Consider a random walk on a two-dimensional surface with circular reflecting boundary conditions (say, of radius 'R'). Here, for a fixed-size area, one finds a larger fraction of the probability density (for the position of the walker) 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 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 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.)



'Focusing' the mass of the Probability Density Function for a Random Walk
Hmm, you're asking for concentration for heat kernels. Over long periods of time, these kernels are dominated by the low-energy eigenfunctions, so basically one needs to construct domains which have concentrated low-energy eigenfunctions. 


Generally one expects in fact that heat kernels become smoother and disperse over time (parabolic regularity). For instance, all the L^p norms of heat kernels are non-increasing in time, so it's going to be harder and harder to concentrate into a small domain as time goes by. There is a substantial theory on controlling heat kernels (using tools such as the Poincare inequality, maximum principle, integration by parts, etc.) though it isn't quite my field; one may have to ask a parabolic PDE person.


