First-Passage-Driven Boundary Recession

TL;DR

This work analyzes a one-dimensional Brownian particle on a semi-infinite line with a moving boundary that recedes after each collision by a distance proportional to the inter-collision time, plus a small cutoff $\epsilon$ to regularize the dynamics. Using first-passage theory and Laplace transforms, it derives the full distribution of successive passage times: the $n$-th passage has a power-law tail $F_n(t|L_0)\sim t^{-(1+2^{-n})}$, with tails becoming fatter as $n$ grows. The authors show the average number of boundary encounters grows as $\langle N(t)\rangle \sim \frac{1}{\ln 2}\ln\ln t$, while the boundary position scales as $\langle L(t)\rangle \sim \frac{\alpha\,t}{\ln 2\,\ln t}$, implying near-ballistic boundary motion despite rare collisions. These results connect to extreme-value and iterated-logarithm phenomena in stochastic resetting contexts and reveal a surprisingly rich dynamics from a simple moving-boundary setup.

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 $t/\ln t$.

Figures (4)

• Figure 1: Illustration of the moving boundary (red line) in a semi-infinite geometry due to first-passage resetting. Each time the particle reaches the boundary, the boundary recedes by a distance $\alpha \tau_i+\epsilon$. The successive first-passage times are denoted $\tau_1,\tau_2,\ldots$.
• Figure 2: The function $g_n(s)$ as a function of $s$ for different values of $n$ with $\alpha=D=1$.
• Figure 3: Average number of boundary encounters $\langle N(t) \rangle$ scaled by $\ln\ln t$ as a function of the inverse of the scaled time for $D=L=\alpha=1$. The theoretical prediction from \ref{['eq:Navgta']} is also shown.
• Figure 4: Rescaled average position of the boundary $\langle L(t) \rangle$ as a function of time for $D=L=\alpha=1$.