SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \int_ {\mathbb {R} ^ {d}} \mathbb {1} _ {x + \sqrt {\gamma} z \in C (x, \gamma)} \langle z, u \rangle^ {2} \varphi (z) d z (89) \\ = \int_ {- \bar {\alpha} / \gamma^ {1 / 2}} ^ {r / \gamma^ {1 / 2}} \int_ {\nabla \Phi (\bar {x}) ^ {\perp}} \left(\langle u, v \rangle + \alpha \alpha_ {u}\right) ^ {2} \mathbb {1} _ {\| v \| ^ {2} \leq \left(\alpha \gamma^ {1 / 2} + \bar {\alpha}\right) / \gamma} \varphi (v) \varphi (\alpha) d v d \alpha (90) \\ \geq \int_ {0} ^ {r / \gamma^ {1 / 2}} \int_ {\nabla \Phi (\bar {x}) ^ {\perp}} \left(\langle u, v \rangle^ {2} + \alpha^ {2} \alpha_ {u} ^ {2}\right) \mathbb {1} _ {\| v \| ^ {2} \leq \left(\alpha \gamma^ {1 / 2} + \bar {\alpha}\right) / \gamma} \varphi (v) \varphi (\alpha) d v d \alpha (91) \\ \geq \alpha_ {u} ^ {2} \int_ {0} ^ {r / \gamma^ {1 / 2}} \alpha^ {2} \varphi (\alpha) \mathrm {d} \alpha + \int_ {- \bar {\alpha} / \gamma^ {1 / 2}} ^ {r / \gamma^ {1 / 2}} \int_ {\nabla \Phi (\bar {x}) ^ {\perp}} \langle u, v \rangle^ {2} \mathbb {1} _ {\| v \| ^ {2} \leq (\alpha \gamma^ {1 / 2} + \bar {\alpha}) / \gamma} \varphi (v) \varphi (\alpha) \mathrm {d} v \mathrm {d} \alpha . (92) \\ \end{array}
$$
Using Cauchy-Schwarz inequality, we have
$$
\int_ {0} ^ {r / \gamma^ {1 / 2}} \alpha^ {2} \varphi (\alpha) d \alpha = (1 / 2) - \int_ {r / \gamma^ {1 / 2}} ^ {+ \infty} \alpha^ {2} \varphi (\alpha) d \alpha \geq (1 / 2) - 3 \Phi (r ^ {2} / \gamma , 1) ^ {1 / 2}. \tag {93}
$$
In addition, using the Cauchy-Schwarz inequality, we have that
$$
\begin{array}{l} \int_ {- \bar {\alpha} / \gamma^ {1 / 2}} ^ {r / \gamma^ {1 / 2}} \int_ {\nabla \Phi (\bar {x}) ^ {\perp}} \langle u, v \rangle^ {2} \mathbb {1} _ {\| v \| ^ {2} \leq (\alpha \gamma^ {1 / 2} + \bar {\alpha}) / \gamma} \varphi (v) \varphi (\alpha) \mathrm {d} v \mathrm {d} \alpha (94) \\ = \int_ {\nabla \Phi (\bar {x}) ^ {\perp}} \langle u, v \rangle^ {2} \varphi (v) d v \int_ {- \bar {\alpha} / \gamma^ {1 / 2}} ^ {r / \gamma^ {1 / 2}} \varphi (\alpha) d \alpha (95) \\ - \int_ {- \bar {\alpha} / \gamma^ {1 / 2}} ^ {r / \gamma^ {1 / 2}} \int_ {\nabla \Phi (\bar {x}) ^ {\perp}} \langle u, v \rangle^ {2} \mathbb {1} _ {\| v \| ^ {2} \geq (\alpha \gamma^ {1 / 2} + \bar {\alpha}) / \gamma} \varphi (v) \varphi (\alpha) d v d \alpha (96) \\ \geq \left(\| u \| ^ {2} - \alpha_ {u} ^ {2}\right) \left(\left(1 / 2\right) - \Phi \left(r ^ {2} / \gamma , 1\right)\right) (97) \\ - \sqrt {3} (d - 1) \| u \| ^ {2} \int_ {0} ^ {+ \infty} \Phi \left(\alpha / \gamma^ {1 / 2}, d - 1\right) ^ {1 / 2} \varphi \left(\alpha - \bar {\alpha} / \gamma^ {1 / 2}\right) \mathrm {d} \alpha . (98) \\ \end{array}
$$
Combining this result, (93), (92) and Lemma 9 there exists  $\bar{\gamma} >0$  such that for any  $\gamma \in (0,\bar{\gamma} ]$  and  $u\in \mathbb{R}^d$
$$
\int_ {\mathbb {R} ^ {d}} \mathbb {1} _ {x + \sqrt {\gamma} z \in \mathrm {C} (x, \gamma)} \langle z, u \rangle^ {2} \varphi (z) \mathrm {d} z \geq (1 / 4) \| u \| ^ {2}, \tag {99}
$$
which concludes the proof of (85). Finally, using Theorem 7-(e), we have that for any  $x \in \overline{\mathcal{M}}$  if  $d(x, \partial \mathcal{M}) \leq R$  then  $x \in V$ . Let  $r = \min(R, 1/(2C))$  with  $C = \sup\{\|\nabla^2 \Phi(x)\| : x \in \overline{\mathcal{M}}\}$ . We have that for any  $x \in \overline{\mathcal{M}}$  such that  $d(x, \partial \mathcal{M}) \leq r$
$$
\left\| \nabla \Phi (x) \right\| \geq \left\| \nabla \Phi (\bar {x} _ {0}) \right\| - C r \geq 1 / 2, \tag {100}
$$
where  $\bar{x}_0$  is such that  $\| x - \bar{x}_0\| \leq r$  and  $\bar{x}_0\in \partial \mathcal{M}$ . Combining this result and (99) concludes the proof upon letting  $\varepsilon = 1 / 16$ .
Finally, we investigate the behavior of the normalized drift near the boundary.
Proposition 21. For any  $\bar{x}_0\in \partial \mathcal{M}$  and  $\varepsilon >0$  , there exist  $\bar{\gamma},r,M > 0$  such that for any  $x\in \overline{\mathcal{M}}$  and  $\gamma \in (0,\bar{\gamma})$  with  $\| x - \bar{x}_0\| \leq r$  and  $\| \hat{b}^{\gamma}(x)\| \geq M$
$$
\left\| \hat {b} ^ {\gamma} (x) / \langle \hat {b} ^ {\gamma} (x), \nabla \Phi (x) \rangle - \nabla \Phi (\bar {x} _ {0}) \right\| \leq \varepsilon . \tag {101}
$$
Proof. Let  $\bar{\gamma}$  be given by Proposition 18. Let  $\psi$  given by Lemma 9 and  $M_0 = \sup \{\psi (\gamma) / \gamma^{1 / 2}:\gamma \in (0,\bar{\gamma})\} < + \infty$ . Let  $M = 16M_0 / (c\varepsilon^{1 / 2})$  with  $c$  given in Proposition 18. Let  $R > 0$  given by Theorem 7-(e) such that for any  $x\in \overline{\mathcal{M}}$  with  $d(x,\partial \mathcal{M})$  there exist  $\bar{x}\in \partial \mathcal{M}$  and  $\alpha \in [0,c\varepsilon /(4C)]$  such that  $x = \bar{x} +\alpha \nabla \Phi (\bar{x})$  with  $C = \sup \{\| \nabla^2\Phi (x)\| :x\in \overline{\mathcal{M}}\}$  and  $c$  given in Proposition 18. Let  $r = \min (\bar{r},c\varepsilon /4,R)$  and  $x\in \overline{M}$  with  $\| x - \bar{x}_0\| \leq r$ . First, since  $d(x,\partial \mathcal{M})\leq R$ , there exist  $\bar{x}\in \partial \mathcal{M}$  and  $\alpha \in [0,\varepsilon /(4C)]$  such that  $x = \bar{x} +\alpha \nabla \Phi (\bar{x})$ . Therefore, we get that  $\| \bar{x} -\bar{x}_0\| \leq \varepsilon /(2C)$  and therefore  $\| \nabla \Phi (\bar{x}_0) - \nabla \Phi (\bar{x})\| \leq \varepsilon /2$ . In addition, we have that
$$
\begin{array}{l} \left\| \hat {b} ^ {\gamma} (x) / \left\langle \hat {b} ^ {\gamma} (x), \nabla \Phi (x) \right\rangle - \hat {b} ^ {\gamma} (x) / \left\langle \hat {b} ^ {\gamma} (x), \nabla \Phi (\bar {x}) \right\rangle \right\| (102) \\ \leq \| \hat {b} ^ {\gamma} (x) \| ^ {2} \| \nabla \Phi (x) - \nabla \Phi (\bar {x}) \| / \left(\langle \hat {b} ^ {\gamma} (x), \nabla \Phi (x) \rangle \langle \hat {b} ^ {\gamma} (x), \nabla \Phi (\bar {x}) \rangle\right). (103) \\ \end{array}
$$
Using Proposition 18, we get that
$$
\left\| \hat {b} ^ {\gamma} (x) / \langle \hat {b} ^ {\gamma} (x), \nabla \Phi (x) \rangle - \hat {b} ^ {\gamma} (x) / \langle \hat {b} ^ {\gamma} (x), \nabla \Phi (\bar {x}) \rangle \right\| \leq \varepsilon / 4. \tag {104}
$$
In what follows, we show that
$$
\left\| \hat {b} ^ {\gamma} (x) / \langle \hat {b} ^ {\gamma} (x), \nabla \Phi (\bar {x}) \rangle - \nabla \Phi (\bar {x}) \right\| ^ {2} \leq \varepsilon / 2. \tag {105}
$$
In particular, we show that for any  $u \in \nabla \Phi(\bar{x})^\perp$  with  $\| u \| = 1$ ,
$$
\langle \hat {b} ^ {\gamma} (x), u \rangle^ {2} \leq (\varepsilon / 1 6) \langle \hat {b} ^ {\gamma} (x), \nabla \Phi (\bar {x}) \rangle^ {2}. \tag {106}
$$
Assuming (106), letting  $u = (\hat{b}^{\gamma}(x) - \langle \hat{b}^{\gamma}(x),\nabla \Phi (\bar{x})\rangle) / (\| \hat{b}^{\gamma}(x)\|^{2} - \langle \hat{b}^{\gamma}(x),\nabla \Phi (\bar{x})\rangle^{2})^{1 / 2}$  and using that  $\hat{b}^{\gamma}(x) = \langle \hat{b}^{\gamma}(x),u\rangle u + \langle \hat{b}^{\gamma}(x),\nabla \Phi (\bar{x})\rangle \nabla \Phi (\bar{x})$  we have
$$
\begin{array}{l} \left\| \hat {b} ^ {\gamma} (x) / \left\langle \hat {b} ^ {\gamma} (x), \nabla \Phi (x) \right\rangle - \nabla \Phi (\bar {x}) \right\| \leq \left\| \hat {b} ^ {\gamma} (x) / \left\langle \hat {b} ^ {\gamma} (x), \nabla \Phi (\bar {x}) \right\rangle - \nabla \Phi (\bar {x}) \right\| (107) \\ + \| \hat {b} ^ {\gamma} (x) / \langle \hat {b} ^ {\gamma} (x), \nabla \Phi (x) \rangle - \hat {b} ^ {\gamma} (x) / \langle \hat {b} ^ {\gamma} (x), \nabla \Phi (\bar {x}) \rangle \| (108) \\ \leq \left| \langle \hat {b} ^ {\gamma} (x), u \rangle / \langle \hat {b} ^ {\gamma} (x), \nabla \Phi (\bar {x}) \rangle \right| + \varepsilon / 4 \leq \varepsilon / 2, (109) \\ \end{array}
$$
which concludes the proof. Let  $u \in \nabla \Phi(\bar{x})^{\perp}$  with  $\| u \| = 1$  and  $\{e_i\}_{i=1}^{d-1}$  an orthonormal basis of  $\nabla \Phi(\bar{x})^{\perp}$ . There exist  $\{a_i\}_{i=1}^{d-1}$  such that  $\sum_{i=1}^{d-1} a_i^2 = 1$  and  $u = \sum_{i=1}^{d-1} a_i e_i$ . Using Theorem 7-(b), we have that for any  $i \in \{1, \ldots, d-1\}$
$$
\begin{array}{l} \int_ {\mathbb {R} ^ {d}} \mathbb {1} _ {x + \sqrt {\gamma} z \in C (x, \gamma)} \langle z, e _ {i} \rangle \varphi (z) \mathrm {d} z = \int_ {- \bar {\alpha} / \gamma^ {1 / 2}} ^ {r / \gamma^ {1 / 2}} \int_ {\nabla \Phi (\bar {x}) ^ {\perp}} \mathbb {1} _ {\| v \| ^ {2} \leq (\gamma^ {1 / 2} \alpha + \bar {\alpha}) / \gamma} \langle v, e _ {i} \rangle \varphi (v) \varphi (\alpha) \mathrm {d} v \mathrm {d} \alpha (110) \\ = \int_ {- \bar {\alpha} / \gamma^ {1 / 2}} ^ {r / \gamma^ {1 / 2}} \int_ {\nabla \Phi (\bar {x}) ^ {\perp}} \mathbb {1} _ {\| v \| ^ {2} \geq (\gamma^ {1 / 2} \alpha + \bar {\alpha}) / \gamma} \langle v, e _ {i} \rangle \varphi (v) \varphi (\alpha) \mathrm {d} v \mathrm {d} \alpha (111) \\ \end{array}
$$
$$
\begin{array}{l} \left(\int_ {\mathbb {R} ^ {d}} \mathbb {1} _ {x + \sqrt {\gamma} z \in \mathsf {C} (x, \gamma)} \langle z, e _ {i} \rangle \varphi (z) \mathrm {d} z\right) ^ {2} = \left(\int_ {- \bar {\alpha} / \gamma^ {1 / 2}} ^ {r / \gamma^ {1 / 2}} \int_ {\nabla \Phi (\bar {x}) ^ {\perp}} \mathbb {1} _ {\| v \| ^ {2} \geq (\gamma^ {1 / 2} \alpha + \bar {\alpha}) / \gamma} \langle v, e _ {i} \rangle \varphi (v) \varphi (\alpha) \mathrm {d} v \mathrm {d} \alpha\right) ^ {2} (112) \\ \leq \int_ {\nabla \Phi (\bar {x}) ^ {\perp}} \left\langle v, e _ {i} \right\rangle^ {2} \varphi (v) d v \left(\int_ {- \bar {\alpha} / \gamma^ {1 / 2}} ^ {r / \gamma^ {1 / 2}} \Psi \left(\left(\bar {\alpha} + \alpha \gamma^ {1 / 2}\right) / \gamma , d - 1\right) ^ {1 / 2} \varphi (\alpha) d \alpha\right) ^ {2}. (113) \\ \end{array}
$$
Hence, we get that
$$
\sum_ {i = 1} ^ {d - 1} a _ {i} ^ {2} \left(\int_ {\mathbb {R} ^ {d}} \mathbb {1} _ {x + \sqrt {\gamma} z \in C (x, \gamma)} \langle z, e _ {i} \rangle \varphi (z) \mathrm {d} z\right) ^ {2} \leq \| u \| ^ {2} \psi^ {2} (\gamma), \tag {114}
$$
with  $\psi$  given by Lemma 9. Recalling that  $\| \hat{b}^{\gamma}(x)\| \geq M$  we have
$$
\langle \hat {b} ^ {\gamma} (x), u \rangle^ {2} \leq 1 6 \psi (\gamma) ^ {2} / \gamma \leq c ^ {2} (\varepsilon / 1 6) M ^ {2} \leq (\varepsilon / 1 6) \langle \hat {b} ^ {\gamma} (x), \nabla \Phi (\bar {x}) \rangle^ {2}, \tag {115}
$$
D.7 Submartingale problem and weak solution
We are now ready to conclude the proof.
Theorem 22. There exists  $\mathbb{P}^{\star}$  a distribution on  $\mathrm{D}([0,T],\overline{\mathcal{M}})$  such that  $\lim_{\gamma \to 0}\hat{\mathbb{P}}^{\gamma} = \mathbb{P}^{\star}$ . In addition, for any  $f\in \mathrm{C}^{1,2}([0,T]\times \overline{\mathcal{M}},\mathbb{R})$  with  $\langle \nabla \Phi (\bar{x}),\nabla f(x)\rangle \geq 0$  for any  $t\in [0,T]$  and  $x\in \partial \mathcal{M}$ , we have that the process  $(f(t,\omega (t)))_{t\in [0,T]}$  given for any  $t\in [0,T]$
$$
f (t, \omega (t)) - \int_ {0} ^ {t} \left(\partial_ {s} f (s, \omega (s) + \frac {1}{2} \Delta f (s, \omega (s))) \mathbb {1} _ {\mathcal {M}} (\omega (s)) \mathrm {d} s, \right. \tag {116}
$$
is a  $\mathbb{P}$  submartingale.
Proof. Condition (A) (Stroock et al., 1971, p.197) is a consequence of Proposition 13. Condition (B) (Stroock et al., 1971, p.197) is a consequence of Proposition 18. Condition (C) (Stroock et al., 1971, p.198) is a consequence of Corollary 16. Condition (D) (Stroock et al., 1971, p.198) is a consequence of Proposition 14. We fix  $\rho = 0$  and condition (1) (Stroock et al., 1971, p.203) is a consequence of Proposition 19. Condition (2)-(iii) (Stroock et al., 1971, p.203) is a consequence of Proposition 20. Condition (2)-(iv) (Stroock et al., 1971, p.203) is a consequence of Proposition 21. We conclude upon using (Stroock et al., 1971, Theorem 6.3) and (Stroock et al., 1971, Theorem 5.8).
We finally conclude the proof of Theorem 6 upon using the results of (Kang et al., 2017) which establish the link between a weak solution to the reflected SDE and the solution to a submartingale problem.
Theorem 23. For any  $T \geq 0$ ,  $(\hat{\mathbf{X}}_t^\gamma)_{t \in [0,T]}$  weakly converges to  $(\mathbf{X}_t)_{t \in [0,T]}$  such that for any  $t \in [0,T]$
$$
\mathbf {X} _ {t} = x + \mathbf {B} _ {t} - \mathbf {k} _ {t}, \qquad | \mathbf {k} | _ {t} = \int_ {0} ^ {t} \mathbb {1} _ {\mathbf {X} _ {s} \in \partial \mathcal {M}} \mathrm {d} | \mathbf {k} | _ {s}, \qquad \mathbf {k} _ {t} = \int_ {0} ^ {t} \mathbf {n} (\mathbf {X} _ {s}) \mathrm {d} | \mathbf {k} | _ {s}. \qquad (1 1 7)
$$
Proof. Using Theorem 22 and (Kang et al., 2017, Theorem 1, Proposition 2.12), we have that  $\mathbb{P}$  in Theorem 22 is associated with a solution to the extended Skorokhod problem. We conclude that a solution to the extended Skorokhod problem is a solution to the Skorokhod problem using (Ramanan, 2006, Corollary 2.10).
D.8 Extension to the Metropolis process
We recall that the Metropolis process is defined as follows. Let  $(X_k^\gamma)_{k\in \mathbb{N}}$  given for any  $\gamma >0$  and  $k\in \mathbb{N}$  by  $X_0^\gamma = x\in \overline{\mathcal{M}}$  and for  $X_{k + 1}^{\gamma} = X_{k}^{\gamma} + \sqrt{\gamma} Z_{k}$  if  $X_{k}^{\gamma} + \sqrt{\gamma} Z_{k}^{\gamma}\in \overline{\mathcal{M}}$  and  $X_{k}^{\gamma}$  otherwise,  $Z_{k}\sim \mathrm{N}(0,\mathrm{Id})$ . We recall that  $\hat{b}^{\gamma},\hat{\Sigma}^{\gamma}$  and  $\hat{\Delta}^{\gamma}$  are given by (48), (51) and (55). In particular, denoting  $\hat{\mathrm{K}}^{\gamma}$  the Markov kernel associated with  $(\hat{X}_k^\gamma)_{k\in \mathbb{N}}$ , i.e.  $\hat{\mathrm{K}}^{\gamma}:\mathcal{M}\times \mathcal{B}(\mathcal{M})\to [0,1]$  such that for any  $x\in \mathcal{M}$ ,  $\hat{\mathrm{K}}^{\gamma}(x,\cdot)$  is a probability measure, for any  $\mathsf{A}\in \mathcal{B}(\mathcal{M})$ ,  $\hat{\mathrm{K}}^{\gamma}(\cdot ,\mathsf{A})$  is a measurable function and  $\mathbb{E}[\mathbb{1}_{\mathsf{A}}(\hat{X}_1^\gamma)\mid \hat{X}_0^\gamma = x] = \hat{\mathrm{K}}^\gamma (x,\mathsf{A})$ . We have that for any  $\gamma >0$  and  $x\in \mathcal{M}$
$$
\hat {b} ^ {\gamma} (x) = (1 / \gamma) \int_ {\mathcal {M}} (y - x) \hat {\mathrm {K}} ^ {\gamma} (x, \mathrm {d} y), \tag {118}
$$
$$
\hat {\Sigma} ^ {\gamma} (x) = \left(1 / \gamma\right) \int_ {\mathcal {M}} (y - x) ^ {\otimes 2} \hat {\mathrm {K}} ^ {\gamma} (x, \mathrm {d} y), \tag {119}
$$
$$
\hat {\Delta} ^ {\gamma} (x) = (1 / \gamma) \int_ {\mathcal {M}} \| y - x \| ^ {4} \hat {\mathrm {K}} ^ {\gamma} (x, \mathrm {d} y). \tag {120}
$$
In what follows, we denote  $a^\gamma(x) = \mathbb{E}[\mathbb{1}_{x + \sqrt{\gamma}Z_0 \in \mathcal{M}}]$ . Denote  $\mathrm{K}^\gamma$  the kernel associated with  $(X_k^\gamma)_{k \in \mathbb{N}}$ . We have that for any  $\mathsf{A} \in \mathcal{B}(\mathcal{M})$ ,  $\gamma > 0$  and  $x \in \mathcal{M}$
$$
\begin{array}{l} \mathrm {K} ^ {\gamma} (x, \mathsf {A}) = \mathbb {E} \left[ \mathbb {1} _ {X _ {k + 1} ^ {\gamma} \in \mathsf {A}} \mathbb {1} _ {x + \sqrt {\gamma} Z _ {k + 1} \in \mathcal {M}} \right] + (1 - a ^ {\gamma} (x)) \mathbb {1} _ {\mathsf {A}} (x) (121) \\ = a ^ {\gamma} (x) \hat {\mathrm {K}} ^ {\gamma} (x, \mathrm {A}) + \left(1 - a ^ {\gamma} (x)\right) \mathbb {1} _ {\mathrm {A}} (x). (122) \\ \end{array}
$$
We define for any  $\gamma > 0$  and  $x \in \mathcal{M}$
$$
b ^ {\gamma} (x) = \left(1 / \gamma\right) \int_ {\mathcal {M}} (y - x) \mathrm {K} ^ {\gamma} (x, \mathrm {d} y), \tag {123}
$$
$$
\Sigma^ {\gamma} (x) = (1 / \gamma) \int_ {\mathcal {M}} (y - x) ^ {\otimes 2} \mathrm {K} ^ {\gamma} (x, \mathrm {d} y), \tag {124}
$$
$$
\Delta^ {\gamma} (x) = \left(1 / \gamma\right) \int_ {\mathcal {M}} \| y - x \| ^ {4} \mathrm {K} ^ {\gamma} (x, \mathrm {d} y). \tag {125}
$$
Using (122), we get that for any  $\gamma > 0$  and  $x \in \mathcal{M}$
$$
b ^ {\gamma} (x) = a ^ {\gamma} (x) \hat {b} ^ {\gamma} (x), \quad \Sigma^ {\gamma} (x) = a ^ {\gamma} (x) \hat {\Sigma} ^ {\gamma} (x), \quad \Delta^ {\gamma} (x) = a ^ {\gamma} (x) \hat {\Delta} ^ {\gamma} (x). \tag {126}
$$
Using Lemma 12, we have that for any  $\gamma \in (0,\bar{\gamma})$  and  $x\in \mathcal{M}$ $a^{\gamma}(x)\geq 1 / 4$
In order to conclude for the convergence of the Metropolis process we adapt Theorem 22 and Theorem 23. We define  $\mathbf{X}^{\gamma}:\mathbb{R}_{+}\to \overline{\mathcal{M}}$  given for any  $k\in \mathbb{N}$  by  $\mathbf{X}_{k\cdot \gamma}^{\gamma} = X_{k}^{\gamma}$  and for any  $t\in [k\gamma ,(k + 1)\gamma)$ ,  $\mathbf{X}_t^\gamma = X_k^\gamma$ . Note that  $(\mathbf{X}_t)_{t\in [0,T]}$  is a  $\mathrm{D}([0,T],\overline{\mathcal{M}})$  valued random variable, where  $\mathrm{D}([0,T],\overline{\mathcal{M}})$  is the space of right-continuous with left-limit processes which take values in  $\overline{\mathcal{M}}$ . We denote  $\mathbb{P}^{\gamma}$  the distribution of  $(\mathbf{X}_t)_{t\in [0,T]}$  on  $\mathrm{D}([0,T],\overline{\mathcal{M}})$ .
Theorem 24. There exists  $\mathbb{P}^{\star}$  a distribution on  $\mathrm{D}([0,T],\overline{\mathcal{M}})$  such that  $\lim_{\gamma \to 0}\mathbb{P}^{\gamma} = \mathbb{P}^{\star}$ . In addition, for any  $f\in \mathrm{C}^{1,2}([0,T]\times \overline{\mathcal{M}},\mathbb{R})$  with  $\langle \nabla \Phi (\bar{x}),\nabla f(x)\rangle \geq 0$  for any  $t\in [0,T]$  and  $x\in \partial \mathcal{M}$ , we have that the process  $(f(t,\omega (t)))_{t\in [0,T]}$  given for any  $t\in [0,T]$
$$
f (t, \omega (t)) - \int_ {0} ^ {t} \left(\partial_ {s} f (s, \omega (s) + \frac {1}{2} \Delta f (s, \omega (s))) \mathbb {1} _ {\mathcal {M}} (\omega (s)) \mathrm {d} s, \right. \tag {127}
$$
Proof. Condition (A) (Stroock et al., 1971, p.197) is a consequence of Proposition 13 and (126). Condition (B) (Stroock et al., 1971, p.197) is a consequence of Proposition 18 and (126). Condition (C) (Stroock et al., 1971, p.198) is a consequence of Corollary 16 and (126). Condition (D) (Stroock et al., 1971, p.198) is a consequence of Proposition 14 and (126). We fix  $\rho = 0$  and condition (1) (Stroock et al., 1971, p.203) is a consequence of Proposition 19 and that  $\lim_{\gamma \to 0}a^{\gamma} = 1$  uniformly on compact subsets  $K\subset \mathcal{M}$ . Condition (2)-(iii) (Stroock et al., 1971, p.203) is a consequence of Proposition 20 and (126). Condition (2)-(iv) (Stroock et al., 1971, p.203) is a consequence of Proposition 21 and (126). We conclude upon using (Stroock et al., 1971, Theorem 6.3) and (Stroock et al., 1971, Theorem 5.8).
Theorem 25. For any  $T \geq 0$ ,  $(\mathbf{X}_t^\gamma)_{t \in [0,T]}$  weakly converges to  $(\mathbf{X}_t)_{t \in [0,T]}$  such that for any  $t \in [0,T]$
$$
\mathbf {X} _ {t} = x + \mathbf {B} _ {t} - \mathbf {k} _ {t}, \quad | \mathbf {k} | _ {t} = \int_ {0} ^ {t} \mathbb {1} _ {\mathbf {X} _ {s} \in \partial \mathcal {M}} \mathrm {d} | \mathbf {k} | _ {s}, \quad \mathbf {k} _ {t} = \int_ {0} ^ {t} \mathbf {n} (\mathbf {X} _ {s}) \mathrm {d} | \mathbf {k} | _ {s}. \tag {128}
$$
Proof. The proof is identical to Theorem 23.  $\square$
E Modelling geospatial data within non-convex boundaries
To demonstrate the ability of the proposed method to model distributions whose support is restricted to manifolds with highly non-convex boundaries, we derived a geospatial dataset based on the historical wildfire incidence rate within the continental United States (described in in Appendix E.1) and, using the corresponding country borders, trained a constrained diffusion model by adapting the point-in-spherical-polytope conditions outlined in (Ketzner et al., 2022) (described in Appendix E.2).
E.1 Derivation of bounded geospatial dataset
Specifically, we retrieved the rasterised version of the wildfire data provided by Welty et al. (2020), converted it to a spherical geodetic coordinate system using the CARTOPY library (Met Office, 2010 - 2015), and drew a weighted subsample of size  $1 \times 10^{6}$ . We then retrieved the country borders of the continental United States from (Natural Earth, 2023) and mapped them to the same geodetic reference frame as the wildfire data. A visualization of the resulting dataset is presented in Figure 9.
E.2 Point-in-spherical-polytope algorithms
The support of the data-generating distribution we aim to approximate is thus restricted to a highly non-convex spherical polytope  $\mathbb{P} \in S^2$  given by the country borders of the continental United States. To determine whether a query point  $q \in S^2$  is within  $\mathbb{P}$ , we adapt an efficient reformulation of the point-in-spherical-polygon algorithm (Bevis et al., 1989) presented in (Ketzner et al., 2022). The algorithm requires the provision of a reference point  $r \in S^2$  known to be located in  $\mathbb{P}$  and determines
Figure 9: Orthographic projection of the wildfire dataset described in Appendix E. The projection is aligned with the centroid of the continental United States and zoomed in ten-fold for visual clarity. All visualisations of geospatial data were generated using the GEOVIEWS (Rudiger et al., 2023) and DATASHADER (Bednar et al., 2023) libraries.
whether  $q$  is inside or outside the polygon by checking whether the geodesic between  $r$  and  $q$  crosses the polygon an even or odd number of times. Letting  $\hat{x} \in \mathbb{R}^3$  denote the Cartesian coordinates of a point  $x \in S^2$ , (Ketzner et al., 2022) rely on a Cartesian reference coordinate system  $\hat{Q}$  (with its  $z$ -axis given by  $\hat{r}$ ) and the corresponding spherical coordinate system  $Q$  to decompose the edge-crossing condition of Bevis et al. (1989) into two efficiently computable parts. That is, the geodesic between  $q$  and  $r$  crosses an edge  $e_i = (v_i, v_j)$  of the polygon if:
(i) the longitude of  $q$  in  $Q$  is bounded by the longitudes of  $v_{i}$  and  $v_{j}$  in  $Q$ , i.e.
$$
\phi_ {Q} (q) \in [ \min  (\phi_ {Q} (v _ {i}), \phi_ {Q} (v _ {j})), \max  (\phi_ {Q} (v _ {i}), \phi_ {Q} (v _ {j})) ],
$$
(ii) the plane specified by the normal vector  $\hat{p}_i = \hat{v}_i \times \hat{v}_j$  represents an equator that separates  $q$  and  $r$  into two different hemispheres, i.e.
$$
\operatorname {s i g n} \left(\left\langle \hat {p} _ {i}, \hat {r} \right\rangle \cdot \left\langle \hat {p} _ {i}, \hat {q} \right\rangle\right) = - 1.
$$
Especially when  $\mathbb{P}$  is fixed and the corresponding coordinate transformations and normal vectors can be precomputed for each edge, this algorithm affords an efficient and parallelisable approach to determining whether any given point on  $S^2$  is contained by a spherical polytope.
F.1 Evaluating log-barrier and Euclidean models
Following (Fishman et al., 2023), we approached the empirical evaluation of our Metropolis model by computing the maximum mean discrepancy (MMD) (Gretton et al., 2012) between samples from the true distribution and the trained diffusion models. The MMD is a statistic that quantifies the similarity of two samples by computing the distance of their respective mean embeddings in a reproducing kernel Hilbert space. For this, we use an RBF kernel with the same length scales as the standard deviations of the normal distributions used to generate the synthetic distribution. We sum these RBF kernels by the weights of the corresponding components of the synthetic Gaussian mixture model.
This is essential to be able to include the log-barrier in the comparison since the log-barrier methods suffer severe instabilities around the boundary, as the space is stretched to more and more. These instabilities cause the problems in fitting the log-barrier model and in computing the likelihood using the log-barrier model.