# Why does Brownian motion cluster around singularities of the potential?

Suppose I give you the potential plotted on the left (with toroidal boundaries). On the right, I've plotted the associated Gibbs measure, which is how I'd naively expect a Brownian particle to spend its time.

However, when I do the simulations (with Hamiltonian MC), I get something that looks very different:

The particle spends most of its time near the singularity at the origin.

What's going on here? I've written up an explanation in terms of looking at the discrete dynamics of a random walker restricted to the set of minimal points (singularities act as a kind of "trap" for random motion), but I'd like to see the actual math for the continuous case.

Has anyone studied this kind of phenomenon? How would you go about predicting the true equilibrium distribution?