Zentralblatt MATH (zbMATH)
Heat Kernel Method and Its Applications, (Birkhaeser, 2015) The heart of the book is the development of a short-time asymptotic expansion for the heat kernel. This is explained in detail and explicit examples of some advanced calculations are given. In addition some advanced methods and extensions, including path integrals, jump diffusion and others are presented. The book consists of four parts: Analysis, Geometry, Perturbations and Applications. The first part shortly reviews of some background material and gives an introduction to PDEs. The second part is devoted to a short introduction to various aspects of differential geometry that will be needed later. The third part and heart of the book presents a systematic development of effective methods for various approximation schemes for parabolic differential equations. The last part is devoted to applications in financial mathematics, in particular, stochastic differential equations. Although this book is intended for advanced undergraduate or beginning graduate students in, it should also provide a useful reference for professional physicists, applied mathematicians as well as quantitative analysts with an interest in PDEs.
“The main theme of the monograph is the development of a short-time expansion for the heat kernel. … The monograph is well organized; each chapter starts with an abstract and ends with a section of notes, that can be effectively used to navigate through the contents. Thanks to an extensive presentation of background material, the book is well suited for undergraduate and graduate students, but can be of interest also for applied mathematicians and physicists.” (Paolo Musolino, zbMATH 1342.35001, 2016)
Mathematical Reviews
MR3410283 35-02 35C20 35K08 35Q91 35R60 58J35
Avramidi, Ivan G. ( 1-NMMT; Socorro, NM )
Heat kernel method and its applications.
Springer, Cham, 2015. xix+390 pp. ISBN 978-3-319-26265-9; 978-3-319-26266-6
Heat kernels are central objects in analysis, partial differential equations, geometry, stochastic analysis and applied mathematics. Their basic role is to give integral representations of solutions to heat equations. Among other applications, we could mention the following important ones. J. F. Nash Jr. found estimates for heat kernels which allowed him to prove Hölder continuity of weak solutions to parabolic equations in divergence form with bounded measurable coefficients [see Amer. J. Math. 80 (1958), 931–954; MR0100158]. E. M. Stein showed in his monograph [Topics in harmonic analysis related to the Littlewood-Paley theory, Annals of Mathematics Studies, No. 63, Princeton Univ. Press, Princeton, NJ, 1970; MR0252961] how heat semigroups can be used in the analysis of non-trigonometric, classical orthogonal expansions. More recently, J. L. Torrea and the reviewer have shown that heat kernels are essential tools in the analysis of fractional nonlocal partial differential equations [see Comm. Partial Differential Equations 35 (2010), no. 11, 2092–2122; MR2754080].
The present volume mainly focuses on the modern analysis of asymptotic expansions of heat kernels and their applications to different stochastic models appearing in financial mathematics.
The book is divided into four parts. Part I is an account of basic notions of analysis, heat semigroups, partial differential equations and heat kernels. Part II collects concepts and results from differential geometry. The core of the book begins with Part III. First, singular perturbation techniques in linear evolution equations are presented. Second, the short-time asymptotic expansion of the heat kernel on smooth manifolds and the corresponding expansion of the Green function are studied. The latter is the main chapter of the book. In particular, the author analyzes the Minakshisundaram-Pleijel expansion, and computes the expansion coefficients in different geometric scenarios. Part III ends with other advanced (mostly algebraic in nature) methods for the calculation of heat kernels and an introduction to path integrals. Finally, Part IV has two chapters. The first one is an introduction to probability, stochastic calculus and Itô’s Formula. The second one regards applications of the results previously developed to explicitly compute solutions to several important stochastic volatility models of mathematical finance.
Every chapter in this book contains a lot of information. Suggested further reading and precise references can be found at the end of each one. As the author explains in his preface, this book is written in a sometimes rather informal style. It is aimed towards non-specialists, allowing them to quickly enter into the heart of the matters described above. This volume will be very useful to those interested in asymptotic expansions of heat kernels and stochastic models from financial mathematics.
Pablo Raúl Stinga
Amazon.com
Heat Kernel and Quantum Gravity, (Springer, 2000) This book tackles quantum gravity via the so-called background field method and its effective action functional. The author presents an explicitly covariant and effective technique to calculate the De Witt coefficients and to analyze the Schwinger-De Witt asymptotic expansion of the effective action. He also investigates the ultraviolet behaviour of higher-derivative quantum gravity. The book addresses theoretical physicists, graduate students as well as researchers, but should also be of interest to physicists working in mathematical or elementary particle physics.
"This monograph rightly belongs to a series ‘Lecture notes in Physics’, as it represents a well-written review of main results by the author, who is a recognized expert on heat kernel techniques in quantum gravity. [...] The results exposed in this book reflect the major contributions of the author to differential geometry and the theory of differential operators. They have many applications in quantum field theory with background fields, and indeed, the book can be used as a text for a short graduate course in the heat kernel techniques and their quantum gravity." (Mathematical Reviews 2003a)
"Spectral theory for the heat equation represents one of the more exciting points of interaction between math and physics: It also serves as a deep link, via spectral theory, between geometry(math), and quantum gravity(physics). While the subject has roots far back, this lovely book presents some of the more exciting developments in the past decade. One of the success stories in interdisciplinary theoretical science! It is well written, and will be a great source for grad students. This very nice book further points toward the research trends of the future. Moreover, the results presented in the book are timeless. The book will be of value also ten years from now. Being an acknowledged authority in the subject, this author is in a unique position to write a book on the central themes and theories in the subject". (Palle Jorgensen, Amazon.com)