Boundedness of level lines for two-dimensional random fields
Boundedness of level lines for two-dimensional random fields
复制标题
二维随机场的水平线有界性
DOI：
10.1214/aop/1041903201
复制
发表时间：
1996
影响因子：
2.3
通讯作者：
K. S. Alexander
中科院分区：
文献类型：
--
作者：
K. S. Alexander
每个二维不可压缩流都遵循R2上某个标量函数ψ的水平线；流的输运性质部分取决于所有水平线是否有界。研究了ψ为平稳随机场时的能级线结构。我们证明了在温和的假设下，除了有界水平线之外，只有一种可能的选择：树状随机场，对于a的某个区间，它在每个水平a有唯一的无界水平线，这条线缠绕在平面的每个区域。如果随机场具有FKG属性，则只有有界水平线是可能的。对于协方差函数在∞处衰减为0的平稳C2高斯随机场，如果谱的绝对连续部分的密度在∞处的衰减速度慢于指数衰减，则树状性质是有界能级线唯一的替代选择，如果协方差函数为非负，则只有有界能级线是可能的。
Every two-dimensional incompressible flow follows the level lines of some scalar function ψ on 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 ψ 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 ∞, 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 ∞ slower than exponentially, and only bounded level lines are possible if the covariance function is nonnegative.