The angle a rare event chooses

Let B be planar Brownian motion and Y(r) = ‖B(r)‖². For 0 < s < t ask for both Y(s) > u and Y(t) > bu at once. Writing the driving Gaussian vector in ℝ⁴ as Z = RU with R² ∼ χ²₄ and U uniform on the sphere, both constraints become a single radial one, R > √u · q(U), where the cost q = max(q₁, q₂) is the larger of the two required radii. The exponential rate is ½ min q²; the prefactor is decided by the local geometry where the minimum is attained. Move b and watch the conditional angular cloud pass through the three mechanisms: a slack constraint leaves a full Gaussian blob, a weakly contacting one asymptotically cuts it in half, and a genuinely switching pair moves the minimiser off and costs a factor u^−1/2. From E. Hashorva, A radial–angular Laplace principle for homogeneous functionals (Theorem C.1).

the conditional angular cloud in the normal chart w = ρ₂(cos ψ, sin ψ), where ρ₂ is the weight on the increment directions and ψ the angle between the two; the orange curve is where the two required radii are equal. At large u, slack gives a full blob away from it; weak contact gives half a blob against it. Switching: the cloud clings to it, long in the quadratic directions and thin across the linear one — that anisotropy is the factor u^−1/2.

the cost q(U)² over the angular chart (ψ, χ), ρ₂ = sin χ; dark is cheap. The minimiser is marked: the edge χ = 0 when b ≤ 1, and one point in this chart when b > 1. In the full sphere each represents a circle of common directions; the edge is a coordinate degeneracy.

the exponential rate I_b = ½ min q² against b, with the five regimes; dots are Monte Carlo values of −log p_b(u)/u at the current u

Monte Carlo p_b(u)

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Asymptotic

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Ratio

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→ 1 for fixed b as u → ∞; the approximation is not uniform near the two transition values.

Threshold ratio b = 1.00 Level u = 60 Second time t = 1.60 (s = 1)

The radial–angular identity p_b(u) = E[ F̄_χ²₄(u q(U)²) ] is exact, and for four degrees of freedom F̄(x) = (1 + x/2)e^−x/2. The displayed Monte Carlo values average this function over 40 000 uniform directions. This avoids counting rare hits, but still has sampling error, especially when the angular mass is concentrated. The effective sample size (ESS) and estimated relative standard error are shown below the estimate. The cloud resamples these finitely many directions with weight F̄(u q(U)²), approximating the exact conditional angular law. Half-clouds and Gaussian shapes describe the large-u limit, not exact finite-u shapes.