We construct covariant random vector fields over 4-dimensional space-time as solutions of a system of first order coupled stochastic partial differential equations, best interpreted as equations for quaternionic valued random fields. The fields are covariant under the proper Euclidean transformations. We give necessary and sufficient conditions in terms of a given source of the infinitely divisible type, for the fields to be covariant also under reflections. In the case of a Gaussian white noise source the fields are Euclidean free electromagnetic potential fields and have the global Markov property. The fields with Poisson white noise source can be used as approximation of the Gaussian fields, with better support properties.