Climbing down Gaussian peaks (Q2412668)

Let \(X(t)\) be a continuous Gaussian field defined on a compact set \(T\subset\mathbb{R}^d\) which is homotopic to a ball. For \(u\in\mathbb{R}\) and \(r\in(0,1]\), consider the probability that there exists a ball \(B\) entirely in \(T\) such that \(X(t)>u\) for all \(t\) on the boundary of \(B\) and \(X(s)<ru\) for some \(s\in B\) (alternatively, for \(s\) being the centre of \(B\)). Such probabilities are special cases of a more general scheme that amounts to the existence of compact sets \(K_1,K_2\subset T\) such that \(X(t)>u\) for all \(t\in K_1\) and \(X(t)<ru\) for each \(t\in K_2\). Using the large deviation approach, the authors obtain results concerning the logarithmic behaviour of such probabilities as \(u\to\infty\).