Subdiffusivity of Brownian motion among a Poissonian field of moving traps

TL;DR

This paper studies the subdiffusive behavior of a Brownian particle avoiding moving traps in a Poisson field, providing bounds on its displacement and revealing how microscopic dynamics influence macroscopic subdiffusivity.

Contribution

It improves existing bounds on the particle's maximal displacement and links microscopic time scale behavior to overall subdiffusivity, offering insights into optimal survival strategies.

Findings

Upper bounds on maximal displacement are improved.

Microscopic time scale behavior influences macroscopic subdiffusivity.

Particle exhibits subdiffusive motion even on microscopic scales.

Abstract

Our model consists of a Brownian particle $X$ moving in $\mathbb{R}$, where a Poissonian field of moving traps is present. Each trap is a ball with constant radius, centered at a trap point, and each trap point moves under a Brownian motion independently of others and of the motion of $X$. Here, we investigate the 'speed' of $X$ on the time interval $[0,t]$ and on 'microscopic' time scales given that $X$ avoids the trap field up to time $t$. Firstly, following the earlier work of Athreya et al. [Math. Phys. Anal. Geom. 20:1 (2017)], we obtain bounds on the maximal displacement of $X$ from the origin. Our upper bound is an improvement of the corresponding bound therein. Then, we prove a result showing how the speed on microscopic time scales affect the overall macroscopic subdiffusivity on $[0,t]$. Finally, we show that $X$ moves subdiffusively even on certain microscopic time scales, in the bulk of $[0,t]$. The results are stated so that each gives an 'optimal survival strategy' for the system. We conclude by giving several related open problems.