SPDE bridges with observation noise and their spatial approximation

This paper introduces SPDE bridges with observation noise and contains an analysis of their spatially semidiscrete approximations. The SPDEs are considered in the form mild solutions abstract Hilbert space framework suitable for parabolic equations. They assumed to be linear additive a cylindrical Wiener process. observational is also formulated via conditional distributions Gaussian random variables spaces. A general spatial discretization these bridge processes introduced. Explicit convergence rates derived spectral finite element based method. It shown that sufficiently rough noise, essentially same as those corresponding original SPDE.