Lagrangian measure convergence conjecture for reflected Brownian flow

Let $D_1$ be the torus and let $D_1\setminus{\mathcal B}(0,r)$ be the domain obtained by deleting the ball of radius $r$. For the reflected Brownian flow $X_t^x$, define the random measure

$${\bf Q}_{r,t}(A)=\left|\{x\in D:X_t^x\in A\}\right|.$$

Here $|A|$ denotes Lebesgue measure.

Lagrangian measure convergence conjecture. The measures ${\bf Q}_{r,t}$ converge to a random measure ${\bf Q}_r$ on $D_1\setminus{\mathcal B}(0,r)$ as $t\to\infty$, in the sense of weak convergence of random measures. The random measures ${\bf Q}_r$ converge weakly to the uniform measure on $D_1$ as $r\to0$, in probability.

The conjecture concerns long-time limits of the spatial distribution generated by the reflected Brownian flow and its small-hole limit. Neither convergence assertion is established in the supplied text.