Geography of the level sets of the Brownian sheet
We describe geometric properties of \(\{W>\alpha\}\), where \(W\) is a standard real-valued Brownian sheet, in the neighborhood of the first hit \(P\) of the level set \(\{W>\alpha\}\) along a straight line or smooth monotone curve \(L\). In such a neighborhood we use a decomposition of the form \(W(s,t)=\alpha-b(s)+B(t)+x(s,t)\), where \(b(s)\) and \(B(t)\) are particular diffusion processes and \(x(s,t)\) is comparatively small, to show that \(P\) is not on the boundary of any connected component of \(\{W>\alpha\}\). Rather, components of this set form clusters near \(P\). An integral test for thorn-shaped neighborhoods of \(L\) with tip at \(P\) that do not meet \(\{W>\alpha\}\) is given. We then analyse the position and size of clusters and individual connected components of \(\{W>\alpha\}\) near such a thorn, giving upper bounds on their height, width and the space between clusters. This provides a local picture of the level set. Our calculations are based on estimates of the length of excursions of \(B\) and \(b\) and an accounting of the error term \(x\).
scientific article; zbMATH DE number 569086
scientific article