  The link between Gaussian random fields and Markov random fields is well
established based on a stochastic partial differential equation in Euclidean
spaces, where the Mat\'ern covariance functions are essential. However, the
Mat\'ern covariance functions are not always positive definite on circles and
spheres. In this manuscript, we focus on the extension of this link to circles,
and show that the link between Gaussian random fields and Markov random fields
on circles is valid based on the circular Mat\'ern covariance function instead.
First, we show that this circular Mat\'ern function is the covariance of the
stationary solution to the stochastic differential equation on the circle with
a formally defined white noise space measure. Then, for the corresponding
conditional autoregressive model, we derive a closed form formula for its
covariance function. Together with a closed form formula for the circular
Mat\'ern covariance function, the link between these two random fields can be
established explicitly. Additionally, it is known that the estimator of the
mean is not consistent on circles, we provide an equivalent Gaussian measure
explanation for this non-ergodicity issue.
