Regarding a particle exhibiting brownian motion within a box...
...In contrast, if the particle had merely diffused from its given initial state, while its expectation position would relax towards the center of the box, its maximum likelihood position would remain stationary.
My intuition here is that for as time progresses the expected and maximum likelihood positions both tend towards the centre of the box. And for small t they are both nearer the starting point.
That is, after a long period of time the particle will have been restricted from moving beyond the boxes bounds many times such that the bias it has towards the starting position is decayed - towards the centre of the box.
So how is it that the maximum likelihood estimate remains stationary?