Classical Potential Theory and Its Probabilistic Counterpart [Paperback]

From the reviews: Here is a momumental work by Doob, one of the masters, in which Part 1 develops the potential theory associated with Laplace's equation and the heat equation, and Part 2 develops those parts (martingales and Brownian motion) of stochastic process theory which are closely related to Part 1 . --G.E.H. Reuter in Short Book Reviews (1985)

From the reviews: This huge book written in several years by one of the few mathematicians able to do it, appears as a precise and impressive study (not very easy to read) of this bothsided question that replaces, in a coherent way, without being encyclopaedic, a large library of books and papers scattered without a uniform language. Instead of summarizing the author gives his own way of exposition with original complements. This requires no preliminary knowledge. ...The purpose which the author explains in his introduction, i.e. a deep probabilistic interpretation of potential theory and a link between two great theories, appears fulfilled in a masterly manner .
M. Brelot in Metrika (1986)I Introduction to the Mathematical Background of Classical Potential Theory.- II Basic Properties of Harmonic, Subharmonic, and Superharmonic Functions.- III Infima of Families of Superharmonic Functions.- IV Potentials on Special Open Sets.- V Polar Sets and Their Applications.- VI The Fundamental Convergence Theorem and the Reduction Operation.- VII Green Functions.- VIII The Dirichlet Problem for Relative Harmonic Functions.- IX Lattices and Related Classes of Functions.- X The Sweeping Operation.- XI The Fine Topology.- XII The Martin Boundary.- XIII Classical Energy and Capacity.- XIV One-Dimensional Potential Theory.- XV Parabolic Potential Theory: Basic Facts.- XVI Subparabolic, Superparabolic, and Parabolic Functions on a Slab.- XVII Parabolic Potential Theory (Continued).- XVIII The Parabolic Dirichlet Problem, Sweeping, and Exceptional Sets.- XIX The Martin Boundary in the Parabolic Context.- Il%