## The Annals of Probability

### Boundedness of level lines for two-dimensional random fields

Kenneth S. Alexander

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 24, Number 4 (1996), 1653-1674.

Dates
First available in Project Euclid: 6 January 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1041903201

Digital Object Identifier
doi:10.1214/aop/1041903201

Mathematical Reviews number (MathSciNet)
MR1415224

Zentralblatt MATH identifier
0866.60084

#### Citation

Alexander, Kenneth S. Boundedness of level lines for two-dimensional random fields. Ann. Probab. 24 (1996), no. 4, 1653--1674. doi:10.1214/aop/1041903201. https://projecteuclid.org/euclid.aop/1041903201

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• LOS ANGELES, CALIFORNIA 90089-1113 E-MAIL: alexandr@math.usc.edu