Suppose that D is a bounded planar domain and consider two reflected Brownian motions X(t) and Y(t) in D. Assume that the two processes are driven by the same Brownian motion. ( X(t) - Y(t) does not change on the time intervals when both processes stay away from the boundary of D). I will address the following problem. Does the distance between X(t) and Y(t) go to zero when t goes to infinity? The answer depends on the topology and geometry of the domain. Some quantitative results on the ``Lyapunov exponent'' will be presented. Time permitting, motivation, history of the problem, related results on reflected Brownian motion, and open problems will be also discussed. This is joint work with Z.-Q. Chen and P. Jones.