Kazuaki Taira Livres

Focusing on the intersection of heat equations and differential geometry, this monograph employs functional analysis and the Weyl-Hörmander calculus to explore heat Green operators under various boundary conditions. It emphasizes constructive methods in partial differential equations, providing detailed examples and applications related to elliptic and parabolic problems. The text is designed for a wide audience, presenting complex ideas clearly to facilitate understanding of initial boundary value problems and their implications in mathematics and probability.

The book offers a comprehensive exploration of the functional analytic methods used to construct Markov processes under Ventcel' boundary conditions within probability theory. It highlights recent advancements in the theory of singular integrals, making complex concepts accessible to readers. The focus on both theoretical and practical aspects provides valuable insights for those interested in advanced probability and mathematical analysis.

The book serves as an accessible reference that connects functional analysis, partial differential equations, and probability, focusing on the mathematics essential for understanding diffusion processes. It explores the interplay between macroscopic, mesoscopic, and microscopic approaches, offering a deep stochastic perspective on elliptic boundary value problems. This integration of diverse mathematical fields aims to enhance comprehension of complex diffusion phenomena.

Focusing on advanced real analysis methods, this book explores the construction of Markov processes influenced by boundary conditions in probability theory. It examines the behavior of a Markovian particle governed by the Waldenfels operator and adhering to the Ventcel boundary condition, which encompasses various phenomena like diffusion, absorption, and jumps. The study delves into first-order Ventcel boundary value problems for second-order elliptic operators, employing Calderón-Zygmund theory to establish existence and uniqueness theorems in Sobolev and Besov spaces.

This is a thorough and accessible exposition on the functional analytic approach to the problem of construction of Markov processes with Ventcel’ boundary conditions in probability theory. It presents new developments in the theory of singular integrals.

This volume will be of great appeal to both advanced students and researchers. For the former, it serves as an effective introduction to three interrelated subjects of semigroups, Markov processes and elliptic boundary value problems. For the latter, it provides a new method for the analysis of Markov processes, a powerful method clearly capable of extensive further development.

This book is an easy-to-read reference providing a link between partial differential equations (pde), stochastic analysis, and index theory. Most mathematicians working in pde are only vaguely familiar with the powerful ideas of stochastic analysis. On the other hand, the additional intuition which Taira´s book conveys might provide better insight and be helpful for their work.In addition, the book provides a nice compendium for a large variety of facts from differential geometry, functional analysis, pseudodifferential operators, and Markov processes - for quickly looking up a theorem.