In this book, Professor Pinsky gives a self-contained account of the theory of positive harmonic functions for second order elliptic operators, using an integrated probabilistic and analytic approach. The book begins with a treatment of the construction and basic properties of diffusion processes. This theory then serves as a vehicle for studying positive harmonic funtions. Starting with a rigorous treatment of the spectral theory of elliptic operators with nice coefficients on smooth, bounded domains, the author then develops the theory of the generalized principal eigenvalue, and the related criticality theory for elliptic operators on arbitrary domains. Martin boundary theory is considered, and the Martin boundary is explicitly calculated for several classes of operators. The book provides an array of criteria for determining whether a diffusion process is transient or recurrent. Also introduced are the theory of bounded harmonic functions, and Brownian motion on manifolds of negative curvature. Many results that form the folklore of the subject are here given a rigorous exposition, making this book a useful reference for the specialist, and an excellent guide for the graduate student.
• Author is expert on subject • Possible supplementary text for graduate course
1. Existence and uniqueness for diffusion processes; 2. The basic properties of diffusion processes; 3. The spectral theory of elliptic operators on smooth bounded domains; 4. Generalized spectral theory for operators on arbitrary domains; 5. Applications to the one-dimensional case and the radially symmetric multi-dimensional case; 6. Criteria for transience or recurrence and explosion or non-explosion of diffusion processes; 7. Positive harmonic functions and the Martin boundary: general theory; 8. Positive harmonic functions and the Martin boundary: applications to certain classes of operators; 9. Bounded harmonic functions and applications to Brownian motion and the Laplacian on a manifold of non-positive curvature.
Review of the hardback: 'This book is reasonably self contained, accessible even to a graduate student but maintaining a high level of mathematical rigor.' European Mathematical Society Newsletter