Ornstein--Uhlenbeck spirals: Absorption Probabilities and Conic Intrinsic Volumes
Abstract
Let (Ut)t∈ℝ be a stationary Ornstein--Uhlenbeck process in ℝd whose coordinate processes (Ut(i))t∈ℝ, 1≤ i≤ d, are independent, centered, and have covariance function 𝔼 [Us(i)Ut(i)]=e−|t−s|/2. For every T>0, we obtain an explicit formula for the probability that the convex hull conv{Ut:0≤ t≤ T} contains the origin. We obtain a similar formula for a circular Ornstein--Uhlenbeck loop whose covariance is the normalized T-periodization of this kernel. For each fixed dimension d, we determine the precise long-time asymptotics of the non-absorption probabilities, which decay as explicit constants times Td−1 e−T/2 as T→∞. We also show that, as d→∞, the absorption probabilities in both models undergo a transition from $0$ to $1$ around T=2d, with a common Gaussian limiting profile on the scale √d. We also obtain explicit formulas for the probability that σZ belongs to the convex hull of Brownian motion or Brownian bridge on [0,1] in ℝd, where Z is an independent standard Gaussian vector and σ≥0. The large-σ asymptotics recover the known formulas for the expected volumes of these convex hulls. Finally, we compute the conic intrinsic volumes of the closed positive hulls of deterministic Ornstein--Uhlenbeck spirals and their circular analogues in Hilbert spaces. Their Grassmann angles coincide with the corresponding Ornstein--Uhlenbeck absorption probabilities. After suitable rescaling, the small-time limits of these conic intrinsic volumes recover the formulas of Gao--Vitale and Gao for the intrinsic volumes of the Brownian body and its Brownian-bridge analogue.
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BibTeX
@misc{kabluchko2026ornsteinuhlenbeck,
title = {{Ornstein--Uhlenbeck spirals: Absorption Probabilities and Conic Intrinsic Volumes}},
author = {Zakhar Kabluchko},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37911},
eprint = {2610.37911},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37911},
}
RIS
TY - UNPB
TI - Ornstein--Uhlenbeck spirals: Absorption Probabilities and Conic Intrinsic Volumes
AU - Zakhar Kabluchko
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37911
UR - https://arxiv.org/abs/2610.37911
AB - Let (U_t)_t∈ℝ be a stationary Ornstein--Uhlenbeck process in ℝ^d whose coordinate processes (U_t^(i))_t∈ℝ, 1≤ i≤ d, are independent, centered, and have covariance function 𝔼 [U_s^(i)U_t^(i)]=e^−|t−s|/2. For every T>0, we obtain an explicit formula for the probability that the convex hull conv{U_t:0≤ t≤ T} contains the origin. We obtain a similar formula for a circular Ornstein--Uhlenbeck loop whose covariance is the normalized T-periodization of this kernel. For each fixed dimension d, we determine the precise long-time asymptotics of the non-absorption probabilities, which decay as explicit constants times T^d−1 e^−T/2 as T→∞. We also show that, as d→∞, the absorption probabilities in both models undergo a transition from $0$ to $1$ around T=2d, with a common Gaussian limiting profile on the scale √d. We also obtain explicit formulas for the probability that σZ belongs to the convex hull of Brownian motion or Brownian bridge on [0,1] in ℝ^d, where Z is an independent standard Gaussian vector and σ≥0. The large-σ asymptotics recover the known formulas for the expected volumes of these convex hulls. Finally, we compute the conic intrinsic volumes of the closed positive hulls of deterministic Ornstein--Uhlenbeck spirals and their circular analogues in Hilbert spaces. Their Grassmann angles coincide with the corresponding Ornstein--Uhlenbeck absorption probabilities. After suitable rescaling, the small-time limits of these conic intrinsic volumes recover the formulas of Gao--Vitale and Gao for the intrinsic volumes of the Brownian body and its Brownian-bridge analogue.
ER -