Capacitary Trapping and Brownian Escape: A Sharp Hardy Principle for Absorbing Media
Abstract
We study Brownian motion killed upon exiting arbitrary open sets Ω⊂ℝn, n∈[3,∞)∩ℕ. Let α∈(0,1]. The infimal radius dα(x) at which the absorber near x carries an α-fraction of the Newtonian capacity of the ball of the same radius provides a local measure of electrostatic trapping. We prove that this capacitary loading forces an absorption probability of at least cnα within a dimension-dependent multiple of the diffusive time scale dα(x)2. Independently, if p∈ (0,1) and τp(x) is the time by which Brownian motion started at x has been absorbed with probability at least p, then ∫Ω|u(x)|2/τp(x) d x ≤2/p2∫Ω|∇ u(x)|2 d x for every u∈ Cc∞(Ω). Together these statements give a capacitary Hardy inequality with constant Cnα−2 on every open set, answering a problem of Maz'ya, where Cn is a positive constant and, moreover, the exponent of α−2 is sharp. As applications, we derive corresponding estimates for survival probabilities, a corresponding spectral lower bound (including a recovery of the Maz'ya--Shubin lower bound), heat dissipation, Schrödinger forms, and Fisher information.
Related
- Dynamic programming principle for controlled systems driven by fractional Brownian motions and related backward stochastic Riccati equations
- Capacitary domination and Fefferman--Phong inequalities for measure potentials on $2$-PI spaces
- Commutative rings with square-difference factor absorbing factorization
- Polynomial Decay in Absorbing Continuous-Time Markov Chains
- Conditional Stable Laws and Rare-Event Limits for Absorbing Markov Chains
BibTeX
@misc{chen2026capacitary,
title = {{Capacitary Trapping and Brownian Escape: A Sharp Hardy Principle for Absorbing Media}},
author = {Yiqun Chen and Jie Xiao and Dachun Yang and Wen Yuan and Yangyang Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.26663},
eprint = {2608.26663},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.26663},
}
RIS
TY - UNPB
TI - Capacitary Trapping and Brownian Escape: A Sharp Hardy Principle for Absorbing Media
AU - Yiqun Chen
AU - Jie Xiao
AU - Dachun Yang
AU - Wen Yuan
AU - Yangyang Zhang
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2608.26663
UR - https://arxiv.org/abs/2608.26663
AB - We study Brownian motion killed upon exiting arbitrary open sets Ω⊂ℝ^n, n∈[3,∞)∩ℕ. Let α∈(0,1]. The infimal radius d_α(x) at which the absorber near x carries an α-fraction of the Newtonian capacity of the ball of the same radius provides a local measure of electrostatic trapping. We prove that this capacitary loading forces an absorption probability of at least c_nα within a dimension-dependent multiple of the diffusive time scale d_α(x)^2. Independently, if p∈ (0,1) and τ_p(x) is the time by which Brownian motion started at x has been absorbed with probability at least p, then ∫_Ω|u(x)|^2/τ_p(x) d x ≤2/p^2∫_Ω|∇ u(x)|^2 d x for every u∈ C_c^∞(Ω). Together these statements give a capacitary Hardy inequality with constant C_nα^−2 on every open set, answering a problem of Maz'ya, where C_n is a positive constant and, moreover, the exponent of α^−2 is sharp. As applications, we derive corresponding estimates for survival probabilities, a corresponding spectral lower bound (including a recovery of the Maz'ya--Shubin lower bound), heat dissipation, Schrödinger forms, and Fisher information.
ER -