Before moving to the proof of Theorem 6, let us recall here (cf. Section 3)
that $V(y)=1+|y|^{2}$ is a Lyapunov function for (25) satisfying a geometric
drift condition, i.e., there are constants $A_{\mu},\,A_{\nu}>0$ and
$B_{\mu},\,B_{\nu}$, independent of $n$ (but depending on $x$ and $T$), such
that

(206) $\mathcal{L}_{\psi^{\star}}V(y)\;\vee\;\mathcal{L}_{\psi^{n}}V(y)\;\leq
B_{\nu}-A_{\nu}\,V(y)\;\text{ and
}\;\mathcal{L}_{\varphi^{\star}}V(y)\;\vee\;\mathcal{L}_{\varphi^{n}}V(y)\;\leq
B_{\mu}-A_{\mu}\,V(y)\;,$

where $\mathcal{L}_{h}\coloneqq\Delta/2-\frac{1}{2}\langle T^{-1}(y-x)+\nabla
h(y),\,\nabla\rangle$ is the generator associated to (25). The possibility of
choosing parameters $A_{\mu},\,A_{\nu},\,B_{\mu},\,B_{\nu}$ independently from
$n\in\mathbb{N}$ follows from the pointwise convergence of the gradients of
Theorem 3 and the convergences
$\alpha_{\mu,n}\uparrow\alpha_{\varphi^{\star}}$,
$\alpha_{\nu,n}\uparrow\alpha_{\psi^{\star}}$ stated in Theorem 7.

###### Proof of Theorem 6.

For sake of notations let us introduce the constant

(207) $C_{x}\coloneqq\max\\{1,\,C_{1}\,T^{-1}\,|x|+C_{2}\\}\;,$

with $C_{1},\,C_{2}>0$ as in Section 5.3. We will proceed as in Section 4,
this time considering a distorted Wasserstein semi-distance

(208)
$\mathbf{W}_{f_{x}^{\nu}}(\cdot,\cdot)\coloneqq\inf_{\pi\in\Pi(\cdot,\cdot)}\mathbb{E}_{(Y,Z)\sim\pi}\biggl{[}f_{x}^{\nu}(|Y-Z|)\,(1+\varepsilon
V(Y)+\varepsilon V(Z))\biggr{]}\;.$

In the above definition $V(y)\coloneqq 1+|y|^{2}$ and we consider the bounded
concave function $f_{x}^{\nu}$ and the parameter $\varepsilon\in(0,1)$ built
in Appendix B, by following [27, Theorem 2.2]. Let us just state here that
there is no dependence from the index $n\in\mathbb{N}$, $\varepsilon\in(0,1)$
is taken small enough such that (249) will hold, $\mathbf{W}_{f_{x}^{\nu}}$
contracts along the semigroup $(P^{n}_{t})_{t\geq 0}$ associated to the SDE

(209) $\
\mathrm{d}Y_{t}^{n}=-\biggl{(}\frac{Y_{t}^{n}-x}{2T}+\frac{1}{2}\nabla\psi^{n}(Y^{n}_{t})\biggr{)}\mathrm{d}t+\mathrm{d}B_{t}\;,$

and there exists a rate $\lambda>0$ (independent from $n\in\mathbb{N}$,
explicitly given at (265)) such that for any
$\mu_{1},\,\mu_{2}\in\mathcal{P}(\mathbb{R}^{d})$

(210) $\mathbf{W}_{f^{\nu}_{x}}(\mu_{1}\,P^{n}_{t},\,\mu_{2}\,P^{n}_{t})\leq
e^{-\lambda\,t}\,\mathbf{W}_{f^{\nu}_{x}}(\mu_{1},\,\mu_{2})\;.$

Moreover, $\mathbf{W}_{f_{x}^{\nu}}$ satisfies a weak triangle inequality,
i.e., for any $\mu_{1},\,\mu_{2},\,\mu_{3}\in\mathcal{P}(\mathbb{R}^{d})$

(211) $\mathbf{W}_{f_{x}^{\nu}}(\mu_{1},\mu_{2})\leq
C_{\Delta}(\mathbf{W}_{f_{x}^{\nu}}(\mu_{1},\mu_{3})+\mathbf{W}_{f_{x}^{\nu}}(\mu_{3},\mu_{2}))\;,$

with $C_{\Delta}=\max\\{3,\,2+2\,R_{2}^{2}\\}$, with the radius $R_{2}>0$
given in (248) such that

(212) $\begin{split}\frac{C_{I}}{2}\,r\leq f_{x}^{\nu}(r)\leq r&\quad\text{
and }\quad\frac{1}{2}\leq(f_{x}^{\nu})^{\prime}(r)\leq
1\quad\forall\,r<R_{2}\;,\\\ f_{x}^{\nu}(r)=f_{x}^{\nu}(R_{2})&\quad\text{ and
hence }\quad(f_{x}^{\nu})^{\prime}(r)=0\quad\forall\,r>R_{2}\;,\end{split}$

for a positive constant $C_{I}\in(0,1)$ given in (252). This particularly
implies that for any coupling
$(Y,Z)\sim\pi\in\Pi(\pi^{x,\psi^{\star}}_{T},\,\pi^{x,\psi^{n}}_{T})$ we have

(213)
$\begin{split}T^{-2}|Y-Z|\,(|Y|+|Z|&+C_{x})\leq\frac{C_{x}}{T^{2}\,\varepsilon^{1/2}}|Y-Z|\,(1+\varepsilon^{1/2}\,|Y|+\varepsilon^{1/2}\,|Z|)\\\
\leq&\begin{cases}\frac{2}{C_{I}}\,\frac{C_{x}}{T^{2}\,\varepsilon^{1/2}}\,f_{x}^{\nu}(|Y-Z|)\,(1+\varepsilon^{1/2}\,|Y|+\varepsilon^{1/2}\,|Z|)\quad&\text{
if }|Y-Z|\leq R_{2}\,,\\\
\frac{C_{x}}{T^{2}\,\varepsilon}\,f_{x}^{\nu}(R_{2})^{-1}\,f^{\nu}_{x}(R_{2})\,(1+\varepsilon^{1/2}\,|Y|+\varepsilon^{1/2}\,|Z|)^{2}\quad&\text{
if }|Y-Z|>R_{2}\end{cases}\\\
\leq&\frac{10}{3}\,\frac{C_{x}}{T^{2}\,\varepsilon^{1/2}}\,\biggl{(}\frac{2}{C_{I}}\vee\frac{f_{x}^{\nu}(R_{2})^{-1}}{\varepsilon^{1/2}}\biggr{)}\,f_{x}^{\nu}(|Y-Z|)\,(1+\varepsilon\,V(Y)+\varepsilon\,V(Z))\;.\end{split}$

Using (200) and by minimizing the above over
$\pi\in\Pi(\pi^{x,\psi^{\star}}_{T},\,\pi^{x,\psi^{n}}_{T})$, we deduce then
that

(214)
$\|\nabla^{2}\varphi^{n+1}-\nabla^{2}\varphi^{\star}\|_{\mathrm{F}}(x)\leq\frac{10}{3}\,\frac{C_{x}}{T^{2}\,\varepsilon^{1/2}}\,\biggl{(}\frac{2}{C_{I}}\vee\frac{f_{x}^{\nu}(R_{2})^{-1}}{\varepsilon^{1/2}}\biggr{)}\,\mathbf{W}_{f^{\nu}_{x}}(\pi^{x,\psi^{\star}}_{T},\,\pi^{x,\psi^{n}}_{T})\;.$

Now, recall that $\pi^{x,\psi^{\star}}_{T}$ corresponds to the invariant
probability of the SDE

(215)
$\mathrm{d}Y_{t}^{\star}=-\biggl{(}\frac{Y_{t}^{\star}-x}{2T}+\frac{1}{2}\nabla\psi^{\star}(Y^{\star}_{t})\biggr{)}\mathrm{d}t+\mathrm{d}B_{t}$

and let $(P^{\star}_{t})_{t\geq 0}$ be its corresponding semigroup.
Particularly we have
$\pi^{x,\psi^{\star}}_{T}\,P^{\star}_{t}=\pi^{x,\psi^{\star}}_{T}$ for any
$t\geq 0$. Similarly, for any $t\geq 0$ it holds
$\pi^{x,\psi^{n}}_{T}\,P^{n}_{t}=\pi^{x,\psi^{n}}_{T}$, with
$(P^{n}_{t})_{t\geq 0}$ being the semigroup introduced above.

Given the above premises, from the weak triangle inequality (211) we deduce
that

(216)
$\begin{split}\mathbf{W}_{f^{\nu}_{x}}(\pi^{x,\psi^{\star}}_{T},\,\pi^{x,\psi^{n}}_{T})=&\,\mathbf{W}_{f^{\nu}_{x}}(\pi^{x,\psi^{\star}}_{T}\,P^{\star}_{t},\,\pi^{x,\psi^{n}}_{T}\,P^{n}_{t})\\\
\leq&\,C_{\Delta}\,\biggl{(}\mathbf{W}_{f^{\nu}_{x}}(\pi^{x,\psi^{\star}}_{T}\,P^{\star}_{t},\,\pi^{x,\psi^{\star}}_{T}\,P^{n}_{t})+\mathbf{W}_{f^{\nu}_{x}}(\pi^{x,\psi^{\star}}_{T}\,P^{n}_{t},\,\pi^{x,\psi^{n}}_{T}\,P^{n}_{t})\biggr{)}\\\
\overset{\eqref{contrazione}}{\leq}&\,C_{\Delta}\,\mathbf{W}_{f^{\nu}_{x}}(\pi^{x,\psi^{\star}}_{T}\,P^{\star}_{t},\,\pi^{x,\psi^{\star}}_{T}\,P^{n}_{t})+C_{\Delta}\,e^{-\lambda\,t}\,\mathbf{W}_{f^{\nu}_{x}}(\pi^{x,\psi^{\star}}_{T},\,\pi^{x,\psi^{n}}_{T})\;.\end{split}$

Therefore for any $t>\lambda^{-1}\,\log C_{\Delta}$ it holds

(217)
$\mathbf{W}_{f^{\nu}_{x}}(\pi^{x,\psi^{\star}}_{T},\,\pi^{x,\psi^{n}}_{T})\leq\frac{C_{\Delta}}{1-C_{\Delta}\,e^{-\lambda\,t}}\,\mathbf{W}_{f^{\nu}_{x}}(\pi^{x,\psi^{\star}}_{T}\,P^{\star}_{t},\,\pi^{x,\psi^{\star}}_{T}\,P^{n}_{t})\;.$

We will show that
$\mathbf{W}_{f^{\nu}_{x}}(\pi^{x,\psi^{\star}}_{T}\,P^{\star}_{t},\,\pi^{x,\psi^{\star}}_{T}\,P^{n}_{t})$
is exponentially small as $n\uparrow+\infty$. Hence consider the diffusion
processes

(218)
$\begin{cases}\mathrm{d}Y_{t}^{\star}=-\biggl{(}\frac{Y_{t}^{\star}-x}{2T}+\frac{1}{2}\nabla\psi^{\star}(Y^{\star}_{t})\biggr{)}\mathrm{d}t+\mathrm{d}B_{t}\;,\\\
\mathrm{d}Z_{t}=-\biggl{(}\frac{Z_{t}-x}{2T}+\frac{1}{2}\nabla\psi^{n}(Z_{t})\biggr{)}\mathrm{d}t+\mathrm{d}B_{t}\;,\\\
Y^{\star}_{0}=Z_{0}\sim\pi^{x,\psi^{\star}}_{T}\;,\end{cases}$

where $(B_{t})_{t\geq 0}$ is the same $d$-dimensional Brownian motion. Notice
that $Y_{t}^{\star}\sim\pi^{x,\psi^{\star}}_{T}P^{\star}_{t}$ whereas
$Z_{t}\sim\pi^{x,\psi^{\star}}_{T}P^{n}_{t}$, which yields to

(219)
$\begin{split}\mathbf{W}_{f^{\nu}_{x}}(\pi^{x,\psi^{\star}}_{T}\,P^{\star}_{t},\,\pi^{x,\psi^{\star}}_{T}\,P^{n}_{t})\leq\mathbb{E}_{Y_{t}^{\star},Z_{t}}\biggl{[}f_{x}^{\nu}(|Y^{\star}_{t}-Z_{t}|)\,(1+\varepsilon
V(Y_{t}^{\star})+\varepsilon V(Z_{t}^{\star}))\biggr{]}\;.\end{split}$

By construction, if we introduce $r_{t}\coloneqq|Y^{\star}_{t}-Z_{t}|$, we
immediately get

(220)
$\mathrm{d}r_{t}^{2}=-\frac{r_{t}^{2}}{T}\,\mathrm{d}t-\left\langle\nabla\psi^{\star}(Y_{t}^{\star})-\nabla\psi^{n}(Z_{t}),\,Y^{\star}_{t}-Z_{t}\right\rangle\,\mathrm{d}t\;.$

By reasoning as in the proof of Theorem 9, fix $\delta>0$ and consider the
function $\rho_{\delta}(r)\coloneqq\sqrt{r+\delta}$. Then it holds

(221) $\displaystyle\mathrm{d}\rho_{\delta}(\bar{r}_{t}^{2})=$
$\displaystyle\;-(2T)^{-1}\,\frac{r_{t}^{2}}{\rho_{\delta}(r_{t}^{2})}\mathrm{d}t-(2\;\rho_{\delta}(\bar{r}_{t}^{2}))^{-1}\left\langle\nabla\psi^{\star}(Y_{t}^{\star})-\nabla\psi^{n}(Z_{t}),\,Y^{\star}_{t}-Z_{t}\right\rangle\,\mathrm{d}t\;,$

and hence we deduce that it holds

(222)
$\begin{split}&\mathrm{d}f_{x}^{\nu}(\rho_{\delta}(r_{t}^{2}))=-\frac{(f^{\nu}_{x})^{\prime}(\rho_{\delta}(r_{t}^{2}))}{2}\biggl{(}T^{-1}\frac{r_{t}^{2}}{\rho_{\delta}(r_{t}^{2})}+(\rho_{\delta}(r_{t}^{2})^{-1}\left\langle\nabla\psi^{\star}(Y_{t}^{\star})-\nabla\psi^{n}(Z_{t}),\,Y^{\star}_{t}-Z_{t}\right\rangle\biggr{)}\,\mathrm{d}t\\\
&\leq-(T^{-1}+\kappa_{\psi^{n}}(r_{t}))\,\frac{(f^{\nu}_{x})^{\prime}(\rho_{\delta}(r_{t}^{2}))}{2}\,\frac{r_{t}^{2}}{\rho_{\delta}(r_{t}^{2})}\,\mathrm{d}t+\frac{(f^{\nu}_{x})^{\prime}(\rho_{\delta}(r_{t}^{2}))}{2}\,\frac{r_{t}}{\rho_{\delta}(r_{t}^{2})}\,|\nabla\psi^{\star}-\nabla\psi^{n}|(Y^{\star}_{t})\mathrm{d}t\\\
&\overset{\eqref{eq_bound_gio_teo:psi}}{\leq}-(T^{-1}+\alpha_{\nu,n}-r_{t}^{-1}{\tilde{g}}_{\nu}(r_{t}))\,\frac{(f^{\nu}_{x})^{\prime}(\rho_{\delta}(r_{t}^{2}))}{2}\,\frac{r_{t}^{2}}{\rho_{\delta}(r_{t}^{2})}\,\mathrm{d}t+\,\frac{(f^{\nu}_{x})^{\prime}(\rho_{\delta}(r_{t}^{2}))}{2}\,\frac{r_{t}}{\rho_{\delta}(r_{t}^{2})}\,|\nabla\psi^{\star}-\nabla\psi^{n}|(Y^{\star}_{t})\mathrm{d}t\,.\end{split}$

Since $\alpha_{\nu,n}>\alpha_{\nu}-T^{-1}$ (cf. Theorem 7), the sublinearity
${\tilde{g}_{\nu}}(r)\leq{\tilde{G}}_{\nu}\,r$ and the upper bound (212) for
$(f_{x}^{\nu})^{\prime}$, imply

(223)
$\begin{split}\mathrm{d}f_{x}^{\nu}(\rho_{\delta}(r_{t}^{2}))\leq&({\tilde{G}}_{\nu}-\alpha_{\nu})^{+}\,\frac{(f^{\nu}_{x})^{\prime}(\rho_{\delta}(r_{t}^{2}))}{2}\,\rho_{\delta}(r_{t}^{2})\,\mathrm{d}t+\frac{1}{2}\,|\nabla\psi^{\star}-\nabla\psi^{n}|(Y^{\star}_{t})\mathrm{d}t\\\
\overset{\eqref{new:f:eq}}{\leq}&C_{I}^{-1}\,({\tilde{G}}_{\nu}-\alpha_{\nu})^{+}\,f_{x}^{\nu}(\rho_{\delta}(r_{t}^{2}))\,\mathrm{d}t+\frac{1}{2}\,|\nabla\psi^{\star}-\nabla\psi^{n}|(Y^{\star}_{t})\mathrm{d}t\;.\end{split}$

Gronwall Lemma and the monotonicity of $f_{x}^{\nu}$ (i.e.,
$(f_{x}^{\nu})^{{}^{\prime}}>0$, cf. (212)) finally yield for any $t\geq 0$ to

(224) $\begin{split}f_{x}^{\nu}(|Y^{\star}_{t}-Z_{t}|)&\leq
f_{x}^{\nu}(\rho_{\delta}(r_{t}^{2}))\leq\exp(C_{I}^{-1}\,({\tilde{G}}_{\nu}-\alpha_{\nu})^{+}\,t)\,\frac{1}{2}\,\int_{0}^{t}|\nabla\psi^{\star}-\nabla\psi^{n}|(Y^{\star}_{s})\,\mathrm{d}s\\\
&\overset{\eqref{eq:pointwisebound}}{\leq}\exp(C_{I}^{-1}\,({\tilde{G}}_{\nu}-\alpha_{\nu})^{+}\,t)\,\frac{1}{2}\,\int_{0}^{t}(A|Y^{\star}_{s}|+B)\mathrm{d}\,s\prod_{k=0}^{n-1}\frac{{\hat{\gamma}_{k}^{\mu}}\;{\hat{\gamma}_{k}^{\nu}}}{T^{2}}\;.\end{split}$

where the last step follows from Theorem 3. By recalling (219) and taking
expectation, so far we have proven that

(225)
$\mathbf{W}_{f^{\nu}_{x}}(\pi^{x,\psi^{\star}}_{T}\,P^{\star}_{t},\pi^{x,\psi^{\star}}_{T}\,P^{n}_{t})\leq\frac{1}{2}\,e^{C_{I}^{-1}({\tilde{G}}_{\nu}-\alpha_{\nu})^{+}t}\int_{0}^{t}\mathbb{E}\biggl{[}(A|Y^{\star}_{s}|+B)(1+\varepsilon
V(Y_{t}^{\star})+\varepsilon
V(Z_{t}))\biggr{]}\mathrm{d}s\,\prod_{k=0}^{n-1}\frac{{\hat{\gamma}_{k}^{\mu}}\;{\hat{\gamma}_{k}^{\nu}}}{T^{2}}\,.$

Next, we claim that the above integral is bounded by a constant independent
from $n\in\mathbb{N}$. From Young’s inequality and the stationarity of
$Y_{s}^{\star}\sim\pi^{x,\psi^{\star}}_{T}$ we have

(226)
$\begin{split}&\int_{0}^{t}\,\mathbb{E}\biggl{[}(A|Y^{\star}_{s}|+B)(1+\varepsilon
V(Y_{t}^{\star})+\varepsilon V(Z_{t}))\biggr{]}\,\mathrm{d}s\\\ &\leq
Bt(1+\varepsilon\mathbb{E}[V(Y_{0}^{\star})]+\varepsilon\mathbb{E}[V(Z_{t})])+A\,t\,\mathbb{E}[|Y^{\star}_{0}|]+\frac{A\varepsilon
t}{2}(2\,\mathbb{E}[|Y^{\star}_{0}|^{2}]+\mathbb{E}[V(Y^{\star}_{0})^{2}]+\mathbb{E}[V(Z_{t})^{2}])\;.\end{split}$

At this point it is enough noticing that the geometric drift condition (78)
obtained in the proof Section 3 guarantees the finiteness of the fourth
moments of the random variables appearing in the last display and hence the
finiteness of the above expected values. For exposition’s clarity we provide a
proof of this last statement in Appendix C in the Appendix, where we show in
(284) that $U(t,\nu,x,A,B,T)$, the upper-bounding constant, can be chosen
independently from $n\in\mathbb{N}$.

As a byproduct of (214), (217) and (225), and by minimizinge over
$t>\lambda^{-1}\,\log C_{\Delta}$ we have proven that

(227)
$\|\nabla^{2}\varphi^{n+1}-\nabla^{2}\varphi^{\star}\|_{\mathrm{F}}(x)\leq
C(x,T,\nu,A,B)\,\prod_{k=0}^{n-1}\frac{\hat{\gamma}_{k}^{\mu}\;\hat{\gamma}_{k}^{\nu}}{T^{2}}\;,$

where the above constant is equal to

(228) $C(x,T,\nu,A,B)\coloneqq\inf_{t>\lambda^{-1}\log
C_{\Delta}}\frac{10}{3}\frac{C_{x}}{T^{2}\varepsilon^{1/2}}\biggl{(}\frac{2}{C_{I}}\vee\frac{f_{x}^{\nu}(R_{2})^{-1}}{\varepsilon^{1/2}}\biggr{)}\frac{C_{\Delta}}{1-C_{\Delta}e^{-\lambda
t}}\frac{1}{2}e^{C_{I}^{-1}({\tilde{G}}_{\nu}-\alpha_{\nu})^{+}t}U(t,\nu,x,A,B,T)\,.$

By following the same line of reasoning, it is possible proving that

(229) $\|\nabla^{2}\psi^{n+1}-\nabla^{2}\psi^{\star}\|_{\mathrm{F}}(x)\leq
C(x,T,\mu,A,B)\,\frac{T}{\hat{\gamma}^{\mu}_{n}}\prod_{k=0}^{n}\frac{\hat{\gamma}_{k}^{\mu}\;\hat{\gamma}_{k}^{\nu}}{T^{2}}\;,$

and the constant $C(x,T,\mu,A,B)$ can be built in analogy to (228). ∎

##  Appendix A Explicit rates for strongly log-concave marginals

In this section we further develop the discussion started in Section 2.2.
Therefore assume the validity of A 1 with
${\tilde{g}}_{\mu}={\tilde{g}}_{\nu}={g}_{\mu}={g}_{\nu}\equiv 0$. Let us
firslty observe that Theorem 7 in this particular setting simply reads as

######  Theorem 10.

Assume A 1 with ${\tilde{g}}_{\mu}={\tilde{g}}_{\nu}={g}_{\mu}={g}_{\nu}\equiv
0$. Then there exist two monotone increasing sequences
$(\alpha_{\mu,n})_{n\in\mathbb{N}}\subseteq(\alpha_{\mu}-T^{-1},\alpha_{\mu}-T^{-1}+(\beta_{\nu}\,T^{2})^{-1}]$
and
$(\alpha_{\nu,n})_{n\in\mathbb{N}}\subseteq(\alpha_{\nu}-T^{-1},\alpha_{\nu}-T^{-1}+(\beta_{\mu}\,T^{2})^{-1}]$
such that for any $n\geq 1$ for any $n\in\mathbb{N}$ it holds $r>0$ it holds

(230) $\kappa_{\varphi^{n}}(r)\geq\alpha_{\mu,n}\quad\text{ and
}\quad\kappa_{\psi^{n}}(r)\geq\alpha_{\nu,n}\;.$

These two sequences are defined as

(231) $\begin{cases}\alpha_{\mu,0}\coloneqq\alpha_{\mu}-T^{-1}\;,\\\
\alpha_{\mu,n+1}\coloneqq\alpha_{\mu}-T^{-1}+\biggl{(}T^{2}\,\beta_{\nu}+(\alpha_{\mu,n}+T^{-1})^{-1}\biggr{)}^{-1}\;,\quad
n\in\mathbb{N}\;,\end{cases}$

and

(232) $\begin{cases}\alpha_{\nu,0}\coloneqq\alpha_{\nu}-T^{-1}\;,\\\
\alpha_{\nu,n+1}\coloneqq\alpha_{\nu}-T^{-1}+\biggl{(}T^{2}\,\beta_{\mu}+(\alpha_{\nu,n}+T^{-1})^{-1}\biggr{)}^{-1}\;,\quad
n\in\mathbb{N}\;.\end{cases}$

Moreover, both sequences converge respectively to

(233)
$\alpha_{\varphi^{\star}}\coloneqq\frac{1}{2}\biggl{(}\alpha_{\mu}+\sqrt{\alpha_{\mu}^{2}+4\alpha_{\mu}/(T^{2}\beta_{\nu})}\biggr{)}-T^{-1}\;\text{
and
}\;\alpha_{\psi^{\star}}\coloneqq\frac{1}{2}\biggl{(}\alpha_{\nu}+\sqrt{\alpha_{\nu}^{2}+4\alpha_{\nu}/(T^{2}\beta_{\mu})}\biggr{)}-T^{-1}\,,$

and for any $r>0$ it holds

(234) $\kappa_{\varphi^{\star}}(r)\geq\alpha_{\varphi^{\star}}\quad\text{ and
}\quad\kappa_{\psi^{\star}}(r)>\alpha_{\psi^{\star}}\;,$

where $\varphi^{\star}$ and $\psi^{\star}$ are the Schrödinger potentials
introduced in (1).

###### Proof.

This is a particular instance of Theorem 7 when
${\tilde{g}}_{\mu}={\tilde{g}}_{\nu}={g}_{\mu}={g}_{\nu}\equiv 0$. The only
statement that does not follow from that theorem is the identification of the
limit values $\alpha_{\varphi^{\star}}$ and $\alpha_{\psi^{\star}}$ in (233).
We will only prove the first one since the second identity can be proven in
the same way. From Theorem 7 we already know that
$\alpha_{\mu,n}\uparrow\alpha_{\varphi^{\star}}\in(\alpha_{\mu}-T^{-1},\alpha_{\mu}-T^{-1}+(\beta_{\nu}\,T^{2})^{-1}]$.
Consider the shifted sequence
$\theta^{\mu}_{n}\coloneqq\alpha_{\mu,n}+T^{-1}$. Clearly
$\theta^{\mu}_{n}>0$,
$\theta^{\mu}_{n}\uparrow\theta^{\mu}_{\infty}\coloneqq\alpha_{\varphi^{\star}}+T^{-1}$
and the latter limit value can be seen as a fixed point for the iteration

(235)
$\theta^{\mu}_{n+1}=\alpha_{\nu}+(T^{2}\,\beta_{\nu}+(\theta^{\mu}_{n})^{-1})^{-1}\;.$

A straightforward computation shows that the there are just two possible fixed
point solutions, namely

(236)
$\frac{1}{2}\biggl{(}\alpha_{\mu}-\sqrt{\alpha_{\mu}^{2}+4\alpha_{\mu}/(T^{2}\beta_{\nu})}\biggr{)}\;\text{
and
}\;\frac{1}{2}\biggl{(}\alpha_{\mu}+\sqrt{\alpha_{\mu}^{2}+4\alpha_{\mu}/(T^{2}\beta_{\nu})}\biggr{)}\;.$

Since one solution is negative, whereas $(\theta_{n}^{\mu})_{n\in\mathbb{N}}$
is a positive increasing sequence, we immediately deduce that
$\theta_{\infty}^{\mu}$ equals the largest (and positive) fixed point. This
proves (233). ∎

From the previous result we immediately deduce explicit expressions for the
rates of convergence appearing in Section 2.2. Indeed Theorem 10 implies the
strict convexity of the Schrödinger potentials, which allows to perform all
our contraction estimates without relying on reflection coupling. Namely, by
relying on synchronous coupling, we obtain the validity of Section 4 this time
with the rates appearing in Theorem 10, i.e.,

(237)
$\mathbf{W}_{1}(\pi_{T}^{x,\psi^{n}},\pi_{T}^{x,\psi^{\star}})\leq\gamma_{n}^{\nu}\int\left|\nabla\psi^{n}-\nabla\psi^{\star}\right|\mathrm{d}\pi_{T}^{x,\psi^{\star}}\quad\text{with
}\gamma_{n}^{\nu}=(\theta_{n}^{\nu})^{-1}=(\alpha_{\nu,n}+T^{-1})^{-1}\;,$

and

(238)
$\mathbf{W}_{1}(\pi_{T}^{x,\varphi^{n+1}},\pi_{T}^{x,\varphi^{\star}})\leq\gamma_{n}^{\mu}\int\left|\nabla\varphi^{n+1}-\nabla\varphi^{\star}\right|\mathrm{d}\pi_{T}^{x,\varphi^{\star}}\quad\text{with
}\gamma_{n}^{\mu}=(\theta_{n}^{\mu})^{-1}=(\alpha_{\mu,n}+T^{-1})^{-1}\;.$

Therefore the rates sequences can satisfy

(239) $\begin{cases}\gamma_{0}^{\mu}\coloneqq\alpha_{\mu}^{-1}\\\
\gamma^{\mu}_{k+1}\coloneqq(\alpha_{\mu}+(T^{2}\,\beta_{\nu}+\gamma^{\mu}_{k})^{-1})^{-1}\end{cases}\;\text{
and }\quad\begin{cases}\gamma_{0}^{\nu}\coloneqq\alpha_{\nu}^{-1}\\\
\gamma^{\nu}_{k+1}\coloneqq(\alpha_{\nu}+(T^{2}\,\beta_{\mu}+\gamma^{\nu}_{k})^{-1})^{-1}\;,\end{cases}$

and hence are increasing sequences which converge to the limits

(240) $\gamma^{\mu}_{\infty}\coloneqq
2\biggl{(}\alpha_{\mu}+\sqrt{\alpha_{\mu}^{2}+4\alpha_{\mu}/(T^{2}\beta_{\nu})}\biggr{)}^{-1}\quad\text{
and }\quad\gamma^{\nu}_{\infty}\coloneqq
2\biggl{(}\alpha_{\nu}+\sqrt{\alpha_{\nu}^{2}+4\alpha_{\nu}/(T^{2}\beta_{\mu})}\biggr{)}^{-1}\;.$

As a consequence, by solving $T>\theta\,\gamma_{\infty}^{\mu}$ and
$T>\theta^{-1}\,\gamma_{\infty}^{\nu}$, we deduce that for any
$\theta\in(0,\infty)$ it holds

(241) $\displaystyle T>\theta\,\alpha_{\mu}^{-1}-\theta^{-1}\beta_{\nu}^{-1}$
$\displaystyle\Leftrightarrow\quad T>\theta\,\gamma_{\infty}^{\mu}\;,$
$\displaystyle T>\theta^{-1}\alpha_{\nu}^{-1}-\theta\,\beta_{\mu}^{-1}$
$\displaystyle\Leftrightarrow\quad T>\theta^{-1}\,\gamma_{\infty}^{\nu}\;,$

and therefore if

(242)
$\begin{split}T>&\inf_{\theta\in(0,\infty)}\max\\{\theta\,\alpha_{\mu}^{-1}-\theta^{-1}\beta_{\nu}^{-1},\,\theta^{-1}\alpha_{\nu}^{-1}-\theta\,\beta_{\mu}^{-1}\\}=\frac{\alpha_{\mu}^{-1}\alpha_{\nu}^{-1}-\beta_{\mu}^{-1}\beta_{\nu}^{-1}}{\sqrt{(\alpha_{\mu}^{-1}+\beta_{\mu}^{-1})(\alpha_{\nu}^{-1}+\beta_{\nu}^{-1})}}\\\
=&\frac{\beta_{\mu}\beta_{\nu}-\alpha_{\mu}\alpha_{\nu}}{\sqrt{\alpha_{\mu}\,\beta_{\mu}\,\alpha_{\nu}\,\beta_{\nu}\,(\alpha_{\mu}+\beta_{\mu})(\alpha_{\nu}+\beta_{\nu})}}\;,\end{split}$

then we are guaranteed that
$T^{2}>\gamma_{\infty}^{\mu}\,\gamma_{\infty}^{\nu}$, and hence the
exponential convergence of Sinkhorn’s algorithm in Theorem 2.

Similarly, we may also give explicit expressions for the exponential rates
$({\hat{\gamma}_{k}^{\mu}})_{k\in\mathbb{N}},\,({{\hat{\gamma}_{k}^{\nu}}})_{k\in\mathbb{N}}$
appearing in Theorem 3, which read as

(243)
$\displaystyle{\hat{\gamma}_{k}^{\mu}}\coloneqq\,\gamma_{k}^{\mu}\;\max\biggl{\\{}T^{-1}\,\gamma_{\infty}^{\mu}\;,\biggl{(}1+\gamma^{\mu}_{\infty}\,\frac{A(1+|\nabla\varphi^{\star}(0)|)}{B}\biggr{)}\biggr{\\}}\;,$
$\displaystyle{{\hat{\gamma}_{k}^{\nu}}}\coloneqq\,\gamma_{k}^{\nu}\;\max\biggl{\\{}T^{-1}\,\gamma_{\infty}^{\nu}\;,\biggl{(}1+\gamma^{\nu}_{\infty}\,\frac{A(1+|\nabla\psi^{\star}(0)|)}{B}\biggr{)}\biggr{\\}}\;.$

Finally, when $\psi^{0}=U_{\nu}$ (i.e., $\varphi^{0}=0$) the above expressions
read as

(244) $\displaystyle{\hat{\gamma}_{k}^{\mu}}\coloneqq$
$\displaystyle\,\,\gamma_{k}^{\mu}\;\max\biggl{\\{}T^{-1}\,\gamma_{\infty}^{\mu}\;,\,2+T^{-1}\,\gamma_{\infty}^{\mu}\biggr{\\}}=\,\gamma_{k}^{\mu}\;(2+T^{-1}\,\gamma_{\infty}^{\mu})\;,$
$\displaystyle{{\hat{\gamma}_{k}^{\nu}}}\coloneqq$
$\displaystyle\,\,\gamma_{k}^{\nu}\;\max\biggl{\\{}T^{-1}\,\gamma_{\infty}^{\nu}\;,\biggl{(}1+\biggl{(}T^{-1}\,\gamma_{\infty}^{\nu}+\frac{\gamma_{\infty}^{\nu}}{\gamma_{\infty}^{\mu}}\biggr{)}\frac{1+|\nabla\psi^{\star}(0)|}{1+|\nabla\varphi^{\star}(0)|}\biggr{)}\biggr{\\}}\;.$

Notice that, if for instance we assume the validity of (241) for $\theta=1$,
then the asymptotic rates read as

(245) $\displaystyle\hat{\gamma}_{\infty}^{\mu}=$
$\displaystyle\,\gamma_{\infty}^{\mu}\;(2+T^{-1}\,\gamma_{\infty}^{\mu})<3\,\gamma^{\mu}_{\infty}\;,$
$\displaystyle\hat{\gamma}_{\infty}^{\nu}=$
$\displaystyle\,\gamma_{\infty}^{\nu}\,\biggl{(}1+\biggl{(}T^{-1}\,\gamma_{\infty}^{\nu}+\frac{\gamma_{\infty}^{\nu}}{\gamma_{\infty}^{\mu}}\biggr{)}\frac{1+|\nabla\psi^{\star}(0)|}{1+|\nabla\varphi^{\star}(0)|}\biggr{)}<\,\gamma_{\infty}^{\nu}\,\biggl{(}1+M\,\frac{1+|\nabla\psi^{\star}(0)|}{1+|\nabla\varphi^{\star}(0)|}\biggr{)}\;,$
$\displaystyle\text{with }M=\,1+\sup_{s\geq
0}\frac{\alpha_{\mu}\,s+\sqrt{\alpha_{\mu}^{2}\,s^{2}+4\alpha_{\mu}/\beta_{\nu}}}{\alpha_{\nu}\,s+\sqrt{\alpha_{\nu}^{2}\,s^{2}+4\alpha_{\nu}/\beta_{\mu}}}<+\infty\;.$

Therefore Sinkhorn’s algorithm converges exponentially fast if for instance it
holds

(246)
$T>\max\biggl{\\{}3\,\alpha_{\mu}^{-1}-3^{-1}\beta_{\nu}^{-1},\;\biggl{(}1+M\,\frac{1+|\nabla\psi^{\star}(0)|}{1+|\nabla\varphi^{\star}(0)|}\biggr{)}\,\alpha_{\nu}^{-1}-\biggl{(}1+M\,\frac{1+|\nabla\psi^{\star}(0)|}{1+|\nabla\varphi^{\star}(0)|}\biggr{)}^{-1}\beta_{\mu}^{-1}\biggr{\\}}\;.$

##  Appendix B Explicit construction and contractive properties of
$\mathbf{W}_{f^{\nu}_{x}}$

Here we carry out the explicit construction of $f_{x}^{\nu}$ and with the
proof of (210). The following result follows form [27, Theorem 2.2].

Firstly, notice the geometric drift condition (206) implies

(247)
$\begin{split}&\mathcal{L}_{\psi^{n}}V(z)+\mathcal{L}_{\psi^{n}}V(y)<0\quad\forall(z,y)\not\in\mathrm{B_{2d}}(2B_{\nu}/A_{\nu})\;,\\\
&\varepsilon\mathcal{L}_{\psi^{n}}V(z)+\varepsilon\mathcal{L}_{\psi^{n}}V(y)<-\frac{A_{\nu}}{2}(1\wedge
4B_{\nu}\varepsilon)(1+\varepsilon V(z)+\varepsilon
V(y))\quad\forall(z,y)\not\in\mathrm{B_{2d}}(4B_{\nu}(1+A_{\nu}^{-1}))\;,\end{split}$

where $\mathrm{B_{2d}}(r^{2})\subseteq\mathbb{R}^{2d}$ denotes the centered
Euclidean ball of radius $r$. For later convenience let us also define the
radii

(248)
$\begin{split}R_{1}&\coloneqq\sup\\{|x-y|\;\colon\;(x,y)\in\mathrm{B_{2d}}(2B_{\nu}/A_{\nu})\\}\;,\\\
R_{2}&\coloneqq\sup\\{|x-y|\;\colon\;(x,y)\in\mathrm{B_{2d}}(4B_{\nu}(1+A_{\nu}^{-1}))\\}\;.\end{split}$

Now, take $\varepsilon\in(0,1)$ satisfying the condition

(249)
$(4B_{\nu}\varepsilon)^{-1}\geq\int_{0}^{R_{1}}\int_{0}^{s}\exp\biggl{(}\frac{{\tilde{G}}_{\nu}-\alpha_{\nu})^{+}}{4}(s^{2}-r^{2})+2\,\varepsilon^{1/2}(s-r)\biggr{)}\mathrm{d}r\,\mathrm{d}s\;,$

which is always possible since the left hand side diverges as $\varepsilon$
vanishes, whereas the right hand side is bounded.

Finally, define

(250) $\begin{split}f_{x}^{\nu}(r)\coloneqq\int_{0}^{r\wedge
R_{2}}\phi(s)g(s)\,\mathrm{d}s\;,\quad\text{with
}\;\phi(r)\coloneqq\exp\biggl{(}-\frac{({\tilde{G}}_{\nu}-\alpha_{\nu})^{+}}{8}\,r^{2}-2\,\varepsilon^{1/2}\,r\biggr{)}\;,\\\
\Phi(r)\coloneqq\int_{0}^{r}\phi(s)\;\mathrm{d}s\quad\text{ and }\quad
g(r)\coloneqq 1-\frac{\int_{0}^{r\wedge
R_{1}}\frac{\Phi(s)}{\phi(s)}\mathrm{d}s}{4\int_{0}^{R_{1}}\frac{\Phi(s)}{\phi(s)}\mathrm{d}s}-\frac{\int_{0}^{r\wedge
R_{2}}\frac{\Phi(s)}{\phi(s)}\mathrm{d}s}{4\int_{0}^{R_{2}}\frac{\Phi(s)}{\phi(s)}\mathrm{d}s}\,.\end{split}$

Let us also consider the positive quantities

(251)
$\xi^{-1}\coloneqq\int_{0}^{R_{1}}\frac{\Phi(s)}{\phi(s)}\mathrm{d}s\quad\text{
and
}\quad\beta^{-1}\coloneqq\int_{0}^{R_{2}}\frac{\Phi(s)}{\phi(s)}\mathrm{d}s\;,$

and notice that (249) equivalently reads as $4B_{\nu}\varepsilon\leq\xi$. The
function $f^{\nu}_{x}$ is clearly bounded, increasing and concave. Moreover,
from the above definitions we immediately deduce the validity of the
properties stated at (212) with

(252) $C_{I}\coloneqq\phi(R_{2})\;,$

and the inequality regarding the first derivative can actually be strengthen
since for any $r<R_{2}$ it holds

(253) $(f_{x}^{\nu})^{\prime}(r)\;\phi(r)^{-1}=g(r)\in[1/2,\,1]\;.$

Finally, a straightforward differentiation shows that for any
$r\in(0,R_{1})\cup(R_{1},R_{2})$ it holds

(254)
$\begin{split}(f_{x}^{\nu}){{}^{\prime\prime}}(r)=&-\biggl{(}\frac{({\tilde{G}_{\nu}}-\alpha_{\nu})^{+}}{4}\,r+2\,\varepsilon^{1/2}\biggr{)}\,\phi(r)g(r)-\frac{\mathbf{1}_{\\{r<R_{1}\\}}\xi+\beta}{4}\,\Phi(r)\\\
\leq&-\biggl{(}\frac{({\tilde{G}_{\nu}}-\alpha_{\nu})^{+}}{4}\,r+2\,\varepsilon^{1/2}\biggr{)}\,(f_{x}^{\nu}){{}^{\prime}}(r)-\frac{\mathbf{1}_{\\{r<R_{1}\\}}\xi+\beta}{4}\,f_{x}^{\nu}(r)\;.\end{split}$

We conclude with the proof of the triangle inequality and the contractive
property for the distorted Wasserstein semi-distance introduced at (208)

(255)
$\mathbf{W}_{f_{x}^{\nu}}(\cdot,\cdot)\coloneqq\inf_{\pi\in\Pi(\cdot,\cdot)}\mathbb{E}_{(Y,Z)\sim\pi}\biggl{[}f_{x}^{\nu}(|Y-Z|)\,(1+\varepsilon
V(Y)+\varepsilon V(Z))\biggr{]}\;.$

###### Proof of the weak triangle inequality (211).

The following proof is an adaptation of [37, Lemma 4.14]. It is enough showing
that there exists $C_{\Delta}>0$ such that for any
$y,\,z,\,p\in\mathbb{R}^{d}$ it holds

(256) $f_{x}^{\nu}(|y-z|)(1+\varepsilon V(y)+\varepsilon V(z))\leq
C_{\Delta}\;\bigl{[}f_{x}^{\nu}(|y-p|)(1+\varepsilon V(y)+\varepsilon
V(p))+f_{x}^{\nu}(|p-z|)(1+\varepsilon V(p)+\varepsilon V(z))\bigr{]}\;.$

Firstly, notice that for any $y,z\in\mathbb{R}^{d}$ such that $|y-z|\leq
R_{2}$ it holds

(257) $V(y)\leq\max\\{2,\,1+2\,R_{2}^{2}\\}\,V(z)\;.$

Without loss of generalities assume that $|z|\leq|y|$ (and hence $V(z)\leq
V(y)$). Since $f_{x}^{\nu}(r)\leq f_{x}^{\nu}(R_{2})$, if $|y-p|\geq R_{2}$
then

(258) $\begin{split}f_{x}^{\nu}(|y-z|)(1+\varepsilon V(y)+\varepsilon
V(z))\leq f_{x}^{\nu}(R_{2})(1+2\,\varepsilon
V(y))=f_{x}^{\nu}(|y-p|)(1+2\varepsilon V(y))\\\ \leq
2\,f_{x}^{\nu}(|y-p|)(1+\varepsilon V(y)+\varepsilon V(p))\;.\end{split}$

On the other hand, if $|y-p|\leq R_{2}$ then (257) and the subadditivity of
$f_{x}^{\nu}$ (which is guaranteed by its concavity) imply

(259) $\begin{split}f_{x}^{\nu}(|y-z|)(1+\varepsilon V(y)+\varepsilon
V(z))\leq(f_{x}^{\nu}(|y-p|)+f_{x}^{\nu}(|p-z|))(1+\varepsilon
V(y)+\varepsilon V(z))\\\ \leq f_{x}^{\nu}(|y-p|)(1+2\varepsilon
V(y))+f_{x}^{\nu}(|p-z|)(1+\max\\{2,\,1+2\,R_{2}^{2}\\}\,\varepsilon
V(z)+\varepsilon V(z))\;.\end{split}$

As a byproduct of (258) and (259) we get the validity of (256) (and hence of
(211)) with

(260) $C_{\Delta}\coloneqq\max\\{3,\,2+2\,R_{2}^{2}\\}\;.$

∎

###### Proof of the contraction (210).

Fix two probability measure $\mu_{1},\,\mu_{2}\in\mathcal{P}(\mathbb{R}^{d})$
and let $\pi\in\Pi(\mu_{1},\mu_{2})$ be a coupling between them. Our proof
starts by considering the coupling by reflection, i.e., the diffusion
processes

(261)
$\begin{cases}\mathrm{d}Z_{t}=-\biggl{(}\frac{Z_{t}-x}{2T}+\frac{1}{2}\nabla\psi^{n}(Z_{t})\biggr{)}\mathrm{d}t+\mathrm{d}B_{t}\;,\\\
\mathrm{d}Y_{t}=-\biggl{(}\frac{Y_{t}-x}{2T}+\frac{1}{2}\nabla\psi^{n}(Y_{t})\biggr{)}\mathrm{d}t+\mathrm{d}\hat{B}_{t}\quad\forall\;t\in[0,\tau)\;\text{
and }Y_{t}=Z_{t}\quad\forall\;t\geq\tau\;,\\\
(Z_{0},Y_{0})\sim\pi\;,\end{cases}$

where $\tau\coloneqq\inf\\{s\geq 0:Z_{s}=Y_{s}\\}$, and $(\hat{B}_{t})_{t\geq
0}$ is defined as

(262) $\mathrm{d}\hat{B}_{t}\coloneqq(\operatorname{Id}-2\,e_{t}\,e_{t}^{{\sf
T}}\,\mathbf{1}_{\\{t<\tau\\}})\,\mathrm{d}B_{t}\qquad\text{where}\quad
e_{t}\coloneqq\begin{cases}\frac{Z_{t}-Y_{t}}{|Z_{t}-Y_{t}|}\quad&\text{ when
}|Z_{t}-Y_{t}|>0\,,\\\ u\quad&\text{ when }|Z_{t}-Y_{t}|=0\,.\end{cases}$

with $u\in\mathbb{R}^{d}$ being a fixed (arbitrary) unit-vector. By Lévy’s
characterization, $(\hat{B}_{t})_{t\geq 0}$ is a $d$-dimensional Brownian
motion, hence $Z_{t}\sim\mu_{1}P^{n}_{t}$ and $Y_{t}\sim\mu_{2}P^{n}_{t}$, and
finally $\mathrm{d}W_{t}\coloneqq e_{t}^{{\sf T}}\mathrm{d}B_{t}$ is a one-
dimensional Brownian motion. By setting $r_{t}=|Z_{t}-Y_{t}|$ and by applying
Ito-Tanaka formula as in the proof of Theorem 9, the trivial bound
$\alpha_{\nu,n}>\alpha_{\nu}-T^{-1}$ (cf. Theorem 7) and the sublinearity
${\tilde{g}}_{\nu}(r)\leq{\tilde{G}}_{\nu}\,r$ imply that

(263)
$\begin{split}\mathrm{d}{f^{\nu}_{x}}(r_{t})\leq&\biggl{(}2\,{(f^{\nu}_{x})}{{}^{\prime\prime}}(r_{t})-\frac{r_{t}\,(f_{x}^{\nu}){{}^{\prime}}(r_{t})}{2}(\alpha_{\nu,n}+T^{-1}-r_{t}^{-1}{\tilde{g}}_{\nu}(r_{t}))\biggr{)}\;\mathrm{d}t+2\;(f_{x}^{\nu}){{}^{\prime}}(r_{t})\;\mathrm{d}W_{t}\\\
\leq&\biggl{(}2\,{(f^{\nu}_{x})}{{}^{\prime\prime}}(r_{t})+\frac{r_{t}\,(f_{x}^{\nu}){{}^{\prime}}(r_{t})}{2}({\tilde{G}}_{\nu}-\alpha_{\nu})^{+}\biggr{)}\;\mathrm{d}t+2\;(f_{x}^{\nu}){{}^{\prime}}(r_{t})\;\mathrm{d}W_{t}\\\
\overset{\eqref{harris:diff}}{\leq}&-4\,\varepsilon^{1/2}\,(f_{x}^{\nu}){{}^{\prime}}(r_{t})\mathrm{d}t-\mathbf{1}_{\\{r_{t}<R_{1}\\}}\frac{\xi}{2}f_{x}^{\nu}(r_{t})\,\mathrm{d}t-\mathbf{1}_{\\{r_{t}<R_{2}\\}}\frac{\beta}{2}f_{x}^{\nu}(r_{t})\,\mathrm{d}t+2\;(f_{x}^{\nu}){{}^{\prime}}(r_{t})\;\mathrm{d}W_{t}\;.\end{split}$

The validity of the above Ito-Tanaka formula for $f_{x}^{\nu}$ can be proven
similarly to what we have done in the proof of Theorem 9 and for this reason
we omit it; we refer the reader to [27, Formula (5.15)] for a detailed
discussion on that.

Next, we notice that from the geometric drift condition (206) and the
definition of $R_{1},\,R_{2}$ it follows that

(264) $\begin{split}\mathrm{d}&(1+\varepsilon V(Z_{t})+\varepsilon
V(Y_{t}))=\varepsilon(\mathcal{L}_{\psi^{n}}V(Z_{t})+\mathcal{L}_{\psi^{n}}V(Y_{t}))\,\mathrm{d}t+2\varepsilon\,\langle
Z_{t}+Y_{t},\,\mathrm{d}B_{t}\rangle-4\varepsilon\,\langle
Y_{t},\,e_{t}\rangle\,\mathrm{d}W_{t}\\\ &\leq
2\varepsilon\,B_{\nu}\,\mathrm{d}t-\varepsilon\,A_{\nu}(V(Z_{t})+V(Y_{t}))\,\mathrm{d}t+2\varepsilon\,\langle
Z_{t}+Y_{t},\,\mathrm{d}B_{t}\rangle-4\varepsilon\,\langle
Y_{t},\,e_{t}\rangle\,\mathrm{d}W_{t}\\\
&\begin{split}\overset{\eqref{cond:eps}}{\leq}\mathbf{1}_{\\{r_{t}<R_{1}\\}}\biggl{(}\xi/2-\varepsilon\,A_{\nu}(V(Z_{t})+V(Y_{t}))\biggr{)}\mathrm{d}t+\varepsilon\,\mathbf{1}_{\\{r_{t}\in[R_{1},R_{2})\\}}(\mathcal{L}_{\psi^{n}}V(Z_{t})+\mathcal{L}_{\psi^{n}}V(Y_{t}))\,\mathrm{d}t\\\
+\varepsilon\,\mathbf{1}_{\\{r_{t}\geq
R_{2}\\}}(\mathcal{L}_{\psi^{n}}V(Z_{t})+\mathcal{L}_{\psi^{n}}V(Y_{t}))\,\mathrm{d}t+2\varepsilon\,\langle
Z_{t}+Y_{t},\,\mathrm{d}B_{t}\rangle-4\varepsilon\,\langle
Y_{t},\,e_{t}\rangle\,\mathrm{d}W_{t}\end{split}\\\
&\overset{\eqref{conseguenzaR}}{\leq}\biggl{(}\mathbf{1}_{\\{r_{t}<R_{1}\\}}\,\xi/2-\mathbf{1}_{\\{r_{t}\geq
R_{2}\\}}\,\lambda\,(1+\varepsilon V(Z_{t})+\varepsilon
V(Y_{t}))\biggr{)}\,\mathrm{d}t+2\varepsilon\,\langle
Z_{t}+Y_{t},\,\mathrm{d}B_{t}\rangle-4\varepsilon\,\langle
Y_{t},\,e_{t}\rangle\,\mathrm{d}W_{t}\end{split}$

where in the last step we have taken

(265)
$\lambda\coloneqq\min\\{\beta,\,A_{\nu},\,4A_{\nu}B_{\nu}\varepsilon\\}/2\;.$

Finally, notice that the choice of coupling by reflection gives to the
covariation between $f_{x}^{\nu}(r_{t})$ and $1+\varepsilon
V(Z_{t})+\varepsilon V(Y_{t})$ the expression

(266) $\mathrm{d}[f_{x}^{\nu}(r_{\cdot}),\,1+\varepsilon
V(Z_{\cdot})+\varepsilon
V(Y_{\cdot})]_{t}=4\varepsilon\,r_{t}\,(f_{x}^{\nu}){{}^{\prime}}(r_{t})\,\mathrm{d}t<4\varepsilon^{1/2}\,(1+\varepsilon
V(y)+\varepsilon V(z))\,(f_{x}^{\nu}){{}^{\prime}}(r_{t})\,\mathrm{d}t\;.$

where the last step follows from the trivial series of inequalities

(267) $\begin{split}4\varepsilon\,|y-z|\leq&4\varepsilon(1+\varepsilon
V(y)+\varepsilon V(z))\biggl{(}\frac{|y|}{1+\varepsilon
V(y)}+\frac{|z|}{1+\varepsilon V(z)}\biggr{)}\\\
\leq&4\varepsilon(1+\varepsilon V(y)+\varepsilon
V(z))\,\sup_{y\in\mathbb{R}^{d}}\frac{2|y|}{1+\varepsilon
V(y)}=4(1+\varepsilon V(y)+\varepsilon
V(z))\,\sqrt{\frac{\varepsilon}{1+\varepsilon}}\;.\end{split}$

If for sake of notation we set $F_{V}(z,y)\coloneqq
f_{x}^{\nu}(|y-z|)(1+\varepsilon V(z)+\varepsilon V(y))$ the inequalities
(263), (264) and (266) gives

(268)
$\begin{split}&\begin{split}\mathrm{d}F_{V}(Z_{t},\,Y_{t})=(1+\varepsilon
V(Z_{t})+\varepsilon
V(Y_{t}))\,\mathrm{d}f_{x}^{\nu}(r_{t})+f_{x}^{\nu}(r_{t})\,\mathrm{d}(1+\varepsilon
V(Z_{t})+\varepsilon V(Y_{t}))\\\
+\mathrm{d}[f_{x}^{\nu}(r_{\cdot}),\,1+\varepsilon V(Z_{\cdot})+\varepsilon
V(Y_{\cdot})]_{t}\end{split}\\\
&\leq-\mathbf{1}_{\\{r_{t}<R_{1}\\}}\frac{\xi}{2}\,(F_{V}(Z_{t},Y_{t})-f_{x}^{\nu}(r_{t}))\,\mathrm{d}t-\biggl{(}\mathbf{1}_{\\{r_{t}<R_{2}\\}}\frac{\beta}{2}+\mathbf{1}_{\\{r_{t}\geq
R_{2}\\}}\lambda\biggr{)}\,F_{V}(Z_{t},Y_{t})\,\mathrm{d}t+\mathrm{d}M_{t}\\\
&\leq-\lambda\,F_{V}(Z_{t},Y_{t})\,\mathrm{d}t+\mathrm{d}M_{t}\;,\end{split}$

where

(269) $\mathrm{d}M_{t}\coloneqq
2\;(f_{x}^{\nu}){{}^{\prime}}(r_{t})(1+\varepsilon V(Z_{t})+\varepsilon
V(Y_{t}))\;\mathrm{d}W_{t}+2\varepsilon\,f_{x}^{\nu}(r_{t})\,\langle
Z_{t}+Y_{t},\,\mathrm{d}B_{t}\rangle-4\varepsilon\,f_{x}^{\nu}(r_{t})\,\langle
Y_{t},\,e_{t}\rangle\,\mathrm{d}W_{t}$

is a local martingale. Hence $e^{\lambda t}F_{V}(Z_{t},Y_{t})$ is a local-
supermartingale.

Now, for any $N\in\mathbb{N}$ consider now the stopping time

(270) $\tau_{N}\coloneqq\inf\\{t\geq 0\colon|Z_{t}-Y_{t}|\leq N^{-1}\;\text{
or }\;|Y_{t}|\vee|Z_{t}|\geq N\\}\;,$

and notice that $T_{N}\uparrow+\infty$ as $N$ grows (cf. Section 3). Then the
previous discussion, Gronwall Lemma and Fatou Lemma give

(271) $\begin{split}e^{\lambda
t}\,\mathbb{E}[F_{V}(Z_{t},Y_{t})]\leq\liminf_{M\to+\infty}\mathbb{E}[\mathbf{1}_{\\{t<\tau_{M}\\}}\,e^{\lambda
t}F_{V}(Z_{t},Y_{t})]=\mathbb{E}[F_{V}(Z_{0},Y_{0})]\;.\end{split}$

In conclusion, since $Z_{t}\sim\mu_{1}P^{n}_{t}$ and
$Y_{t}\sim\mu_{2}P^{n}_{t}$ we have

(272)
$\mathbf{W}_{f_{x}^{\nu}}(\mu_{1}\,P^{n}_{t},\mu_{2}\,P^{n}_{t})\leq\mathbb{E}[F_{V}(Z_{t},Y_{t})]\leq
e^{-\lambda\,t}\,\mathbb{E}_{\pi}[F_{V}(Z_{0},Y_{0})]\;.$

By minimizing the above bound over $\pi\in\Pi(\mu_{1},\mu_{2})$ concludes the
proof of (210). ∎

##  Appendix C Technical results and moment bounds

######  Lemma  (Exponential integrability of the marginals).

Let $\zeta\in\mathcal{P}_{\mathrm{alc}}(\mathbb{R}^{d})$ associated with
$U:\mathbb{R}^{d}\to\mathbb{R}$ satisfying (18). Then for any
$\sigma\in(0,\alpha_{U}/2)$ it holds
$\int\exp(\sigma|x|^{2})\mathrm{d}\zeta(x)<+\infty$.

###### Proof.

It is enough noticing that for any $x\in\mathbb{R}^{d}$ it holds

(273) $\begin{split}\langle\nabla U(x),\,x\rangle=\langle\nabla U(x)-\nabla
U(0),\,x\rangle+\langle\nabla
U(0),\,x\rangle\geq\kappa_{U}(|x|)\,|x|^{2}-|\nabla U(0)|\,|x|\\\
\geq\alpha_{U}\,|x|^{2}-({\tilde{g}}_{U}(|x|)+|\nabla
U(0)|)|x|\geq\alpha_{U}\,|x|^{2}-\bar{G}_{U}\,|x|\end{split}$

where above we have set $\bar{G}_{U}=\|{\tilde{g}}_{U}\|_{\infty}+|\nabla
U(0)|$. Therefore for any $x\in\mathbb{R}^{d}$ it holds

(274) $U(x)=U(0)+\int_{0}^{1}\langle\nabla U(tx),x\rangle\,\mathrm{d}t\geq
U(0)+\int_{0}^{1}(\alpha_{U}\,t\,|x|^{2}-\bar{G}_{U}\,|x|)\,\mathrm{d}t=U(0)+\frac{\alpha_{U}}{2}\,|x|^{2}-\bar{G}_{U}\,|x|\;.$

Finally we may deduce for any $\sigma\in(0,\alpha_{U}/2)$

(275)
$\begin{split}\|\exp(\sigma|x|^{2})\|_{\mathrm{L}^{1}(U)}=\int_{\\{|x|\leq
R\\}}\exp(\sigma|x|^{2})\mathrm{d}\zeta(x)+\int_{\\{|x|>R\\}}\exp(\sigma|x|^{2})\mathrm{d}\zeta(x)\\\
\leq\mathrm{e}^{\sigma|R|^{2}}\,U(\\{|x|\leq
R\\})+\mathrm{e}^{-U(0)}\int_{\\{|x|>R\\}}\exp(-(\alpha_{U}/2-\sigma)|x|^{2}+\bar{G}_{U}|x|)\,\mathrm{d}x\\\
\leq\mathrm{e}^{\sigma|R|^{2}}\,U(\\{|x|\leq
R\\})+\mathrm{e}^{-U(0)}\int_{\\{|x|>R\\}}\exp(-(\alpha_{U}/4-\sigma/2)|x|^{2})\,\mathrm{d}x<+\infty\;,\\\
\end{split}$

where above we have set $R\coloneqq 2\bar{G}_{U}\,(\alpha_{U}/2-\sigma)^{-1}$.
∎

Finally, let us conclude by giving explicit upper-bounds for the fourth
moments appearing in the proof of Theorem 6, under the geometric-drift
condition (78) obtained in the proof of Section 3. More precisely if for any
even $p\geq 2$ we set $V_{p}(y)=1+|y|^{p}$, similarly to what happened in
(206), Section 3 and (78), the pointwise convergence of the gradients of
Theorem 3 and the convergences
$\alpha_{\mu,n}\uparrow\alpha_{\varphi^{\star}}$,
$\alpha_{\nu,n}\uparrow\alpha_{\psi^{\star}}$ (stated in Theorem 7) imply the
existence of constants $A_{\mu,p},\,A_{\nu,p}>0$ and $B_{\mu,p},\,B_{\nu,p}$,
independent of $n$ (but depending on $x$ and $T$), such that

(276)
$\mathcal{L}_{\psi^{\star}}V_{p}(y)\;\vee\;\mathcal{L}_{\psi^{n}}V_{p}(y)\;\leq
B_{\nu,p}-A_{\nu,p}\,V_{p}(y)\;\text{ and
}\;\mathcal{L}_{\varphi^{\star}}V_{p}(y)\;\vee\;\mathcal{L}_{\varphi^{n}}V_{p}(y)\;\leq
B_{\mu,p}-A_{\mu,p}\,V_{p}(y)\;,$

where $\mathcal{L}_{h}\coloneqq\Delta/2-\frac{1}{2}\langle T^{-1}(y-x)+\nabla
h(y),\,\nabla\rangle$ is the generator associated to (25). We will bound the
moments appearing in the proof of Theorem 6 in terms of the above constants
$A_{\mu,p},\,A_{\nu,p}>0$ and $B_{\mu,p},\,B_{\nu,p}$.

######  Lemma .

Take $p\geq 2$ and set $V_{p}(y)=1+|y|^{p}$. Let $(Y^{\star}_{t})_{t\geq 0}$
and $(Z_{t})_{t\geq 0}$ be defined as in (218) in the proof of Theorem 6.
Recall that $Z_{0}=Y^{\star}_{0}\sim\pi^{x,\psi^{\star}}_{T}$,
$Y_{t}^{\star}\sim\pi^{x,\psi^{\star}}_{T}P^{\star}_{t}=\pi^{x,\psi^{\star}}_{T}$
whereas $Z_{t}\sim\pi^{x,\psi^{\star}}_{T}P^{n}_{t}$. Then for any $t\geq 0$
it holds

(277)
$\begin{split}\mathbb{E}[V_{p}(Y^{\star}_{t})]=&\mathbb{E}[V_{p}(Y^{\star}_{0})]\leq
B_{\nu,p}/A_{\nu,p}\\\
\mathbb{E}[V_{p}(Z_{t})]\leq&(B_{\nu,p}+B_{\nu,p}/A_{\nu,p})\exp(t\,A_{\nu,p})\;.\end{split}$

###### Proof.

By choosing $h=\psi^{\star}$ we immediately deduce that

(278)
$\mathrm{d}V_{p}(Y^{\star}_{t})=\mathcal{L}_{\psi^{\star}}V_{p}(Y^{\star}_{t})\,\mathrm{d}t+4|Y^{\star}_{t}|^{2}\langle
Y^{\star}_{t},\,\mathrm{d}B_{t}\rangle\leq-
A_{\nu,p}V_{p}(Y^{\star}_{t})\,\mathrm{d}t+B_{\nu,p}\,\mathrm{d}t+p\,|Y^{\star}_{t}|^{p-2}\langle
Y^{\star}_{t},\,\mathrm{d}B_{t}\rangle\;.$

Therefore, up to considering a stopping time as already detailed in the proof
of the contraction (210), by taking expectation and integrating over time,
from the stationarity of the process
$Y_{t}^{\star}\sim\pi^{x,\psi^{\star}}_{T}$ we deduce

(279) $\mathbb{E}[V_{p}(Y^{\star}_{t})]\leq
B_{\nu,p}/A_{\nu,p}\quad\forall\,t\geq 0\;.$

Similarly, when considering $h=\psi^{n}$ we get

(280)
$\mathrm{d}V_{p}(Z_{t})=\mathcal{L}_{\psi^{\star}}V_{p}(Z_{t})\,\mathrm{d}t+4|Z_{t}|^{2}\langle
Z_{t},\,\mathrm{d}B_{t}\rangle\leq-
A_{\nu,p}V_{p}(Z_{t})\,\mathrm{d}t+B_{\nu,p}\,\mathrm{d}t+p\,|Z_{t}|^{p-2}\langle
Z_{t},\,\mathrm{d}B_{t}\rangle\;.$

Up to considering again a stopping time, taking expectation and integrating
over time yields to

(281)
$\mathbb{E}[V_{p}(Z_{t})]\leq\mathbb{E}[V_{p}(Z_{0})]+B_{\nu,p}\,t-A_{\nu,p}\int_{0}^{t}\mathbb{E}[V_{p}(Z_{s})]\,\mathrm{d}s\;.$

By recalling that $Z_{0}=Y^{\star}_{0}$, the previous bound (279), from
Gronwall lemma we finally deduce

(282)
$\mathbb{E}[V_{p}(Z_{t})]\leq(B_{\nu,p}\;t+B_{\nu,p}/A_{\nu,p})\exp(t\,A_{\nu,p})\quad\forall\,t\geq
0\;.$

∎

Then, from (226), (172) and Appendix C (for $p=2,\,4$) we finally conclude
that

######  Corollary .

Let $(Y^{\star}_{t})_{t\geq 0}$ and $(Z_{t})_{t\geq 0}$ be defined as in (218)
in the proof of Theorem 6. Then (226) can be bounded as

(283) $\int_{0}^{t}\,\mathbb{E}\biggl{[}(A|Y^{\star}_{s}|+B)(1+\varepsilon
V_{2}(Y_{t}^{\star})+\varepsilon V_{2}(Z_{t}))\biggr{]}\,\mathrm{d}s\leq
U(t,\nu,x,A,B,T)\;,$

with

(284)
$\begin{split}&U(t,\nu,x,A,B,T)\coloneqq\,Bt\biggl{(}1+\varepsilon\frac{B_{\nu,2}}{A_{\nu,2}}(1+e^{t\,A_{\nu,2}})+\varepsilon
B_{\nu,2}\,t\,e^{t\,A_{\nu,2}}\biggr{)}\\\
&+\frac{A\,t}{\alpha_{\psi^{\star}}+T^{-1}}\,(T^{-1}|x|+1+\|{\tilde{g}_{\nu}}\|_{\infty}+|\nabla\psi^{\star}(0)|)+A\varepsilon\,t\biggl{(}\frac{B_{\nu,2}}{A_{\nu,2}}+\frac{B_{\nu,4}}{A_{\nu,4}}(1+e^{t\,A_{\nu,4}})+B_{\nu,4}\,t\,e^{t\,A_{\nu,4}}\biggr{)}\;.\end{split}$

###  C.1. Proof of Section 2.2

We will only prove the case $h=\psi^{n}$ since the other cases can be proven
with the same argument. The proof will run as in [18, Proposition 5.2], once
we have noticed that A 1 and Appendix C guarantees the validity of
$\exp(\sigma_{\nu}|x|^{2})\in\mathrm{L}^{1}(\nu)$ for some positive
$\sigma_{\nu}>0$. Therefore from (4) we know that

(285)
$\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\exp\biggl{(}\sigma_{\nu}|y|^{2}-\varphi^{n}(x)-\psi^{n}(y)-\frac{|x-y|^{2}}{2T}\biggr{)}\,\mathrm{d}x\,\mathrm{d}y=\int_{\mathbb{R}^{d}}\sigma_{\nu}|y|^{2}\,\mathrm{d}\nu(y)<+\infty\;,$

and hence there exists at least one point $\bar{x}\in\mathbb{R}^{d}$ such that

(286)
$\int_{\mathbb{R}^{d}}\exp\biggl{(}\sigma_{\nu}|y|^{2}-\psi^{n}(y)-\frac{|\bar{x}-y|^{2}}{2T}\biggr{)}\,\mathrm{d}y<+\infty\;.$

Since for any $x\in\mathbb{R}^{d}$ we can always write

(287) $|x-y|^{2}=|\bar{x}-y|^{2}-2\langle
x-\bar{x},y\rangle+|x|^{2}-|\bar{x}|^{2}\geq|\bar{x}-y|^{2}-2|x-\bar{x}|\,|y|+|x|^{2}-|\bar{x}|^{2}\;,$

for any $\bar{\sigma}<\sigma_{\nu}$ we have

(288)
$\int_{\mathbb{R}^{d}}\exp\biggl{(}\bar{\sigma}|y|^{2}-\psi^{n}(y)-\frac{|\bar{x}-y|^{2}}{2T}\biggr{)}\,\mathrm{d}y<+\infty\quad\forall\,x\in\mathbb{R}^{d}\;.$

This allows to differentiate under the integral sign in

(289) $\log P_{T}\exp(-\psi^{n})(x)=-\frac{d}{2}\log(2\pi
T)+\log\int\exp\biggl{(}-\psi^{n}(y)-\frac{|x-y|^{2}}{2T}\biggr{)}\mathrm{d}y$

and get the validity of (24) for $h=\psi^{n}$, i.e.

(290) $\nabla\log
P_{T}\exp(-\psi^{n})(x)=T^{-1}\int(y-x)\,\pi^{x,\psi^{n}}_{T}(\mathrm{d}y)=-\frac{x}{T}+\frac{1}{T}\,\frac{\int
y\exp(-\psi^{n}(y)-\frac{|x-y|^{2}}{2T})\mathrm{d}y}{\int\exp(-\psi^{n}(y)-\frac{|x-y|^{2}}{2T})\mathrm{d}y}\;.$

The bound (288) guarantees to differentiate again the above integral and
finally deduce our thesis.

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# Automated object detection for muon tomography data analysis

Anzori Sh. Georgadze,11footnotetext: Corresponding author.

###### Abstract

In recent years, there have been ongoing efforts to improve screening
technologies to improve security and prevent terrorist threats. The most
widely used technologies for scanning shipping containers are gamma and x-ray
radiography, which can be harmful to operators and the environment. Muon
tomography screening systems are considered as a potential tool to enhance
border security and prevent terrorist threats or smuggling, especially in the
context of shipping container inspections. Muon tomography is a technique that
uses naturally occurring cosmic ray muons to create detailed images of the
inside of objects, such as shipping containers, without the need for physical
intervention. In this paper, we describe the algorithms that allow automatic
detection of illegal dangerous items hidden inside legal cargo in shipping
containers. We used the Point of Closest Approach (PoCA) reconstruction
algorithm to reconstruct the 3D image of the shipping container and applied
the nearest neighbor filtering method to separate and differentiate the
filament materials from the complex structure of the surrounding common
materials. The use of these methods makes it possible to identify implicit
information in the data and visualize the contents of transport containers
with precise localization of threats or contraband materials.

##  1 Introduction

The rapid growth in international commerce has presented a significant
challenge for cargo security, and one of the primary issues is the limited
capacity to inspect and verify the contents of containers. Only a small
percentage of containers are typically scanned or inspected thoroughly at
ports and border crossings which leaves open the possibility of security
breaches and smuggling activities. The cargo imaging inspection systems are
aimed at finding illegal materials, such as weapons, explosives, drugs or
Special Nuclear Materials (SNM) through the imaging of large objects, such as
cargo containers, unoccupied vehicles, trains, trucks or boats. Traditional
methods of cargo inspection, such as X-ray scanning and manual inspections,
are often time-consuming and may not provide a comprehensive view of the
contents of containers or cargo. Cosmic-ray tomography is an innovative method
for cargo security and inspection that has the potential to address the
challenges posed by the rapid growth in international commerce. Muons are
subatomic particles that are naturally produced when cosmic rays interact with
the Earth’s atmosphere. They have the unique property of being highly
penetrating, which means they can pass through thick materials like concrete
and metal. This property makes them suitable for cargo inspection, as they can
be used to create detailed images of the interior of containers or cargo
without the need for physical intrusion. Muons interact with matter primarily
through the electromagnetic force which makes muons to deflect from their
original trajectory (Coulomb scattering) and causes the reduction in muon
energy (mainly via ionisation and excitation energy loss mechanism). Cosmic-
ray muons are part of the natural background radiation environment that humans
are exposed to continuously and hence, will not harm or alter the objects
being investigated, making them ideal for non-invasive inspection and imaging.
Cosmic-ray muons have energies at sea level about 3–4 GeV and are able to
penetrate much deeper into the materials compared to X-rays. Cosmic-ray
tomography [1, 2, 3, 4, 5, 6] is considered to be a new inspection technology,
complementary to the X-ray imaging which is most widely used to scan cargo at
borders. It is expected that CRT will help to increase the number and fraction
of scanned cargoes.

This paper investigates the possibility of applying cosmic-ray tomography for
scanning shipping containers and lorries for contraband hidden inside legal
cargo within acceptable measuring time. We have used GEANT4 toolkit [7] for
simulating realistic scenarios of illegal materials hidden inside legal cargo.

##  2 GEANT4 simulation and tomographic reconstruction

Simulation model of the muon tomography system consisting of two tracking
modules above and below the shipping container was constructed with the GEANT4
toolkit [7]. Each tracking module, having two position-sensitive detectors,
was modeled as a plane of plastic scintillators with a detection efficiency of
100% and an area of 8 m $\times$4 m and fit that of a shipping container.

As a possible tracking detector concept we consider technology based on
plastic scintillating fiber arrays readout with Silicon Photomultipliers
(SiPMs) [8]. The tracker consists of two planes, with the two layers of fibers
in each plane oriented perpendicularly to those in the adjacent plane, which
allow three-dimensional reconstruction of the muon track (figure 1).

Figure 1: (a) Layout of the plastic scintillator fiber tracker in GEANT4 with
optical photons generated in fibers by muon; (b) plastic scintillator fiber
tracker with an area around $30\times 30$ cm2 constructed in [8]; (c)
illustration cosmic ray tomography (CRT) system.

The Cosmic-Ray Shower Library (CRY) [9] was used to generate muons for the
latitude 50° and altitude at sea-level. The origin points of generated muons
we sampled from a horizontal plane surface of 10 m $\times$ 10 m. Generated
muons are interfaced to GEANT4 to simulate the interaction of the muons with
the detector, shipping container and cargo. The simulated data samples were
produced with an exposure time ranging from 5 to 10 minutes (1 million muons
correspond to 1 minute of scanning time). The recorded data on hit position on
each detector layer are used to produce two tracks. Cosmic ray muon scattering
tomography is achieved by tracking the trajectories of both incoming and
outgoing muons using a set of position-sensitive detectors. We calculate
scattering angle between the trajectories of incoming ($\vec{v}_{1}$) and
outgoing ($\vec{v}_{2}$) muons using formula [10]:

$\theta_{scatt}=arccos\left(\frac{\vec{}v_{1}\cdot\vec{}v_{2}}{\left|v_{1}\right|\left|v_{2}\right|}\right).$
(2.1)

Since some of the hits create statistical noise due to the applied detector
resolution, several selection criteria were implemented to select only
relevant scattering points for track recovery - The muon tracks must pass
through all detectors. The scattering point must be located inside the RoI.
The angle between the tracks constructed from any pair of detectors in the top
and bottom set must not exceed 10 mrad. We use the standard physics list for
high-energy transport named "FTFP_BERT". Synthetic datasets were analyzed
using ROOT data analysis package [11].

For the reconstruction of tomographic images we have used Point-of-Closest-
Approach algorithm (POCA) [12], which is based on the assumption that muon
scattering occurs in a single point. In order to take into account detector
position resolution we reconstruct the POCA points with hit positions in the
tracking detector smeared simultaneously in x and y directions by Gaussian
with the resolution $\sigma$ of 0.15 mm obtained in [8].

In order to obtain a clearer image with better signal-to-background ratio and
determine positions of hidden illegal goods and an estimate of their density
besides image filtering we perform spatial cuts, removing PoCA points outside
shipping container area. Next we subtract the image of an empty container
(figure 2 ) from the 3D image of a loaded container. Strict standards of
shipping containers make the procedure of image subtraction quite accurate.
After subtraction we apply additional spatial cut of an image to dimensions
$5.85\times 2.35\times 2.35$ m3 which correspond to internal part of the
container.

We apply statistical filtering method to remove noise by finding a median
value among voxels in 3D image. Median filtering removes noise by replacing
the pixel or voxel value at a given location with the median value of the
pixels in a local neighborhood around that location. This method allow to
clear image of areas with lower scattering density comparing comparing to
large scattering density generated by most of illegal materials. In some
cases, hidden contraband materials has lower density then material matrix of
legal cargo. In this case hidden contraband will be visualized on image as a
gap or hole and will be recognized as anomaly and case for additional
investigation. If illegal material has a higher density than legal cargo, it
can be detected by the difference in scattering density of materials. Since
illegal materials are hidden in massive surrounding cargo material we use an
algebraic filtering algorithm to remove voxels that represent material of
cargo to focus on anomalies that might indicate the presence of illegal
materials. To do that the nearest neighboring voxels are examined. The
filtering algorithm compares the value of a voxel with a threshold value that
has a functional dependence on the average scattering density around the
voxel. This dependence is a function of the variance of the scattering density
around a given voxel in a 3D grid. In detail, each voxel has 26 adjacent
voxels in the immediate surrounding layer. Moving to the second layer around
the given voxel, there would be 80 voxels, not including the original voxel or
the immediate neighbors. The applied algorithm evaluates up to ten layers
around a given voxel. This multilevel estimation allows the threshold to vary
depending on the scattering density in the far environment around the voxel.
This algorithm is particularly effective at removing background from medium-Z
materials such as steel, and aggregating voxels containing high-Z materials
when searching for smuggled Special Nuclear Materials (SNMs) in cargo.

##  3 Simulation in a real scenarios

To evaluate the ability of muon tomography to detect illegal materials in a
real-world environment, we generated geometries in Geant4 of shipping
container loaded with legal cargo and hidden inside illegal goods. The sheer
volume and complexity of the flow of commercial goods make it impractical to
model every possible smuggling scenario. Simulation of a limited number of
scenarios serves as the basis for the development of artificial intelligence
tools designed to identify illegal goods among legal cargo. This approach is
based on the principles of machine learning and artificial intelligence, where
models are trained on representative data sets to generalize patterns and make
predictions.

According to research [13] the mean cargo density is expected to be about or
below 0.2 g/cm3, which means that most of the commerce goods traffic consist
of low-Z materials that contain mainly C, H, O and N. Illegal materials like
drugs or explosives also consist of the same elements which make the task of
their detection challenging when they are hidden among the legal goods,
because the Coulomb scattering of muons for these materials is small,
resulting in a small deviation of the muon trajectory from a straight line.

There are several tips for shipping container loading to ensure safe
transportation. A master case is a bag with bulk materials or carton box that
contains multiple units, and sometimes also includes inner packs, which is
another level of packaging. To facilitate handling and storage, cargo usually
placed on pallets. A pallet is a shipping platform in which multiple master
cases are shipped on. The maximum weight of 20 ft container (with load) is
30,480 kg and maximum loading capacity is 28,200 kg.

Figure 2:  Visualization of CRT system with shipping container using GEANT4
(a), XY (b) and YZ (c) projections of tomography image of empty shipping
container

To simulate realistic smuggling scenarios we have developed in Geant4 detailed
model of shipping container ($609.6\times 259.1\times 243.8$ $cm^{3}$) loaded
with cloths in cardboxes on standard pallets (b) figures 3 (a). On Figure 3
(b) shows 2D projection of tomographic images after applying angle cuts. The
image is characterized by uniform distribution of scattering density and each
palette can be resolved.

We consider the scenario of illegal material hidden among boxes with legal
goods. For this in each palette one randomly selected box of is loaded with
explosive material (density 1.812 g/cm3)(figure 4(b).

Figure 3: (a) GEANT4 visualization of shipping container loaded cardboard
boxes with cloths arranged on pallets; XY projection of PoCA scattering
density of container loaded with cloths before (b) and after applying proper
processing (c).

Figure 4: (a) GEANT4 visualization of shipping container loaded cardboard
boxes with cloths arranged on pallets; (b) XY projection of PoCA scattering
density of container loaded with cloths;(c). GEANT4 visualization of the
container in which one bag in each pallet is loaded with explosive (bags in
blue color).

Figure 4(b) shows the 2D projection of the tomographic image and some
suspicious objects can be identified on this figure. After image processing to
remove noise from the cargo material, objects with high scattering density are
increased in contrast (figure 4(c)). For automatic detection of hidden
explosive we have applied two-dimensional peak search functions from ROOT
package which perform automatic peak searches in two-dimensional data. Most of
peaks are detected. Besides real peaks of there were detected ghost peaks due
to remaining noise from cargo material.

To test applying algorithms in different environments we simulated three
scenarios: explosive hidden inside legal cargoes such as cloths, bananas and
dry pasta shown in Figure 5(a),(b),(c), correspondingly. Reconstruction of
hidden explosive is well done in cloths, worse in bananas, and unreliable for
case of dry pasta. Figure 6 shows for comparison radiographic images of the
same geometries simulated with Geant4 using energy spectrum of 9 MeV X-rays.
As can be seen from comparing Figures 5 and 6 sensitivity of the two
techniques is similar.

However, muon tomography has advantage on providing three-dimensional data.
Figure 7 shows the results of simulated scenario of explosive inside bulk dry
past. Dividing the tomographic image on two slices and applying image
filtering techniques all to 10 boxes (marked with white numbers) can be
identified.

Figure 5: Reconstructed muon tomographic images for scenarios of explosive
hidden in container among cloths (a), bananas (b) and dry pasta (c).

Figure 6: Radiographic images simulated in Geant4 using 9 MeV X-rays for the
same scenarios of explosive hidden in container among cloths (a), bananas (b)
and dry pasta (c).

Figure 7: (a) 2d projection of upper slice of tomographic image; (b) 2d
projection of lower slice of tomographic image; (c) GEANT4 visualization of a
transport container with the location of boxes with explosives, numbered with
white numbers.

Cigarette smuggling, also informally referred to as "buttlegging", is the
illicit transportation of cigarettes or cigars from an administrative division
with low taxation to a division with high taxation for sale and consumption
[14, 15]. It often involves diversion from legal trade channels of entire
container loads, each carrying about 10 million cigarettes. In case of such
contraband cigarettes are declared as towel paper. To simulate a cigarette
smuggling scenario, we develop a container geometry half loaded with cardboard
boxes of paper towels and half loaded with boxes of cigarettes. The bulk
density of cigarettes is $\approx$ 0.18 g/cm3, the density of packaged bulk
paper towels is 0.11 g/cm3. The reconstructed tomographic image of this
scenario is illustrated on figure 8. It can be seen that the difference in
scattering density between cigarettes and paper towels is clearly visible,
allowing the detection of contraband cigarettes.

Figure 8: (a) GEANT4 visualization of a shipping container half loaded with
cardboard boxes of cigarettes (red) and half loaded with cardboard boxes of
paper towels (blue); (b) tomographic reconstruction of a container with
cigarettes and a towel; (c) 2D projection of the tomographic image after image
processing.

It is already have been proven that muon tomography is effective at detecting
high atomic mass materials, such as Special Nuclear Materials (SNM). We
simulated scenario of SNM concealed within a shipping container loaded with
dry pasta in boxes (loading of container in weight is 27 tones). Figure 9(a)
shows the positioning of 10 cubes of SNM, each with dimensions of 10 cm3
inside boxes with dry pasta. Please note that cardboxes and container wall
have been made invisible in the figure to enhance clarity. Results of
simulation and filtered tomographic reconstruction shows (figure 9 (b)) that
all 10 cubes of SNM for this scenario can be successfully identified. Applied
data processing algorithms allow fully suppress noise from container and cargo
materials. For automate detection of positions of SNM cubes in the filtered
tomographic image we have used the three-dimensional peak search functions
from ROOT package which searches for peaks in the 3three-dimensional
tomographic data. We have generated 1000 data samples for this scenario and in
all samples 10 cubes were successfully detected using peak search function and
their positions localized 9(c). Only in several cases noise signals were
detected as high-Z objects.

Figure 9: (a) Tomographic imaging of the simulated scenario obtained with POCA
of 10 cubes of SNM hidden inside of dry pasta loaded the container; (b)
filtered tomographic image using adaptive threshold with detected objects
using 3d peak search function. Rad triangles shows detected objects; (c)
detected positions of high-Z objects in 1000 simulated data samples using 3D
peak search function.

##  4  Conclusions

In this work, various scenarios of smuggling illegal low-Z and high-Z
materials were investigated through Monte Carlo simulations. We tested
performance of image processing tools in both cases low-Z and high-Z materials
using POCA reconstruction. The tests demonstrate the validity of image
processing tools and reconstruction techniques used, confirming the
effectiveness of the muon tomography in combating smuggling both low- and
high-Z illegal materials. The developed algorithms are fast and suitable to be
used in a real time processing of a tomographic images. The crucial aspect of
developed approach is adapting threshold dependent on the scattering density
observed across multiple layers around given voxel when removing the
background from cargo. It implies a dynamic threshold setting that accounts
for variations in scattering density in the wider environment. The idea of
dynamically setting thresholds accounts for the variations in scattering
density observed across different layers. This adaptability ensures that the
algorithm remains responsive to changes in density patterns, providing a more
accurate identification of anomalies. Noise removal using the developed
algorithms allows the use of peak search function from the ROOT data analysis
package to automatically detect anomalies in tomographic images caused by
hidden objects. Automatic anomaly detection can then trigger an alarm and aims
to improve the detection and localization of hidden objects in tomographic
data. In this work we have demonstrated that muon scattering tomography is a
promising technique for combating smuggling using shipping containers.

##  5 Acknowledgements

The authors acknowledge partial funding from the EU Horizon 2020 Research and
Innovation Programme under grant agreement no. 101021812 (“SilentBorder”).

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