The Wiener filter for locally stationary stochastic processes is rarely locally stationary

The Wiener filter (i.e., linear minimum mean squared error filter) for wide-sense stationary stochastic processes is translation-invariant, i.e., its impulse response, like the covariance function, is only a function of the time-shift. We investigate whether there is a generalization of this result to continuous-time stochastic processes that are locally stationary in Silverman´s sense: Is the optimal filter for locally stationary processes locally stationary itself? The answer is surprisingly negative: Even though the optimal filter can be locally stationary in special cases, it rarely is, even when the covariance functions have Gaussian shape.