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

2023/08/09 by Frank Aurzada, Aurzada, Frank, Martin Kolb +3
Economics, Econometrics and Finance · Mathematics · #60J55 #60J65 #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2308.04866

openalex publication_date 2023/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that a Brownian motion on ℝ_{≥ 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 → ∞.

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