This e-book grew out of a graduate path at ETH Zürich in the course of the spring 2011 time period. It explores a number of hyperlinks among such notions as profession instances of Markov chains, Gaussian loose fields, Poisson aspect tactics of Markovian loops, and random interlacements, that have been the thing of in depth learn over the past few years. those notions are constructed within the handy setup of finite weighted graphs endowed with killing measures. This ebook first discusses parts of continuous-time Markov chains, Dirichlet types, strength concept, including a few outcomes for Gaussian loose fields. subsequent, isomorphism theorems and generalized Ray-Knight theorems, which relate career instances of Markov chains to Gaussian unfastened fields, are offered. Markovian loops are developed and a few in their key homes derived. the sector of profession instances of Poisson aspect strategies of Markovian loops is investigated. Of detailed curiosity are its connection to the Gaussian unfastened box, and a formulation of Symanzik. ultimately, hyperlinks among random interlacements and Markovian loops are mentioned, and a few additional connections with Gaussian unfastened fields are pointed out. A book of the eu Mathematical Society (EMS). allotted in the Americas via the yankee Mathematical Society.
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