Sharp small-time asymptotics for the occupation time of cones by Brownian motion (the Meyre–Werner conjectures (13)–(14))

Let $B$ be a standard $d$-dimensional Brownian motion ($d ≥ 2$) started at the vertex $0$ of a closed cone $C ⊂ ℝ^d$ (a scale-invariant set: $x ∈ C$ implies $tx ∈ C$ for every $t ≥ 0$), and let $A_1 = ∫_0^1 1_C(B_s) ds$ be the occupation time, the time $B$ spends in $C$ up to time $1$. Meyre and Werner (1995) proved two-sided power bounds $k_1 u^{1/ξ} ≤ ℙ(A_1 < u) ≤ k_2 u^{1/ξ}$ for closed convex cones, where the cone exponent $ξ = ξ(C)$ is determined by the principal Dirichlet eigenvalue of $−Δ/2$ on the spherical cross-section of the complement of $C$, and they conjectured the sharp one-term form $ℙ(A_1 < u) ∼ k(C) u^{1/ξ}$ as $u → 0$ with a constant $k(C) ∈ (0,∞)$, together with the density asymptotics $g_C(x) ∼ (k(C)/ξ) x^{1/ξ − 1}$ as $x → 0$ (and the matching statement at $1$, governed by the exponent of the complement). These conjectures remain open — even for the planar quadrant — as do the endpoint-singularity thresholds they imply (the 1988 Bingham–Doney questions: in the plane, the occupation-time density of a sector of angle $ω$ is unbounded at both endpoints exactly when $π/2 < ω < 3π/2$; for $d ≥ 3$, unbounded at $1$ exactly when $ξ(C^c) > 1$) and the extension beyond convex cones, where only the logarithmic form $lim_{u→0} log ℙ(A_1 < u)/log u = 1/ξ$ (Bass–Burdzy, 1996) is known. The problem is to prove or refute these conjectures and to settle the geometric scope of their validity.

Background

A standard $d$-dimensional Brownian motion $B = (B_s)_{s ≥ 0}$ is the $ℝ^d$-valued stochastic process with continuous paths and stationary independent Gaussian increments, started at the origin. A closed cone with vertex $0$ is a closed set $C ⊂ ℝ^d$ with $x ∈ C ⇒ tx ∈ C$ for all $t ≥ 0$; basic examples are half-spaces through the origin, planar sectors of opening angle $ω$ (wedges), and the positive orthant. The occupation time of $C$ is $A_t = ∫_0^t 1_C(B_s) ds ∈ [0,t]$. Brownian scaling together with the scale-invariance of cones gives that $A_t$ has the same distribution as $tA_1$, so the law of $A_1$ on $[0,1]$ encodes all times, and the behavior of $ℙ(A_1 < u)$ as $u → 0$ is the natural local question about this law. Throughout, $B$ starts at the vertex $0 ∈ ∂C$ — the regime in which the problem is posed.

In dimension one, with $C = [0,∞)$, the classical arcsine law of Lévy identifies the law of $A_1$: it has density $z ↦ 1/(π√{z(1−z)})$ on $(0,1)$, so $ℙ(A_1 < u) ∼ (2/π)√u$ and the density is unbounded at both endpoints. Bingham and Doney (1988) posed the higher-dimensional analogues: for $B$ started at the vertex of a cone $C$ in $ℝ^d$, $d ≥ 2$, what is the law of $A_1$, and when does that law — or its density, when it exists — blow up at the endpoints $0$ and $1$? They computed the first moments for the planar quadrant (their third moment was subsequently corrected by Desbois, and a formal correction note by Zhou appeared in 2008); no explicit formula for the law is known in any genuinely multidimensional case, and Ernst and Shepp (2017) give evidence that transform methods cannot produce one.

The power in the small-$u$ tail is the cone exponent. Let $S^{d−1}$ be the unit sphere, $C^c$ the complement of $C$, and $λ > 0$ the principal (smallest) Dirichlet eigenvalue of $−Δ/2$ on the spherical domain $S^{d−1} ∩ C^c$ (the smallest $λ$ for which $−Δφ = 2λφ$ holds in the domain with $φ = 0$ on its boundary). With $ν = d/2 − 1$ (this parameter is written $u$ in Meyre–Werner; it is renamed here so that $u$ stays free for the time variable), set

$$ξ(C) = (√{ν^2 + 2λ} + ν) / λ,$$

equivalently, $ξ$ is the positive root of $λξ^2 − 2νξ − 2 = 0$; it depends only on the connected component of $C^c$ carrying the smallest eigenvalue. For a planar cone of magnitude (opening angle) $ω$ this gives $ξ = 2(2π − ω)/π$: a half-plane has $ξ = 2$, recovering the arcsine exponent $1/2$ (a half-space reduces to its one-dimensional projection), and the quadrant has $ξ = 3$, hence exponent $1/3$. In $d = 2$ one always has $ξ ≤ 4$, with $ξ < 2$ possible for reflexive cones ($ω > π$); for $d ≥ 3$, $ξ$ is unbounded as $C$ shrinks toward a ray, because single points are polar — never hit by the spherical process — on $S^{d−1}$ when $d ≥ 3$. The same spectral data governs exit times from cones (DeBlassie 1987) and the almost-sure lower limits of $A_t/t$ (Mountford 1990 in the plane; Meyre 1991 in general dimension; Hobson 1994 settles the half-space case by one-dimensional excursion theory, the Itô decomposition of a real Brownian path into its excursions away from $0$).

Meyre and Werner (1995) proved: if $C$ is a closed convex cone, then $k_1 u^{1/ξ} ≤ ℙ(A_1 < u) ≤ k_2 u^{1/ξ}$ for all $u ∈ (0,1]$ with constants $k_1, k_2 ∈ (0,∞)$ depending on $C$ — for the quadrant, $ℙ(A_1 < u)$ is caught between constant multiples of $u^{1/3}$. The lower bound needs only that $C$ is closed; the upper bound uses convexity through the invariance $x + C ⊂ C$ for $x ∈ C$ and a dyadic induction. Bass and Burdzy (1996), via eigenvalue expansions for Brownian motion killed on leaving domains, proved for every closed cone, with no convexity, the logarithmic form $lim_{u→0} log ℙ(A_1 < u)/log u = 1/ξ$: this pins the exponent but gives no two-sided bounds and no constant. Nakayama (1997) established analogues of the two-sided estimates in general dimension. Moments are known in special cases — Bingham and Doney (quadrant), Desbois (2007, wedge moments and scaling analysis), Grünbaum and McGrouther (2007, study of the wedge occupation-time distribution), Zhou (2009) — and Aurzada, Döring and Pitters (2024; published 2026) obtained a random-sampling representation equating the moments of cone occupation times with persistence probabilities (the probabilities that a process stays in a prescribed set up to a given time) of random walks in cones, while stating explicitly that characterizing the distribution of occupation times of planar and higher-dimensional Brownian motion in cones is still open.

The gap between the exponent and the law is the problem. The bounds leave open whether $lim_{u→0} ℙ(A_1 < u)/u^{1/ξ}$ exists in $(0,∞)$; no density asymptotics are known in any genuinely multidimensional case; the Bingham–Doney threshold questions have only answers conditional on the conjectured density behavior; and beyond convex cones not even two-sided power bounds are available. In their closing remarks, Meyre and Werner conjectured the sharp tail and density asymptotics — items (a) and (b) below, labeled (13) and (14) in the paper — and noted that their proof 'can not be generalized to general cones'; the companion paper on the density that they announced never appeared. Desbois (2007) wrote that 'new ideas are needed' for the occupation-time problem away from one dimension, and Ernst and Shepp (2017) concluded that an explicit law is probably out of reach, which makes the sharp asymptotic the natural target.

Problem Statement

Let $d ≥ 2$, let $C ⊂ ℝ^d$ be a closed cone with vertex $0$ such that $C ≠ ℝ^d$ and $C$ has nonempty interior, let $B$ be a standard $d$-dimensional Brownian motion with $B_0 = 0$, and let $A_1 = ∫_0^1 1_C(B_s) ds$. Let $ξ = ξ(C)$ be the cone exponent defined in the background — the positive root of $λξ^2 − 2νξ − 2 = 0$, where $ν = d/2 − 1$ and $λ$ is the principal Dirichlet eigenvalue of $−Δ/2$ on $S^{d−1} ∩ C^c$ — and let $C^c$ denote the complement of $C$ (itself a cone, generally open and possibly disconnected, with $ξ(C^c)$ defined by the same recipe; for a planar cone of magnitude $ω$, $ξ(C^c) = 2ω/π$). The common objective is to determine the sharp small-time behavior of the law of the occupation time $A_1$ and of its density: the power $1/ξ$ is already pinned by the known bounds and by the Bass–Burdzy logarithmic limit, and what is at stake is the one-term asymptotics, the constants, and the geometric scope. Prove or disprove:

(a) (Meyre–Werner conjecture (13).) For every closed convex cone $C$, there exists a constant $k(C) ∈ (0,∞)$ such that

$$ℙ(A_1 < u) ∼ k(C) u^{1/ξ} (u → 0+).$$

(b) (Meyre–Werner conjecture (14).) The law of $A_1$ admits a density $g_C$ on $(0,1)$, and

$$g_C(x) ∼ (k(C)/ξ) x^{1/ξ − 1} (x → 0+),$$

with the corresponding statement at the other endpoint: $g_C(x) ∼ c_+(C) (1 − x)^{1/ξ(C^c) − 1}$ as $x → 1−$ for some constant $c_+(C) ∈ (0,∞)$, expected to equal $k(C^c)/ξ(C^c)$ whenever the complement $C^c$ lies in the class settled in (a) (in particular for planar cones, where $C^c$ is again a cone).

(c) (Bingham–Doney endpoint thresholds, as consequences of (b).) In the plane, for the sector of angle $ω$, $0 < ω < 2π$: the density $g_C$ is unbounded at $0$ and at $1$ if and only if $π/2 < ω < 3π/2$, so the infimum of angles with both endpoints singular is $π/2$ — the answer predicted by Meyre and Werner for the 1988 question of Bingham and Doney, who asked for that infimum; the set of such angles is open at its lower edge (at $ω = π/2$ the density is unbounded at $0$ but bounded at $1$). For $d ≥ 3$: the density of $A_1$ is unbounded at $1$ if and only if $ξ(C^c) > 1$.

(d) (Scope beyond convexity.) Determine the class of closed cones for which the two-sided power bounds of Meyre and Werner — and hence the asymptotics (a)–(b) — hold; in particular, whether convexity can be weakened. Note that for arbitrary closed cones the logarithmic form is known (Bass–Burdzy), but no two-sided power bounds are.

The conjectures are refuted by a cone within the claimed scope — convex, unless the question is directed only at part (d) — for which the limit in (a) fails to exist in $(0,∞)$, or for which the density asymptotics in (b) fails. They are posed for the full class of closed convex cones, and the half-space ($ξ = 2$, $k = 2/π$ by the arcsine law) is the trivially one-dimensional boundary of the problem.

References

1. T. Meyre and W. Werner, On the occupation times of cones by Brownian motion, Probability Theory and Related Fields 101 (1995) 409–419. DOI 10.1007/BF01200504, https://doi.org/10.1007/BF01200504. (Primary source; Section 6 states conjectures (13)–(14).)
2. N. H. Bingham and R. A. Doney, On higher-dimensional analogues of the arc-sine law, Journal of Applied Probability 25 (1988) 120–131. DOI 10.2307/3214239, https://doi.org/10.2307/3214239. (Origin of the higher-dimensional occupation-time questions and of the endpoint-threshold problem.)
3. R. D. DeBlassie, Exit times from cones in R^n of Brownian motion, Probability Theory and Related Fields 74 (1987) 1–29. DOI 10.1007/BF01845637, https://doi.org/10.1007/BF01845637. (Exit-time asymptotics with the same spectral exponent.)
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11. P. A. Ernst and L. A. Shepp, On occupation times of the first and third quadrants for planar Brownian motion, Journal of Applied Probability 54 (2017), no. 1, 337–342. DOI 10.1017/jpr.2016.104, https://doi.org/10.1017/jpr.2016.104; arXiv identifier 1602.07605, https://arxiv.org/abs/1602.07605. (Attempt on the non-convex hourglass — the union of the first and third quadrants — via a Feynman–Kac/Laplace-transform reduction and Kontorovich–Lebedev-type representations, intractable beyond its Equation (23).)
12. C. Profeta, A remark on positive sojourn times of symmetric processes, Journal of Applied Probability 55 (2018), no. 1, 69–81. DOI 10.1017/jpr.2018.6, https://doi.org/10.1017/jpr.2018.6. (Survey of the cone occupation-time problem and arcsine-type results for symmetric, one-dimensionally reducible settings; open-access full text at HAL, https://hal.science/hal-04346499.)
13. F. Aurzada, L. Döring and H. H. Pitters, Occupation times and areas derived from random sampling, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 62 (2026), no. 3, 1622–1638. DOI 10.1214/25-aihp1561, https://doi.org/10.1214/25-aihp1561; arXiv identifier 2406.09886, https://arxiv.org/abs/2406.09886. (Random-sampling representation of occupation-time moments — their Remark 1: the m-th moment of the cone occupation time equals the probability that a d-dimensional random walk stays in the cone up to time m — and the explicit statement that the cone occupation-time distribution problem is open.)
14. Y. Zhou, On the occupation time in a cone for two-dimensional Brownian motion, Journal of Fudan University (Natural Science) 48 (2009), no. 6, 741–746. zbMATH 1224.60212, https://zbmath.org/1224.60212. (Planar cone occupation times: first and second moments in the positive quadrant, shown not to follow a generalized arc-sine law.)
15. Y. Zhou, Correction to Bingham and Doney (1988), Journal of Applied Probability 45 (2008), no. 4, 1204–1206. DOI 10.1239/jap/1231340244, https://doi.org/10.1239/jap/1231340244. (Formal correction note to Bingham–Doney (1988).)