Functional Analytic Techniques For Diffusion Processes
Download Functional Analytic Techniques For Diffusion Processes PDF/ePub or read online books in Mobi eBooks. Click Download or Read Online button to get Functional Analytic Techniques For Diffusion Processes book now. This website allows unlimited access to, at the time of writing, more than 1.5 million titles, including hundreds of thousands of titles in various foreign languages.
Functional Analytic Techniques for Diffusion Processes
This book is an easy-to-read reference providing a link between functional analysis and diffusion processes. More precisely, the book takes readers to a mathematical crossroads of functional analysis (macroscopic approach), partial differential equations (mesoscopic approach), and probability (microscopic approach) via the mathematics needed for the hard parts of diffusion processes. This work brings these three fields of analysis together and provides a profound stochastic insight (microscopic approach) into the study of elliptic boundary value problems. The author does a massive study of diffusion processes from a broad perspective and explains mathematical matters in a more easily readable way than one usually would find. The book is amply illustrated; 14 tables and 141 figures are provided with appropriate captions in such a fashion that readers can easily understand powerful techniques of functional analysis for the study of diffusion processes in probability. The scope of the author’s work has been and continues to be powerful methods of functional analysis for future research of elliptic boundary value problems and Markov processes via semigroups. A broad spectrum of readers can appreciate easily and effectively the stochastic intuition that this book conveys. Furthermore, the book will serve as a sound basis both for researchers and for graduate students in pure and applied mathematics who are interested in a modern version of the classical potential theory and Markov processes. For advanced undergraduates working in functional analysis, partial differential equations, and probability, it provides an effective opening to these three interrelated fields of analysis. Beginning graduate students and mathematicians in the field looking for a coherent overview will find the book to be a helpful beginning. This work will be a major influence in a very broad field of study for a long time.
Stochastic Processes and Applications
This book presents various results and techniques from the theory of stochastic processes that are useful in the study of stochastic problems in the natural sciences. The main focus is analytical methods, although numerical methods and statistical inference methodologies for studying diffusion processes are also presented. The goal is the development of techniques that are applicable to a wide variety of stochastic models that appear in physics, chemistry and other natural sciences. Applications such as stochastic resonance, Brownian motion in periodic potentials and Brownian motors are studied and the connection between diffusion processes and time-dependent statistical mechanics is elucidated. The book contains a large number of illustrations, examples, and exercises. It will be useful for graduate-level courses on stochastic processes for students in applied mathematics, physics and engineering. Many of the topics covered in this book (reversible diffusions, convergence to equilibrium for diffusion processes, inference methods for stochastic differential equations, derivation of the generalized Langevin equation, exit time problems) cannot be easily found in textbook form and will be useful to both researchers and students interested in the applications of stochastic processes.
Real Analysis Methods for Markov Processes
Zusammenfassung: This book is devoted to real analysis methods for the problem of constructing Markov processes with boundary conditions in probability theory. Analytically, a Markovian particle in a domain of Euclidean space is governed by an integro-differential operator, called the Waldenfels operator, in the interior of the domain, and it obeys a boundary condition, called the Ventcel (Wentzell) boundary condition, on the boundary of the domain. Most likely, a Markovian particle moves both by continuous paths and by jumps in the state space and obeys the Ventcel boundary condition, which consists of six terms corresponding to diffusion along the boundary, an absorption phenomenon, a reflection phenomenon, a sticking (or viscosity) phenomenon, and a jump phenomenon on the boundary and an inward jump phenomenon from the boundary. More precisely, we study a class of first-order Ventcel boundary value problems for second-order elliptic Waldenfels integro-differential operators. By using the Calderón-Zygmund theory of singular integrals, we prove the existence and uniqueness of theorems in the framework of the Sobolev and Besov spaces, which extend earlier theorems due to Bony-Courrège-Priouret to the vanishing mean oscillation (VMO) case. Our proof is based on various maximum principles for second-order elliptic differential operators with discontinuous coefficients in the framework of Sobolev spaces. My approach is distinguished by the extensive use of the ideas and techniques characteristic of recent developments in the theory of singular integral operators due to Calderón and Zygmund. Moreover, we make use of an Lp variant of an estimate for the Green operator of the Neumann problem introduced in the study of Feller semigroups by me. The present book is amply illustrated; 119 figures and 12 tables are provided in such a fashion that a broad spectrum of readers understand our problem and main results