Positive Harmonic Functions and Diffusion: An Integrated Analytic and Probabilistic Approach (Hardback)

Condition

VERY GOOD: This book is in very good condition, showing only slight signs of use and wear.

Product Details

This is #45 in theCambridge studies in advanced mathematics series from Cambridge University Press.

From the back cover: "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 functions. 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 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."

Chapters

Each chapter ends with a set of exercises.

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 elliptic 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 operator
9. Bounded harmonic functions and applications to Brownian motion and the Laplacian on a manifold of non-positive curvature

The book concludes with References and Index.