Stationary Random Fields Arising From Second-Order Partial Differential Equations on Compact Lie Groups

Abstract

Wide sense stationary processes are a mainstay of classical signal processing. It is well known that they can be obtained by solving ordinary differential equations with constant coefficients whose right-hand side is a white noise. This paper addresses the extension of this construction to random fields defined on compact Lie groups. On an underlying compact Lie group, the paper studies left invariant second-order elliptic partial differential equations whose right-hand side is a spatial white noise. Quite often, the solution of a partial differential equation is not defined as a function but as a distribution. To adapt to this situation, the paper introduces a definition of wide sense stationary distributions on a compact Lie group. This is shown to be consistent with the more restricted definition of wide sense stationary fields given in a classic paper by Yaglom. It is proved that the solution of a partial differential equation, of the kind being studied, is a wide sense stationary distribution whose covariance structure is determined by the fundamental solution of the equation. As a concrete example, this paper describes the fundamental solution of the Helmholtz equation on the rotation group and the resulting covariance structure.