Persistence of a Brown–Resnick process

A stationary Brown–Resnick process is the upper envelope X(t) = sup_i P_i Z⁽ⁱ⁾(t) of countably many spectral curves, each a log-normal Z(t) = exp(√2 B_H(t) − t^2H) scaled by a point P_i of a Poisson process. Watch the construction assemble on the left. Two very different questions about a level u: how many curves exceed it somewhere on [0,T] — Poisson with mean ℋ_V(T)/u, the Pickands functional, growing linearly in T — and how many stay above it throughout — Poisson with mean 𝒢_V(T)/u, the expected infimum E inf_[0,T] Z, which decays like 2Ψ(√γ_V(T)) for H > ½ and with an extra factor T⁻¹ for Brownian motion. Several curves can together keep X above u without any one of them doing so — the picture makes that plain. From Dębicki, Hashorva & Novikov, Expected infimum and persistence probabilities of log-normal stationary Brown–Resnick processes.

log-scale: spectral curves P_iZ⁽ⁱ⁾(t) (dim), the envelope of the shown curves (dark), level u (dashed); shown curves above u at every grid point in orange, curves exceeding u at a grid point marked blue at their maximum; crossings between grid points can be missed

Grid estimates of 𝒢_V(T) = E inf_[0,T] Z (dots, Monte Carlo) against T, with the endpoint bound 2Ψ(√γ_V(T)) (dashed); and ℋ_V(T) = E sup_[0,T] Z (blue) for contrast

Shown curves above u somewhere

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Shown curves above u throughout

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Shown envelope above u on grid?

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grid minimum of the partial envelope > u

Hurst index H = 0.75 Interval length T = 2.0 Level u (log₁₀) = 0.0

Left: the first 80 Poisson points P_i = 1/Γ_i; omitted curves can change both counts and the envelope. The Poisson laws above describe the complete infinite system. All sampled intervals include 0 and T on a grid of spacing 1/32. Right: 20 000 fBm paths; the grid biases 𝒢 upwards and ℋ downwards, with additional Monte Carlo error (especially large for ℋ at high H and T). These plots do not certify continuous-time persistence or asymptotic convergence. At H = 1 we use exactly B₁(t) = tN. Here γ_V(T) = T^2H/2.