16 problems

Simultaneous uniqueness conjecture. The asserted simultaneous cluster-containment property holds for all three pairs of random variables and all quadruples in $\mathcal D$. This co…

Relaxation conjecture. For $m\geq 4$, this condition can be further relaxed while still permitting a simultaneous coupling constructed by a similar recipe to achieve the stated sim…

Let $K$ be a Markov kernel and let the method use a coupling of the corresponding Markov chains, with an expected improvement measured through the effective sample size. In the per…

Let an empirical effective sample size be computed for a system of coupled Markov chains, and consider its degradation under the proposed weight-harmonization method. In practice,…

General Gaussian coupling conjecture. For more general ${\bf \Omega}$ and $\widehat{{\bf \Omega}}$, the coupling measure used to generate the approximate and perfect knockoff varia…

Let $\kappa=M/m$ be the condition number associated with a strongly log-concave target, and let kinetic Langevin Monte Carlo use a non-synchronous coupling. Improved kinetic Langev…

Let $\{\{\tilde{X}_{\alpha,t}\}_{t}\}_{\alpha\in\mathbb{R}}$ be a grand coupling of Brownian motions $\{\mathrm{BM}(\alpha)\}_{\alpha\in\mathbb{R}}$. A failure probability bound is…

For $i \in \{1,\ldots,n\}$, let $(\mathcal{L}_{i},\mathcal{D}(\mathcal{L}_i))$ be generators of ergodic Feller semigroups. Let $A_k^i$, $B^i$, and $\mathcal{J}$ arise from decompos…

Let $\mathcal{G}$ be the set of coupling operators and let $\mathcal{G}^0$ be the subset for which the corresponding Feller coupling has the required invariant measure. Convexity c…

Let $(\bar{E},\bar{\tau})$ be the product state space, let $(\mathcal{L}_i,\mathcal{D}(\mathcal{L}_i))$ be the generators of the marginal Feller semigroups, and let $\mathcal{G}$ d…

Deterministic coupling conjecture.

Asymptotically deterministic coupling conjecture. Assume $H(X)=H(Y)$. Then

The Kolmogorov diffusion is a smooth non-trivial nilpotent diffusion, and a Markovian maximal coupling is a Markovian coupling that achieves the maximal possible coupling probabili…

Efficiency conjecture. In the case where the coupling is suboptimal, the synchronous coupling is efficient in the minimisation problem, and the mirror coupling is efficient in the…

Let ${\cal V}$ be the common distribution of the i.i.d. local velocities, and consider pairs of processes with equal densities under the weak normalization. Nearly successful coupl…

Let $D$ be a bounded convex domain, and consider reflected Brownian motion in $D$. A coupling of two reflected Brownian motions is shy if the coupled processes remain separated by…