Brownian Motion and Potential Theory, Modern and Classical (Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes Book 9) Kindle Edition

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.Readership: Advanced undergraduate and graduate students, researchers, and practitioners in mathematics, statistics, finance, and engineering, especially those focusing on probability theory, stochastic processes, and mathematical analysis. It is suitable for upper-level and graduate courses in mathematics, as well as physics, engineering, theoretical computer science, probability, and statistics, and is also perfect for self-study. Readers with interest in current developments in pure and applied mathematics, students and specialists alike. Read more