Brownian motion conditioned to spend limited time outside a bounded interval -- an extreme example of entropic repulsion

TL;DR

This paper demonstrates that a Brownian motion constrained to spend limited total time outside a bounded interval remains within it, illustrating an extreme case of entropic repulsion, and provides asymptotic probability estimates.

Contribution

It establishes that Brownian motion with limited outside time cannot leave the interval and derives exact asymptotics for the probability as time grows large.

Findings

Brownian motion with limited outside time stays within the interval.

Explicit asymptotic behavior of the probability as T approaches infinity.

Illustration of entropic repulsion in a stochastic process.

Abstract

We show that a Brownian motion on $\mathbb{R}_{\ge 0}$ which is allowed to spend a total of $s > 0$ time units outside a bounded interval does not leave the interval at all. This can be seen as an extreme example of entropic repulsion. Moreover, we explicitly determine the exact asymptotic behaviour of the probability that a Brownian motion on $[0,T]$ spends limited time outside a bounded interval, as $T \to \infty$.