Boundary local time on wedges and prefractal curves

TL;DR

This paper analyzes boundary local time on polygonal and fractal boundaries, deriving theoretical properties and developing a multi-scale Monte Carlo simulation to study diffusion reactions on complex geometries.

Contribution

It introduces a detailed analysis of boundary local time on wedges, derives moments, and proposes an efficient simulation method for complex polygonal boundaries like the Koch snowflake.

Findings

Mean boundary local time and variance on wedges are characterized.

Coupled PDEs for higher moments are established.

A multi-scale Monte Carlo method effectively simulates boundary interactions.

Abstract

We investigate the boundary local time on polygonal boundaries such as finite generations of the Koch snowflake. To reveal the role of angles, we first focus on wedges and obtain the mean boundary local time, its variance, and the asymptotic behavior of its distribution. Moreover, we establish the coupled partial differential equations for higher-order moments. Next, we propose an efficient multi-scale Monte Carlo approach to simulate the boundary local time, as well as the escape duration and position of the associated reaction event on a polygonal boundary. This numerical approach combines the walk-on-spheres algorithm in the bulk with an approximate solution of the escape problem from a sector. We apply it to investigate how the statistics of the boundary local time depends on the geometric complexity of the Koch snowflake. Eventual applications to diffusion-controlled reactions on partially reactive boundaries, including the asymptotic behavior of the survival probability, are discussed.