Statistics of boundary encounters by a particle diffusing outside a compact planar domain

We consider a particle diffusing outside a compact planar set and investigate its boundary local time $ell_t$, i.e., the rescaled number of encounters between the particle and the boundary up to time $t$. In the case of a disk, this is also the (rescaled) number of encounters of two diffusing circular particles in the plane. For that case, we derive explicit integral representations for the probability density of the boundary local time $ell_t$ and for the probability density of the first-crossing time of a given threshold by $ell_t$. The latter density is shown to exhibit a very slow long-time decay due to extremely long diffusive excursions between encounters. We briefly discuss some practical consequences of this behavior for applications in chemical physics and biology.