  Computer simulations of differential equations require a time discretization,
which inhibits to identify the exact solution with certainty. Probabilistic
simulations take this into account via uncertainty quantification. The
construction of a probabilistic simulation scheme can be regarded as Bayesian
filtering by means of probabilistic numerics. Gaussian prior based filters,
specifically Gauss-Markov priors, have successfully been applied to simulation
of ordinary differential equations (ODEs) and give rise to filtering problems
that can be solved efficiently. This work extends this approach to partial
differential equations (PDEs) subject to periodic boundary conditions and
utilizes continuous Gaussian processes in space and time to arrive at a
Bayesian filtering problem structurally similar to the ODE setting. The usage
of a process that is Markov in time and statistically homogeneous in space
leads to a probabilistic spectral simulation method that allows for an
efficient realization. Furthermore, the Bayesian perspective allows the
incorporation of methods developed within the context of information field
theory such as the estimation of the power spectrum associated with the prior
distribution, to be jointly estimated along with the solution of the PDE.
