Absorption Probabilities for Brownian Convex Hulls in Arbitrary Dimension
Abstract
For every dimension d≥1, we determine the probability that a prescribed point in ℝd belongs to the convex hull over [0,1] of a standard Brownian motion starting at the origin or of a Brownian bridge from $0$ to $0$. We obtain explicit one-dimensional integral formulas involving only standard special functions. For example, in dimension three the Brownian-motion absorption probability p3B(r) for a prescribed point at distance r>0 from the origin is p3B(r) =r/(√2 π3/2) ∫0∞ (π/(2√(1+v2))−1) e−r2(1+v2)/4 (K1−K0) ((r2(1+v2))/4) d v, where Kν denotes the modified Bessel function of the second kind. For each fixed dimension d≥1 and each prescribed nonzero point in ℝd, we derive precise asymptotics of the non-absorption probabilities for the convex hulls of Brownian motion on [0,T] and a Brownian bridge from $0$ to $0$ over the same interval, as T→∞. The corresponding probabilities are asymptotic to explicit positive constants times T−1/2(log T)d−1 and T−1(log T)d−1, respectively. We also consider the first time τd at which the growing convex hull of a standard Brownian motion in ℝd absorbs a prescribed point at distance one from the origin. We determine the Mellin and Laplace transforms of τd explicitly. Although 𝔼[τd]=∞ in every dimension, we prove (logτd−2d)/2√d xrightarrowdN(0,1), d→∞. In particular, τd1/d→e2 in probability, as d→∞. Surprisingly, the same limit holds for Sd1/d, where Sd is the first time at which the Brownian convex hull contains the entire unit ball.
Related
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- Ornstein--Uhlenbeck spirals: Absorption Probabilities and Conic Intrinsic Volumes
- Coverage probability of the planar Brownian convex hull
- A quadratic comparison of Neumann eigenvalues on thin convex domains in arbitrary dimenstion
- Graph distance and percolation dimension of three-dimensional Brownian motion
BibTeX
@misc{kabluchko2026absorption,
title = {{Absorption Probabilities for Brownian Convex Hulls in Arbitrary Dimension}},
author = {Zakhar Kabluchko},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00381},
eprint = {2611.00381},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00381},
}
RIS
TY - UNPB
TI - Absorption Probabilities for Brownian Convex Hulls in Arbitrary Dimension
AU - Zakhar Kabluchko
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00381
UR - https://arxiv.org/abs/2611.00381
AB - For every dimension d≥1, we determine the probability that a prescribed point in ℝ^d belongs to the convex hull over [0,1] of a standard Brownian motion starting at the origin or of a Brownian bridge from $0$ to $0$. We obtain explicit one-dimensional integral formulas involving only standard special functions. For example, in dimension three the Brownian-motion absorption probability p_3^B(r) for a prescribed point at distance r>0 from the origin is p_3^B(r) =r/(√2 π^3/2) ∫_0^∞ (π/(2√(1+v^2))−1) e^−r^2(1+v^2)/4 (K_1−K_0) ((r^2(1+v^2))/4) d v, where K_ν denotes the modified Bessel function of the second kind. For each fixed dimension d≥1 and each prescribed nonzero point in ℝ^d, we derive precise asymptotics of the non-absorption probabilities for the convex hulls of Brownian motion on [0,T] and a Brownian bridge from $0$ to $0$ over the same interval, as T→∞. The corresponding probabilities are asymptotic to explicit positive constants times T^−1/2(log T)^d−1 and T^−1(log T)^d−1, respectively. We also consider the first time τ_d at which the growing convex hull of a standard Brownian motion in ℝ^d absorbs a prescribed point at distance one from the origin. We determine the Mellin and Laplace transforms of τ_d explicitly. Although 𝔼[τ_d]=∞ in every dimension, we prove (logτ_d−2d)/2√d xrightarrowdN(0,1), d→∞. In particular, τ_d^1/d→e^2 in probability, as d→∞. Surprisingly, the same limit holds for S_d^1/d, where S_d is the first time at which the Brownian convex hull contains the entire unit ball.
ER -