In this book, potential theory is presented in an inclusive and accessible manner, with the emphasis reaching from classical to modern, from analytic to probabilistic, and from Newtonian to abstract or axiomatic potential theory (including Dirichlet spaces). The reader is guided through stochastic analysis featuring Brownian motion in its early chapters to potential theory in its latter sections. This path covers the following themes: martingales, diffusion processes, semigroups and potential operators, analysis of super harmonic functions, Dirichlet problems, balayage, boundaries, and Green functions. The wide range of applications encompasses random walk models, especially reversible Markov processes, and statistical inference in machine learning models. However, the present volume considers the analysis from the point of view of function space theory, using Dirchlet energy as an inner product. This present volume is an expanded and revised version of an original set of lectures in the Aarhus University Mathematics Institute Lecture Note Series. Key Features: oInterdisciplinary, including classical and modern, and the key tools from analysis as well as probability oIt is accessible, suitable for both beginning and more advanced courses oStudent friendly oIncludes many exercises, with hints oPoints readers to new directions, and to new and related areas oAddresses core questions in classical and modern potential theory oPuts in perspective tools from probability (of Markov transitions) and from stochastic analysis. The tools in the book from stochastic analysis have proved to be of independent interest in both pure and applied mathematics, including in harmonic analysis, in dynamical systems and diffusion theory, in PDE, and in financial mathematics, e.g., hedging and pricing formulas for derivative securities oCourses for upper-level students, graduate students; primarily mathematics departments, and also including departments of physics, engineering, theoretical computer science, probability and statistics oSuitable for self-study

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