Consider a random walk on a two-dimensional surface with a circular reflecting boundary conditions (say, of radius 'R'). Here, for a fixed-size area, one finds that a larger fraction of the probability density function (for the position of the walkercan ) near the midpoint of the circle than near its contour.
Given this example, my question is - for a discrete/continuous random walk in a two-dimensional (or higher dimensional) space, now with arbitrary reflecting boundary conditions, how 'well' can one do in restricting restrict/focus the mass of the probability density function to the smallest possible area relative to the total surface area available to the walker?
In other words, how effectively can one define construct a 'trap' (I'm using this term very loosely) for such a walker, given random initial conditions?
(I obviously welcome any help to ask this question in a more appropriate manner.)