Reducibility and unobservability of Markov processes: the linear system case

A vector Markov process will be called stochastically unobservable by the measurement process if there exists an initial distribution such that some marginal conditional distributions equal the corresponding unconditional ones. It will be called reducible if there exists an invertible transformation such that the transformed process is stochastically unobservable. Necessary and sufficient conditions are derived in the context of linear diffusions. It is also shown that reducibility can be regarded as a natural extension of the concept of estimability, defined for linear stochastic systems.<>