Random Gaussian Fields and Systems of Stochastic Partial Differential Equations

In this work we consider certain systems of Stochastic Partial Differential Equations, that allow us to generate multivariate Gaussian random fields (GF), \(x(s) = (x_1(s),x_2(s))\) . We consider a theoretical case were we have observations of a vector field that displays spatial dependency given by a GRF \(x(s)\) which is approximated by a Gaussian Markov random field (GMRF) \(\mathbf {x}\) , applying the Finite Element Method. Considering a hierarchical model for the observations with a latent Gaussian model for x, under a Bayesian framework, we can obtain the posterior distribution of the GMRF \(\mathbf {x}\) . The main goal of this work is to explicitly present the calculations needed to obtain the posterior distributions for the multivariate case, as they are not gathered all together, neither fully detailed, in a single source in the literature. This can prove to be very useful for new future applications of this methodology.