This publication grew out of a graduate direction at ETH Zürich through the spring 2011 time period. It explores a number of hyperlinks among such notions as profession occasions of Markov chains, Gaussian loose fields, Poisson element procedures of Markovian loops, and random interlacements, which were the item of in depth learn during the last few years. those notions are constructed within the handy setup of finite weighted graphs endowed with killing measures. This publication first discusses components of continuous-time Markov chains, Dirichlet types, capability thought, including a few results for Gaussian unfastened fields. subsequent, isomorphism theorems and generalized Ray-Knight theorems, which relate profession occasions of Markov chains to Gaussian unfastened fields, are provided. Markovian loops are developed and a few in their key houses derived. the sector of career instances of Poisson element approaches of Markovian loops is investigated. Of unique curiosity are its connection to the Gaussian loose box, and a formulation of Symanzik. ultimately, hyperlinks among random interlacements and Markovian loops are mentioned, and a few extra connections with Gaussian unfastened fields are pointed out. A booklet of the eu Mathematical Society (EMS). disbursed in the Americas by means of the yank Mathematical Society.
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