| This thesis studies and extends constructions of Gaussian fields from Markov processes introduced by Dynkin and Diaconis-Evans. These constructions provide simple recipes for constructing Gaussian fields on complicated spaces, which can otherwise be a challenging task. The Gaussian fields have several attractive properties, which in particular facilitates their use as priors in prediction and design problems. Dynkin's construction gives rise to Gaussian fields with all non-negative covariances, while Diaconis-Evans' construction gives rise to Gaussian fields with all non-positive covariances. We extend Dynkin's and Diaconis-Evans' constructions to allow general covariance sign patterns, while preserving their useful properties.;We observe that the Gaussian fields constructed above arise in other areas of the literature, and these connections can be put to good use. Using ideas from the proof of Dynkin's results, we prove that the joint Laplace transform of the occupation times of a skip-free Markov process on N ∪ {0} before hitting a state n (starting at 0) has a very simple form. We investigate the properties of this Laplace transform and make connections to permanental vectors and extensions of Dynkin's Gaussian fields which take complex values.;It was observed by Diaconis-Evans that the central limit theorem for Markov chains (by Gordin and Lifsic), and Markov processes (by Bhattacharya) show that the Gaussian fields obtained by Dynkin's constructions can be realized as distributional limits of additive functionals of Markov chains and Markov processes. We extend these central limit theorems (with appropriate modifications) for Markov chains and Markov processes with a sign-structure, and show that the limiting Gaussian fields are the extended Dynkin's Gaussian fields.;A variety of semi-supervised learning algorithms can in fact be understood as estimating a partially observed Gaussian field with covariance structure arising from Dynkin's construction. We provide a conservative modification of these algorithms which provides a way to smooth the estimates and protect against bad choices of the covariance matrix. The proposed algorithm can be understood in terms of reinforced random walks, which in turn helps in the implementation of the algorithm. |
