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An Introduction to the Analysis of Paths on a Riemannian Manifold

Daniel W. Stroock Massachusetts Institute of Technology, Cambridge, MA
Available Formats:
Electronic ISBN: 978-1-4704-1301-9
Product Code: SURV/74.S.E
List Price: $49.00 MAA Member Price:$44.10
AMS Member Price: $39.20 Click above image for expanded view An Introduction to the Analysis of Paths on a Riemannian Manifold Daniel W. Stroock Massachusetts Institute of Technology, Cambridge, MA Available Formats:  Electronic ISBN: 978-1-4704-1301-9 Product Code: SURV/74.S.E  List Price:$49.00 MAA Member Price: $44.10 AMS Member Price:$39.20
• Book Details

Mathematical Surveys and Monographs
Volume: 742000; 269 pp
MSC: Primary 60;

This book aims to bridge the gap between probability and differential geometry. It gives two constructions of Brownian motion on a Riemannian manifold: an extrinsic one where the manifold is realized as an embedded submanifold of Euclidean space and an intrinsic one based on the “rolling” map. It is then shown how geometric quantities (such as curvature) are reflected by the behavior of Brownian paths and how that behavior can be used to extract information about geometric quantities. Readers should have a strong background in analysis with basic knowledge in stochastic calculus and differential geometry.

Professor Stroock is a highly-respected expert in probability and analysis. The clarity and style of his exposition further enhance the quality of this volume. Readers will find an inviting introduction to the study of paths and Brownian motion on Riemannian manifolds.

Graduate students, research mathematicians, and physicists interested in probability theory and stochastic analysis; theoretical physicists; electrical engineers.

• Chapters
• 1. Brownian motion in Euclidean space
• 2. Diffusions in Euclidean space
• 3. Some addenda, extensions, and refinements
• 4. Doing it on a manifold, an extrinsic approach
• 5. More about extrinsic Riemannian geometry
• 6. Bochner’s identity
• 7. Some intrinsic Riemannian geometry
• 8. The bundle of orthonormal frames
• 9. Local analysis of Brownian motion
• 10. Perturbing Brownian paths

• Reviews

• From one of the major players in modern probability theory … a welcome addition to the existing literature … I made the fateful decision to read the book in the way most mathematical books either are not meant or do not deserve to be read, namely page by page. It turned out to be a decision … that paid off handsomely; otherwise I would have certainly missed many interesting facts and perspicacious observations scattered throughout the whole book. Having finished in this fashion, I can now confidently share with the general public my appreciation of this unique work … [a] highly informative book.

Mathematical Reviews
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Volume: 742000; 269 pp
MSC: Primary 60;

This book aims to bridge the gap between probability and differential geometry. It gives two constructions of Brownian motion on a Riemannian manifold: an extrinsic one where the manifold is realized as an embedded submanifold of Euclidean space and an intrinsic one based on the “rolling” map. It is then shown how geometric quantities (such as curvature) are reflected by the behavior of Brownian paths and how that behavior can be used to extract information about geometric quantities. Readers should have a strong background in analysis with basic knowledge in stochastic calculus and differential geometry.

Professor Stroock is a highly-respected expert in probability and analysis. The clarity and style of his exposition further enhance the quality of this volume. Readers will find an inviting introduction to the study of paths and Brownian motion on Riemannian manifolds.

Graduate students, research mathematicians, and physicists interested in probability theory and stochastic analysis; theoretical physicists; electrical engineers.

• Chapters
• 1. Brownian motion in Euclidean space
• 2. Diffusions in Euclidean space
• 3. Some addenda, extensions, and refinements
• 4. Doing it on a manifold, an extrinsic approach
• 5. More about extrinsic Riemannian geometry
• 6. Bochner’s identity
• 7. Some intrinsic Riemannian geometry
• 8. The bundle of orthonormal frames
• 9. Local analysis of Brownian motion
• 10. Perturbing Brownian paths
• From one of the major players in modern probability theory … a welcome addition to the existing literature … I made the fateful decision to read the book in the way most mathematical books either are not meant or do not deserve to be read, namely page by page. It turned out to be a decision … that paid off handsomely; otherwise I would have certainly missed many interesting facts and perspicacious observations scattered throughout the whole book. Having finished in this fashion, I can now confidently share with the general public my appreciation of this unique work … [a] highly informative book.

Mathematical Reviews
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