Sojourn of a bivariate Gaussian field above a high level

A centered, unit-variance stationary Gaussian field X(t) = (X₁(t), X₂(t)) on the unit square [0,1]². Each component has covariance r(h) = exp(−(h₁²+h₂²)/(2ℓ²)), and Cov(X₁(s),X₂(t)) = ρ r(t−s). The sojourn set is where both coordinates exceed the level u; its area is L_u = ∫_[0,1]² 1{X₁(t) > u, X₂(t) > u} dt. Tail probabilities of rescaled sojourn areas are studied in Dębicki, Hashorva & Michna (arXiv:2510.06930). Here the threshold vector is b = (1,1). Raise u to see the excursion set of this one realisation shrink.

X₁(t)

X₂(t)

sojourn set {X₁ > u, X₂ > u}

Level u = 1.00 Cross-correlation ρ = 0.5 Correlation length ℓ = 0.12

Grid estimate of sojourn area

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The 120×120 grid is sampled from the Gaussian covariance by numerical factorisation. We use X₁ = A and X₂ = ρA + √(1−ρ²)B with independent copies A and B; changing ρ keeps these copies fixed. The displayed area is a grid approximation and can miss small excursions. The bottom strip follows the area of this one realisation as u increases. The paper studies probabilities across random realisations, P{θ(u)Lᵤ > x}, as u → ∞; this curve is not such a probability.