A universal exponent for Brownian entropic repulsion

TL;DR

This paper explores the universality of Brownian entropic repulsion, proposing an exponent to measure its strength and demonstrating its consistency across various approximations, suggesting a universal value of 3.

Contribution

It introduces an exponent to quantify Brownian entropic repulsion and provides evidence that this exponent is universal across different approximation methods.

Findings

The exponent κ=3 for several natural approximations.

Speed of the process is highly sensitive to approximation methods.

Proposes conjecture that κ=3 is a universal constant.

Abstract

We investigate the extent to which the phenomenon of Brownian entropic repulsion is universal. Consider a Brownian motion conditioned on the event $\mathcal{E}$ -- that its local time is bounded everywhere by 1. This event has probability zero and so must be approximated by events of positive probability. We prove that several natural quantities, in particular the speed of the process, are highly sensitive to the approximation procedure, and hence are not universal. However, we also propose an exponent $\kappa$ -- which measures the strength of the entropic repulsion by evaluating the probability that a particular point comes close to violating the condition $\mathcal{E}$. We show that $\kappa=3$ for several natural approximations of $\mathcal{E}$, and conjecture that $\kappa=3$ is universal in a sense that we make precise.