A spectral gap for Brownian loops on the round two-sphere
Let $S^2$ carry its unit round metric, fix $o \in S^2$, and let $\mu$ be the law of the Brownian bridge of duration one from $o$ to $o$, Brownian motion having generator $\tfrac12 \Delta_{S^2}$; it is a probability measure on $\{\sigma \in C([0,1]; S^2) : \sigma(0) = \sigma(1) = o\}$. Let $\parallel_t : T_o S^2 \to T_{\sigma(t)} S^2$ be stochastic Levi-Civita parallel translation along $\sigma$, and let $H_{00}$ be the space of $h \in H^1([0,1]; T_o S^2)$ with $h(0) = h(1) = 0$ and norm $\|h\|_{H_{00}}^2 = \int_0^1 |\dot h(t)|^2\, dt$. A smooth cylindrical function $F(\sigma) = f(\sigma(t_1), \dots, \sigma(t_k))$, with $f \in C^\infty((S^2)^k)$ and $0 < t_1 < \dots < t_k < 1$, has derivative $d_H F(\sigma)[h] = \sum_{i=1}^k \langle \nabla_i f(\sigma(t_1), \dots, \sigma(t_k)), \parallel_{t_i} h(t_i) \rangle$, whose norm in $H_{00}^*$ is given by $$\|d_H F(\sigma)\|_{H_{00}^*}^2 = \sum_{i,j=1}^k \bigl(\min(t_i, t_j) - t_i t_j\bigr) \langle a_i(\sigma), a_j(\sigma) \rangle, \qquad a_i(\sigma) = \parallel_{t_i}^{-1} \nabla_i f(\sigma(t_1), \dots, \sigma(t_k)).$$ For loop spaces over simply connected compact manifolds with positive Ricci curvature, Chen, Li and Wu proved, with this gradient, the weak Poincaré inequality $\operatorname{Var}_\mu(F) \le s^{-\alpha} \int \|d_H F\|_{H_{00}^*}^2\, d\mu + s \|F\|_\infty^2$ for every small $\alpha > 0$ and all $s \in (0, s_0)$. Eberle gave a Riemannian metric on a manifold diffeomorphic to a sphere whose loop space has no spectral gap. Chen, Li and Wu proved a spectral gap for loop spaces over hyperbolic space, and Aida for a class of manifolds with a pole including hyperbolic spaces.
Problem. Is there a finite constant $C$ such that $$\operatorname{Var}_\mu(F) \le C \int \|d_H F(\sigma)\|_{H_{00}^*}^2\, \mu(d\sigma)$$ for every smooth cylindrical function $F$, with $C$ independent of $k$, of the evaluation times and of $f$?
Source: Elworthy and Li (2006), Section 4, who ask whether loop spaces over non-flat manifolds, such as spheres, ever have a spectral gap. Stated by AIM — Open Applied Problems for the round two-sphere, with the Brownian bridge measure, Levi-Civita parallel translation and the H¹₀ gradient.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, loop spaces, Poincaré inequalities, spectral gaps
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@incollection{elworthy2006,
author = {Elworthy, K. David and Li, Xue-Mei},
title = {Geometric stochastic analysis on path spaces},
booktitle = {International Congress of Mathematicians. {V}ol. {III}},
publisher = {Eur. Math. Soc.},
address = {Z{\"u}rich},
year = {2006},
pages = {575--594},
doi = {10.4171/022-3/30},
eprint = {1911.09764},
archivePrefix = {arXiv},
}
@article{chen2011,
author = {Chen, Xin and Li, Xue-Mei and Wu, Bo},
title = {A concrete estimate for the weak {P}oincar{\'e} inequality on loop space},
journal = {Probab. Theory Related Fields},
volume = {151},
number = {3-4},
year = {2011},
pages = {559--590},
doi = {10.1007/s00440-010-0308-5},
eprint = {0910.4846},
archivePrefix = {arXiv},
}
@article{eberle2002,
author = {Eberle, Andreas},
title = {Absence of spectral gaps on a class of loop spaces},
journal = {J. Math. Pures Appl. (9)},
volume = {81},
number = {10},
year = {2002},
pages = {915--955},
doi = {10.1016/S0021-7824(02)01260-6},
}
@article{chen2010,
author = {Chen, Xin and Li, Xue-Mei and Wu, Bo},
title = {A {P}oincar{\'e} inequality on loop spaces},
journal = {J. Funct. Anal.},
volume = {259},
number = {6},
year = {2010},
pages = {1421--1442},
doi = {10.1016/j.jfa.2010.05.006},
eprint = {0905.3007},
archivePrefix = {arXiv},
}
@article{aida2015,
author = {Aida, Shigeki},
title = {Asymptotics of spectral gaps on loop spaces over a class of {R}iemannian manifolds},
journal = {J. Funct. Anal.},
volume = {269},
number = {12},
year = {2015},
pages = {3714--3764},
doi = {10.1016/j.jfa.2015.09.023},
eprint = {1401.6739},
archivePrefix = {arXiv},
}
@misc{aim-a-spectral-gap-for-brownian-loops-on-the-round-two-sphere,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A spectral gap for Brownian loops on the round two-sphere}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/995-a-spectral-gap-for-brownian-loops-on-the-round-two-sphere.md},
note = {Entry 995 at commit 4f5b165, 2026-11-04},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01142, "A spectral gap for Brownian loops on the round two-sphere", clopen.solutions.