A moving Gaussian field ft, and the Euler-characteristic heuristic for its supremum
A smooth Gaussian field ft over a non-convex Ω, pinned to 0 on ∂Ω (a Brownian pillow): white noise smoothed by the Dirichlet heat flow, flowing in time by its stationary Ornstein–Uhlenbeck dynamics.
the moving field — Dirichlet-heat-smoothed white noise, Ornstein–Uhlenbeck in time
The Euler-characteristic heuristic (Adler–Taylor, eq. 14.0.2)
the excursion set and its Euler characteristic — a random functional of the field ft
its mean approximates the exceedance probability of the maximum: