Open Access
October 1996 Boundedness of level lines for two-dimensional random fields
Kenneth S. Alexander
Ann. Probab. 24(4): 1653-1674 (October 1996). DOI: 10.1214/aop/1041903201


Every two-dimensional incompressible flow follows the level lines of some scalar function $\psi$ on $\mathbb{R}^2$; transport properties of the flow depend in part on whether all level lines are bounded. We study the structure of the level lines when $\psi$ is a stationary random field. We show that 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 $\infty$, 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 $\infty$ "slower than exponentially," and only bounded level lines are possible if the covariance function is nonnegative.


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Kenneth S. Alexander. "Boundedness of level lines for two-dimensional random fields." Ann. Probab. 24 (4) 1653 - 1674, October 1996.


Published: October 1996
First available in Project Euclid: 6 January 2003

zbMATH: 0866.60084
MathSciNet: MR1415224
Digital Object Identifier: 10.1214/aop/1041903201

Primary: 60K35
Secondary: 82B43 , 82C70

Keywords: incompressible flow , Lagrangian trajectory , level line , Minimal spanning tree , percolation , Random field , statistical topography

Rights: Copyright © 1996 Institute of Mathematical Statistics

Vol.24 • No. 4 • October 1996
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