Last February, Mike Cranston spoke in this Seminar about a polymer model based on three-dimensional Brownian motion conditioned to hit (and keep returning to) the origin. I will discuss the construction and certain properties of this conditioned Brownian motion from two points of view (i) Dirichlet forms, and (ii) excursion theory. The latter gives a nice interpretation of the Johnson-Helms example from martingale theory. It turns out that this diffusion process is not a semimartingale, even though its radial part is just a one-dimensional Brownian motion reflected at the origin.
This is based on joint work with Liping Li of Fudan University.