Convergence of synchronous reflected-Brownian couplings

Definitions and mathematical statement

Let $D\subseteq\mathbb R^2$ be connected and open with smooth boundary, and let $\mathbf n$ denote its inward unit normal. For distinct starting points $x,y\in\overline D$, a synchronous reflected-Brownian coupling is the pair solving \[ X_t=x+B_t+\int_0^t\mathbf n(X_s)\,dL_s^X, \qquad Y_t=y+B_t+\int_0^t\mathbf n(Y_s)\,dL_s^Y, \] with the same standard planar Brownian driver $B$. The paths stay in $\overline D$. Each $L$ is continuous and nondecreasing, starts at zero, and increases only when its corresponding path is on $\partial D$; these are the boundary pushing terms in the Skorokhod reflection equations.

First, does there exist a bounded such domain and a pair of distinct initial points for which \[ \mathbb P\left(\limsup_{t\to\infty}|X_t-Y_t|>0\right)>0? \] Second, does this positive-probability assertion hold for distinct initial points when $D$ is the exterior of a nondegenerate closed Euclidean disk?

No independence between $X$ and $Y$ is intended: their dependence is fixed by the common driver. The event means that the distance fails to converge to zero and is positive along arbitrarily late times by some sample-dependent amount. It is weaker than asking the two paths to remain uniformly separated at all times. The bounded-domain question allows holes, while the exterior-disk question concerns an unbounded domain with one compact boundary component.

Short English statement

Can synchronously driven reflected Brownian motions fail to approach each other in a bounded planar domain or outside a disk?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 9500005 (AMR-094-0005), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Burdzy - My favorite open problems (2009), Problem 5 (parts i and ii). Original source indexed by AMR-094-0005. https://sites.math.washington.edu/~burdzy/open_mathjax.php.