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.Read more
- Author is expert on subject
- Possible supplementary text for graduate course
Reviews & endorsements
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
Not yet reviewed
Be the first to review
Review was not posted due to profanity×
- Date Published: April 2008
- format: Paperback
- isbn: 9780521059831
- length: 492 pages
- dimensions: 228 x 152 x 30 mm
- weight: 0.773kg
- availability: Available
Table of Contents
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.
Sorry, this resource is locked
Please register or sign in to request access. If you are having problems accessing these resources please email firstname.lastname@example.orgRegister Sign in
You are now leaving the Cambridge University Press website. Your eBook purchase and download will be completed by our partner www.ebooks.com. Please see the permission section of the www.ebooks.com catalogue page for details of the print & copy limits on our eBooks.Continue ×
Are you sure you want to delete your account?
This cannot be undone.
Thank you for your feedback which will help us improve our service.
If you requested a response, we will make sure to get back to you shortly.×