  In many applications it is important to be able to sample paths of SDEs
conditional on observations of various kinds. This paper studies SPDEs which
solve such sampling problems. The SPDE may be viewed as an infinite dimensional
analogue of the Langevin SDE used in finite dimensional sampling. Here the
theory is developed for conditioned Gaussian processes for which the resulting
SPDE is linear. Applications include the Kalman-Bucy filter/smoother. A
companion paper studies the nonlinear case, building on the linear analysis
provided here.
