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

Abstract

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.

References in corpus (7)

• Paradigm shift in diffusion-mediated surface phenomena
• Occupation Time Statistics in the Quenched Trap Model
• Diffusive escape through a narrow opening: new insights into a classic problem
• Probability distribution of the boundary local time of reflected Brownian motion in Euclidean domains
• Surface Hopping Propagator: An Alternative Approach to Diffusion-Influenced Reactions
• Encounter times in overlapping domains: application to epidemic spread in a population of territorial animals
• First-encounter time of two diffusing particles in confinement