Exact winding shells of planar Brownian motion: $L^p$ and almost-sure convergence beyond the $L^2$ shell theorem, and the second moment of the Brownian loop's winding shells

This problem record concerns Werner's 1994 study of the points around which planar Brownian motion turns many times. For fixed time $t$, let $A_n^t$ denote the area of the set of points around which the motion has wound between $n$ and $n+1$ full turns (the $n$-th winding shell). Werner proved that $n^2 A_n^t \to t/(2\pi)$ in $L^2$ and explicitly posed what his method could not reach: the $L^p$ extension for $p>2$ (Section 5.2, where he sketches a $p$-point analysis and predicts moment asymptotics at order $n^{-2p}$), the almost-sure version (blocked, he notes, because his estimates cannot be made fine enough for Kolmogorov's criterion), and the second moment of the winding shells of the planar Brownian loop (Section 5.3). The cumulative analogues are resolved: Sauzedde (2022) proved $N D_N \to 1/(2\pi)$ in every $L^p$, $p\in[2,\infty)$, and almost surely, where $D_N$ is the area of the set of points wound at least $N$ times — but these provably do not reach the exact shells, since the cumulative error induces a divergent error on $N^2 A_N$. The loop shell first moment is known exactly, $\mathbb{E}[A_n]=1/(2\pi n^2)$ (Garban-Trujillo Ferreras, 2006), while the second moment is open; Sauzedde's December 2024 conjectures on the renormalised Amperean area (arXiv:2412.16781) still treat the second-order behaviour of the winding shells as conjectural. This problem asks for the shell-level $L^p$ ($p>2$) and almost-sure convergence of $n^2 A_n^t$, and the second-moment asymptotics of the loop shells.

Background

Planar Brownian motion — the scaling limit of random walks in the plane — is a central object of probability theory, and a natural way to quantify how its path coils around the plane is by windings. For a point $y$ not on the path, the winding of the motion around $y$ up to time $t$ is the total accumulated angle $\theta_t^y$ of the vector from $y$ to the moving point, counted continuously and with sign (counterclockwise positive); since planar Brownian motion started away from a point almost surely never hits it, this is well defined. Spitzer (1958) computed the large-time distribution of the winding around a fixed point, and Yor (1980) computed the explicit law of the winding index of the planar Brownian loop — the Brownian bridge from a point back to itself over unit time, a closed curve whose winding around any point off its trace is an integer. Werner's 1994 study of the points around which planar Brownian motion turns many times introduced the winding shells: for an integer $n$, $A_n^t$ is the area (Lebesgue measure) of the set of points $y$ with $2\pi n \le \theta_t^y < 2\pi(n+1)$ — the points around which the path has wound between $n$ and $n+1$ full turns. He proved (Théorème 1) that for every fixed $t>0$, $n^2 A_n^t \to t/(2\pi)$ in $L^2$ as $|n|\to\infty$ — the areas of very high winding shells concentrate, after scaling by $n^2$, around the deterministic value $t/(2\pi)$ — and the cumulative analogue (Théorème 15): for $D_N^t$, the area of the set of points wound at least $N$ full turns, $N D_N^t \to t/(2\pi)$ in $L^2$.

The final section of Werner's paper formulates what the method could not reach. For the $L^p$ extension, $p>2$, he indicates that the proof should generalize at the cost of analyzing the joint winding law around $p$ points at once — the dominant contribution coming from trajectories that make one single ascent ('une et une seule montée') near each of the $p$ points — and predicts the resulting moment asymptotics of $A_n^t$ at order $n^{-2p}$, but no complete proof is given. For almost-sure convergence he states explicitly that his estimates cannot be made fine enough for Kolmogorov's convergence criterion. And for the Brownian loop he hopes for the second moment of the loop shells $A_n$ — the areas of the sets of points with winding index exactly $n$ — of which only the first moment was known.

Subsequent progress has settled the cumulative analogues but not the shells. Garban and Trujillo Ferreras (2006) computed exactly the expected area of the filled planar Brownian loop (the filling, or hull, of the loop: the set of points the curve disconnects from infinity), obtaining $\pi/5$, and, as their Theorem 5.2, the exact loop-shell first moment $\mathbb{E}[A_n] = 1/(2\pi n^2)$, from the Mansuy-Yor explicit form of the loop-winding law. Sauzedde (2022) proved that for unit-time Brownian motion $N D_N \to 1/(2\pi)$ in every $L^p$, $p\in[2,\infty)$, and almost surely, with rates $O(N^{-\delta})$ for every $\delta<1/2$ — the strongest available results — and developed a stochastic Green's formula (2026) making winding expectations exactly computable, together with a two-path winding calculus (2021) for pairs of motions. But the cumulative bounds are provably too coarse for the shells: since $A_N = D_N - D_{N+1}$, the error $O(N^{-\delta})$ on $N D_N - 1/(2\pi)$ induces an error of order $N^{1-\delta}$ on $N^2 A_N$, which diverges. Finally, in his 2024 construction of the renormalised Amperean area — the winding-area functional of a closed Brownian curve that appears in the Symanzik (Feynman-Kac-type) representation of the two-dimensional Abelian Yang-Mills-Higgs field — Sauzedde must still state as a conjecture the almost-sure and $L^p$ convergence of the centred, $k^2$-weighted sums of the shell areas. The shell-level $L^p$ and almost-sure questions, and the loop second moment, thus remain the open core of Werner's program.

Problem Statement

Let $B=(B^1,B^2)$ be a standard planar Brownian motion started at the origin, and for $y\ne 0$ and $t>0$ let $\theta_t^y$ denote the winding of $B$ around $y$ up to time $t$: the total accumulated continuous argument of $B_s-y$ over $s\in[0,t]$, counterclockwise positive. Planar Brownian motion started away from a point almost surely never hits it, so $\theta_t^y$ is almost surely well defined for each fixed $y\ne 0$; the winding is undefined only for $y$ on the Brownian trace, which has zero Lebesgue measure (as stated for the loop below), so each shell area $A_n^t$ is well defined up to null sets. For $n\in\mathbb{Z}$ define the $n$-th winding shell

$$A_n^t = \operatorname{Leb}\{y\in\mathbb{R}^2 : 2\pi n \le \theta_t^y < 2\pi(n+1)\},$$

where $\operatorname{Leb}$ is Lebesgue measure: the total area of the set of points around which the path has wound between $n$ and $n+1$ full turns. Werner (1994, Théorème 1) proved that for each fixed $t>0$,

$$n^2 A_n^t \to \frac{t}{2\pi} \quad\text{in } L^2 \quad (|n|\to\infty).$$

The open problem is the extension of this shell-level theorem along three items, which together form the surviving open core of Sections 5.2-5.3 of Werner's paper (the cumulative analogues being settled; see Scope below).

(1) Higher moments. For fixed $t>0$ and $p>2$, prove or disprove that $n^2 A_n^t \to t/(2\pi)$ in $L^p$ as $|n|\to\infty$. (For $p\le 2$ this follows from the $L^2$ theorem.) A resolution includes the moment asymptotics: identify $\lim n^{2p}\,\mathbb{E}[(A_n^t)^p]$ when it exists.

(2) Almost-sure convergence. For fixed $t>0$, prove or disprove that $n^2 A_n^t \to t/(2\pi)$ almost surely as $|n|\to\infty$. Werner notes that his estimates cannot be made fine enough for Kolmogorov's convergence criterion.

(3) Brownian-loop second moment. Let $\ell$ be the standard planar Brownian loop of unit duration: the Brownian bridge from $0$ back to $0$ over $[0,1]$. It is a closed curve; its winding around any point off its trace is an integer $n_\ell(y)$ (the winding index), and the trace has zero area. Define $A_n = \operatorname{Leb}\{y\in\mathbb{R}^2 : n_\ell(y)=n\}$. The first moment is known exactly: $\mathbb{E}[A_n] = 1/(2\pi n^2)$ for $|n|\ge 1$ (Garban-Trujillo Ferreras 2006, Theorem 5.2). Prove or disprove that

$$\mathbb{E}[(A_n)^2] \sim \frac{1}{(2\pi n^2)^2} \quad (|n|\to\infty),$$

equivalently (given the exact first moment) that $n^2 A_n \to 1/(2\pi)$ in $L^2$; more generally, determine whether $\lim n^4\,\mathbb{E}[(A_n)^2]$ exists and its value, which must be at least $1/(4\pi^2)$.

Scope. The cumulative analogues are settled and do not imply the shell statements: for $D_N^t = \operatorname{Leb}\{y : \theta_t^y \ge 2\pi N\} = \sum_{k\ge N} A_k^t$, Sauzedde (2022, Theorem 1.5, unit time; general fixed $t$ by scaling) proved $\mathbb{E}[|N D_N - 1/(2\pi)|^p]^{1/p} \le C N^{-\delta}$ for all $p\in[2,\infty)$ and $\delta<1/2$, and the corresponding almost-sure bounds; since $A_N = D_N - D_{N+1}$, this induces an error of order $N^{1-\delta}$ on $N^2 A_N$, which diverges. Werner's original Section 5.3 statement of item (3) is the chain $\mathbb{E}[(A_n)^2] \sim 1/(2\pi n^2)^2 \sim \mathbb{E}[A_n]^2$; given the exact first moment, this chain is equivalent to the single equivalence stated above (a subtractive reading is impossible since $\mathbb{E}[(A_n)^2] \ge \mathbb{E}[A_n]^2$). Item (3) is the loop analogue of items (1)-(2) at the level $p=2$, and a resolution feeds the conjectured almost-sure and $L^p$ convergence of the centred, $k^2$-weighted shell sums $\sum_k k^2(A_k - \mathbb{E}[A_k])$ in the renormalised Amperean area program (Sauzedde 2024, Conjecture 2) — a related but distinct statement about sums rather than individual shells. Items (1)-(3) are independent of one another.

References

1. Wendelin Werner, 'Sur les points autour desquels le mouvement brownien plan tourne beaucoup', Probability Theory and Related Fields 99(1):111-144 (1994). DOI:10.1007/BF01199592, https://doi.org/10.1007/BF01199592
2. Christophe Garban and José A. Trujillo Ferreras, 'The expected area of the filled planar Brownian loop is π/5', Communications in Mathematical Physics 264(3):797-810 (2006). DOI:10.1007/s00220-006-1555-2, https://doi.org/10.1007/s00220-006-1555-2
3. Isao Sauzedde, 'Lévy area without approximation', Annales de l'Institut Henri Poincaré, Probabilités et Statistiques (2022); arXiv:2101.03992. DOI:10.1214/21-aihp1230, https://arxiv.org/abs/2101.03992
4. Isao Sauzedde, 'Planar Brownian motion winds evenly along its trajectory', Latin American Journal of Probability and Mathematical Statistics (ALEA) 20 (2023); arXiv:2102.12372. DOI:10.30757/alea.v20-50, https://arxiv.org/abs/2102.12372
5. Isao Sauzedde, 'Winding and intersection of Brownian motions', arXiv:2112.01645 (2021). arXiv:2112.01645, https://arxiv.org/abs/2112.01645
6. Isao Sauzedde, 'Brownian windings, stochastic Green's formula and inhomogeneous magnetic impurities', Electronic Journal of Probability (2026); arXiv:2301.00551. DOI:10.1214/26-ejp1584, https://arxiv.org/abs/2301.00551
7. Isao Sauzedde, 'Renormalised Amperean area of Brownian motions and Symanzik representation of the 2D Abelian Yang-Mills-Higgs field', arXiv:2412.16781 (2024). arXiv:2412.16781, https://arxiv.org/abs/2412.16781