F(x)$. Note that
$\mathrm{D}F(x)^{\top}=(\nabla F_{1}(x),\dots,\nabla F_{p}(x))$ is a basis of
$\mathrm{ker}(\mathrm{D}F(x))^{\perp}$, where we recall that
$\mathrm{ker}(\mathrm{D}F(x))$ is the tangent space to $F^{-1}(0)$ at $x$.
Denote by $O(x)=(f_{1}(x),\dots,f_{p}(x))$ the orthonormal basis of
$\mathrm{ker}(\mathrm{D}F(x))^{\perp}$ obtained from $\mathrm{D}F(x)^{\top}$
using the Gram-Schmidt process. There exists a triangular $p\times p$ matrix
$T(x)$ such that $O(x)=\mathrm{D}F(x)^{\top}T(x)$. We also have

$\operatorname{Id}=O(x)^{\top}O(x)=T(x)^{\top}\mathrm{D}F(x)\mathrm{D}F(x)^{\top}T(x)\;.$
(164)

Hence, we get that $\det(T(x))=\mathrm{J}F(x)^{-1}$. We also have that

$\det(\nabla_{\perp}^{2}U(x))=\det(O(x)^{\top}\nabla^{2}U(x)O(x))=\det(T(x)^{\top}\mathrm{D}F(x)\mathrm{D}F(x)^{\top}\mathrm{D}F(x)\mathrm{D}F(x)^{\top}T(x))=\mathrm{J}F(x)^{2}\;,$
(165)

which concludes the proof. ∎

###  A.2 Truncation and lower bounds

######  Lemma A.0.

Assume H 1 and H 2. Then, for any $\bar{\varepsilon}\geq 0$ there exists
$A_{0}\geq 0$ such that for any
$\varepsilon\in\left[0,\bar{\varepsilon}\right]$,
$\mathcal{J}_{\varepsilon}\geq A_{0}m_{0,\Psi}$ with
$m_{0,\Psi}=\inf\\{\Psi(x)\,:\;x\in F^{-1}(0)\\}$ and $A_{0}$ independent from
$\Psi$.

###### Proof.

Since $F(0)=0$ and $F\in\mathrm{C}^{1}(\mathbb{R}^{d},\mathbb{R}^{p})$, there
exists $M\geq 0$ such that for any $x\in\bar{\operatorname{B}}(0,1)$,
$\left\|F(x)\right\|\leq M\left\|x\right\|$. Note that $F^{-1}(0)$ is compact
since $\lim_{\left\|x\right\|\to+\infty}\left\|F(x)\right\|=+\infty$ and
$F\in\mathrm{C}(\mathbb{R}^{d},\mathbb{R}^{p})$. Hence, since for any $x\in
F^{-1}(0)$, $\Psi(x)>0$ and $\Psi\in\mathrm{C}(\mathbb{R}^{d},\mathbb{R}_{+})$
there exists $\eta\in\left(0,1\right)$ such that for any
$x\in\operatorname{B}(0,\eta)\cup F^{-1}(0)$, $\Psi(x)\geq m_{0,\Psi}/2$.
Using this result we have for any $\varepsilon>0$

$\displaystyle\mathcal{J}_{\varepsilon}$
$\displaystyle=\textstyle{\varepsilon^{-d/k}C_{1}^{-1}\int_{\mathbb{R}^{d}}\Psi(x)\exp[-\left\|F(x)\right\|^{k}/\varepsilon]\mathrm{d}x}$
(166)
$\displaystyle\geq\textstyle{\varepsilon^{-d/k}(m_{0,\Psi}/2)C_{1}^{-1}\int_{\operatorname{B}(0,\eta)}\exp[-M^{k}\left\|x\right\|^{k}/\varepsilon]\mathrm{d}x}$
(167) $\displaystyle\textstyle{\geq
C_{1}^{-1}(m_{0,\Psi}/2)M^{-d}\int_{\bar{\operatorname{B}}(0,M\eta/\bar{\varepsilon}^{1/k})}\exp[-\left\|x\right\|^{k}]\mathrm{d}x\;.}$
(168)

Using that $F^{-1}(0)$ is compact, there exists $\mathtt{M}\geq 0$ such that
for any $x\in F^{-1}(0)$, $\mathrm{J}F(x)\leq\mathtt{M}$. Therefore, we get
that

$\textstyle{\mathcal{J}_{0}=\int_{F^{-1}(0)}\Psi(x)\mathrm{J}F(x)^{-1}\mathrm{d}\mathcal{H}^{d-\hat{d}}(x)\geq
m_{0,\Psi}\mathtt{M}^{-1}\mathcal{H}^{d-\hat{d}}(F^{-1}(0))\;.}$ (169)

Since $\mathcal{H}^{d-\hat{d}}(F^{-1}(0))<+\infty$ using Appendix C in the
case where $d\geq p$ and the fact that $\mathcal{H}^{0}(F^{-1}(0))<+\infty$ if
$d\leq p$ (see the first part of the proof of Section A.2), we have that for
any $\varepsilon\in\left[0,\bar{\varepsilon}\right]$

$\displaystyle\mathcal{J}_{\varepsilon}\geq A_{0}m_{0,\Psi}\;,\qquad
A_{0}=\min(A_{0}^{1},A_{0}^{2})\;,$ (170)
$\displaystyle\textstyle{A_{0}^{1}=(1/2)C_{1}^{-1}M^{-d}\int_{\bar{\operatorname{B}}(0,M\eta/\bar{\varepsilon}^{1/k})}\exp[-\left\|x\right\|^{k}]\mathrm{d}x\;,}\quad
A_{0}^{2}=M^{-1}\mathcal{H}^{d-\hat{d}}(F^{-1}(0))\;,$ (171)

which concludes the proof. ∎

######  Lemma A.0.

Assume H 1 and H 2. Let $\varphi:\ \mathbb{R}^{d}\to\mathbb{R}$ and
$C_{\varphi}\geq 0$ such that for any $x\in\mathbb{R}^{d}$

$\textstyle{|\varphi(x)|\leq C_{\varphi}\exp[C_{\varphi}\|x\|^{\upalpha
k}]\;.}$ (172)

Then, for any $\bar{\varepsilon}\in(0,\mathtt{m}^{k}/(1+C_{\varphi,\Psi}))$
and $\mathsf{V}\subset\mathbb{R}^{d}$ open and bounded such that
$F^{-1}(0)\subset\mathsf{V}$ there exist $\beta_{1}>0$ and
$A_{1}\in\mathrm{C}(\mathbb{R}_{+},\mathbb{R}_{+})$ such that for any
$\varepsilon\in\left(0,\bar{\varepsilon}\right)$

$\mathcal{I}_{\varepsilon}^{\mathrm{out}}(\varphi)\leq
A_{1}(C_{\varphi,\Psi})\varepsilon^{-d/k}\exp[-\beta_{1}/\varepsilon]\;,$
(173)

with
$\mathcal{I}_{\varepsilon}^{\mathrm{out}}(\varphi)=\mathcal{I}_{\varepsilon}(\varphi\mathbbm{1}_{\mathsf{V}^{\mathrm{c}}})$,
$C_{\varphi,\Psi}=C_{\varphi}+C_{\Psi}+C_{\varphi}C_{\Psi}$ and
$A_{1},\beta_{1}$ independent functions from $\varphi$ and $\Psi$. Finally,
$A_{1}$ is non-decreasing.

###### Proof.

First using H 2 and (172) there exists $C_{\varphi,\Psi}$ such that for any
$x\in\mathbb{R}^{d}$

$\left|\varphi(x)\right|\Psi(x)\leq
C_{\varphi,\Psi}\exp[C_{\varphi,\Psi}\left\|x\right\|]\;,\qquad
C_{\varphi,\Psi}=C_{\varphi}+C_{\Psi}+C_{\varphi}C_{\Psi}\;.$ (174)

Since $\mathsf{V}$ is bounded there exists $R^{\prime}\geq R$ (where $R$ is
given in H 1) such that
$\mathsf{V}\subset\bar{\operatorname{B}}(0,R^{\prime})$. Note that for any
$\varepsilon>0$, we have

$\displaystyle\mathcal{I}_{\varepsilon}^{\mathrm{out}}(\varphi)=\mathcal{I}_{\varepsilon}^{1}(\varphi)+\mathcal{I}_{\varepsilon}^{2}(\varphi)\;,$
(175)
$\displaystyle\mathcal{I}_{\varepsilon}^{1}(\varphi)=\mathcal{I}_{\varepsilon}(\varphi\mathbbm{1}_{\bar{\operatorname{B}}(0,R^{\prime})^{\mathrm{c}}})\;,\quad\mathcal{I}_{\varepsilon}^{2}(\varphi)=\mathcal{I}_{\varepsilon}(\varphi\mathbbm{1}_{\mathsf{V}^{\mathrm{c}}\cap\bar{\operatorname{B}}(0,R^{\prime})})\;.$
(176)

Let $\varepsilon\in\left(0,\bar{\varepsilon}\right)$, we divide the rest of
the proof into two parts. First, we bound
$\mathcal{I}_{\varepsilon}^{1}(\varphi)$ and then
$\mathcal{I}_{\varepsilon}^{2}(\varphi)$.

  1. (a)

Let $u=(\mathtt{m}^{k}/\varepsilon-C_{\varphi,\Psi})^{1/\upalpha k}$ (which
makes sense, since $\varepsilon<\mathtt{m}^{k}/(C_{\varphi,\Psi}+1)$). Since
$R^{\prime}\geq R$ we have using (18) and that $u\geq 1$

$\displaystyle\mathcal{I}_{\varepsilon}^{1}(\varphi)$
$\displaystyle=\textstyle{C_{1}^{-1}\varepsilon^{-d/k}\int_{\bar{\operatorname{B}}(0,R^{\prime})^{\mathrm{c}}}\varphi(x)\Psi(x)\exp[-\left\|F(x)\right\|^{k}/\varepsilon]\mathrm{d}x}$
(177)
$\displaystyle\leq\textstyle{C_{1}^{-1}C_{\varphi,\Psi}\varepsilon^{-d/k}\int_{\bar{\operatorname{B}}(0,R^{\prime})^{\mathrm{c}}}\exp[-(\mathtt{m}^{k}/\varepsilon-
C_{\varphi,\Psi})\left\|x\right\|^{\upalpha k}]\mathrm{d}x}$ (178)
$\displaystyle\leq\textstyle{C_{1}^{-1}C_{\varphi,\Psi}\varepsilon^{-d/k}\int_{\bar{\operatorname{B}}(0,R^{\prime}u)^{\mathrm{c}}}\exp[-\left\|x\right\|^{\upalpha
k}]\mathrm{d}x\;.}$ (179)

Let $C_{\upalpha}=\int_{\mathbb{R}^{d}}\exp[-\left\|x\right\|^{\upalpha
k}]\mathrm{d}x$. Using that $u=(\mathtt{m}^{k}/\varepsilon-
C_{\varphi,\Psi})^{1/\upalpha k}$, we have

$\displaystyle\mathcal{I}_{\varepsilon}^{1}(\varphi)$
$\displaystyle\textstyle{\leq
C_{1}^{-1}C_{\varphi,\Psi}\varepsilon^{-d/k}\int_{\bar{\operatorname{B}}(0,R^{\prime}u)^{\mathrm{c}}}\exp[-\left\|x\right\|^{\upalpha
k}]\mathrm{d}x}$ (180) $\displaystyle\textstyle{\leq
C_{1}^{-1}C_{\varphi,\Psi}\varepsilon^{-d/k}\int_{\mathbb{R}^{d}}\exp[\left\|x\right\|^{\upalpha
k}/2]\exp[-\left\|x\right\|^{\upalpha
k}]\mathrm{d}x\exp[-(R^{\prime}u)^{\upalpha k}/2]}$ (181) $\displaystyle\leq
2^{d/\upalpha
k}C_{\upalpha}C_{1}^{-1}C_{\varphi,\Psi}\exp[(R^{\prime})^{\upalpha
k}C_{\varphi,\Psi}/2]\varepsilon^{-d/k}\exp[-(R^{\prime})^{\upalpha
k}\mathtt{m}^{k}/(2\varepsilon)]\leq
A_{1}^{1}\varepsilon^{-d/k}\exp[-\beta_{1}^{1}/\varepsilon]\;,$ (182)

with

$A_{1}^{1}=2^{d/\upalpha
k}C_{\upalpha}C_{1}^{-1}C_{\varphi,\Psi}\exp[(R^{\prime})^{\upalpha
k}C_{\varphi,\Psi}/2]\;,\qquad\beta_{1}^{1}=(R^{\prime})^{\upalpha
k}\mathtt{m}^{k}/2\;.$ (183)

  2. (b)

Second, note that
$\mathsf{K}=\mathsf{V}^{\mathrm{c}}\cap\bar{\operatorname{B}}(0,R^{\prime})$
is bounded and closed, i.e. $\mathsf{K}$ is compact. Note that for any
$x\in\mathsf{K}$, $\left\|F(x)\right\|>0$, hence there exists $m>0$ such that
for any $x\in\mathsf{K}$, $\left\|F(x)\right\|\geq m$. In addition, we have
that for any $x\in\mathsf{K}$,

$\left|\varphi(x)\right|\Psi(x)\leq
C_{\varphi,\Psi}\exp[C_{\varphi,\Psi}\left\|x\right\|^{\upalpha k}]\leq
C_{\varphi}\exp[C_{\varphi,\Psi}(R^{\prime})^{\upalpha k}]\;.$ (184)

Therefore, we have

$\mathcal{I}_{\varepsilon}^{2}(\varphi)\leq
C_{1}^{-1}C_{\varphi,\Psi}\varepsilon^{-d/k}\exp[C_{\varphi,\Psi}(R^{\prime})^{\upalpha
k}]\exp[-m^{k}/\varepsilon]\lambda(\mathsf{K})\;,$ (185)

where we recall that $\lambda(\mathsf{K})$ is the Lebesgue measure of
$\mathsf{K}$. Since $\mathsf{K}\subset\bar{\operatorname{B}}(0,R^{\prime})$ we
have

$\mathcal{I}_{\varepsilon}^{2}(\varphi)\leq\uppi^{d/2}(R^{\prime})^{d}\Gamma^{-1}(d/2+1)C_{1}^{-1}C_{\varphi,\Psi}\exp[(R^{\prime})^{\upalpha
k}]\varepsilon^{-d/k}\exp[-m^{k}/\varepsilon]\leq
A_{1}^{2}\varepsilon^{-d/k}\exp[-\beta_{1}^{2}/\varepsilon]\;,$ (186)

where $\Gamma:\ \left(0,+\infty\right)\to\mathbb{R}_{+}$ is given for any
$s\in\left(0,+\infty\right)$ by
$\Gamma(s)=\int_{0}^{+\infty}t^{s-1}\exp[-t]\mathrm{d}t$ and

$A_{1}^{2}=\uppi^{d/2}(R^{\prime})^{d}\Gamma^{-1}(d/2+1)C_{1}^{-1}C_{\varphi,\Psi}\exp[C_{\varphi,\Psi}(R^{\prime})^{\upalpha
k}]\;,\qquad\beta_{1}^{2}=m^{k}\;.$ (187)

We conclude the proof upon combining (182), (186), letting
$\beta_{1}=\min(\beta_{1}^{1},\beta_{1}^{2})$ and $A_{1}=A_{1}^{1}+A_{1}^{2}$.
∎

######  Lemma A.0.

Assume H 1 and that $d\leq p$. Then there exist $N\in\mathbb{N}$,
$\\{x_{0}^{k}\\}_{k=1}^{N}\in(\mathbb{R}^{d})^{N}$ and
$\mathsf{W}_{k}\subset\mathbb{R}^{d}$ open such that for any
$k\in\\{1,\dots,N\\}$, $x_{0}^{k}\in\mathsf{W}_{k}$, $F:\
\bar{\mathsf{W}}_{k}\to F(\bar{\mathsf{W}}_{k})$ is a bi-Lipschitz
homeomorphism, for any $x\in\mathsf{W}_{k}$, $\mathrm{d}F(x)$ is injective and
for any $j\in\\{1,\dots,N\\}$,
$\bar{\mathsf{W}}_{k}\cap\bar{\mathsf{W}}_{j}=\emptyset$. In addition,
$F^{-1}(0)=\cup_{k=1}^{N}\\{x_{0}^{k}\\}$.

###### Proof.

Since, for any $x\in F^{-1}(0)$, $\mathrm{J}F(x)>0$ and $d\leq p$ there exists
$r_{x}>0$ such that for any $y\in\bar{\operatorname{B}}(x,r_{x})$, $F(y)=F(x)$
implies that $y=x$. Since
$\lim_{\left\|x\right\|\to+\infty}\left\|F(x)\right\|=+\infty$ we have that
$F^{-1}(0)$ is compact. Assume that $\mathcal{H}^{0}(F^{-1}(0))=+\infty$.
Then, there exists $(x_{k})_{k\in\mathbb{N}}$ such that for any
$k\in\mathbb{N}$, $F(x_{k})=0$ and for any $j\in\\{0,\dots,k-1\\}$, $x_{j}\neq
x_{k}$. Up to extraction, there exists $x^{\star}\in F^{-1}(0)$ such that
$\lim_{k\to+\infty}x_{k}=x^{\star}$ and for any $k\in\mathbb{N}$, $x_{k}\neq
x^{\star}$. Hence, there exists $k\in\mathbb{N}$ such that
$x_{k}\in\bar{\operatorname{B}}(0,r_{x^{\star}})$ which is absurd. Hence
$\mathcal{H}^{0}(F^{-1}(0))<+\infty$. In what follows, we let
$N=\mathcal{H}^{0}(F^{-1}(0))$ and denote
$\\{x_{0}^{k}\\}_{k=1}^{N}\in(\mathbb{R}^{d})^{N}$ such that
$F^{-1}(0)=\\{x_{0}^{k}\\}_{k=1}^{N}$. There exists
$\\{r_{k}^{\prime}\\}_{k=1}^{N}\in(\mathbb{R}_{+})^{N}$ such that for any
$k,j\in\\{1,\dots,N\\}$,
$\bar{\operatorname{B}}(x_{0}^{k},r_{k}^{\prime})\cap\bar{\operatorname{B}}(x_{0}^{j},r_{j}^{\prime})=\emptyset$.
For any $k\in\\{1,\dots,N\\}$, we let
$\mathsf{V}_{k}=\operatorname{B}(x_{0}^{k},\min(r_{k}^{\prime},r_{x_{0}^{k}}/2))$.
Let $k\in\\{1,\dots,N\\}$. By construction, $F:\ \bar{\mathsf{V}}_{k}\to
F(\bar{\mathsf{V}}_{k})$ is bijective and continuous. Since
$\bar{\mathsf{V}}_{k}$ is compact, we have that $F:\ \bar{\mathsf{V}}_{k}\to
F(\bar{\mathsf{V}}_{k})$ is a homeomorphism. Since for any
$k\in\\{1,\dots,N\\}$, $\mathrm{J}F(x_{0}^{k})>0$, there exists $m>0$ such
that for any $k\in\\{1,\dots,N\\}$ and $v\in\mathbb{R}^{d}$,
$v^{\top}H(x_{0}^{k},x_{0}^{k})v\geq m\|v\|^{2}$ with
$H(x,y)=\mathrm{D}F(x)^{\top}\mathrm{D}F(y)$ for any $x,y\in\mathbb{R}^{d}$.
For any $k\in\\{1,\dots,N\\}$, there exists
$\mathsf{W}_{k}\subset\mathsf{V}_{k}$ such that for any $x,y\in\mathsf{W}_{k}$
we have $\|H(x,y)-H(x_{0}^{k},x_{0}^{k})\|_{2}\leq m/2$. Therefore we have for
any $x,y\in\mathsf{W}_{k}$

$\displaystyle\left\|F(x)-F(y)\right\|^{2}=\textstyle{\int_{0}^{1}\int_{0}^{1}\langle\mathrm{D}F(x+t(y-x))(y-x),\mathrm{D}F(x+s(y-x))(y-x)\rangle\mathrm{d}t\mathrm{d}s}$
(188)
$\displaystyle\qquad\textstyle{=\int_{0}^{1}\int_{0}^{1}(y-x)^{\top}H(x_{s},x_{t})(y-x)\mathrm{d}t\mathrm{d}s}$
(189)
$\displaystyle\qquad\textstyle{=(y-x)^{\top}H(x_{0}^{k},x_{0}^{k})(y-x)+\int_{0}^{1}\int_{0}^{1}(y-x)^{\top}(H(x_{t},x_{s})-H(x_{0}^{k},x_{0}^{k}))(y-x)\mathrm{d}t\mathrm{d}s}$
(190) $\displaystyle\qquad\geq(m/2)\left\|y-x\right\|^{2}\;,$ (191)

where $x_{t}=x+t(y-x)$, which concludes the proof. ∎

###  A.3 Quantitative control of the derivative

######  Lemma A.0.

Under the same assumptions as Section 4.1.1, there exist $\eta>0$ and
$P\in\mathtt{Poly}(4,\mathbb{R}_{+})$ such that for $x\in F^{-1}(0)$ and
$t\in\operatorname{B}_{\infty}(0,\eta)$ we have

$\displaystyle\left\|\mathrm{D}_{t}\chi(t,x)\right\|$
$\displaystyle\leq(1+M_{0,\varphi}+M_{1,\varphi})(1+M_{0,\Psi}+M_{1,\Psi})$
(192) $\displaystyle\quad\times
P(M_{1,F},M_{2,F},M_{3,F},1/m_{1,F})\exp[P(M_{1,F},M_{2,F},M_{3,F},1/m_{1,F})]\;,$
(193)

where we recall that $\chi$ is defined in (84) and for any
$\ell\in\mathbb{N}$, $i\in\\{0,\dots,\ell\\}$ and
$f\in\mathrm{C}^{\ell}(\mathbb{R}^{d},\mathbb{R}^{p})$,
$M_{i,f}=\sup\\{\|\mathrm{D}^{i}f(x)\|\,:\;x\in
F^{-1}(\operatorname{B}_{\infty}(0,\eta))\\}$.

###### Proof.

In this proof, for any $f:\ \mathbb{R}^{m_{0}}\to\mathbb{R}^{m_{1}}$ with
$m_{0},m_{1}\in\mathbb{N}$ differentiable, we denote $\mathrm{d}f$ its
differential. Recall that $\bar{\Phi}_{t}:\ \mathbb{R}^{d}\to\mathbb{R}^{d}$
is defined such that for any $x\in\mathbb{R}^{d}$, $\bar{\Phi}_{t}(x)=x^{(p)}$
with $x^{(0)}=x$ and for any $i\in\\{0,\dots,p-1\\}$,
$x^{(i+1)}=\Phi_{i+1}(t_{i+1},x^{(i)})$, with $\Phi_{i+1}$ given by (81). The
compact sets $\mathsf{K}_{0}$ and $\mathsf{K}_{1}$ are defined in the proof of
Section 4.1.1. Since for any $x\in F^{-1}(0)$,
$|\det(\mathrm{D}\bar{\Phi}_{t}^{-1}(x))|>0$, we assume without loss of
generality that $\det(\mathrm{D}\bar{\Phi}_{t}^{-1}(x))>0$. For ease of
notation we denote $\varphi_{\Psi}=\varphi\times\Psi$. We have

$M_{0,\varphi_{\Psi}}=M_{0,\varphi}M_{0,\Psi}\;,\qquad
M_{1,\varphi_{\Psi}}=M_{0,\varphi}M_{1,\Psi}+M_{1,\varphi}M_{0,\Psi}\;.$ (194)

In addition, we have for any $x\in\mathbb{R}^{d}$ and $t\in\mathsf{K}_{0}$

$\chi(t,x)=\varphi_{\Psi}(\bar{\Phi}_{t}^{-1}(x))\mathrm{J}F(\bar{\Phi}_{t}^{-1}(x))^{-1}\det(\mathrm{D}\bar{\Phi}_{t}^{-1}(x))\;.$
(195)

We now control the first derivative of $\chi$. We divide the rest of the proof
in three steps.

  1. (a)

We start by providing upper bounds for $x\mapsto h_{i,j}(x)$,
$x\mapsto\mathrm{D}h_{i,j}(x)$ and $x\mapsto\mathrm{D}^{2}h_{i,j}(x)$ for any
$i,j\in\\{1,\dots,p\\}$ where we recall that for any $x\in\mathsf{K}_{1}$ we
have

$\\{h_{i,j}(x)\\}_{1\leq i,j\leq
p}=G(x)^{-1}=\mathrm{Adj}(G(x))/\det(G(x))=\mathrm{Adj}(G(x))/\mathrm{J}F(x)^{2}\;,$
(196)

where $\mathrm{Adj}(G(x))$ is the adjugate of $G(x)=\\{\langle\nabla
F_{i}(x),\nabla F_{j}(x)\rangle\\}_{1\leq i,j\leq p}$, where for any
$x\in\mathsf{K}_{1}$ and $i,j\in\\{1,\dots,p\\}$

$\mathrm{Adj}(G(x))=\\{(-1)^{i+j}\det(G^{i,j}(x))\\}_{1\leq i,j\leq p}\;,$
(197)

where for any $i,j\in\\{1,\dots,p\\}$, $\det(G^{i,j}(x))$ is the $(i,j)$ minor
of $G(x)$. Hence, using the Cauchy-Schwarz inequality there exists $D_{0}\geq
0$ such that for any $x\in\mathsf{K}_{1}$ and $i,j\in\\{1,\dots,p\\}$

$\left|h_{i,j}(x)\right|\leq D_{0}M_{1,F}^{2p-2}/m_{1,F}^{2}\;,$ (198)

where we define

$M_{1,F}=\sup\\{\left\|\nabla F_{i}(x)\right\|\,:\;x\in\mathsf{K}_{1},\
i\in\\{1,\dots,p\\}\\}\;,\quad
m_{1,F}=\inf\\{\mathrm{J}F(x)\,:\;x\in\mathsf{K}_{1}\\}\;.$ (199)

Recall that $m_{1,F}>0$ since $\mathsf{K}_{1}$ is compact and for any
$x\in\mathsf{K}_{1}$, $\mathrm{J}F(x)>0$. We have that for any
$x\in\mathsf{K}_{1}$, $u\in\mathbb{R}^{d}$ and $i,j\in\\{1,\dots,p\\}$

$\mathrm{d}h_{i,j}(x)(u)=e_{i}^{\top}\mathrm{d}G(x)^{-1}(u)e_{j}=-e_{i}^{\top}G(x)^{-1}\mathrm{d}G(x)(u)G(x)^{-1}e_{j}\;.$
(200)

In addition, we have for any $x,u\in\mathbb{R}^{d}$ and
$i,j\in\\{1,\dots,p\\}$

$\mathrm{d}G_{i,j}(x)(u)=\mathrm{d}^{2}F_{i}(x)(\nabla
F_{j}(x),u)+\mathrm{d}^{2}F_{j}(x)(\nabla F_{i}(x),u)\;.$ (201)

Combining this result and (200) we get that there exists $C_{1}\geq 0$ such
that for any $x\in\mathsf{K}_{1}$ and $i,j\in\\{1,\dots,p\\}$

$\left\|\mathrm{D}h_{i,j}(x)\right\|\leq
C_{1}M_{1,F}^{4p-3}M_{2,F}/m_{1,F}^{4}\;,$ (202)

where $M_{2,F}=\sup\\{\|\nabla^{2}F_{i}(x)\|\,:\;x\in\mathsf{K}_{1},\
i\in\\{1,\dots,p\\}\\}$. Hence using (79), (198) and (202), there exist
$C_{2},C_{3}\geq 0$ such that for any $x\in\mathsf{K}_{1}$ and
$i\in\\{1,\dots,p\\}$

$\left\|g_{i}(x)\right\|\leq
C_{2}M_{1,F}^{2p-1}/m_{1,F}^{2}\;,\quad\left\|\mathrm{D}g_{i}(x)\right\|\leq
C_{3}M_{2,F}/m_{1,F}^{2}\\{M_{1,F}^{4p-2}/m_{1,F}^{2}+M_{1,F}^{2p-2}\\}\;.$
(203)

Similarly, for any $x\in\mathsf{K}_{1}$, $u,v\in\mathbb{R}^{d}$ and
$i,j\in\\{1,\dots,p\\}$ we have

$\displaystyle\mathrm{d}^{2}h_{i,j}(x)(u,v)$
$\displaystyle=e_{i}^{\top}\mathrm{d}^{2}G(x)^{-1}(u,v)e_{j}$ (204)
$\displaystyle=e_{i}^{\top}\left\\{G(x)^{-1}\mathrm{d}G(x)(u)G(x)^{-1}\mathrm{d}G(x)(v)G(x)^{-1}\right.$
(205)
$\displaystyle\qquad\left.+G(x)^{-1}\mathrm{d}G(x)(v)G(x)^{-1}\mathrm{d}G(x)(u)G(x)^{-1}\right.$
(206)
$\displaystyle\qquad\left.-G(x)^{-1}\mathrm{d}^{2}G(x)(u,v)G(x)^{-1}\right\\}e_{j}\;.$
(207)

We also have for any $x,u,v\in\mathbb{R}^{d}$ and $i,j\in\\{1,\dots,p\\}$

$\displaystyle\mathrm{d}^{2}G_{i,j}(x)$
$\displaystyle=\mathrm{d}^{3}F_{i}(x)(\nabla
F_{j}(x),u,v)+\mathrm{d}^{3}F_{j}(x)(\nabla F_{i}(x),u,v)$ (208)
$\displaystyle\qquad+\mathrm{d}^{2}F_{i}(x)(\mathrm{d}^{2}F_{j}(x)(v),u)+\mathrm{d}^{2}F_{j}(x)(\mathrm{d}^{2}F_{i}(x)(v),u)\;.$
(209)

Hence there exist $P_{3}\in\mathtt{Poly}(4,\mathbb{R}_{+})$ such that for any
$x\in\mathsf{K}_{1}$ we have

$\|\mathrm{D}^{2}h_{i,j}(x)\|\leq P_{3}(M_{1,F},M_{2,F},M_{3,F},1/m_{1,F})\;,$
(210)

where $M_{3,F}=\sup\\{\|\nabla^{3}F_{i}(x)\|\,:\;x\in\mathsf{K}_{1},\
i\in\\{1,\dots,p\\}\\}$. Next, note that for any $i\in\\{1,\dots,p\\}$, we can
choose $g_{i}$ such that
$g_{i}\in\mathrm{C}^{2}(\mathbb{R}^{d},\mathbb{R}^{d})$ (and therefore
$\Phi_{i}\in\mathrm{C}^{3,2}(\mathbb{R}\times\mathbb{R}^{d},\mathbb{R}^{d})$
by Appendix C). Combining this result, (203) and (210), there exist
$P_{1}\in\mathtt{Poly}(2,\mathbb{R}_{+})$,
$P_{2}\in\mathtt{Poly}(3,\mathbb{R}_{+})$ and
$P_{3}\in\mathtt{Poly}(4,\mathbb{R}_{+})$ such that for any
$x\in\mathsf{K}_{1}$ and $i\in\\{1,\dots,p\\}$

$\displaystyle\left\|g_{i}(x)\right\|\leq
P_{1}(M_{1,F},1/m_{1,F})\;,\qquad\left\|\mathrm{D}g_{i}(x)\right\|\leq
P_{2}(M_{1,F},M_{2,F},1/m_{1,F})\;,$ (211)
$\displaystyle\|\mathrm{D}^{2}g_{i}(x)\|\leq
P_{3}(M_{1,F},M_{2,F},M_{3,F},1/m_{1,F})\;.$ (212)

  2. (b)

For any $x\in\mathsf{K}_{1}$, $i\in\\{1,\dots,p\\}$ and $t_{i}\in\mathbb{R}$
such that $\Phi_{i}(t_{i},x)\in\mathsf{K}_{1}$ we have $\left|t_{i}\right|\leq
2\eta$, since $F_{i}(\Phi_{i}(t_{i},x))=F_{i}(x)-t_{i}$. Hence combining this
result, the fact that for any $x\in\mathbb{R}^{d}$, $\Phi_{i}(0,x)=x$, (81)
and (203), for any $x\in\mathsf{K}_{1}$, $i\in\\{1,\dots,p\\}$ and
$t_{i}\in\mathbb{R}$ such that $\Phi_{i}(t_{i},x)\in\mathsf{K}_{1}$ we have

$\textstyle{\left\|\Phi_{i}(t_{i},x)\right\|\leq\left\|x\right\|+\int_{0}^{t_{i}}\left\|\mathrm{D}_{t}\Phi_{i}(s,x)\right\|\mathrm{d}s\leq\left\|x\right\|+2\eta
C_{2}M_{1,F}^{2p-1}/m_{1,F}^{2}\;.}$ (213)

Therefore, there exists $C_{4}\geq 0$ such that for any $x\in\mathsf{K}_{1}$,
$i\in\\{1,\dots,p\\}$ and $t_{i}\in\mathbb{R}$ such that
$\Phi_{i}(t_{i},x)\in\mathsf{K}_{1}$ we have

$\left\|\Phi_{i}(t_{i},x)\right\|\leq
C_{4}(1+M_{1,F}^{2p-1}/m_{1,F}^{2})\;,\quad\left\|\mathrm{D}_{t}\Phi_{i}(t_{i},x)\right\|\leq
C_{2}M_{1,F}^{2p-1}/m_{1,F}^{2}\;.$ (214)

Similarly, for any $x\in\mathsf{K}_{1}$, $i\in\\{1,\dots,p\\}$ and
$t_{i}\in\mathbb{R}$ such that $\Phi_{i}(t_{i},x)\in\mathsf{K}_{1}$ we have

$\displaystyle\left\|\mathrm{D}_{x}\Phi_{i}(t_{i},x)\right\|$
$\displaystyle\textstyle{\leq\left\|\operatorname{Id}\right\|+\int_{0}^{t_{i}}\left\|\mathrm{D}_{t,x}\Phi_{i}(s,x)\right\|\left\|\mathrm{D}_{x}\Phi_{i}(s,x)\right\|\mathrm{d}s}$
(215)
$\displaystyle\textstyle{\leq\left\|\operatorname{Id}\right\|+\int_{0}^{t_{i}}\left\|\mathrm{D}g_{i}(\Phi_{i}(t_{i},x))\right\|\left\|\mathrm{D}_{x}\Phi_{i}(s,x)\right\|\mathrm{d}s}$
(216)
$\displaystyle\textstyle{\leq\left\|\operatorname{Id}\right\|+C_{3}(M_{2,F}/m_{1,F}^{2})\\{M_{1,F}^{4p-2}/m_{1,F}^{2}+M_{1,F}^{2p-2}\\}\int_{0}^{t_{i}}\left\|\mathrm{D}_{x}\Phi_{i}(t_{i},x)\right\|\mathrm{d}s\;.}$
(217)

Hence, using Grönwall’s lemma, for any $x\in\mathsf{K}_{1}$,
$i\in\\{1,\dots,p\\}$ and $t_{i}\in\mathbb{R}$ such that
$\Phi_{i}(t_{i},x)\in\mathsf{K}_{1}$ we have

$\left\|\mathrm{D}_{x}\Phi_{i}(t_{i},x)\right\|\leq\left\|\operatorname{Id}\right\|\exp[2\eta
C_{3}(M_{2,F}/m_{1,F}^{2})\\{M_{1,F}^{4p-2}/m_{1,F}^{2}+M_{1,F}^{2p-2}\\}]\;.$
(218)

Therefore, there exists $C_{5}\geq 0$ such that for any $x\in\mathsf{K}_{1}$,
$i\in\\{1,\dots,p\\}$ and $t_{i}\in\mathbb{R}$ with
$\Phi_{i}(t_{i},x)\in\mathsf{K}_{1}$ we have

$\left\|\mathrm{D}_{x}\Phi_{i}(t_{i},x)\right\|\leq
C_{5}\exp[C_{5}(M_{2,F}/m_{1,F}^{2})\\{M_{1,F}^{4p-2}/m_{1,F}^{2}+M_{1,F}^{2p-2}\\}]\;.$
(219)

Using this result, (81) and (211), there exists $C_{6}\geq 0$ such that for
any $x\in\mathsf{K}_{1}$, $i\in\\{1,\dots,p\\}$ and $t_{i}\in\mathbb{R}$ with
$\Phi_{i}(t_{i},x)\in\mathsf{K}_{1}$ we have

$\left\|\mathrm{D}_{t,x}\Phi_{i}(t_{i},x)\right\|\leq
C_{6}\exp[C_{6}(M_{2,F}/m_{1,F}^{2})\\{M_{1,F}^{4p-2}/m_{1,F}^{2}+M_{1,F}^{2p-2}\\}]\;.$
(220)

Hence combining (214), (219) and (220), there exist
$P_{4}\in\mathtt{Poly}(2,\mathbb{R}_{+})$ and
$P_{5}\in\mathtt{Poly}(3,\mathbb{R}_{+})$ such that for any
$x\in\mathsf{K}_{1}$, $i\in\\{1,\dots,p\\}$, $t_{i}\in\mathbb{R}$ with
$\Phi_{i}(t_{i},x)\in\mathsf{K}_{1}$ we have

$\displaystyle\|\mathrm{D}_{t}\Phi_{i}(t_{i},x)\|+\|\Phi_{i}(t_{i},x)\|\leq
P_{4}(M_{1,F},1/m_{1,F})\;,$ (221)
$\displaystyle\|\mathrm{D}_{x}\Phi_{i}(t_{i},x)\|+\|\mathrm{D}_{t,x}\Phi_{i}(t_{i},x)\|\leq
P_{5}(M_{1,F},M_{2,F},1/m_{1,F})\exp[P_{5}(M_{1,F},M_{2,F},1/m_{1,F})]\;.$
(222)

In addition, using (81) we have for any $i\in\\{1,\dots,p\\}$,
$s\in\mathbb{R}$, $x\in\mathbb{R}^{d}$

$\mathrm{D}_{t}\mathrm{D}_{x}^{2}\Phi_{i}(s,x)=\mathrm{D}g_{i}(\Phi_{i}(s,x))\mathrm{D}_{x}^{2}\Phi_{i}(s,x)+\mathrm{D}^{2}g_{i}(\Phi_{i}(s,x))\mathrm{D}_{x}\Phi_{i}(s,x)\;.$
(223)

Therefore, using (211) and (221), we have for any $x\in\mathsf{K}_{1}$,
$i\in\\{1,\dots,p\\}$ and $t_{i}\in\mathbb{R}$ such that
$\Phi_{i}(t_{i},x)\in\mathsf{K}_{1}$

$\displaystyle\textstyle{\|\mathrm{D}_{x}^{2}\Phi_{i}(t_{i},x)\|\leq\int_{0}^{t_{i}}\\{\|\mathrm{D}g_{i}(\Phi_{i}(s,x))\|\|\mathrm{D}_{x}^{2}\Phi_{i}(s,x)\|+\|\mathrm{D}^{2}g_{i}(\Phi_{i}(s,x))\|\|\mathrm{D}_{x}\Phi_{i}(s,x)\|\\}\mathrm{d}s}$
(224) $\displaystyle\leq 2\eta
P_{3}(M_{1,F},M_{2,F},M_{3,F},1/m_{1,F})P_{5}(M_{1,F},M_{2,F},1/m_{1,F})\exp[P_{5}(M_{1,F},M_{2,F},1/m_{1,F})]$
(225)
$\displaystyle\qquad\textstyle{+P_{2}(M_{1,F},M_{2,F},1/m_{1,F})\int_{0}^{t_{i}}\|\mathrm{D}_{x}^{2}\Phi_{i}(s,x)\|\mathrm{d}s\;.}$
(226)

Hence, using Grönwall’s lemma, there exists
$P_{6}\in\mathtt{Poly}(4,\mathbb{R}_{+})$ such that for any
$x\in\mathsf{K}_{1}$, $i\in\\{1,\dots,p\\}$ and $t_{i}\in\mathbb{R}$ such that
$\Phi_{i}(t_{i},x)\in\mathsf{K}_{1}$ we have

$\|\mathrm{D}_{x}^{2}\Phi_{i}(t_{i},x)\|\leq
P_{6}(M_{1,F},M_{2,F},M_{3,F},1/m_{1,F})\exp[P_{6}(M_{1,F},M_{2,F},M_{3,F},1/m_{1,F})]\;.$
(227)

Hence, combining this result and (221), we have for any $x\in\mathsf{K}_{1}$,
$i\in\\{1,\dots,p\\}$ and $t_{i}\in\mathbb{R}$ such that
$\Phi_{i}(t_{i},x)\in\mathsf{K}_{1}$

$\displaystyle\|\mathrm{D}_{t}\Phi_{i}(t_{i},x)\|+\|\Phi_{i}(t_{i},x)\|\leq
P_{4}(M_{1,F},1/m_{1,F})\;,$ (228)
$\displaystyle\|\mathrm{D}_{x}\Phi_{i}(t_{i},x)\|+\|\mathrm{D}_{t,x}\Phi_{i}(t_{i},x)\|\leq
P_{5}(M_{1,F},M_{2,F},1/m_{1,F})\exp[P_{5}(M_{1,F},M_{2,F},1/m_{1,F})]\;,$
(229) $\displaystyle\|\mathrm{D}_{x}^{2}\Phi_{i}(t_{i},x)\|\leq
P_{6}(M_{1,F},M_{2,F},M_{3,F},1/m_{1,F})\exp[P_{6}(M_{1,F},M_{2,F},M_{3,F},1/m_{1,F})]\;.$
(230)

  3. (c)

In what follows, we fix $t\in\operatorname{B}_{\infty}(0,\eta)$ and use (230)
to provide uniform bounds for $\mathrm{D}_{t}\bar{\Phi}^{-1}_{t}$,
$\mathrm{D}_{t}\mathrm{D}\bar{\Phi}^{-1}_{t}$ and
$\mathrm{D}\bar{\Phi}^{-1}_{t}$ on $F^{-1}(0)$. We introduce
$\\{\bar{\Phi}_{i,t}^{-1}\\}_{i=1}^{p}$ such that for any $x\in F^{-1}(0)$ and
$i\in\\{1,\dots,p\\}$

$\bar{\Phi}_{t,i}^{-1}(x)=\Phi_{i}(-t_{i},\bar{\Phi}_{t,i+1}^{-1}(x))\;,\qquad\bar{\Phi}_{t,p+1}^{-1}(x)=x\;.$
(231)

Note that $\bar{\Phi}_{t}^{-1}=\bar{\Phi}_{t,1}^{-1}$. Let
$j\in\\{1,\dots,p\\}$. For any $i\in\\{1,\dots,p\\}$ we distinguish three
cases:

    1. (i)

$i>j$, then for any $x\in F^{-1}(0)$,
$\mathrm{D}_{t_{j}}\bar{\Phi}_{t,i}^{-1}(x)=0$.

    2. (ii)

$i=j$, then for any $x\in F^{-1}(0)$,
$\mathrm{D}_{t_{j}}\bar{\Phi}_{t,i}^{-1}(x)=-\mathrm{D}_{t}\Phi_{j}(-t_{j},\bar{\Phi}_{t,j+1}^{-1}(x))$.

    3. (iii)

$i<j$, then for any $x\in F^{-1}(0)$,
$\mathrm{D}_{t_{j}}\bar{\Phi}_{t,i}^{-1}(x)=\mathrm{D}_{x}\Phi_{i}(-t_{i},\bar{\Phi}_{t,i+1}^{-1}(x))\mathrm{D}_{t_{j}}\bar{\Phi}_{t,i+1}^{-1}(x)$.

Combining these results, (230) and the fact that
$\bar{\Phi}_{t}^{-1}=\bar{\Phi}_{t,1}^{-1}$ we get that there exists
$\bar{P}_{7}\in\mathtt{Poly}(3,\mathbb{R}_{+})$ such that for any $x\in
F^{-1}(0)$, and $j\in\\{1,\dots,p\\}$ we have

$\|\mathrm{D}_{t_{j}}\bar{\Phi}_{t}^{-1}(x)\|\leq\bar{P}_{7}(M_{1,F},M_{2,F},1/m_{1,F})\exp[\bar{P}_{7}(M_{1,F},M_{2,F},1/m_{1,F})]\;.$
(232)

Hence, there exists $P_{7}\in\mathtt{Poly}(3,\mathbb{R}_{+})$ such that for
any $x\in F^{-1}(0)$ we have

$\|\mathrm{D}_{t}\bar{\Phi}_{t}^{-1}(x)\|\leq
P_{7}(M_{1,F},M_{2,F},1/m_{1,F})\exp[P_{7}(M_{1,F},M_{2,F},1/m_{1,F})]\;.$
(233)

Next, note that for any $x\in F^{-1}(0)$,
$t\in\operatorname{B}_{\infty}(0,\eta)$ and $i\in\\{1,\dots,p\\}$ we have

$\mathrm{D}\bar{\Phi}_{t,i}^{-1}(x)=\mathrm{D}_{x}\Phi_{i}(-t_{i},\bar{\Phi}_{t,i+1}^{-1}(x))\mathrm{D}\bar{\Phi}_{t,i+1}^{-1}(x)\;,\qquad\mathrm{D}\bar{\Phi}_{t,p+1}^{-1}(x)=\operatorname{Id}\;.$
(234)

Combining this result and (230), there exists
$P_{8}\in\mathtt{Poly}(3,\mathbb{R}_{+})$ such that for any $x\in F^{-1}(0)$
and $i\in\\{1,\dots,p\\}$

$\|\mathrm{D}\bar{\Phi}_{t,i}^{-1}(x)\|\leq
P_{8}(M_{1,F},M_{2,F},1/m_{1,F})\exp[P_{8}(M_{1,F},M_{2,F},1/m_{1,F})]\;.$
(235)

In particular, since $\bar{\Phi}_{t}=\bar{\Phi}_{t,1}$ we have for any $x\in
F^{-1}(0)$

$\|\mathrm{D}\bar{\Phi}_{t}^{-1}(x)\|\leq
P_{8}(M_{1,F},M_{2,F},1/m_{1,F})\exp[P_{8}(M_{1,F},M_{2,F},1/m_{1,F})]\;.$
(236)

Finally, we give a uniform upper-bound on
$\mathrm{D}_{t}\mathrm{D}\bar{\Phi}_{t}^{-1}$. Let $j\in\\{1,\dots,p\\}$. For
any $i\in\\{1,\dots,p\\}$ we distinguish three cases:

    1. (i)

$i>j$, then for any $x\in F^{-1}(0)$,
$\mathrm{D}_{t_{j}}\mathrm{D}\bar{\Phi}_{t,i}^{-1}(x)=0$.

    2. (ii)

$i=j$, then for any $x\in F^{-1}(0)$,
$\mathrm{D}_{t_{j}}\mathrm{D}\bar{\Phi}_{t,i}^{-1}(x)=-\mathrm{D}_{t,x}\Phi_{j}(-t_{j},\bar{\Phi}_{t,j+1}^{-1}(x))\mathrm{D}\bar{\Phi}_{t,j+1}^{-1}(x)$.

    3. (iii)

$i<j$, then for any $x\in F^{-1}(0)$, we have

$\displaystyle\mathrm{D}_{t_{j}}\mathrm{D}\bar{\Phi}_{t,i}^{-1}(x)$
$\displaystyle=\mathrm{D}_{x}\Phi_{i}(-t_{i},\bar{\Phi}_{t,i+1}^{-1}(x))\mathrm{D}_{t_{j}}\mathrm{D}\bar{\Phi}_{t,i+1}^{-1}(x)$
(237)
$\displaystyle\qquad+\mathrm{D}_{x}^{2}\Phi_{i}(-t_{i},\bar{\Phi}_{t,i+1}^{-1}(x))(\mathrm{D}\bar{\Phi}_{t,i+1}^{-1}(x),\mathrm{D}_{t_{j}}\mathrm{D}\bar{\Phi}_{t,i+1}^{-1}(x))\;.$
(238)

Using (230) and (235), there exists $P_{9}\in\mathtt{Poly}(4,\mathbb{R}_{+})$
such that for any $x\in F^{-1}(0)$ we have

$\|\mathrm{D}_{t}\mathrm{D}\bar{\Phi}_{t}^{-1}(x)\|\leq
P_{9}(M_{1,F},M_{2,F},M_{3,F},1/m_{1,F})\exp[P_{9}(M_{1,F},M_{2,F},M_{3,F},1/m_{1,F})]\;.$
(239)

Therefore, summarizing (233), (236) and (239), there exist
$P_{7},P_{8}\in\mathtt{Poly}(\mathbb{R}_{+},3)$ and
$P_{9}\in\mathtt{Poly}(4,\mathbb{R}_{+})$ such that for any $x\in F^{-1}(0)$
and $t\in\operatorname{B}_{\infty}(0,\eta)$ we have

$\displaystyle\|\mathrm{D}_{t}\bar{\Phi}_{t}^{-1}(x)\|\leq
P_{7}(M_{1,F},M_{2,F},1/m_{1,F})\exp[P_{7}(M_{1,F},M_{2,F},1/m_{1,F})]\;,$
(240) $\displaystyle\|\mathrm{D}\bar{\Phi}_{t}^{-1}(x)\|\leq
P_{8}(M_{1,F},M_{2,F},1/m_{1,F})\exp[P_{8}(M_{1,F},M_{2,F},1/m_{1,F})]\;,$
(241) $\displaystyle\|\mathrm{D}_{t}\mathrm{D}\bar{\Phi}_{t}^{-1}(x)\|\leq
P_{9}(M_{1,F},M_{2,F},M_{3,F},1/m_{1,F})\exp[P_{9}(M_{1,F},M_{2,F},M_{3,F},1/m_{1,F})]\;.$
(242)

  4. (d)

Next, we use (240) to conclude the proof by providing uniform upper-bounds on
the differential $\chi$ on $F^{-1}(0)$ w.r.t. $t$, where $\chi$ given in
(195). For any $x\in F^{-1}(0)$ and $t\in\operatorname{B}_{\infty}(0,\eta)$ we
have

$\displaystyle\mathrm{D}_{t}\chi(t,x)$
$\displaystyle=\mathrm{D}_{t}\varphi_{\Psi}(\bar{\Phi}_{t}^{-1}(x))\mathrm{J}F(\bar{\Phi}_{t}^{-1}(t))^{-1}\det(G(\bar{\Phi}_{t}^{-1}(x)))$
(243)
$\displaystyle\qquad+\varphi_{\Psi}(\bar{\Phi}_{t}^{-1}(x))\mathrm{D}_{t}\mathrm{J}F(\bar{\Phi}_{t}^{-1}(t))^{-1}\det(G(\bar{\Phi}_{t}^{-1}(x)))$
(244)
$\displaystyle\qquad+\varphi_{\Psi}(\bar{\Phi}_{t}^{-1}(x))\mathrm{J}F(\bar{\Phi}_{t}^{-1}(t))^{-1}\mathrm{D}_{t}\det(G(\bar{\Phi}_{t}^{-1}(x)))\;.$
(245)

First, for any $x\in F^{-1}(0)$ and $t\in\operatorname{B}_{\infty}(0,\eta)$ we
have

$|\varphi_{\Psi}(\bar{\Phi}_{t}^{-1}(x))|\leq
M_{0,\varphi_{\Psi}}\;,\qquad\mathrm{J}F(\bar{\Phi}_{t}^{-1}(x))^{-1}\leq
1/m_{1,F}\;.$ (246)

In addition, using (240), there exists
$P_{10}\in\mathtt{Poly}(3,\mathbb{R}_{+})$ such that for any
$x\in\mathsf{K}_{1}$ and $t\in\operatorname{B}_{\infty}(0,\eta)$

$|\det(\mathrm{D}\bar{\Phi}_{t}^{-1}(x))|\leq
P_{10}(M_{1,F},M_{2,F},1/m_{1,F})\exp[P_{10}(M_{1,F},M_{2,F},1/m_{1,F})]\;.$
(247)

For any $x\in F^{-1}(0)$ and $t\in\operatorname{B}_{\infty}(0,\eta)$ we have

$\mathrm{D}_{t}\varphi_{\Psi}(\bar{\Phi}_{t}^{-1}(x))=\mathrm{D}\varphi_{\Psi}(\bar{\Phi}_{t}^{-1}(x))\mathrm{D}_{t}\bar{\Phi}_{t}^{-1}(x)\;.$
(248)

Combining this result and (240), there exists
$P_{11}\in\mathtt{Poly}(3,\mathbb{R}_{+})$ such that for any $x\in F^{-1}(0)$
and $t\in\operatorname{B}_{\infty}(0,\eta)$ we have

$\|\mathrm{D}_{t}\varphi(\bar{\Phi}_{t}^{-1}(x))\|\leq
M_{1,\varphi_{\Psi}}P_{11}(,M_{1,F},M_{2,F},1/m_{1,F})\exp[P_{11}(,M_{1,F},M_{2,F},1/m_{1,F})]\;.$
(249)

For any $x\in F^{-1}(0)$, $t\in\operatorname{B}_{\infty}(0,\eta)$ we have

$\mathrm{D}_{t}\mathrm{J}F(\bar{\Phi}_{t}^{-1}(x))^{-1}=\mathrm{D}_{t}\det(G(\bar{\Phi}_{t}^{-1}(x)))^{1/2}=\mathrm{J}F(\bar{\Phi}_{t}^{-1}(x))^{-1}\mathrm{D}_{t}\det(G(\bar{\Phi}_{t}^{-1}(x))^{-1})\;.$
(250)

In addition, for any $x\in F^{-1}(0)$, $t\in\operatorname{B}_{\infty}(0,\eta)$
and $h\in\mathbb{R}^{p}$ we have

$\mathrm{D}_{t}\det(G(\bar{\Phi}_{t}^{-1}(x)))(h)=\operatorname{Tr}(\mathrm{Adj}(G(\bar{\Phi}_{t}^{-1}(x)))\mathrm{D}G(\bar{\Phi}_{t}^{-1}(x))(\mathrm{D}_{t}\bar{\Phi}_{t}^{-1}(x)(h)))\;.$
(251)

Using this result, (250), (201) and (240), there exists
$P_{12}\in\mathtt{Poly}(3,\mathbb{R}_{+})$ such that for any $x\in F^{-1}(0)$,
$t\in\operatorname{B}_{\infty}(0,\eta)$ we have

$\|\mathrm{D}_{t}\mathrm{J}F(\bar{\Phi}_{t}^{-1}(x))^{-1}\|\leq
P_{12}(M_{1,F},M_{2,F},1/m_{1,F})\exp[P_{12}(M_{1,F},M_{2,F},1/m_{1,F})]\;.$
(252)

Finally, for any $x\in F^{-1}(0)$, $t\in\operatorname{B}_{\infty}(0,\eta)$ and
$h\in\mathbb{R}^{p}$ we have

$\mathrm{D}_{t}\det(\mathrm{D}\bar{\Phi}_{t}^{-1}(x))=\operatorname{Tr}(\mathrm{Adj}(\mathrm{D}\bar{\Phi}_{t}^{-1}(x))\mathrm{D}_{t}\mathrm{D}\bar{\Phi}_{t}^{-1}(x)(h))\;.$
(253)

Hence, using (240), there exists $P_{13}\in\mathtt{Poly}(4,\mathbb{R}_{+})$
such that for any $x\in F^{-1}(0)$ and $t\in\operatorname{B}_{\infty}(0,\eta)$
we have

$\displaystyle\|\mathrm{D}_{t}\det(\mathrm{D}\bar{\Phi}_{t}^{-1}(x))\|$
$\displaystyle\leq P_{13}(M_{1,F},M_{2,F},M_{3,F},1/m_{1,F})$ (254)
$\displaystyle\qquad\times\exp[P_{13}(M_{1,F},M_{2,F},M_{3,F},1/m_{1,F})]\;.$
(255)

We conclude the proof upon combining (243), (246), (247), (249), (252) and
(254). ∎

###  A.4 Regularity results

In the following section we prove a smoothing lemma which is key to extend
Section 4.1.3 to the case where $\Psi$ and $\varphi$ are no longer in
$\mathrm{C}^{1}(\bar{\mathsf{U}},\mathbb{R})$ but are Lipschitz continuous.

######  Lemma A.0.

Let $\mathsf{U}$ be open bounded, $r>0$,
$\mathsf{V}=\mathsf{U}+\operatorname{B}(0,r)$ and
$\varphi\in\mathrm{C}(\mathbb{R}^{d},\mathbb{R})$ which satisfies (18) and is
Lipschitz on $\mathsf{V}$, i.e. there exists $M_{1,\varphi}\geq 0$ such that
for any $x,y\in\mathsf{V}$ we have

$\left|\varphi(x)-\varphi(y)\right|\leq M_{1,\varphi}\left\|x-y\right\|\;.$
(256)

In addition, let $M_{0,\varphi}=\sup\\{|\varphi(x)|\,:\;x\in\mathsf{V}\\}$.
Then there exist $\bar{\delta}>0$ and
$\\{\varphi^{\delta}\,:\;\delta\in(0,\bar{\delta})\\}$ such that the following
hold:

  1. (a)

For any $x\in\mathsf{U}$, $\lim_{\delta\to 0}\varphi^{\delta}(x)=\varphi(x)$.

  2. (b)

For any $\delta\in(0,\bar{\delta})$,
$\varphi^{\delta}\in\mathrm{C}^{1}(\mathbb{R}^{d},\mathbb{R})$ and there
exists $\mathtt{L}_{\delta}\geq 0$ such that for any $x,y\in\mathsf{U}$,

$\left|\varphi^{\delta}(x)-\varphi^{\delta}(y)\right|\leq\mathtt{L}_{\delta}\left\|x-y\right\|\;.$
(257)

Let
$\mathtt{L}_{0}=\sup\\{\mathtt{L}_{\delta}\,:\;\delta\in(0,\bar{\delta})\\}<+\infty$
and we have $\mathtt{L}_{0}\leq
C_{1}(1+M_{0,\varphi}+M_{1,\varphi})(1+C_{\varphi})$, with $C_{1}\geq 0$
independent from $\varphi$.

  3. (c)

For any $\delta\in(0,\bar{\delta})$ there exists $M_{\delta}\geq 0$ such that
for any $x\in\mathsf{U}$, $|\varphi^{\delta}(x)|\leq M_{\delta}$ and
$M_{0}=\sup\\{M_{\delta}\,:\;\delta\in(0,\bar{\delta})\\}<+\infty$. In
addition, $M_{0}\leq C_{2}(1+M_{0,\varphi})(1+C_{\varphi})$ with $C_{2}\geq 0$
independent from $\varphi$.

  4. (d)

For any $\delta\in(0,\bar{\delta})$ there exists $D_{\delta}\geq 0$ such that
for any $x\in\mathbb{R}^{d}$ we have

$|\varphi^{\delta}(x)|\leq D_{\delta}\exp[D_{\delta}\left\|x\right\|^{\upalpha
k}]\;,$ (258)

and $D_{\delta}\leq C_{3}(1+C_{\varphi})\exp[C_{3}C_{\varphi}]$ with $C_{3}$
independent from $\varphi$.

  5. (e)

Assume H 1, that $F^{-1}(0)\subset\mathsf{U}$ and that there exists
$m_{0,\varphi}>0$ such that for any $x\in F^{-1}(0)$, $\varphi(x)\geq
m_{0,\varphi}$. Then there exists $\bar{\delta}^{\star}>0$ such that for any
$\delta\in(0,\bar{\delta}^{\star})$ and $x\in F^{-1}(0)$,
$\varphi^{\delta}(x)\geq m_{0,\varphi}/2$. In addition if $\varphi\geq 0$ then
for any $\delta>0$, $\varphi^{\delta}\geq 0$.

In addition, $\bar{\delta}=f_{1}(C_{\varphi})$ and
$\bar{\delta}^{\star}=f_{2}(M_{0,\varphi},M_{1,\varphi},1/m_{0,\varphi},C_{\varphi})$
with $f_{1}\in\mathrm{C}(\mathbb{R}_{+},\mathbb{R}_{+})$ and
$f_{2}\in\mathrm{C}(\mathbb{R}_{+}^{4},\mathbb{R}_{+})$ and are non-decreasing
w.r.t to each of their variables. Finally, there exists $C_{4}\geq 0$ such
that

$\limsup_{\delta\to 0}M_{\delta}\leq
C_{4}M_{0,\varphi}\;,\quad\limsup_{\delta\to 0}\mathtt{L}_{\delta}\leq
C_{4}M_{1,\varphi}\;,\quad\limsup_{\delta\to 0}D_{\delta}\leq
C_{4}C_{\varphi}\;,$ (259)

with $C_{4}$ independent from $\varphi$.

###### Proof.

Since $\mathsf{U}$ is bounded there exists $R\geq 0$ such that
$\mathsf{U}\subset\bar{\operatorname{B}}(0,R)$. Let $p\in\mathbb{N}^{\star}$
such that $2p>\upalpha k$ and for any $\delta>0$ define $k_{\delta}:\
\mathbb{R}^{d}\to\mathbb{R}_{+}$ such that for any $x\in\mathbb{R}^{d}$ we
have

$\textstyle{k_{\delta}(x)=\exp[-\left\|x\right\|^{2p}/\delta]/\int_{\mathbb{R}^{d}}\exp[-\left\|\tilde{x}\right\|^{2p}/\delta]\mathrm{d}\tilde{x}\;.}$
(260)

Let
$C_{1}=\int_{\mathbb{R}^{d}}\exp[-\left\|\tilde{x}\right\|^{2p}]\mathrm{d}\tilde{x}$
and we have for any $x\in\mathbb{R}^{d}$,
$k_{\delta}(x)=C_{1}^{-1}\delta^{-d/2p}\exp[-\left\|x\right\|^{2p}/\delta]$.
Note that since $2p>\upalpha k$ and using (18) we have that for any
$x\in\mathbb{R}^{d}$
$\int_{\mathbb{R}^{d}}|\varphi(x-y)|k_{\delta}(y)\mathrm{d}y<+\infty$. For any
$\delta>0$ we define $\varphi^{\delta}$ such that for any $x\in\mathbb{R}^{d}$

$\textstyle{\varphi^{\delta}(x)=\int_{\mathbb{R}^{d}}\varphi(x-y)k_{\delta}(y)\mathrm{d}y\;.}$
(261)

We divide the rest of the proof into five parts.

  1. (a)

We have that for any $x\in\mathbb{R}^{d}$, $\lim_{\delta\to
0}\varphi^{\delta}(x)=\varphi(x)$, since
$\varphi\in\mathrm{C}(\mathbb{R}^{d},\mathbb{R})$ and
$\\{k_{\delta}\,:\;\delta>0\\}$ is a mollifier. This concludes the proof of
Section A.4-(a).

  2. (b)

Using that $2p>\upalpha k$ and (18) we have that for any
$\upbeta\in\mathbb{N}$,
$\int_{\mathbb{R}^{d}}\left\|y\right\|^{\upbeta}|\varphi(x-y)|k_{\delta}(y)<+\infty$.
Hence, $\varphi^{\delta}\in\mathrm{C}^{\infty}(\mathbb{R}^{d},\mathbb{R})$.
Let $x\in\mathsf{U}$. Using that
$\int_{\mathbb{R}^{d}}y\|y\|^{2p-1}k_{\delta}(y)\mathrm{d}y=0$, we have that
for any $\delta>0$

$\displaystyle\textstyle{\|\nabla\varphi^{\delta}(x)\|=(2p/\delta)\|\int_{\mathbb{R}^{d}}\varphi(x-y)y\|y\|^{2p-2}k_{\delta}(y)\mathrm{d}y\|}$
(262)
$\displaystyle\quad\textstyle{=(2p/\delta)\int_{\mathbb{R}^{d}}|\varphi(x-y)-\varphi(x)|\|y\|^{2p-1}k_{\delta}(y)\mathrm{d}y}$
(263)
$\displaystyle\quad\textstyle{\leq(2pM_{1,\varphi}/\delta)\int_{\mathbb{R}^{d}}\|y\|^{2p}k_{\delta}(y)\mathrm{d}y+(2p/\delta)\int_{\bar{\operatorname{B}}(0,r/2)^{\mathrm{c}}}|\varphi(x-y)-\varphi(x)|\|y\|^{2p-1}k_{\delta}(y)\mathrm{d}y}$
(264)
$\displaystyle\quad\textstyle{\leq(2pM_{1,\varphi}/C_{1})\int_{\mathbb{R}^{d}}\|y\|^{2p}\exp[-\|y\|^{2p}]\mathrm{d}y+(2p/\delta)\int_{\bar{\operatorname{B}}(0,r/2)^{\mathrm{c}}}|\varphi(x-y)-\varphi(x)|\|y\|^{2p-1}k_{\delta}(y)\mathrm{d}y\;.}$
(265)

We now bound the second term. Let $\bar{\upalpha}_{k}=\lceil\upalpha k\rceil$.
Using (18), we have for any and $x\in\mathsf{U}$ and $y\in\mathbb{R}^{d}$

$\displaystyle|\varphi(x-y)-\varphi(x)|$
$\displaystyle\leq(1+M_{0,\varphi})C_{\varphi}\exp[C_{\varphi}\left\|x-y\right\|^{\upalpha
k}]$ (266)
$\displaystyle\leq(1+M_{0,\varphi})C_{\varphi}\exp[3^{\bar{\upalpha}_{k}-1}C_{\varphi}(1+\|x\|^{\bar{\upalpha}_{k}}+\|y\|^{\bar{\upalpha}_{k}})]$
(267)
$\displaystyle\leq(1+M_{0,\varphi})C_{\varphi}\exp[3^{\bar{\upalpha}_{k}-1}C_{\varphi}(1+R^{\bar{\upalpha}_{k}})]\exp[3^{\bar{\upalpha}_{k}-1}C_{\varphi}\|y\|^{\bar{\upalpha}_{k}}]\;.$
(268)

Therefore we get that for any $x\in\mathsf{U}$ and
$y\in\bar{\operatorname{B}}(0,r/2)^{\mathrm{c}}$

$|\varphi(x-y)-\varphi(x)|\left\|y\right\|^{2p-1}\leq(1+M_{0,\varphi})C_{\varphi}\exp[3^{\bar{\upalpha}_{k}-1}C_{\varphi}(1+R^{\bar{\upalpha}_{k}})]\exp[(3^{\bar{\upalpha}_{k}}(r/2)^{\bar{\upalpha}_{k}-2p}C_{\varphi}+1)\left\|y\right\|^{2p}]\;.$
(269)

Therefore there exists $C_{a}\geq 0$ such that for any $x\in\mathsf{U}$ and
$y\in\mathbb{R}^{d}$ we have

$|\varphi(x-y)-\varphi(x)|\left\|y\right\|^{2p-1}\leq
C_{a}(1+M_{0,\varphi})C_{\varphi}\exp[C_{a}(1+C_{\varphi})\left\|y\right\|^{2p}]\;,$
(270)

with $C_{a}\geq 0$ independent from $\varphi$. Using this result, we have for
any $x\in\mathsf{U}$ and $\delta\in\left(0,1/(2C_{a}(1+C_{\varphi}))\right)$

$\displaystyle\textstyle{(2p/\delta)\int_{\bar{\operatorname{B}}(0,r/2)^{\mathrm{c}}}|\varphi(x-y)-\varphi(x)|\left\|y\right\|^{2p-1}k_{\delta}(y)\mathrm{d}y}$
(271) $\displaystyle\qquad\qquad\qquad\textstyle{\leq
2pC_{a}(1+M_{0,\varphi})C_{\varphi}\delta^{-d/2p-1}C_{1}^{-1}\int_{\bar{\operatorname{B}}(0,r/2)^{\mathrm{c}}}\exp[(C_{a}(1+C_{\varphi})-1/\delta)\left\|y\right\|^{2p}]\mathrm{d}y}$
(272) $\displaystyle\qquad\qquad\qquad\textstyle{\leq
2^{1+d(1+2p)/2p}pC_{a}(1+M_{0,\varphi})C_{\varphi}/(\delta
C_{1})\int_{\bar{\operatorname{B}}(0,r/(2^{1+2p}\delta)^{1/2p})^{\mathrm{c}}}\exp[-\left\|y\right\|^{2p}]\mathrm{d}y\;.}$
(273)

We have for any $\delta\in\left(0,1/(2C_{a}(1+C_{\varphi}))\right)$

$\textstyle{\int_{\bar{\operatorname{B}}(0,r/(2^{1+2p}\delta)^{1/2p})^{\mathrm{c}}}\exp[-\left\|y\right\|^{2p}]\mathrm{d}y/C_{1}\leq\exp[-r^{2p}/(2^{2+2p}\delta)]\int_{\mathbb{R}^{d}}\exp[-\left\|y\right\|^{2p}/2]\mathrm{d}y/C_{1}\;.}$
(274)

Therefore, there exists $C_{b}\geq 0$ (independent from $\varphi$) such that
for any $x\in\mathsf{U}$ and $\delta\in(0,1/(2C_{a}(1+C_{\varphi})))$ we have

$\textstyle{(2p/\delta)\int_{\bar{\operatorname{B}}(0,r/2)^{\mathrm{c}}}|\varphi(x-y)-\varphi(x)|\left\|y\right\|^{2p-1}k_{\delta}(y)\mathrm{d}y\leq
C_{b}(1+M_{0,\varphi})C_{\varphi}\delta^{-1}\exp[-r^{2p}/(2^{2+2p}\delta)]\;.}$
(275)

Combining this bound and (265), we get that for any
$\delta\in(0,1/(2C_{a}(1+C_{\varphi})))$ there exists $\mathtt{L}_{\delta}\geq
0$ such that for any $x\in\mathsf{U}$,
$\|\nabla\varphi^{\delta}(x)\|\leq\mathtt{L}_{\delta}$. Let
$\mathtt{L}_{0}=\sup\\{\mathtt{L}_{\delta}\,:\;\delta\in(0,\bar{\delta})\\}<+\infty$
and using, (265), (275) and that for any $t\geq 0$,
$t\exp[-r^{2p}t/2^{2+2p}]\leq 2^{2+2p}/(\mathrm{e}r^{2p})$ we have

$\textstyle{\mathtt{L}_{0}\leq(2pM_{1,\varphi}/C_{1})\int_{\mathbb{R}^{d}}\|y\|^{2p}\exp[-\|y\|^{2p}]\mathrm{d}y+2^{2+2p}C_{b}(1+M_{0,\varphi})C_{\varphi}/(\mathrm{e}r^{2p})\;.}$
(276)

This concludes the proof of Section A.4-(b).

  3. (c)

For any $x\in\mathsf{U}$ and $\delta\in(1/(2C_{a}(1+C_{\varphi})))$ we have

$\displaystyle|\varphi(x)|$
$\displaystyle\textstyle{\leq\int_{\bar{\operatorname{B}}(0,r/2)}|\varphi(y-x)|k_{\delta}(y)\mathrm{d}y+\int_{\bar{\operatorname{B}}(0,r/2)^{\mathrm{c}}}|\varphi(y-x)|k_{\delta}(y)\mathrm{d}y}$
(277) $\displaystyle\textstyle{\leq
M_{0,\varphi}+\int_{\bar{\operatorname{B}}(0,r/2)^{\mathrm{c}}}|\varphi(y-x)|k_{\delta}(y)\mathrm{d}y\;.}$
(278)

Similarly to (275), there exists $c\geq 0$ (independent from $\varphi$) such
that for $x\in\mathsf{U}$ and $\delta\in(1/(2C_{a}(1+C_{\varphi})))$

$\textstyle{\int_{\bar{\operatorname{B}}(0,r/2)^{\mathrm{c}}}|\varphi(y-x)|k_{\delta}(y)\mathrm{d}y\leq(C_{\varphi}/c)\exp[-c/\delta]\;.}$
(279)

Combining this result and (277) for any
$\delta\in\left(0,1/(2C_{a}(1+C_{\varphi}))\right)$ there exists
$M_{\delta}>0$ such that for any $x\in\mathsf{U}$, $|\varphi^{\delta}(x)|\leq
M_{\delta}$. Let
$M_{0}=\sup\\{M_{\delta}\,:\;\delta\in(0,\bar{\delta})\\}<+\infty$. We have
that

$M_{0}\leq M_{0,\varphi}+(C_{\varphi}/c)\exp[-c(1+C_{\varphi})]\;,$ (280)

which concludes the proof of Section A.4-(c).

  4. (d)

Using that for any $a,b\geq 0$ and $p\geq 0$, $(a+b)^{p}\leq
2^{\min(p-1,0)}(a^{p}+b^{p})$ we have that for any $x\in\mathbb{R}^{d}$ and
$\delta>0$

$\textstyle{|\varphi^{\delta}(x)|\leq
C_{\varphi}\int_{\mathbb{R}^{d}}\exp[C_{\varphi}\beta_{\upalpha,k}(\left\|x\right\|^{\upalpha
k}+\left\|y\right\|^{\upalpha k})]k_{\delta}(y)\mathrm{d}y\;,}$ (281)

where $\beta_{\upalpha,k}=2^{\min(\upalpha k-1,0)}$. Hence, using this result
we have for any $\delta\in\left(0,1/(2\beta_{\upalpha,k}C_{\varphi})\right)$

$\displaystyle|\varphi^{\delta}(x)|$ $\displaystyle\textstyle{\leq
C_{\varphi}/(C_{1}\delta^{d/2p})\exp[C_{\varphi}\beta_{\upalpha,k}\left\|x\right\|^{\upalpha
k}]\int_{\mathbb{R}^{d}}\exp[(C_{\varphi}\beta_{\upalpha,k}-1/\delta)\left\|y\right\|^{2p}]\mathrm{d}y}$
(282) $\displaystyle\textstyle{\leq
C_{\varphi}/(C_{1}\delta^{d/2p})\exp[C_{\varphi}\beta_{\upalpha,k}\left\|x\right\|^{\upalpha
k}]\int_{\mathbb{R}^{d}}\exp[-\left\|y\right\|^{2p}/(2\delta)]\mathrm{d}y\;}$
(283) $\displaystyle\textstyle{\leq
2^{d/2p}C_{\varphi}\exp[C_{\varphi}\beta_{\upalpha,k}\left\|x\right\|^{\upalpha
k}]\;.}$ (284)

Therefore, for any
$\delta\in\left(0,1/(2\beta_{\upalpha,k}C_{\varphi})\right)$ there exists
$D_{\delta}\geq 0$ such that for any $x\in\mathbb{R}^{d}$, $\varphi(x)\leq
D_{\delta}\exp[D_{\delta}\left\|x\right\|^{\upalpha k}]$. In addition, there
exists $C_{d}\geq 0$ (independent of $\varphi$) such that $D_{\delta}\leq
C_{d}(1+C_{\varphi})\exp[C_{d}C_{\varphi}]$, which concludes the proof of
Section A.4-(d).

  5. (e)

If $\varphi\geq 0$ then for any $\delta>0$, $\varphi^{\delta}\geq 0$ using
(261). For any $x\in\mathsf{U}$ we have

$\textstyle{\varphi^{\delta}(x)=\varphi(x)+\int_{\mathbb{R}^{d}}(\varphi(y-x)-\varphi(x))k_{\delta}(y)\mathrm{d}y\geq
m_{0,\varphi}-\int_{\mathbb{R}^{d}}\left|\varphi(y-x)-\varphi(x)\right|k_{\delta}(y)\;.}$
(285)

For any $x\in\mathsf{U}$ we have

$\textstyle{\int_{\mathbb{R}^{d}}\left|\varphi(y-x)-\varphi(x)\right|k_{\delta}(y)\leq
M_{1,\varphi}\delta^{1/k}+\int_{\bar{\operatorname{B}}(0,r/2)^{\mathrm{c}}}|\varphi(y-x)-\varphi(x)|k_{\delta}(y)\mathrm{d}y\;.}$
(286)

Similarly to (275), there exists $C_{d}\geq 0$ independent from $\varphi$ such
that for any $x\in\mathsf{U}$

$\textstyle{\int_{\bar{\operatorname{B}}(0,r/2)^{\mathrm{c}}}|\varphi(y-x)-\varphi(x)|k_{\delta}(y)\mathrm{d}y\leq
C_{d}(1+M_{0,\varphi})C_{\varphi}\exp[-1/(C_{d}\delta)]\;.}$ (287)

Let $\bar{\delta}_{1}=(m_{0,\varphi}/(4M_{1,\varphi}))^{k}$ and
$\bar{\delta}_{2}=C_{d}^{-1}(\log(4C_{d}C_{\varphi}(1+M_{0,\varphi}))-\min(0,\log(m_{0,\varphi})))^{-1}$.
Then for any $\delta\in(0,\min(\bar{\delta}_{1},\bar{\delta}_{2}))$ we have
that for any $x\in\mathsf{U}$

$\textstyle{\int_{\mathbb{R}^{d}}\left|\varphi(y-x)-\varphi(x)\right|k_{\delta}(y)\leq
m_{0,\varphi}/2\;,}$ (288)

which concludes the proof of Section A.4-(e) upon combining this result with
(285).

∎

##  Appendix B Technical results for Section 4.3.1

In this section, we derive a quantitative parametric theory for Laplace-type
expansion in Section B.1 and Section B.2. We start by deriving technical
bounds in Section B.1. Our main result, Section B.2, is presented in Section
B.2 along with a quantitative Morse lemma. Finally, in Section B.3 we derive
some moments bounds.

Let $u:\ \mathbb{R}^{d}\times\mathsf{Z}\to\mathbb{R}$ with $\mathsf{Z}$ a
topological space. We consider the following assumption.

######  H 5.

$u\in\mathrm{C}(\mathbb{R}^{d}\times\mathsf{Z},\mathbb{R})$, for any
$z\in\mathsf{Z}$, $u(\cdot,z)\in\mathrm{C}^{2}(\mathbb{R}^{d},\mathbb{R})$ and
the following hold:

  1. (a)

$\mathsf{Z}$ is compact.

  2. (b)

There exist $N\in\mathbb{N}$,
$\\{x_{0}^{k}\\}_{k=1}^{N}\in\mathrm{C}(\mathsf{Z},\mathbb{R}^{d})$ such that
$\operatorname*{arg\,min}\\{u(x,z)\,:\;x\in\mathbb{R}^{d}\\}\subset\\{x_{0}^{k}(z)\\}_{k=1}^{N}$
and $\operatorname*{arg\,min}\\{u(x,z)\,:\;x\in\mathbb{R}^{d}\\}\neq\emptyset$
for any $z\in\mathsf{Z}$.

  3. (c)

There exists $\mathtt{M}\geq 0$ such that for any
$x_{1},x_{2}\in\mathbb{R}^{d}$ and $z\in\mathsf{Z}$,
$\|\nabla^{k}_{x}u(x_{1},z)-\nabla^{k}_{x}u(x_{2},z)\|\leq\mathtt{M}\|x_{1}-x_{2}\|$.

  4. (d)

There exists $\mathtt{m},\upalpha>0$ and $R\geq 0$ such that for any
$x\in\mathbb{R}^{d}$ with $\|x\|\geq R$,
$u(x,z)\geq\mathtt{m}\|x\|^{\upalpha}$.

We denote $u^{\star}:\ \mathsf{Z}\to\mathbb{R}$ such that for any
$z\in\mathsf{Z}$, $u^{\star}(z)=\min\\{u(x,z)\,:\;x\in\mathbb{R}^{d}\\}$.
Under H 5 we have that for any $z\in\mathsf{Z}$,
$u^{\star}=\min\\{u(x_{0}^{k}(z),z)\,:\;k\in\\{1,\dots,N\\}\\}$ and therefore
$u^{\star}\in\mathrm{C}(\mathsf{Z},\mathbb{R})$. For any $\varphi:\
\mathbb{R}^{d}\to\mathbb{R}_{+}$ and $\varepsilon>0$ we define

$\displaystyle\textstyle{\mathcal{I}_{\varepsilon}(\varphi,z)=C_{\varepsilon}^{-1}\int_{\mathbb{R}^{d}}\varphi(x)\exp[-(u(x,z)-u^{\star}(z))/\varepsilon]\mathrm{d}x\;,\quad\mathcal{J}_{\varepsilon}(z)=\mathcal{I}_{\varepsilon}(1,z)\;,}$
(289)
$\displaystyle\textstyle{C_{\varepsilon}=\int_{\mathbb{R}^{d}}\exp[-\left\|x\right\|^{2}/\varepsilon]\mathrm{d}x=(\uppi\varepsilon)^{d/2}\;.}$
(290)

In addition, we define

$\textstyle{\mathcal{I}_{0}(\varphi,z)=\int_{\operatorname*{arg\,min}u(\cdot,z)}\varphi(x)\det(\nabla^{2}_{x}u(x,z))^{-1/2}\mathrm{d}\mathcal{H}^{0}(x)\;,\quad\mathcal{J}_{0}(z)=\mathcal{I}_{0}(1,z)\;,}$
(291)

where $\det(\nabla^{2}_{x}u(x,z))^{-1/2}=+\infty$ if $\nabla^{2}_{x}u(x,z)$ is
not invertible. If $\varphi:\ \mathbb{R}^{d}\to\mathbb{R}$ and
$I_{\varepsilon}(\left|\varphi\right|)<+\infty$ for some $\varepsilon\geq 0$
we define $\mathcal{I}_{\varepsilon}(\varphi)$ similarly as in (289), (291).
For any $\varepsilon\geq 0$ and $\varphi:\ \mathbb{R}^{d}\to\mathbb{R}^{p}$
such that it is defined we let
$\updelta_{z}\mathrm{S}_{\varepsilon}[\varphi]=\mathcal{I}_{\varepsilon}(\varphi,z)/\mathcal{J}_{\varepsilon}(z)$.
We emphasize that these definitions are the parametric counterparts to the
ones introduced in Section 4.1.

In what follows, we define $\sigma:\
\mathbb{R}^{d}\times\mathsf{Z}\to\left[0,+\infty\right]$ such that for any
$z\in\mathsf{Z}$ and $x\in\mathbb{R}^{d}$ $\sigma(x,z)$ is the inverse of the
minimum eigenvalue of $\nabla^{2}_{x}u(x,z)$. In addition, for any
$\upbeta>0$, we define $\sigma^{\star}_{\upbeta}:\
\mathsf{Z}\to\left[0,+\infty\right]$ such that for any $z\in\mathsf{Z}$,
$\sigma^{\star}_{\upbeta}(z)=\int_{\operatorname*{arg\,min}u(\cdot,z)}\sigma(x,z)^{\upbeta}\mathrm{d}\mathcal{H}^{0}(x)$.

###  B.1 Parametric lower and truncation bounds

######  Lemma B.0.

Assume H 5. Then, for any $\bar{\varepsilon}\geq 0$ there exists $A_{0}>0$
such that for any $z\in\mathsf{Z}$ and
$\varepsilon\in\left[0,\bar{\varepsilon}\right]$ we have
$\mathcal{J}_{\varepsilon}(z)\geq A_{0}$.

###### Proof.

Let $z\in\mathsf{Z}$ and $\varepsilon>0$. Let $x^{\star}(z)$ be a global
minimizer of $x\mapsto u(x,z)$ which exists since
$\operatorname*{arg\,min}\\{u(x,z)\,:\;x\in\mathbb{R}^{d}\\}\neq\emptyset$.
Using the change of variable $x\mapsto x+x^{\star}(z)$ we have

$\textstyle{\mathcal{J}_{\varepsilon}(z)=C_{\varepsilon}^{-1}\int_{\mathbb{R}^{d}}\exp[-(u(x,z)-u^{\star}(z))/\varepsilon]\mathrm{d}x=C_{\varepsilon}^{-1}\int_{\mathbb{R}^{d}}\exp[-(u(x^{\star}(z)+x,z)-u^{\star}(z))/\varepsilon]\mathrm{d}x\;.}$
(292)

For any $x\in\mathbb{R}^{d}$ we have

$\textstyle{u(x^{\star}(z)+x,z)-u^{\star}(z)=\int_{0}^{1}\langle\nabla_{x}u(x^{\star}(z)+tx,z)-\nabla_{x}u(x^{\star}(z),z),x\rangle\mathrm{d}t\leq\mathtt{M}\|x\|^{2}/2\;.}$
(293)

Combining this result and (292) we get
$\mathcal{J}_{\varepsilon}(z)\geq\int_{\mathbb{R}^{d}}\exp[-(\mathtt{M}/2)\|x\|^{2}/\varepsilon]\mathrm{d}x/C_{\varepsilon}\geq(2/\mathtt{M})^{d/2}$.
In addition, we have that for any $x\in\mathbb{R}^{d}$,
$\mathcal{J}_{0}(z)=\int_{\operatorname*{arg\,min}u(\cdot,z)}\det(\nabla^{2}_{x}u(x,z))^{-1/2}\mathrm{d}\mathcal{H}^{0}(x)$.
Note that if $\sigma_{1}^{\star}(z)=+\infty$, then there exists
$\tilde{x}(z)\in\operatorname*{arg\,min}\\{u(x,z)\,:\;x\in\mathbb{R}^{d}\\}$
such that $\sigma(\tilde{x}(z),z)=+\infty$ and therefore
$\mathcal{J}_{0}(z)=+\infty$. We have that for any $x\in\mathbb{R}^{d}$,
$\det(\nabla^{2}_{x}u(x,z))\leq\mathtt{M}^{d}$ and therefore,
$\mathcal{J}_{0}(z)\geq N\mathtt{M}^{-d/2}$. We conclude upon letting
$A_{0}=\min(N,2^{d/2})/\mathtt{M}^{d/2}$. ∎

######  Lemma B.0.

Assume H 5. Let $\varphi:\ \mathbb{R}^{d}\to\mathbb{R}$ and $C_{\varphi}\geq
0$ such that for any $x\in\mathbb{R}^{d}$

$\left|\varphi(x)\right|\leq
C_{\varphi}\exp[C_{\varphi}\left\|x\right\|^{\upalpha}]\;.$ (294)

Let $\bar{\varepsilon}\in(0,\mathtt{m}/(1+C_{\varphi}))$ and $z\in\mathsf{Z}$.
Assume that there exists $\mathsf{V}(z)\subset\mathbb{R}^{d}$ open and bounded
such that

$\operatorname*{arg\,min}\\{u(x,z)\,:\;x\in\mathbb{R}^{d}\\}\subset\mathsf{V}(z)\subset\operatorname*{arg\,min}\\{u(x,z)\,:\;x\in\mathbb{R}^{d}\\}+\bar{\operatorname{B}}(0,1)\;.$
(295)

Then, there exist $\beta_{1}>0$ and
$A_{1}\in\mathrm{C}(\mathbb{R}_{+},\mathbb{R}_{+})$ such that for any
$\varepsilon\in\left(0,\bar{\varepsilon}\right)$

$\mathcal{I}_{\varepsilon}^{\mathrm{out}}(\varphi,z)\leq
A_{1}(C_{\varphi})\varepsilon^{-d/2}\\{\exp[-m(z)/\varepsilon]+\exp[-\beta_{1}/\varepsilon]\\}\;,$
(296)

with
$m(z)=\inf\\{u(x,z)\,:\;x\in\mathbb{R}^{d}\backslash\mathsf{V}(z)\\}-u^{\star}(z)$,
$\mathcal{I}_{\varepsilon}^{\mathrm{out}}(\varphi,z)=\mathcal{I}_{\varepsilon}(\varphi\mathbbm{1}_{\mathsf{V}(z)^{\mathrm{c}}},z)$
and $A_{1},\beta_{1}$ independent from $\varphi$, $z$, with $A_{1}$ non-
decreasing.

###### Proof.

First, we have that

$\mathsf{V}(z)\subset\cup_{k=1}^{N}\\{x_{0}^{k}(z)\,:\;z\in\mathsf{Z}\\}+\bar{\operatorname{B}}(0,1)\;.$
(297)

Since $\mathsf{Z}$ is compact and that for any $k\in\\{1,\dots,N\\}$,
$x_{0}^{k}\in\mathrm{C}(\mathsf{Z},\mathbb{R}^{d})$, there exists
$R^{\prime}\geq 0$ (independent from $z$) such that $R^{\prime}\geq R$ (where
$R$ is given in H 5) and
$\mathsf{V}(z)\subset\bar{\operatorname{B}}(0,R^{\prime})$. Note that for any
$\varepsilon>0$, we have

$\displaystyle\mathcal{I}_{\varepsilon}^{\mathrm{out}}(\varphi,z)=\mathcal{I}_{\varepsilon}^{1}(\varphi,z)+\mathcal{I}_{\varepsilon}^{2}(\varphi,z)\;,$
(298)
$\displaystyle\mathcal{I}_{\varepsilon}^{1}(\varphi,z)=\mathcal{I}_{\varepsilon}(\varphi\mathbbm{1}_{\bar{\operatorname{B}}(0,R^{\prime})^{\mathrm{c}}},z)\;,\quad\mathcal{I}_{\varepsilon}^{2}(\varphi,z)=\mathcal{I}_{\varepsilon}(\varphi\mathbbm{1}_{\mathsf{V}^{\mathrm{c}}\cap\bar{\operatorname{B}}(0,R^{\prime})},z)\;.$
(299)

Let $\varepsilon\in\left(0,\bar{\varepsilon}\right)$. The rest of the proof is
similar to the one of Section A.2 but is given for completeness. We divide the
proof into two parts. First, we bound
$\mathcal{I}_{\varepsilon}^{1}(\varphi,z)$ and then
$\mathcal{I}_{\varepsilon}^{2}(\varphi,z)$.

  1. (a)

Let $w=(\mathtt{m}/\varepsilon-C_{\varphi})^{1/\upalpha}$ (which makes sense,
since $\varepsilon<\mathtt{m}/(C_{\varphi}+1)$). Since $R^{\prime}\geq R$,
using (294) and the fact that $u\geq 1$, we have

$\displaystyle\mathcal{I}_{\varepsilon}^{1}(\varphi,z)$
$\displaystyle\textstyle{=C_{1}^{-1}\varepsilon^{-d/2}\int_{\bar{\operatorname{B}}(0,R^{\prime})^{\mathrm{c}}}\varphi(x)\exp[-(u(x,z)-u^{\star}(z))/\varepsilon]\mathrm{d}x}$
(300) $\displaystyle\textstyle{\leq
C_{1}^{-1}C_{\varphi}\varepsilon^{-d/2}\exp[\bar{u}^{\star}]\int_{\bar{\operatorname{B}}(0,R^{\prime})^{\mathrm{c}}}\exp[-(\mathtt{m}/\varepsilon-
C_{\varphi})\left\|x\right\|^{\upalpha}]\mathrm{d}x}$ (301)
$\displaystyle\leq\textstyle{C_{1}^{-1}C_{\varphi}\exp[\bar{u}^{\star}]\varepsilon^{-d/2}\int_{\bar{\operatorname{B}}(0,R^{\prime}w)^{\mathrm{c}}}\exp[-\left\|x\right\|^{\upalpha}]\mathrm{d}x\;,}$
(302)

where $\bar{u}^{\star}=\sup\\{u^{\star}(z)\,:\;z\in\mathsf{Z}\\}$. Let
$C_{\upalpha}=\int_{\mathbb{R}^{d}}\exp[-\left\|x\right\|^{\upalpha}]\mathrm{d}x$.
Using that $w=(\mathtt{m}/\varepsilon-C_{\varphi})^{1/\upalpha}$, we have

$\displaystyle\mathcal{I}_{\varepsilon}^{1}(\varphi,z)$
$\displaystyle\leq\textstyle{C_{1}^{-1}C_{\varphi}\exp[\bar{u}^{\star}]\varepsilon^{-d/2}\int_{\bar{\operatorname{B}}(0,R^{\prime}w)^{\mathrm{c}}}\exp[-\left\|x\right\|^{\upalpha}]\mathrm{d}x}$
(303)
$\displaystyle\leq\textstyle{C_{1}^{-1}C_{\varphi}\exp[\bar{u}^{\star}]\varepsilon^{-d/2}\int_{\mathbb{R}^{d}}\exp[-\left\|x\right\|^{\upalpha}/2]\mathrm{d}x\exp[-(R^{\prime}u)^{\upalpha}/2]}$
(304)
$\displaystyle\leq\textstyle{2^{d/\upalpha}C_{\upalpha}C_{1}^{-1}C_{\varphi}\exp[\bar{u}^{\star}]\exp[(R^{\prime})^{\upalpha}C_{\varphi}/2]\varepsilon^{-d/2}\exp[-(R^{\prime})^{\upalpha}\mathtt{m}/(2\varepsilon)]\leq
A_{1}^{1}\varepsilon^{-d/2}\exp[-\beta_{1}^{1}/\varepsilon]\;,}$ (305)

with

$A_{1}^{1}=2^{d/\upalpha}C_{\upalpha}C_{1}^{-1}C_{\varphi}\exp[\bar{u}^{\star}]\exp[(R^{\prime})^{\upalpha}C_{\varphi}/2]\;,\qquad\beta_{1}^{1}=(R^{\prime})^{\upalpha}\mathtt{m}/2\;.$
(306)

  2. (b)

Second, let
$\mathsf{K}(z)=\mathsf{V}^{\mathrm{c}}(z)\cap\bar{\operatorname{B}}(0,R^{\prime})$.
Note that for any $x\in\mathsf{K}(z)$, $u(x,z)-u^{\star}(z)\geq m(z)$. Hence,
we have

$\mathcal{I}_{\varepsilon}^{2}(\varphi,z)\leq
C_{1}^{-1}C_{\varphi}\varepsilon^{-d/2}\exp[C_{\varphi}(R^{\prime})^{\upalpha}]\exp[-m(z)/\varepsilon]\lambda(\mathsf{K}(z))\;,$
(307)

where we recall that $\lambda(\mathsf{K}(z))$ is the Lebesgue measure of
$\mathsf{K}(z)$. Since
$\mathsf{K}(z)\subset\bar{\operatorname{B}}(0,R^{\prime})$ we have

$\mathcal{I}_{\varepsilon}^{2}(\varphi,z)\leq\uppi^{d/2}(R^{\prime})^{d}\Gamma^{-1}(d/2+1)C_{1}^{-1}C_{\varphi}\exp[(R^{\prime})^{\upalpha}]\varepsilon^{-d/2}\exp[-m(z)/\varepsilon]\leq
A_{1}^{2}\varepsilon^{-d/2}\exp[-m(z)/\varepsilon]\;,$ (308)

where $\Gamma:\ \left(0,+\infty\right)\to\mathbb{R}_{+}$ is given for any
$s\in\left(0,+\infty\right)$ by
$\Gamma(s)=\int_{0}^{+\infty}t^{s-1}\exp[-t]\mathrm{d}t$ and

$A_{1}^{2}=\uppi^{d/2}(R^{\prime})^{d}\Gamma^{-1}(d/2+1)C_{1}^{-1}C_{\varphi}\exp[C_{\varphi}(R^{\prime})^{\upalpha}]\;.$
(309)

We conclude the proof upon combining (305), (308) and letting
$\beta_{1}=\beta_{1}^{1}$ and $A_{1}=A_{1}^{1}+A_{1}^{2}$. ∎

###  B.2 Quantitative Morse lemma and parametric Laplace-type results

We begin by recalling a quantitative version of the Morse lemma, see [51,
Theorem 4.2].

######  Proposition B.0.

Let $\mathsf{U}\subset\mathbb{R}^{d}$ be open with $x_{0}\in\mathsf{U}$ and
$f\in\mathrm{C}^{k}(\mathsf{U},\mathbb{R})$ with $k\in\mathbb{N}$ and $k\geq
3$. Assume that $\nabla f(x_{0})=0$ and let $K\geq 0$ such that
$K\geq\max_{j\in\\{1,\dots,k\\}}\sup\\{\|\mathrm{D}^{j}f(x)\|\,:\;x\in\mathsf{U}\\}<+\infty$.
Let $\sigma$ be the minimal eigenvalue of $\nabla^{2}f(x_{0})$ and assume that
$\sigma>0$. Let $\sigma_{0}=\min(\sigma,1)$. Then, there exist $c_{0}>0$,
$\delta=c_{0}\sigma_{0}^{4}$ and $\Phi:\
\operatorname{B}(0,\delta)\to\Phi(\operatorname{B}(0,\delta))$ such that the
following hold:

  1. (a)

$\Phi\in\mathrm{C}^{k-1}(\operatorname{B}(0,\delta),\Phi(\operatorname{B}(0,\delta)))$
is a diffeomorphism.

  2. (b)

For any $x\in\operatorname{B}(0,\delta)$, $f(\Phi(x))=f(x_{0})+\|x\|^{2}$.

  3. (c)

There exist $c_{1},\upbeta>0$ such that
$\max_{j\in\\{1,\dots,k-1\\}}\sup\\{\|\mathrm{D}^{j}\Phi(x)\|\,:\;x\in\operatorname{B}(0,\delta)\\}\leq
c_{1}\sigma_{0}^{-\upbeta}$.

In addition, $c_{0},c_{1}$ and $\upbeta$ depend only on $k$, $d$ and $K$.

Note that in [51, Theorem 4.2], the constant $c_{1}\sigma_{0}^{-\upbeta}$ is
replaced by $M(K,\sigma_{0},k)$ where $M:\
\mathbb{R}_{+}^{3}\to\mathbb{R}_{+}$. However, a close examination of the
proof reveals that the dependency of $M(K,\sigma_{0},k)$ with respect to
$\sigma_{0}^{-1}$ is of order $\sigma_{0}^{-\upbeta}$ for some $\upbeta>0$
which can be made explicit. Using Section B.2 we derive the following
parametric version of Section 4.1.2.

This proposition relies on the notion of _thermodynamic barrier_ associated
with $u$ which we define as follows. Assume that
$\operatorname*{arg\,min}\\{u(x,z)\,:\;x\in\mathbb{R}^{d}\\}\neq\emptyset$ for
any $z\in\mathsf{Z}$. For any $z\in\mathsf{Z}$ we introduce $\mathsf{A}(z)$
such that $\mathsf{A}(z)=\emptyset$ if there are no other minimizers than the
global minimizers and $\mathsf{A}(z)=\\{u(x,z)\,:\;\text{$x$ is a local
minimizer of $u(\cdot,z)$ but not a global minimizer}\\}$ otherwise. Then we
define $c^{\star}:\ \mathsf{Z}\to\mathbb{R}$ such that for any
$z\in\mathsf{Z}$ we have

$c^{\star}(z)=\inf\mathsf{A}(z)-\inf\\{u(x,z)\,:\;x\in\mathbb{R}^{d}\\}\;,$
(310)

with the convention that $\inf\emptyset=+\infty$. In words, the _thermodynamic
barrier_ constant quantifies how close the value of the local minimizers are
from the global ones. We refer to Section 3.2.3 for a discussion on the
importance of _thermodynamic barrier_ when establishing parametric Laplace-
type expansions.

######  Proposition B.0.

Assume H 5. Let $\varphi\in\mathrm{C}(\mathbb{R}^{d},\mathbb{R})$ and
$z\in\mathsf{Z}$, and assume that $\sigma_{1}^{\star}(z)<+\infty$. There
exists $\mathsf{V}(z)$ open such that

$\operatorname*{arg\,min}\\{u(x,z)\,:\;x\in\mathbb{R}^{d}\\}\subset\mathsf{V}(z)\subset\operatorname*{arg\,min}\\{u(x,z)\,:\;x\in\mathbb{R}^{d}\\}+\bar{\operatorname{B}}(0,1)\;.$
(311)

We have that $\lim_{\varepsilon\to
0}|\mathcal{I}_{\varepsilon}^{\mathrm{in}}(\varphi,z)-\mathcal{I}_{0}(\varphi,z)|=0$,
with
$\mathcal{I}_{\varepsilon}^{\mathrm{in}}(\varphi,z)=\mathcal{I}_{\varepsilon}(\varphi\mathbbm{1}_{\mathsf{V}(z)},z)$.
Assume that $\varphi\in\mathrm{C}^{1}(\mathbb{R}^{d},\mathbb{R})$. Then there
exist $B_{1}\geq 0$ and $\upbeta>0$ such that for any $\varepsilon>0$ we have

$|\mathcal{I}_{\varepsilon}^{\mathrm{in}}(\varphi,z)-\mathcal{I}_{0}(\varphi,z)|\leq
B_{1}\sigma_{\upbeta}^{\star}(z)(1+M_{0,\varphi}(z)+M_{1,\varphi}(z))\varepsilon^{1/2}\;,$
(312)

with $B_{1}$ independent from $\varphi$ and $z$ and for any $i\in\\{0,1\\}$,
$M_{i,\varphi}(z)=\sup\\{\|\nabla^{i}\varphi(x)\|\,:\;x\in\mathsf{V}(z)\\}$.
In addition, there exist $c_{1},\upgamma>0$ (independent from $z$) such that
$m(z)\geq\min(c_{1}/\sigma_{\upgamma}^{\star}(z),c^{\star}(z))$, with
$m(z)=\inf\\{u(x,z)\,:\;x\in\mathbb{R}^{d}\backslash\mathsf{V}(z)\\}-u^{\star}(z)$.

###### Proof.

Let
$\\{x_{\star}^{k}(z)\\}_{k=1}^{M}=\operatorname*{arg\,min}\\{u(x,z)\,:\;x\in\mathbb{R}^{d}\\}$.
Note that for any $k\in\\{1,\dots,M\\}$ there exists $i_{k}\in\\{1,\dots,N\\}$
such that $x_{\star}^{k}(z)=x_{0}^{i_{k}}(z)$. We recall that
$\mathsf{K}=\bigcup_{k=1}^{N}\\{x_{0}^{k}(z)\,:\;z\in\mathsf{Z}\\}$ is
compact. Therefore there exists $R\geq 0$ such that for any $z\in\mathsf{Z}$,
$\\{x_{\star}^{k}(z)\\}_{k=1}^{M}\subset\bar{\operatorname{B}}(0,R)$. We
define $K_{\mathrm{global}}\geq 0$ such that

$K_{\mathrm{global}}=\sup\\{\|\nabla_{x}^{j}u(x,z)\|\,:\;x\in\mathsf{K},\
z\in\mathsf{Z},\ j\in\\{1,2,3\\}\\}\;.$ (313)

Let $i\in\\{1,\dots,M\\}$. Using that $\sigma_{1}^{\star}(z)<+\infty$ and
Section B.2 with $K\leftarrow K_{\mathrm{global}}$ and $k=3$, there exist
$\Phi_{i}^{z}\in\mathrm{C}^{2}(\operatorname{B}(0,\delta_{i}(z)),\Phi_{i}^{z}(\operatorname{B}(0,\delta_{i}(z))))$
which is a diffeomorphism, with $\Phi_{i}^{z}(0)=x_{\star}^{i}(z)$,
$\delta_{i}(z)=c_{0}\min(\sigma^{4}(x_{\star}^{i}(z),z),1)$ and for any
$x\in\operatorname{B}(0,\delta_{i}(z))$ we have
$u(\Phi_{i}^{z}(x),z)=u^{\star}+\|x\|^{2}$.

We let $\delta_{0}(z)=\min\\{\delta_{i}(z)\,:\;i\in\\{1,\dots,M\\}\\}$. Using
Section B.2, there exist $c_{0}^{\prime}\geq 0$ and $\upalpha>0$ independent
from $z$ such that for any $\ell\in\\{1,\dots,M\\}$ and
$x\in\operatorname{B}(0,\delta_{0}(z))$,
$\left\|\mathrm{d}\Phi_{\ell}^{z}(x)\right\|\leq
c_{0}^{\prime}\sigma_{\upalpha}^{\star}(z)$. Let
$\delta(z)=\min(\delta_{0}(z),1/(c_{0}^{\prime}\sigma_{\upalpha}^{\star}(z)))$.
We have that for any $\ell\in\\{1,\dots,N\\}$,
$\Phi_{\ell}(\operatorname{B}(0,\delta(z)))\subset\operatorname{B}(x_{\star}^{\ell}(z),c_{0}^{\prime}\sigma_{\upalpha}^{\star}(z)\delta(z))\subset\operatorname*{arg\,min}\\{u(x,z)\,:\;x\in\mathbb{R}^{d}\\}+\bar{\operatorname{B}}(0,1)$.
We let
$\mathsf{V}(z)=\cup_{i=1}^{M}\Phi_{\ell}(\operatorname{B}(0,\delta(z)))$.

We now show that for any $i,j\in\\{1,\dots,M\\}$,
$\Phi_{i}^{z}(\operatorname{B}(0,\delta(z)))\cap\Phi_{j}^{z}(\operatorname{B}(0,\delta(z)))=\emptyset$.
Let $i,j\in\\{1,\dots,M\\}$ and for ease of notation let
$\mathsf{W}_{\ell}=\Phi_{\ell}^{z}(\operatorname{B}(0,\delta(z)))$ for any
$\ell\in\\{1,\dots,M\\}$. Assume that
$\mathsf{W}_{i}\cap\mathsf{W}_{j}\neq\emptyset$. Then, since $\mathsf{W}_{i}$
and $\mathsf{W}_{j}$ are connected,
$\mathsf{W}=\mathsf{W}_{i}\cup\mathsf{W}_{j}$ is connected as well. In
addition, note that $\Phi_{i}^{z}(0)\notin\mathsf{W}_{j}$ and
$\Phi_{j}^{z}(0)\notin\mathsf{W}_{i}$. There exists
$\gamma\in\mathrm{C}(\left[0,1\right],\mathsf{W})$ such that
$\gamma(0)=\Phi_{i}^{z}(0)$ and $\gamma(1)=\Phi_{j}^{z}(0)$. Denote
$t^{\star}=\inf\\{t\in\left[0,1\right]\,:\;\gamma(t)\in\mathsf{W}_{j}\\}$. We
have that $\gamma(t^{\star})\in\bar{\mathsf{W}}_{j}\backslash\mathsf{W}_{j}$.
Hence $u(\gamma(t^{\star}),z)=\delta(z)^{2}+u^{\star}(z)$. But
$\gamma(t^{\star})\in\mathsf{W}_{i}$ and therefore
$u(\gamma(t^{\star}),z)<\delta(z)^{2}+u^{\star}(z)$. This is absurd hence for
any $i,j\in\\{1,\dots,M\\}$, $\mathsf{W}_{i}\cap\mathsf{W}_{j}=\emptyset$.

Let $\varepsilon>0$ and $\mathcal{I}_{0,\varepsilon}^{\mathrm{in}}$ be given
by

$\textstyle{\mathcal{I}_{0,\varepsilon}^{\mathrm{in}}(\varphi,z)=\sum_{\ell=1}^{M}\int_{\operatorname{B}(0,\delta(z)/\varepsilon^{1/2})}\exp[-\left\|x\right\|^{2}]\mathrm{d}x\varphi(x_{\star}^{\ell}(z))\det(\nabla^{2}_{x}u(x_{\star}^{\ell}(z),z))^{-1/2}/C_{1}\;.}$
(314)

Recall that
$\mathcal{I}_{0}(\varphi,z)=\sum_{\ell=1}^{M}\varphi(x_{\star}^{\ell}(z))\det(\nabla^{2}_{x}u(x_{\star}^{\ell}(z),z))^{-1/2}$.
Therefore, we have

$\displaystyle\textstyle{\left|\mathcal{I}_{0}(\varphi,z)-\mathcal{I}_{0,\varepsilon}^{\mathrm{in}}(\varphi,z)\right|}$
$\displaystyle\leq\textstyle{\sigma_{1/2}^{\star}M_{0,\varphi}\exp[-\delta(z)^{2}/(2\varepsilon)]\int_{\mathbb{R}^{d}}\exp[-\|x\|^{2}/2]\mathrm{d}x/C_{1}}$
(315) $\displaystyle\leq
2^{d/2}\sigma_{1/2}^{\star}M_{0,\varphi}\exp[-\delta(z)^{2}/(2\varepsilon)]\;.$
(316)

Finally, using (314), that for any $\ell\in\\{1,\dots,M\\}$ and
$x\in\operatorname{B}(0,\delta(z))$,
$u(\Phi_{\ell}^{z}(x),z)=u^{\star}(z)+\|x\|^{2}$, that for any
$\ell\in\\{1,\dots,M\\}$,
$\det(\mathrm{d}\Phi_{\ell}^{z}(0))=\det(\nabla^{2}_{x}u(x_{\star}^{\ell}(z),z))^{-1/2}$
and $\Phi_{\ell}^{z}(0)=x_{\star}^{\ell}(z)$, we have

$\displaystyle\textstyle{|\mathcal{I}_{0,\varepsilon}^{\mathrm{in}}(\varphi,z)-\mathcal{I}_{\varepsilon}^{\mathrm{in}}(\varphi,z)|}$
$\displaystyle\textstyle{=\sum_{\ell=1}^{M}\int_{\operatorname{B}(0,\delta(z)/\varepsilon^{1/2})}\left|\vphantom{\varphi(\Phi_{\ell}^{z}(\varepsilon^{1/2}x))\det(\mathrm{d}\Phi_{\ell}^{z}(\varepsilon^{1/2}x))}\varphi(\Phi_{\ell}^{z}(0))\det(\mathrm{d}\Phi_{\ell}^{z}(0))\right.}$
(317)
$\displaystyle\qquad\qquad\textstyle{\left.-\varphi(\Phi_{\ell}^{z}(\varepsilon^{1/2}x))\det(\mathrm{d}\Phi_{\ell}^{z}(\varepsilon^{1/2}x))\right|\exp[-\left\|x\right\|^{2}]\mathrm{d}x/C_{1}\;,}$
(318)

which concludes the first part of the proof upon combining this result and the
dominated convergence theorem. Next assume that
$\varphi\in\mathrm{C}^{1}(\mathbb{R}^{d},\mathbb{R})$ and for any
$\ell\in\\{1,\dots,M\\}$, let $\chi_{\ell}^{z}:\
\operatorname{B}(0,\delta(z))\to\mathbb{R}$ given for any
$x\in\operatorname{B}(0,\delta(z))$ by

$\chi_{\ell}^{z}(x)=\varphi(\Phi_{\ell}^{z}(x))\det(\mathrm{d}\Phi_{\ell}^{z}(x))$
(319)

We have that for any $\ell\in\\{1,\dots,M\\}$,
$\chi_{\ell}^{z}\in\mathrm{C}^{1}(\operatorname{B}(0,\delta))$. Therefore, we
have that for any $\ell\in\\{1,\dots,M\\}$ and
$x\in\operatorname{B}(0,\delta(z))$ and $h\in\mathbb{R}^{d}$

$\mathrm{d}\chi_{\ell}(x)(h)=\mathrm{d}\varphi(\Phi_{\ell}^{z}(x))\mathrm{d}\Phi_{\ell}^{z}(x)(h)\det(\mathrm{d}\Phi_{\ell}^{z}(x))+\varphi(\Phi_{\ell}^{z}(x))\operatorname{Tr}(\mathrm{Adj}(\mathrm{d}\Phi_{\ell}^{z}(x))\mathrm{d}^{2}\Phi_{\ell}^{z}(x)(h))\;.$
(320)

Therefore, using Section B.2 we have that there exist $C\geq 0$ and
$\upbeta>0$ such that for any $\ell\in\\{1,\dots,M\\}$ and
$x\in\operatorname{B}(0,\delta(z))$, $\|\mathrm{d}\chi_{\ell}^{z}(x)\|\leq
C(1+M_{0,\varphi}+M_{1,\varphi})\sigma^{\star}_{\upbeta}(z)$. Using this
result in (317) we get that

$\textstyle{\left|\mathcal{I}_{0,\varepsilon}^{\mathrm{in}}(\varphi,z)-\mathcal{I}_{\varepsilon}^{\mathrm{in}}(\varphi,z)\right|\leq
C\sigma^{\star}_{\upbeta}(z)\varepsilon^{1/2}(1+M_{0,\varphi}+M_{1,\varphi})\int_{\mathbb{R}^{d}}\left\|x\right\|\exp[-\left\|x\right\|^{2}]\mathrm{d}x/C_{1}\;,}$
(321)

which concludes the proof of (312) upon combining this result, (315), and the
fact that $\exp[-t]\leq 1/t$ for any $t>0$.

Next, we show that there exist $c_{1},\upbeta>0$ such that $m(z)\geq
c_{1}\min(1/\sigma_{\beta}^{\star}(z),c^{\star}(z))$ with $c_{1},\upbeta>0$
independent from $z$. Since $\lim_{\|x\|\to+\infty}u(x,z)=+\infty$, there
exists $\tilde{x}(z)$ which minimizes $x\mapsto u(x,z)$ on
$\mathsf{E}=\mathbb{R}^{d}\backslash\mathsf{V}(z)$. We distinguish two cases.
If $\tilde{x}(z)\in\mathrm{int}(\mathsf{E})$ then $\tilde{x}(z)$ is a local
minimizer of $x\mapsto u(x,z)$. Hence $m(z)\geq c^{\star}(z)$. If
$\tilde{x}(z)\in\mathsf{E}\backslash\mathrm{int}(\mathsf{E})$ then
$\tilde{x}(z)\in\bar{\mathsf{V}}(z)\backslash\mathsf{V}(z)$ and we have that
$m(z)\geq\delta(z)^{2}$, which concludes the proof. ∎

Finally using Section B.1, Section B.1 and Section B.2 we establish our main
result.

######  Proposition B.0.

Assume H 5. Let $\varphi:\ \mathbb{R}^{d}\to\mathbb{R}$ be a
$M_{1,\varphi}$-Lipschitz function, $M_{1,\varphi},C_{\varphi}\geq 0$ and
$z\in\mathsf{Z}$. Assume that $\sigma_{1}^{\star}(z)<+\infty$ and that for any
$x\in\mathbb{R}^{d}$, $\left|\varphi(x)\right|\leq
C_{\varphi}\exp[C_{\varphi}\left\|x\right\|^{\upalpha}]$. Then, there exist
$B_{2}\in\mathrm{C}(\mathbb{R}_{+},\mathbb{R}_{+})$ and $\upbeta>0$ such that

$\left|\updelta_{z}\mathrm{S}_{\varepsilon}[\varphi]-\updelta_{z}\mathrm{S}_{0}[\varphi]\right|\leq
B_{2}(C_{\varphi})(1+M_{0,\varphi}+M_{1,\varphi})(1+\sigma_{\upbeta}^{\star}(z))\\{\varepsilon^{1/2}+\varepsilon^{-d/2}\exp[-c^{\star}(z)/\varepsilon]\\}\;,$
(322)

with $M_{0,\varphi}=\sup\\{|\varphi(x)|\,:\;x\in\mathsf{K}\\}$, $\mathsf{K}$,
$B_{2}$ and $\upbeta$ independent from $z$ and $B_{2}$ non-decreasing.

###### Proof.

In this proof we assume that
$\varphi\in\mathrm{C}^{1}(\mathbb{R}^{d},\mathbb{R})$. The extension to
Lipschitz function is similar to the proof of Theorem 2, see Section 4.1.3,
i.e. we use the smoothing Section A.4. Let
$\bar{\varepsilon}\in\left(0,\mathtt{m}/(1+C_{\varphi})\right)$. First, let
$\mathsf{V}(z)$ be given by Section B.2 and set $\mathsf{K}$ such that

$\mathsf{K}=\\{x_{0}^{k}(z)\,:\;z\in\mathsf{Z},\
k\in\\{1,\dots,N\\}\\}+\bar{\operatorname{B}}(0,1)\;.$ (323)

Note that $\mathsf{V}(z)\subset\mathsf{K}$. Applying Section B.1 there exists
$A_{0}\geq 0$ such that for any
$\varepsilon\in\left[0,\bar{\varepsilon}\right]$,

$\mathcal{J}_{\varepsilon}(z)\geq A_{0}\;.$ (324)

In addition, using Section B.1 we have that there exist $\beta_{1}>0$ and
$A_{1}\in\mathrm{C}(\mathbb{R}_{+},\mathbb{R}_{+})$ such that for any
$\varepsilon\in\left(0,\bar{\varepsilon}\right)$ we have

$\mathcal{I}_{\varepsilon}^{\mathrm{out}}(\varphi,z)\leq
A_{1}(C_{\varphi})\varepsilon^{-d/2}\\{\exp[-m(z)/\varepsilon]+\exp[-\beta_{1}/\varepsilon]\\}\;.$
(325)

Using Section B.2 there exist $c_{1},\upgamma>0$ (independent from $z$) such
that $m(z)\geq\min(c_{1}/\sigma_{\upgamma}^{\star}(z),c^{\star}(z))$. Hence,
combining this result, (325) and the fact that there exists $c_{3}$ such that
for any $t>0$, $\exp[-1/t]\leq c_{3}t^{(d+1)/2}$, we get

$\mathcal{I}_{\varepsilon}^{\mathrm{out}}(\varphi,z)\leq
A_{1}(C_{\varphi})\varepsilon^{-d/2}\\{\exp[-c^{\star}(z)/\varepsilon]+(\sigma_{\upbeta}^{\star}(z))^{(d+1)/2}\varepsilon^{(d+1)/2}/c_{3}+\exp[-\beta_{1}/\varepsilon]\\}\;.$
(326)

Therefore, there exist $\upbeta^{\prime}>0$ and
$A_{2}\in\mathrm{C}(\mathbb{R}_{+},\mathbb{R}_{+})$ (non-increasing) such that
for any $\varepsilon\in\left(0,\bar{\varepsilon}\right)$ we have

$\mathcal{I}_{\varepsilon}^{\mathrm{out}}(\varphi,z)\leq
A_{2}(C_{\varphi})\\{(1+\sigma_{\upbeta^{\prime}}^{\star}(z))\varepsilon^{1/2}+\varepsilon^{-d/2}\exp[-c^{\star}(z)/\varepsilon]\\}\;.$
(327)

In addition, using Section B.2, there exist $B_{1}\geq 0$ and
$\upbeta^{\prime\prime}>0$ such that for any $\varepsilon>0$ we have

$|\mathcal{I}_{\varepsilon}^{\mathrm{in}}(\varphi,z)-\mathcal{I}_{0}(\varphi,z)|\leq
B_{1}\sigma_{\upbeta^{\prime\prime}}^{\star}(z)(1+M_{0,\varphi}(z)+M_{1,\varphi}(z))\varepsilon^{1/2}\;,$
(328)

Note that $M_{i,\varphi}(z)\leq M_{i,\varphi}$ for $i\in\\{0,1\\}$. Hence,
combining (327) and (328), there exist
$A_{3}\in\mathrm{C}(\mathbb{R}_{+},\mathbb{R}_{+})$ (non-increasing) and
$\upbeta_{0}>0$ such that for any
$\varepsilon\in\left(0,\bar{\varepsilon}\right)$ we have

$|\mathcal{I}_{\varepsilon}(\varphi,z)-\mathcal{I}_{0}(\varphi,z)|\leq
A_{3}(C_{\varphi})(1+M_{0,\varphi}+M_{1,\varphi})\\{(1+\sigma_{\upbeta_{0}}^{\star}(z))\varepsilon^{1/2}+\varepsilon^{-d/2}\exp[-c^{\star}(z)/\varepsilon]\\}\;,$
(329)

Similar results hold if $\varphi$ is replaced by $1$ and we get that there
exist $A_{4}\geq 0$, $\upbeta_{1}>0$ such that for any
$\varepsilon\in\left(0,\bar{\varepsilon}\right)$ we have

$\displaystyle|\mathcal{I}_{\varepsilon}(\varphi,z)-\mathcal{I}_{0}(\varphi,z)|\leq
A_{3}(C_{\varphi})(1+M_{0,\varphi}+M_{1,\varphi})\\{(1+\sigma_{\upbeta_{1}}^{\star}(z))\varepsilon^{1/2}+\varepsilon^{-d/2}\exp[-c^{\star}(z)/\varepsilon]\\}\;,$
(330) $\displaystyle|\mathcal{J}_{\varepsilon}(z)-\mathcal{J}_{0}(z)|\leq
A_{4}\\{(1+\sigma_{\upbeta_{1}}^{\star}(z))\varepsilon^{1/2}+\varepsilon^{-d/2}\exp[-c^{\star}(z)/\varepsilon]\\}\;.$
(331)

In addition, we have that

$\left|\updelta_{z}\mathrm{S}_{\varepsilon}(\varphi)-\updelta_{z}\mathrm{S}_{0}(\varphi)\right|\leq(\mathcal{I}_{\varepsilon}(\varphi,z)-\mathcal{I}_{0}(\varphi,z))/\mathcal{J}_{\varepsilon}(z)+\mathcal{I}_{0}(\varphi,z)(\mathcal{J}_{0}(z)-\mathcal{J}_{\varepsilon}(z))/(\mathcal{J}_{0}(z)\mathcal{J}_{\varepsilon}(z))\;.$
(332)

Finally, we have that $\mathcal{I}_{0}(\varphi,z)\leq
M_{0,\varphi}\sigma_{1}^{\star}(z)$. Combining this result (324), (330) and
(332) concludes the proof.

∎

###  B.3 Control of the moments

In order to derive the uniform stability of the limiting measure, we first
need to control the moments of $\updelta_{z^{1:n}}\mathrm{S}_{\varepsilon}$
uniformly in w.r.t $\varepsilon$, $z$ and $n$. Let $z^{1:n}\in\mathsf{Z}^{n}$
and $\varepsilon>0$. We consider the Langevin diffusion
$(\mathbf{X}_{t}^{\varepsilon}(z^{1:n}))_{t\geq 0}$ given by the following
Stochastic Differential Equation (SDE):
$\mathbf{X}_{0}^{\varepsilon}(z^{1:n})\in\mathbb{R}^{d}$ and

$\mathrm{d}\mathbf{X}_{t}^{\varepsilon}(z^{1:n})=-\nabla_{x}U_{n}(\mathbf{X}_{t}^{\varepsilon}(z^{1:n}),z^{1:n})\mathrm{d}t+\sqrt{2\varepsilon}\mathrm{d}\mathbf{B}_{t}\;,$
(333)

where $(\mathbf{B}_{t})_{t\geq 0}$ is a $d$-dimensional Brownian motion with
filtration $(\mathcal{F}_{t})_{t\geq 0}$. We recall that for any
$n\in\mathbb{N}$, $z^{1:n}\in\mathsf{Z}^{n}$ and $x\in\mathbb{R}^{d}$ we have
$U_{n}(x,z^{1:n})=(1/n)\sum_{i=1}^{n}u(x,z_{i})$. Therefore, under H 3($n$) we
have that $(\mathbf{X}_{t}^{\varepsilon}(z^{1:n}))_{t\geq 0}$ is well-defined
and admits $\updelta_{z^{1:n}}\mathrm{S}_{\varepsilon}$ as an invariant
measure, see [61] for instance.

######  Lemma B.0.

Let $n\in\mathbb{N}$ and assume H 3($n$) and H 4($n$). Let
$z^{1:n}\in\mathsf{Z}^{n}$ and assume that
$\sigma_{1}^{\star}(z^{1:n})<+\infty$. Then there exist $\bar{\varepsilon}>0$
such that for any $k\in\mathbb{N}$ there exists $C_{k}\geq 0$ such that for
any $\varepsilon\in\left[0,\bar{\varepsilon}\right)$

$\textstyle{\int_{\mathbb{R}^{d}}\left\|x\right\|^{2k}\mathrm{S}_{\varepsilon}(z^{1:n},\mathrm{d}x)\leq
C_{k}\;,}$ (334)

with $C_{k}$ and $\bar{\varepsilon}$ independent from $n\in\mathbb{N}$ and
$z^{1:n}\in\mathsf{Z}^{n}$.

###### Proof.

Let $\varepsilon>0$. First, since $x\mapsto\nabla_{x}U_{n}(x,z^{1:n})$ is
Lipschitz continuous we have that
$(\mathbf{X}_{t}(z^{1:n})^{\varepsilon})_{t\geq 0}$ is well-defined and is a
continuous semi-martingale using [45, Theorem 2.3, Theorem 2.4 , Chapter 4]
such that for any $t\geq 0$

$\textstyle{\mathbf{X}_{t}^{\varepsilon}(z^{1:n})=\mathbf{X}_{0}^{\varepsilon}(z^{1:n})-\int_{0}^{t}\nabla_{x}U_{n}(\mathbf{X}_{s}^{\varepsilon}(z^{1:n}),z^{1:n})\mathrm{d}s+\sqrt{2\varepsilon}\mathbf{B}_{t}\;.}$
(335)

Let $\mathtt{m}_{0}>0$. Using Itô’s formula, see [45, Theorem 5.1] we have
that for any $t\geq 0$ and $k\in\mathbb{N}$ with $k\geq 2$

$\displaystyle\|\mathbf{X}_{t}^{\varepsilon}(z^{1:n})\|^{2k}\exp[\mathtt{m}_{0}t]=\textstyle{\mathbf{X}_{0}^{\varepsilon}(z^{1:n})+2k\int_{0}^{t}\langle\mathbf{X}_{s}^{\varepsilon}(z^{1:n}),\nabla_{x}U_{n}(\mathbf{X}_{s}^{\varepsilon}(z^{1:n}),z^{1:n})\rangle\|\mathbf{X}_{s}^{\varepsilon}(z^{1:n})\|^{2(k-1)}\exp[\mathtt{m}_{0}s]\mathrm{d}s}$
(336)
$\displaystyle\qquad\qquad\textstyle{+\mathtt{m}_{0}\int_{0}^{t}\|\mathbf{X}_{s}^{\varepsilon}(z^{1:n})\|^{2k}\exp[\mathtt{m}_{0}s]\mathrm{d}s+4\varepsilon
k(2k-1)\int_{0}^{t}\|\mathbf{X}_{s}^{\varepsilon}(z^{1:n})\|^{2(k-1)}\exp[\mathtt{m}_{0}s]\mathrm{d}s+\mathbf{M}_{t}^{\varepsilon}(z^{1:n})\;,}$
(337)

with $(\mathbf{M}_{t}^{\varepsilon}(z^{1:n}))_{t\geq 0}$ a
$\mathcal{F}_{t}$-martingale such that
$\mathbf{M}_{0}^{\varepsilon}(z^{1:n})=0$. First, using Fubini-Tonelli
theorem, we have that for any $t\geq 0$ and $k\in\mathbb{N}$

$\displaystyle\textstyle{\mathbb{E}[\|\mathbf{X}_{t}^{\varepsilon}(z^{1:n})\|^{2k}]\exp[\mathtt{m}_{0}t]\leq\mathbf{X}_{0}^{\varepsilon}(z^{1:n})+\left|2k\mathtt{m}-\mathtt{m}_{0}\right|\int_{0}^{t}\mathbb{E}[\|\mathbf{X}_{s}^{\varepsilon}(z^{1:n})\|^{2k}]\exp[\mathtt{m}_{0}s]\mathrm{d}s}$
(338) $\displaystyle\qquad\textstyle{+(4\varepsilon
k(2k-1)+2k\mathtt{c})(1+\int_{0}^{t}\mathbb{E}[\|\mathbf{X}_{s}^{\varepsilon}(z^{1:n})\|^{2k}]\exp[\mathtt{m}_{0}s]\mathrm{d}s)\;.}$
(339)

Using Grönwall’s lemma we have that for any $t\geq 0$ and $k\in\mathbb{N}$,
$\mathbb{E}[\|\mathbf{X}_{t}^{\varepsilon}(z^{1:n})\|^{2k}]<+\infty$. Hence,
using this result, the Fubini theorem and H 3($n$), we have that for any
$t\geq 0$ and $k\in\mathbb{N}$

$\displaystyle\textstyle{\mathbb{E}[\|\mathbf{X}_{t}^{\varepsilon}(z^{1:n})\|^{2k}]\exp[\mathtt{m}_{0}t]\leq\mathbf{X}_{0}^{\varepsilon}(z^{1:n})-(2k\mathtt{m}-\mathtt{m}_{0})\int_{0}^{t}\mathbb{E}[\|\mathbf{X}_{s}^{\varepsilon}(z^{1:n})\|^{2k}]\exp[\mathtt{m}_{0}s]\mathrm{d}s}$
(340) $\displaystyle\qquad\textstyle{+(4\varepsilon
k(2k-1)+2k\mathtt{c})\int_{0}^{t}\mathbb{E}[\|\mathbf{X}_{s}^{\varepsilon}(z^{1:n})\|^{2(k-1)}]\exp[\mathtt{m}_{0}s]\mathrm{d}s\;.}$
(341)

Combining this result and the fact that for any $t,a,b>0$ and
$k\in\mathbb{N}$, $at^{2(k-1)}\leq bt^{2k}+a^{k}/b^{k-1}$, we have for any
$t\geq 0$ and $k\in\mathbb{N}$

$\displaystyle\mathbb{E}[\|\mathbf{X}_{t}^{\varepsilon}(z^{1:n})\|^{2k}]\exp[\mathtt{m}_{0}t]$
$\displaystyle\textstyle{\leq\mathbf{X}_{0}^{\varepsilon}(z^{1:n})-(k\mathtt{m}-\mathtt{m}_{0})\int_{0}^{t}\mathbb{E}[\|\mathbf{X}_{s}^{\varepsilon}(z^{1:n})\|^{2k}]\exp[\mathtt{m}_{0}s]\mathrm{d}s}$
(342) $\displaystyle\quad+(4\varepsilon
k(2k-1)+2k\mathtt{c})^{k}/((k\mathtt{m})^{k-1}\mathtt{m}_{0})\exp[\mathtt{m}_{0}t]\;.$
(343)

Therefore, for any $k\in\mathbb{N}$, there exists $C_{k}\geq 0$ such that for
any $t\geq 0$, $\mathbb{E}[\|\mathbf{X}_{t}^{\varepsilon}(z^{1:n})\|^{2k}]\leq
C_{k}$ upon letting $\mathtt{m}_{0}=k\mathtt{m}$ in (342) with $C_{k}$
independent from $\varepsilon$ and $z^{1:n}$. Therefore, using that the
sequence of distributions associated with
$(\mathbf{X}_{t}^{\varepsilon}(z^{1:n}))_{t\geq 0}$ weakly converges towards
$\updelta_{z^{1:n}}\mathrm{S}_{\varepsilon}$, see [61] for instance, and the
monotone convergence theorem we get that
$\int_{\mathbb{R}^{d}}\left\|x\right\|^{2k}\mathrm{S}_{\varepsilon}(z^{1:n},\mathrm{d}x)\leq
C_{k}$. We conclude upon using Section B.2 in the case where $\varepsilon=0$.
∎

Finally, we will make use of the following lemma.

######  Lemma B.0.

Let $\mu,\nu\in\mathscr{P}(\mathbb{R}^{d})$ such that for any $k\in\mathbb{N}$
there exists $C_{k}\geq 0$ such that
$\int_{\mathbb{R}^{d}}\|x\|^{2k}\mathrm{d}\mu(x)+\int_{\mathbb{R}^{d}}\|x\|^{2k}\mathrm{d}\nu(x)\leq
C_{k}$. Then, for any $k\in\mathbb{N}$ we have that

$\mathbf{W}_{2}(\mu,\nu)\leq\sqrt{2C_{k}}\mathbf{W}_{1}(\mu,\nu)^{(k-1)/(2k-1)}\;.$
(344)

###### Proof.

Let $k\in\mathbb{N}$ and $(X,Y)$ be the optimal coupling between $\mu$ and
$\nu$ w.r.t. the Wasserstein distance of order one. Let $Z=X-Y$. Using
Hölder’s inequality we have

$\textstyle{\mathbb{E}[\|Z\|^{2}]\leq\mathbb{E}[\|Z\|^{2k}]^{1/(2k-1)}\mathbb{E}[\|Z\|]^{(2k-2)/(2k-1)}\leq\mathbb{E}[\|Z\|^{2k}]^{1/(2k-1)}\mathbf{W}_{1}(\mu,\nu)^{(2k-2)/(2k-1)}\;.}$
(345)

We conclude upon combining this result and the fact that
$\mathbb{E}[\|Z\|^{2k}]^{1/(2k-1)}\leq 2C_{k}$. ∎

##  Appendix C Basics on flows and geometric measure theory

In this section, we recall basic facts from geometric measure theory. We refer
to [29, 54, 1] for a complete exposition of geometric measure theory concepts.
We begin with a proposition establishing the existence of flows. Then, we
prove useful facts on rectifiable sets. Finally, we state area and coarea
formulas which are central to our analysis.

######  Lemma C.0.

Let $k\in\mathbb{N}^{\star}$ and
$X\in\mathrm{C}^{k}(\mathbb{R}^{d},\mathbb{R}^{d})$. Assume that $X$ is
compactly supported. Then, there exists a unique mapping
$\Phi\in\mathrm{C}^{k+1,k}(\mathbb{R}\times\mathbb{R}^{d},\mathbb{R}^{d})$
such that for any $x\in\mathbb{R}^{d}$, $\Phi(0,x)=x$ and
$\partial_{s}\Phi(s,\mathbb{R}^{d})=X(\Phi(s,\mathbb{R}^{d}))$. In addition,
for any $s\in\mathbb{R}$, $x\mapsto\Phi(s,x)$ is a diffeomorphism.

###### Proof.

The first part of the proposition is an application of [49, Theorem 2.5,
Theorem 2.6]. The second part is an application of [53, Lemma 2.4]. ∎

Note that the previous lemma can be extended to smooth manifolds.

######  Definition C.0.

[1, Definition 2.57] Let $\mathsf{E}\subset\mathbb{R}^{d}$ and
$k\in\mathbb{N}$ with $k\leq d$. $\mathsf{E}$ is countably
$\mathcal{H}^{k}$-rectifiable if there exists $(\psi_{i})_{i\in\mathbb{N}}$
such that for any $i\in\mathbb{N}$, $\psi_{i}:\
\mathbb{R}^{k}\to\mathbb{R}^{d}$ is Lipschitz continuous and

$\mathcal{H}^{k}\left(\mathsf{E}\backslash\cup_{i\in\mathbb{N}}\psi_{i}(\mathbb{R}^{k})\right)=0\;.$
(346)

In what follows, we provide an easy criterion to verify if a given level set
is a countably $\mathcal{H}^{k}$-rectifiable set. We start by recalling the
following proposition.

######  Proposition C.0.

[50, Theorem 10.5] Let $k\in\mathbb{N}$ with $k\leq d$ and $F:\
\mathbb{R}^{d}\to\mathbb{R}^{k}$ be Lipschitz continuous. Then for
$\lambda$-almost every $y\in\mathbb{R}^{k}$, $F^{-1}(\\{y\\})$ is countably
$\mathcal{H}^{d-k}$-rectifiable.

The following lemma ensures the stability of countably
$\mathcal{H}^{k}$-rectifiable sets through diffeomorphisms.

######  Lemma C.0.

Let $k\in\mathbb{N}$ with $k\leq d$, $\mathsf{E}\subset\mathbb{R}^{d}$ be a
compact countably $\mathcal{H}^{k}$-rectifiable set and
$\Phi\in\mathrm{C}^{1}(\mathbb{R}^{d},\mathbb{R}^{d})$ be a diffeomorphism.
Then $\Phi(\mathsf{E})$ is countably $\mathcal{H}^{k}$-rectifiable.

###### Proof.

Since $\mathsf{E}\subset\mathbb{R}^{d}$ is countably
$\mathcal{H}^{k}$-rectifiable there exists $(\psi_{i})_{i\in\mathbb{N}}$ such
that for any $i\in\mathbb{N}$, $\psi_{i}:\ \mathbb{R}^{k}\to\mathbb{R}^{d}$ is
Lipschitz continuous and

$\mathcal{H}^{k}\left(\mathsf{E}\backslash\cup_{i\in\mathbb{N}}\psi_{i}(\mathbb{R}^{k})\right)=0\;.$
(347)

Let
$y\in\Phi(\mathsf{E})\backslash\cup_{i\in\mathbb{N}}(\Phi\circ\psi_{i})(\mathbb{R}^{k})$.
Then there exists $x\in\mathsf{E}$ such that $y=\Phi(x)$ and for any
$i\in\mathbb{N}$ and $y_{i}\in\psi_{i}(\mathbb{R}^{k})$,
$\Phi(x)\neq\Phi(y_{i})$, i.e. $x\neq y_{i}$. Therefore
$\Phi(\mathsf{E})\backslash\cup_{i\in\mathbb{N}}(\Phi\circ\psi_{i})(\mathbb{R}^{k})\subset\Phi(\mathsf{E}\backslash\cup_{i\in\mathbb{N}}\psi_{i}(\mathbb{R}^{k}))$.
$\Phi$ is Lipschitz-continuous on $\mathsf{E}$ with constant
$\mathtt{L}_{\mathsf{E}}\geq 0$ and therefore using [1, Proposition 2.49 (iv)]
we get

$0\leq\mathcal{H}^{k}\left(\Phi(\mathsf{E})\backslash\cup_{i\in\mathbb{N}}(\Phi\circ\psi_{i})(\mathbb{R}^{k}))\right)\leq\mathtt{L}_{\mathsf{E}}^{k}\mathcal{H}^{k}\left(\mathsf{E}\backslash\cup_{i\in\mathbb{N}}\psi_{i}(\mathbb{R}^{k})\right)\leq
0\;,$ (348)

which concludes the proof. ∎

Finally, we show that if a function is regular enough then its level-sets are
countably $\mathcal{H}^{k}$-rectifiable.

######  Lemma C.0.

Let $F\in\mathrm{C}^{1}(\mathbb{R}^{d},\mathbb{R}^{p})$ with $d\geq p$ and
$\mathsf{A}\subset\mathbb{R}^{p}$ compact such that $F^{-1}(\mathsf{A})$ is
compact and $0\in\mathrm{int}(\mathsf{A})$. In addition, assume that for any
$x\in F^{-1}(\\{0\\})$, $|\mathrm{J}F(x)|>0$. Then, there exists $\eta>0$ such
that for any $t\in\bar{\operatorname{B}}_{\infty}(0,\eta)$, $F^{-1}(t)$ is
countably $\mathcal{H}^{d-p}$-rectifiable and
$\mathcal{H}^{d-p}(F^{-1}(t))<+\infty$.

###### Proof.

In this proof, we show that the level sets $F^{-1}(t)$ and $F^{-1}(s)$ are
diffeomorphic for $t,s\in\bar{\operatorname{B}}(0,\eta)$. Then, we conclude
upon combining Appendix C and Appendix C. First, note that $F^{-1}(0)$ is
compact since it is closed in $F^{-1}(\mathsf{A})$ which is compact. There
exists $\eta_{0}>0$ such that for any $x\in
F^{-1}(\bar{\operatorname{B}}_{\infty}(0,\eta_{0}))$,
$\left|\mathrm{J}F(x)\right|>0$. Indeed, if this is not the case then there
exists $(x_{k})_{k\in\mathbb{N}}$ with $\lim_{k\to+\infty}F(x_{k})=0$ and
$\left|\mathrm{J}F(x_{k})\right|=0$. Since $0\in\mathrm{int}(\mathsf{A})$
there exists $\eta^{\prime}>0$ such that
$F^{-1}(\bar{\operatorname{B}}(0,\eta^{\prime}))$ is compact and therefore
there exists $x^{\star}$ such that, up to extraction,
$\lim_{k\to+\infty}x_{k}=x^{\star}$. Then $F(x^{\star})=0$ and
$\mathrm{J}F(x^{\star})=0$, which is absurd. We define
$\mathsf{K}_{0}=F^{-1}(\bar{\operatorname{B}}_{\infty}(0,\eta_{0}))$ and
$\mathsf{K}_{1}=F^{-1}(\bar{\operatorname{B}}_{\infty}(0,\eta))$ with
$\eta=\min(\eta_{0}/2,\eta^{\prime})$. Note that
$\mathsf{K}_{1}\subset\mathrm{int}(\mathsf{K}_{0})$.

Let $G(x)=\mathrm{D}F(x)\mathrm{D}F(x)^{\top}$. Note that for any
$x\in\mathsf{K}_{0}$, $G(x)$ is invertible. We define $\\{f_{i}\\}_{i=1}^{p}$
such that for any $i\in\\{1,\dots,p\\}$, $f_{i}:\
\mathbb{R}^{d}\to\mathbb{R}^{d}$ with for any $x\in\mathsf{K}_{0}$

$\textstyle{f_{i}(x)=\sum_{k=1}^{p}h_{i,k}(x)\nabla F_{k}(x)\;,}$ (349)

with $\\{h_{i,j}(x)\\}_{1\leq i,j\leq p}=G^{-1}(x)$. In addition, we assume
that $f_{i}(x)=0$ for any $x\notin\mathsf{K}_{1}$. For any
$x\in\mathsf{K}_{1}$ and $i,j\in\\{1,\dots,p\\}$ we have

$\textstyle{\langle f_{i}(x),\nabla
F_{j}(x)\rangle=\sum_{k=1}^{p}h_{i,k}(x)\langle\nabla F_{k}(x),\nabla
F_{j}(x)\rangle=\updelta_{i}(j)\;.}$ (350)

In what follows, we let $\\{g_{i}\\}_{i=1}^{p}$ such that for any
$i\in\\{1,\dots,p\\}$, $g_{i}:\ \mathbb{R}^{d}\to\mathbb{R}^{d}$ and
$g_{i}\in\mathrm{C}^{1}(\mathbb{R}^{d},\mathbb{R}^{d})$ such that for any
$x\in\mathsf{K}_{1}$, $g_{i}(x)=f_{i}(x)$ and for any
$x\notin\mathrm{int}(\mathsf{K}_{1})$, $g_{i}(x)=0$, such functions exist
using Whitney extension theorem for instance, see [69]. In what follows, we
fix $t=(t_{0},\dots,t_{p})\in\bar{\operatorname{B}}_{\infty}(0,\eta)$. For any
$i\in\\{1,\dots,p\\}$ let $\Phi_{i}:\
\mathbb{R}\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ given by
$\Phi_{i}(0,\cdot)=\operatorname{Id}$ and for any $s\in\mathbb{R}$ and
$x\in\mathbb{R}^{d}$

$\partial_{s}\Phi_{i}(s,x)=-g_{i}(\Phi_{i}(s,x))\;.$ (351)

For any $i\in\\{1,\dots,p\\}$, $\Phi_{i}$ is well-defined using Appendix C.
Therefore, we have for any $i\in\\{1,\dots,p\\}$ and $s\in\mathbb{R}$,
$x\in\mathbb{R}^{d}$ such that $\Phi(s,x)\in\mathsf{K}_{1}$

$\partial_{s}F(\Phi_{i}(s,x))=-(\langle g_{i}(\Phi_{i}(s,x)),\nabla
F_{1}(\Phi_{i}(s,x))\rangle,\dots,\langle g_{i}(\Phi_{i}(s,x)),\nabla
F_{p}(\Phi_{i}(s,x))\rangle)=-e_{i}\;,$ (352)

where we recall that $\\{e_{i}\\}_{i=1}^{p}$ is the canonical basis of
$\mathbb{R}^{p}$. We define $\bar{\Phi}_{t}:\ \mathbb{R}^{d}\to\mathbb{R}^{d}$
such that for any $x\in\mathbb{R}^{d}$, $\bar{\Phi}_{t}(x)=x^{(p)}$ with
$x^{(0)}=x$ and for any $i\in\\{0,\dots,p-1\\}$,
$x^{(i+1)}=\Phi_{i+1}(t_{i+1},x^{(i)})$. Note that
$\bar{\Phi}_{t}\in\mathrm{C}^{1}(\mathbb{R}^{d},\mathbb{R})$ is a
diffeomorphism using Appendix C. Using Appendix C, there exists
$t_{0}\in\bar{\operatorname{B}}_{\infty}(0,\eta,$) such that $F^{-1}(t_{0})$
is countably $\mathcal{H}^{d-p}$ rectifiable. Let
$s\in\bar{\operatorname{B}}_{\infty}(0,\eta)$. Using Appendix C and that
$\bar{\Phi}_{-s}\circ\bar{\Phi}_{t_{0}}$ is a diffeomorphism between
$F^{-1}(t_{0})$ and $F^{-1}(s)$ we get that $F^{-1}(s)$ is countably
$\mathcal{H}^{d-p}$ rectifiable, which concludes the first part of the proof.

For the second part of the proof, we let $R\geq 0$ such that
$F^{-1}(\mathsf{A})\subset\bar{\operatorname{B}}(0,R)$. Since
$\bar{\operatorname{B}}(0,R)$ is $\mathcal{H}^{d}$-rectifiable we obtain using
the coarea formula Theorem C.1

$\textstyle{\int_{\bar{\operatorname{B}}(0,R)}|\mathrm{J}F(x)|\mathrm{d}x=\int_{\mathbb{R}^{p}}\mathcal{H}^{d-p}(\bar{\operatorname{B}}(0,R)\cap
F^{-1}(t))\mathrm{d}t\geq\int_{\bar{\operatorname{B}}_{\infty}(0,\eta)}\mathcal{H}^{d-p}(F^{-1}(t))\mathrm{d}t\;.}$
(353)

Therefore, there exists $t_{0}\in\bar{\operatorname{B}}_{\infty}(0,\eta)$ such
that $\mathcal{H}^{d-p}(F^{-1}(t_{0}))<+\infty$. Let
$\Psi=\bar{\Phi}_{-t}\circ\bar{\Phi}_{t_{0}}$ and $\mathtt{L}\geq 0$ such that
for any $x\in F^{-1}(t_{0})$ we have
$\left\|\mathrm{d}\Psi(x)\right\|\leq\mathtt{L}$. Then, using [1, Proposition
2.49 (iv)] we have

$\mathcal{H}^{d-p}(F^{-1}(t))=\mathcal{H}^{d-p}(\Psi(F^{-1}(t_{0})))\leq\mathtt{L}^{d-p}\mathcal{H}^{d-p}(F^{-1}(t_{0}))<+\infty\;,$
(354)

which concludes the proof. ∎

We conclude this section with the area and coarea formulae.

######  Theorem C.1.

Let $F:\mathrm{C}^{1}(\mathbb{R}^{d},\mathbb{R}^{p})$ be Lipschitz continuous,
$k\in\mathbb{N}$ with $k\leq d$ and $\mathsf{E}\subset\mathbb{R}^{d}$ be a
countably $\mathcal{H}^{k}$-rectifiable set. Let $\varphi:\
\mathbb{R}^{d}\to\mathbb{R}$ measurable such that

$\textstyle{\int_{\mathsf{E}}\left|\varphi(x)\right|\mathrm{J}F(x)\mathrm{d}\mathcal{H}^{k}(x)<+\infty\;,}$
(355)

or assume that $\varphi:\ \mathbb{R}^{d}\to\left[0,+\infty\right)$. Then, the
following hold:

  * •

(Area formula) If $d\leq p$ then

$\textstyle{\int_{\mathsf{E}}\varphi(x)\mathrm{J}F(x)\mathrm{d}\mathcal{H}^{k}(x)=\int_{\mathbb{R}^{p}}\left(\int_{\mathsf{E}\cap
F^{-1}(y)}\varphi(x)\mathrm{d}\mathcal{H}^{0}(x)\right)\mathrm{d}\mathcal{H}^{k}(y)\;.}$
(356)

  * •

(Coarea formula) If $k\geq p$ then

$\textstyle{\int_{\mathsf{E}}\varphi(x)\mathrm{J}F(x)\mathrm{d}\mathcal{H}^{k}(x)=\int_{\mathbb{R}^{p}}\left(\int_{\mathsf{E}\cap
F^{-1}(y)}\varphi(x)\mathrm{d}\mathcal{H}^{k-p}(x)\right)\mathrm{d}\mathcal{H}^{p}(y)\;.}$
(357)

###### Proof.

These results follow from [1, Theorem 2.91, Theorem 2.93] combined with [1,
Exercicse 2.12]. ∎

In particular, note that if
$F\in\mathrm{C}^{1}(\mathbb{R}^{d},\mathbb{R}^{d})$ is a Lipschitz
diffeomorphism we have using Theorem C.1 that for any $\varphi:\
\mathbb{R}^{d}\to\mathbb{R}$ measurable such that

$\textstyle{\int_{\mathsf{E}}\left|\varphi(x)\right|\left|J\Phi(x)\right|\mathrm{d}\mathcal{H}^{k}(x)<+\infty\;,}$
(358)

the following change of variable formula with respect to $\mathcal{H}^{k}$
holds

$\textstyle{\int_{\mathsf{E}}\varphi(x)\mathrm{J}F(x)\mathrm{d}\mathcal{H}^{k}(x)=\int_{\mathbb{R}^{d}}\left(\int_{\mathsf{E}\cap
F^{-1}(y)}\varphi(x)\mathrm{d}\mathcal{H}^{0}(x)\right)\mathrm{d}\mathcal{H}^{k}(y)=\int_{F(\mathsf{E})}\varphi(F^{-1}(y))\mathrm{d}\mathcal{H}^{k}(y)\;.}$
(359)

Note that in order for (359) to hold, $\mathsf{E}$ needs to be countably
$\mathcal{H}^{k}$-rectifiable.