  This paper addresses the detection of a stochastic process in noise from
irregular samples. We consider two hypotheses. The \emph{noise only} hypothesis
amounts to model the observations as a sample of a i.i.d. Gaussian random
variables (noise only). The \emph{signal plus noise} hypothesis models the
observations as the samples of a continuous time stationary Gaussian process
(the signal) taken at known but random time-instants corrupted with an additive
noise. Two binary tests are considered, depending on which assumptions is
retained as the null hypothesis. Assuming that the signal is a linear
combination of the solution of a multidimensional stochastic differential
equation (SDE), it is shown that the minimum Type II error probability
decreases exponentially in the number of samples when the False Alarm
probability is fixed. This behavior is described by \emph{error exponents} that
are completely characterized. It turns out that they are related with the
asymptotic behavior of the Kalman Filter in random stationary environment,
which is studied in this paper. Finally, numerical illustrations of our claims
are provided in the context of sensor networks.
