Conditioning of Banach Space Valued Gaussian Random Variables: An
  Approximation Approach Based on Martingales
Ingo Steinwart
In this paper we investigate the conditional distributions of two Banach
space valued, jointly Gaussian random variables. These conditional
distributions are again Gaussian and their means and covariances are determined
by a general approximation scheme based upon a martingale idea. We then apply
our general results to the case of Gaussian processes with continuous paths
conditioned to partial observations of their paths.