Efficient couplings in acute triangles

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $D\subset\mathbb R^2$ be the interior of a triangle all of whose angles are strictly below $\pi/2$. Let $\mathcal L$ be the reflected-Brownian generator with Neumann boundary conditions, and let $0=\mu_1<\mu_2\le\cdots$ be the eigenvalues of $-\mathcal L$. Thus $\mu_2$ is its positive spectral gap. With standard Brownian time normalization, $\mathcal L=\tfrac12\Delta$; the Brownian time and spectral eigenvalues use this same normalization. A coupling of reflected Brownian motions is a joint construction of two processes with the respective reflected Brownian marginal laws; normal reflection keeps each process in $\overline D$. Define their first meeting time $\tau=\inf\{t\ge0:X_t=Y_t\}$.

Statement recorded in the supplied source.

Can one construct such a coupling starting from different points, with $\tau<\infty$ almost surely, for which for every $\varepsilon>0$ there is $t_0<\infty$ such that \[\mathbb P(\tau>t)\le\exp[-(\mu_2-\varepsilon)t]\quad(t\ge t_0)?\] [S2]

Short English statement

Can reflected Brownian motions in an acute triangle be coupled to meet with the prescribed nearly spectral exponential tail?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9500004 (AMR-094-0004). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Krzysztof Burdzy, My favorite open problems, Problem 4, Efficient couplings in acute triangles. https://sites.math.washington.edu/~burdzy/open_mathjax.php.