Journal of Applied Mathematics and Stochastic Analysis

Volume 16 (2003), Issue 3, Pages 209-231

doi:10.1155/S1048953303000169

## Mean anisotropy of homogeneous Gaussian random fields and anisotropic norms of linear translation-invariant operators on multidimensional integer lattices

# Phil Diamond,^{1} Peter Kloeden,^{2} and Igor Vladimirov^{3}

^{1}Department of Mathematics, University of Queensland, Brisbane QLD 4072, Australia^{2}Department of Mathematics, Johann Wolfgang Goethe University, Frankfurt D–60054, Germany^{3}Department of Mathematics, University of Queensland, Brisbane QLD 4072, Australia

Received 1 December 2002; Revised 1 August 2003

Copyright © 2003 Phil Diamond et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Sensitivity of output of a linear operator to its input can be quantified in various ways. In Control Theory, the input is usually interpreted as disturbance and the output is to be minimized in some sense. In stochastic worst-case design settings, the disturbance is considered random with imprecisely known probability distribution. The prior set of probability measures can be chosen so as to quantify how far the disturbance deviates from the white-noise hypothesis of Linear Quadratic Gaussian control. Such deviation can be measured by the minimal Kullback-Leibler informational divergence from the Gaussian distributions with zero mean and *scalar* covariance matrices. The resulting *anisotropy* functional is defined for *finite power* random vectors. Originally, anisotropy was introduced for *directionally generic* random vectors as the relative entropy of the normalized vector with respect to the uniform distribution on the unit sphere. The associated *a-anisotropic* norm of a matrix is then its maximum *root mean square* or *average energy* gain with respect to finite power or directionally generic inputs whose anisotropy is bounded above by *mean anisotropy*. Correspondingly, the anisotropic norms of finite matrices are extended to bounded linear translation invariant operators over such fields.