  It is often said that control and estimation problems are in duality.
Recently, in (Aubin-Frankowski,2021), we found new reproducing kernels in
Linear-Quadratic optimal control by focusing on the Hilbert space of controlled
trajectories, allowing for a convenient handling of state constraints and
meeting points. We now extend this viewpoint to estimation problems where it is
known that kernels are the covariances of stochastic processes. Here, the
Markovian Gaussian processes stem from the linear stochastic differential
equations describing the continuous-time dynamics and observations. Taking
extensive care to require minimal invertibility requirements on the operators,
we give novel explicit formulas for these covariances. We also determine their
reproducing kernel Hilbert spaces, stressing the symmetries between a space of
forward-time trajectories and a space of backward-time information vectors. The
two spaces play an analogue role for filtering to Sobolev spaces in variational
analysis, and allow to recover the Kalman estimate through a direct variational
argument. For comparison, we then recover the Kalman filter and smoother
formulas through more classical arguments based on the innovation process.
Extension to discrete-time observations or infinite-dimensional state, tough
technical, would be straightforward.
