'Focusing' the mass of the Probability Density Function for a Random Walk

Question

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.)

Answer 1

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.

Answer 2

I have no idea about the continuous case, which is presumably subtle, but the discrete case has an easy answer ('not very well'). In particular, if the random walk is simple random walk on a finite collection of vertices in the integer lattice in R^{d} (possibly with many self-loops on the boundary to simulate reflection), the stationary distribution is proportional to the degree at every vertex. In other words, the points away from the boundary don't really feel its presence at all.

If you have quantitative questions, a sufficiently clever person may be able to answer them using, say, Billingsley's book on convergence and the fact that many of these questions will be easy in the finite case.

Answer 3

I am not sure that a larger fraction of the probability density concentrates near the midpoint of the surface.

Let us consider a flat disk as the simplest example. As time goes on, the density will approach the lowest eigenfunction of the Laplacian on the disk, i.e. a constant corresponding to the uniform distribution. (I assume that what you mean by a random walk is the Wiener process on the disk).

Another way to see this effect is to consider a square (or hypercube in higher dimensions). Then you can mimic the reflection of the process at the boundary by reflecting the square wrt its edges and tiling the plane with these reflected copies of the square. If you consider the Wiener process on thus ``unfolded'' plane, you will recover the Wiener process with reflection after you fold the tiles back to the original square. Since the distribution of the Wiener process is Gaussian with variance equal to time, for large times the distribution is going to be almost flat which results in the uniform limiting distribution.