Bounded and Unbounded Level Lines in Two-Dimensional Random Fields

Abstract

Every two-dimensional incompressible flow follows the level lines of some scalar function ψ on ∝²; transport properties of the flow depend on the qualitative structure e.g. boundedness—of these level lines. We discuss some recent results on the boundedness of level lines when ψ is a stationary random field. Under mild hypotheses there is only one possible alternative to bounded level lines: the “treelike” random fields, which, for some interval of values of a, have a unique unbounded level line at each level a, with this line “winding through every region of the plane.” If the random field has the FKG property then only bounded level lines are possible. For stationary C ² Gaussian random fields with covariance function decaying to 0 at infinity, the treelike property is the only alternative to bounded level lines provided the density of the absolutely continuous part of the spectral measure decays at infinity “slower than exponentially,” and only bounded level lines are possible if the covariance function is nonnegative.