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

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