I understand that a stochastic process or function is considered a Gaussian process if sampling from it at any point some set of times yields a set of observations that match a Gaussian random variable, and an other way to view this is that it can be described by a mean function with some kind of covariance function specifying the variance in each dimension between pairs of inputs. But I lack an intuition of what this really means or looks like.
Does saying something is a Gaussian process imply that it is a smooth function? Is there a good way to visualize making predictions with kriging in low-dimension, say a 3D plot with two dimensions the inputs and the third the model's guesses?
Edit: This part of this video is really helpful for the visualization question.