Constructing continuous stationary covariances as limits of the second-order stochastic difference equations

In Bayesian statistical inverse problems the a priori probability distributions are often given as stochastic difference equations. We derive a certain class of stochastic partial difference equations by starting from second-order stochastic partial differential equations in one and two dimensions. We discuss discretisation schemes on uniform lattices of these stationary continuous-time stochastic processes and convergence of the discrete-time processes to the continuous-time processes. A special emphasis is given to an analytical calculation of the covariance kernels of the processes. We find a representation for the covariance kernels in a simple parametric form with controllable parameters: correlation length and variance. In the discrete-time processes the discretisation step is also given as a parameter. Therefore, the discrete-time covariances can be considered as discretisation-invariant. In the two-dimensional cases we find rotation-invariant and anisotropic representations of the difference equations and the corresponding continuous-time covariance kernels.