  We consider a combined state and drift estimation problem for the linear
stochastic heat equation. The infinite-dimensional Bayesian inference problem
is formulated in terms of the Kalman-Bucy filter over an extended state space,
and its long-time asymptotic properties are studied. Asymptotic posterior
contraction rates in the unknown drift function are the main contribution of
this paper. Such rates have been studied before for stationary non-parametric
Bayesian inverse problems, and here we demonstrate the consistency of our
time-dependent formulation with these previous results building upon scale
separation and a slow manifold approximation.
