  The analysis of space-time data from complex, real-life phenomena requires
the use of flexible and physically motivated covariance functions. In most
cases, it is not possible to explicitly solve the equations of motion for the
fields or the respective covariance functions. In the statistical literature,
covariance functions are often based on mathematical constructions. We propose
deriving space-time covariance functions by solving "effective equations of
motion", which can be used as statistical representations of systems with
diffusive behavior. In particular, we propose using the linear response theory
to formulate space-time covariance functions based on an equilibrium effective
Hamiltonian. The effective space-time dynamics are then generated by a
stochastic perturbation around the equilibrium point of the classical field
Hamiltonian leading to an associated Langevin equation. We employ a Hamiltonian
which extends the classical Gaussian field theory by including a curvature term
and leads to a diffusive Langevin equation. Finally, we derive new forms of
space-time covariance functions.
