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.
Contents:
IntroductionMartingales and Markov ProcessesBrownian Motion and Ito-CalculusSemi-Groups of Operators, Potentials, and Diffusion EquationsHarmonic Functions, Dynkin, and TransformsSuperharmonic Functions and Riesz MeasuresGreen Functions, Boundary Value Problems, and KernelsPotential Theory, Capacity, Boundaries, Dirichlet Spaces, and ApplicationsAppendix: Kernels and More General Classes of Gaussian Processes
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.
Key Features: Interdisciplinary, including classical and modern, and the key tools from analysis as well as probability It is accessible, suitable for both beginning and more advanced courses Student friendly Includes many exercises, with hints Points readers to new directions, and to new and related areas Addresses core questions in classical and modern potential theory Puts 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 Courses for upper-level students, graduate students; primarily mathematics departments, and also including departments of physics, engineering, theoretical computer science, probability and statistics Suitable for self-study
