We consider a model of Brownian motion on a bounded interval which upon exiting the interval is being redistributed back  into the interval according to a probability measure depending on the exit point, then starting afresh, repeating the above mechanism indefinitely.  It is not hard to show that the process is exponentially ergodic, although characterizing the rate of convergence is non-trivial. In this talk, after providing a general overview of the probabilistic method of coupling and its applications,  I’ll show how to study the ergodicity for the model through coupling, how it leads to an  intuitive and geometric explanation for  the rates of convergence previously obtained analytically, other insights, and more questions. The talk will be accessible to general mathematical audience.