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.