Covariance kernel representations of multidimensional second-order stochastic processes

The dynamics of stationary stochastic processes in space is not exactly analogous to that of stationary stochastic processes in the time domain. This is due to the unilateral nature of the time series that is only influenced by past values as opposed to the dependence in all directions of the spatial process. In this work, we unfold the connection that exits between the covariance kernel of a multi-dimensional second-order autoregressive random process and its underlying discrete random dynamical system. Starting from a discrete random dynamical system, we show that the random process satisfying that system is governed by the modified Helmholtz equation in the continuous limit. We establish the dependence of the correlation constant on the grid size of the discretization. We also show that the random forcing term in the continuous case turns out to be a white noise process. A number of covariance functions are worked out for simple and more complex geometrical domains with various boundary conditions in multi-dimensions. We use both the discrete and the continuous systems in our computations.