A Gaussian process gives a probability distribution over functions that pass through the data points. Is there a way to parameterize the Gaussian process to give a probability distribution over closed curves instead of functions? Can the same approach be generalized to give a distribution over surfaces without boundary?
If you take 2 Gaussian processes, $X(t)$ and $Y(t)$ on the unit interval, and with the property that $X(0)=X(1)$ almost surely (similarly for $Y$), then $(X(t),Y(t))$ will be a random curve in $R^2$.
Is that what you are looking for?