Wide Sense Stationary Solutions of Linear Systems with Additive Noise

Let $z$ be a (wide sense) stationary process with spectral measure $F^z $, acting as noise on \[( * )\qquad \dot x = Ax + Bz,\quad y = Gx.\] The aim of this paper is to describe the set of stationary outputs $y$. A stationary solution $x$ is called hard if it satisfies (*) as a stochastic process, its spectral measure $F^x $ then satisfies \[( * * )\qquad (i\lambda - A)dF^x (\lambda )(i\lambda - A)^ * = dF^z (\lambda ).\] Any solution of (**) is called a soft stationary solution of (*). If $i\lambda \in \sigma (A)$, $\lambda $ is called critical. For $B = G = I$, there exists a soft stationary solution if and only if there exists a hard stationary solution on the original probability space if and only if $(i\lambda - A)^{ - 1} \in L_2 (F^z )$ and ${\operatorname{image}}\Delta F^2 (\lambda ) \subset {\operatorname{image}}(i\lambda - A)$ for critical $\lambda $. The intuitive meaning of this condition is that the critical frequencies of the undisturbed system have to be missing in the noise spectrum in a certain sense. Otherwise we encounter resonance. The soft stationary solution is unique if and only if the hard stationary solution is unique on any probability space if and only if all $\operatorname{Re} \lambda _j (A) \ne 0$. The set of all stationary solutions is described. The results carry over to the observable case for general $C$.