  Gaussian processes provide a compact representation for modeling and
estimating an unknown function, that can be updated as new measurements of the
function are obtained. This paper extends this powerful framework to the case
where the unknown function dynamically changes over time. Specifically, we
assume that the function evolves according to an integro-difference equation
and that the measurements are obtained locally in a spatial sense. In this
setting, we will provide the expressions for the conditional mean and
covariance of the process given the measurements, which results in a
generalized estimation framework, for which we coined the term Dynamic Gaussian
Process (DGP) estimation. This new framework generalizes both Gaussian process
regression and Kalman filtering. For a broad class of kernels, described by a
set of basis functions, fast implementations are provided. We illustrate the
results on a numerical example, demonstrating that the method can accurately
estimate an evolving continuous function, even in the presence of noisy
measurements and disturbances.
