Gaussian Process for Closed Curves

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 $R^2$.
• A simple example is the Brownian bridge with covariance kernel $k(x,y)=min(x,y)-xy$. You can also use Green's Functions to build models with boundary constraints. See page 134 of Rasmussen and Williams "Gaussian Processes for Machine Learning". It's available online. – Placidia Jan 9 '18 at 2:02