First-Passage-Driven Boundary Recession

TL;DR

This paper studies a moving boundary problem for a Brownian particle where the boundary recedes based on collision timing, revealing complex scaling behaviors and long-term dynamics.

Contribution

It introduces a novel boundary movement rule linked to collision times and analyzes the resulting asymptotic scaling laws for first passage times and collision counts.

Findings

First passage probability scales as t^{-(1+2^{-n})}

Number of collisions scales as ln(ln t)

Boundary position varies as t/ln t

Abstract

We investigate a moving boundary problem for a Brownian particle on the semi-infinite line in which the boundary moves by a distance proportional to the time between successive collisions of the particle and the boundary. Phenomenologically rich dynamics arises. In particular, the probability for the particle to first reach the moving boundary for the $n^\text{th}$ time asymptotically scales as $t^{-(1+2^{-n})}$. Because the tail of this distribution becomes progressively fatter, the typical time between successive first passages systematically gets longer. We also find that the number of collisions between the particle and the boundary scales as $\ln\ln t$, while the time dependence of the boundary position varies as $t/\ln t$.