Positive Harmonic Functions and Diffusion (Cambridge Studies in Advanced Mathematics, Series Number 45) (Volume 0)

In this book, Professor Pinsky gives a self-contained account of the construction and basic properties of diffusion processes, including both analytic and probabilistic techniques. He starts with a rigorous treatment of the spectral theory of elliptic operators with nice coefficients on smooth, bounded domains, and then develops the theory of the generalized principal eigenvalue and the related criticality theory for elliptic operators on arbitrary domains. He considers Martin boundary theory and calculates the Martin boundary 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 a manifold. Many results that form the folklore of the subject are given a rigorous exposition, making this book a useful reference for the specialist, and an excellent guide for the graduate student.

In this book Professor Pinsky gives a self contained account of the construction and basic properties of diffusion processes, and includes both analytic and probabilistic techniques. Starting with a rigorous treatment of the spectral theory of elliptic operators with nice coefficients on smooth, bounded domains, the author develops the theory of the generalised 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 and 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 a manifold.Many results that form the folklore of the subject are here given a rigorous exposition, making this book an useful reference for the specialist, and an excellent guide for the graduate student.