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 (23)

• Paradigm shift in diffusion-mediated surface phenomena
• The Local and the Occupation Time of a Particle Diffusing in a Random Medium
• Kinetics of stochastically-gated diffusion-limited reactions and geometry of random walk trajectories
• Occupation Time Statistics in the Quenched Trap Model
• Diffusive escape through a narrow opening: new insights into a classic problem
• Towards a full quantitative description of single-molecule reaction kinetics in biological cells
• The Dirichlet-to-Neumann operator via hidden compactness
• Spectral theory of imperfect diffusion-controlled reactions on heterogeneous catalytic surfaces
• Explicit Construction of the Brownian Self-Transport Operator
• Residence Time Distribution for a Class of Gaussian Markov Processes
• Computation of the Mean First-Encounter Time Between the Ends of a Polymer Chain
• 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
• The two-particle problem in comb-like structures
• Slow Encounters of Particle Pairs in Branched Structures
• Joint distribution of multiple boundary local times and related first-passage time problems with multiple targets
• On the Robin problem in fractal domains
• Residence time statistics for $N$ blinking quantum dots and other stochastic processes
• First-encounter time of two diffusing particles in confinement
• First exit times and residence times for discrete random walks on finite lattices
• On the joint residence time of N independent two-dimensional Brownian motions
• Order statistics of the trapping problem

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• Escape-from-a-layer approach for simulating the boundary local time in Euclidean domains
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