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The link between Gaussian random fields and Markov random fields is wellestablished based on a stochastic partial differential equation in Euclideanspaces, where the Mat\'ern covariance functions are essential. However, theMat\'ern covariance functions are not always positive definite on circles andspheres. 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 fieldson 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 thestationary solution to the stochastic differential equation on the circle witha formally defined white noise space measure. Then, for the correspondingconditional autoregressive model, we derive a closed form formula for itscovariance function. Together with a closed form formula for the circularMat\'ern covariance function, the link between these two random fields can beestablished explicitly. Additionally, it is known that the estimator of themean is not consistent on circles, we provide an equivalent Gaussian measureexplanation for this non-ergodicity issue.

The Circular Matern Covariance Function and its Link to Markov Random Fields on the Circle