Backbone probability of planar Brownian motion

Motivated by critical planar percolation, we investigate a ``backbone’’ event of planar Brownian motion, i.e.~the existence of two disjoint subpaths on the Brownian trajectory connecting the $\varepsilon$-neighborhood of the starting point to a macroscopic distance. We show that the probability of this event is $C(\log|\log\varepsilon|)^{-1}(1+o(1))$ as $\varepsilon\to0$ for some constant $C\in(0,\infty)$.

💡 Research Summary

This paper provides a precise asymptotic analysis of the “backbone” event probability for planar Brownian motion, drawing inspiration from analogous concepts in critical planar percolation. The central object of study is a planar Brownian motion (B_t) starting at the origin and stopped at its first hitting time τ_D of the unit circle. For a small ε>0, the backbone event Bac_ε is defined as the existence of two disjoint continuous subpaths on the trajectory B