Schrödinger Diffusion Processes
Schrödinger Diffusion Processes

Author: Robert Aebi

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 196

ISBN-13: 3034890273

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In 1931 Erwin Schrödinger considered the following problem: A huge cloud of independent and identical particles with known dynamics is supposed to be observed at finite initial and final times. What is the "most probable" state of the cloud at intermediate times? The present book provides a general yet comprehensive discourse on Schrödinger's question. Key roles in this investigation are played by conditional diffusion processes, pairs of non-linear integral equations and interacting particles systems. The introductory first chapter gives some historical background, presents the main ideas in a rather simple discrete setting and reveals the meaning of intermediate prediction to quantum mechanics. In order to answer Schrödinger's question, the book takes three distinct approaches, dealt with in separate chapters: transformation by means of a multiplicative functional, projection by means of relative entropy, and variation of a functional associated to pairs of non-linear integral equations. The book presumes a graduate level of knowledge in mathematics or physics and represents a relevant and demanding application of today's advanced probability theory.

Schrödinger Equations and Diffusion Theory
Schrödinger Equations and Diffusion Theory

Author: M. Nagasawa

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 335

ISBN-13: 3034885687

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Schrödinger Equations and Diffusion Theory addresses the question "What is the Schrödinger equation?" in terms of diffusion processes, and shows that the Schrödinger equation and diffusion equations in duality are equivalent. In turn, Schrödinger's conjecture of 1931 is solved. The theory of diffusion processes for the Schrödinger equation tell us that we must go further into the theory of systems of (infinitely) many interacting quantum (diffusion) particles. The method of relative entropy and the theory of transformations enable us to construct severely singular diffusion processes which appear to be equivalent to Schrödinger equations. The theory of large deviations and the propagation of chaos of interacting diffusion particles reveal the statistical mechanical nature of the Schrödinger equation, namely, quantum mechanics. The text is practically self-contained and requires only an elementary knowledge of probability theory at the graduate level.

Schrödinger Equations and Diffusion Theory
Schrödinger Equations and Diffusion Theory

Author: Masao Nagasawa

Publisher: Springer Science & Business Media

Published: 2012-12-13

Total Pages: 333

ISBN-13: 3034805608

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Schrödinger Equations and Diffusion Theory addresses the question “What is the Schrödinger equation?” in terms of diffusion processes, and shows that the Schrödinger equation and diffusion equations in duality are equivalent. In turn, Schrödinger’s conjecture of 1931 is solved. The theory of diffusion processes for the Schrödinger equation tells us that we must go further into the theory of systems of (infinitely) many interacting quantum (diffusion) particles. The method of relative entropy and the theory of transformations enable us to construct severely singular diffusion processes which appear to be equivalent to Schrödinger equations. The theory of large deviations and the propagation of chaos of interacting diffusion particles reveal the statistical mechanical nature of the Schrödinger equation, namely, quantum mechanics. The text is practically self-contained and requires only an elementary knowledge of probability theory at the graduate level. --- This book is a self-contained, very well-organized monograph recommended to researchers and graduate students in the field of probability theory, functional analysis and quantum dynamics. (...) what is written in this book may be regarded as an introduction to the theory of diffusion processes and applications written with the physicists in mind. Interesting topics present themselves as the chapters proceed. (...) this book is an excellent addition to the literature of mathematical sciences with a flavour different from an ordinary textbook in probability theory because of the author’s great contributions in this direction. Readers will certainly enjoy the topics and appreciate the profound mathematical properties of diffusion processes. (Mathematical Reviews)​

Stochastic Processes in Quantum Physics
Stochastic Processes in Quantum Physics

Author: Masao Nagasawa

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 609

ISBN-13: 3034883838

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From the reviews: "The text is almost self-contained and requires only an elementary knowledge of probability theory at the graduate level. The book under review is recommended to mathematicians, physicists and graduate students interested in mathematical physics and stochastic processes. Furthermore, some selected chapters can be used as sub-textbooks for advanced courses on stochastic processes, quantum theory and quantum chemistry." ZAA

Stochastic Processes and Applications
Stochastic Processes and Applications

Author: Grigorios A. Pavliotis

Publisher: Springer

Published: 2014-11-19

Total Pages: 345

ISBN-13: 1493913239

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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.

Handbooks in Operations Research and Management Science: Financial Engineering
Handbooks in Operations Research and Management Science: Financial Engineering

Author: John R. Birge

Publisher: Elsevier

Published: 2007-11-16

Total Pages: 1026

ISBN-13: 9780080553252

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The remarkable growth of financial markets over the past decades has been accompanied by an equally remarkable explosion in financial engineering, the interdisciplinary field focusing on applications of mathematical and statistical modeling and computational technology to problems in the financial services industry. The goals of financial engineering research are to develop empirically realistic stochastic models describing dynamics of financial risk variables, such as asset prices, foreign exchange rates, and interest rates, and to develop analytical, computational and statistical methods and tools to implement the models and employ them to design and evaluate financial products and processes to manage risk and to meet financial goals. This handbook describes the latest developments in this rapidly evolving field in the areas of modeling and pricing financial derivatives, building models of interest rates and credit risk, pricing and hedging in incomplete markets, risk management, and portfolio optimization. Leading researchers in each of these areas provide their perspective on the state of the art in terms of analysis, computation, and practical relevance. The authors describe essential results to date, fundamental methods and tools, as well as new views of the existing literature, opportunities, and challenges for future research.

Seminar on Stochastic Analysis, Random Fields and Applications
Seminar on Stochastic Analysis, Random Fields and Applications

Author: Erwin Bolthausen

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 392

ISBN-13: 3034870264

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Pure and applied stochastic analysis and random fields form the subject of this book. The collection of articles on these topics represent the state of the art of the research in the field, with particular attention being devoted to stochastic models in finance. Some are review articles, others are original papers; taken together, they will apprise the reader of much of the current activity in the area.

Itô’s Stochastic Calculus and Probability Theory
Itô’s Stochastic Calculus and Probability Theory

Author: Nobuyuki Ikeda

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 425

ISBN-13: 4431685324

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Professor Kiyosi Ito is well known as the creator of the modern theory of stochastic analysis. Although Ito first proposed his theory, now known as Ito's stochastic analysis or Ito's stochastic calculus, about fifty years ago, its value in both pure and applied mathematics is becoming greater and greater. For almost all modern theories at the forefront of probability and related fields, Ito's analysis is indispensable as an essential instrument, and it will remain so in the future. For example, a basic formula, called the Ito formula, is well known and widely used in fields as diverse as physics and economics. This volume contains 27 papers written by world-renowned probability theorists. Their subjects vary widely and they present new results and ideas in the fields where stochastic analysis plays an important role. Also included are several expository articles by well-known experts surveying recent developments. Not only mathematicians but also physicists, biologists, economists and researchers in other fields who are interested in the effectiveness of stochastic theory will find valuable suggestions for their research. In addition, students who are beginning their study and research in stochastic analysis and related fields will find instructive and useful guidance here. This volume is dedicated to Professor Ito on the occasion of his eightieth birthday as a token of deep appreciation for his great achievements and contributions. An introduction to and commentary on the scientific works of Professor Ito are also included.

Schrodinger
Schrodinger

Author: Walter Moore

Publisher: Cambridge University Press

Published: 2015-10-06

Total Pages: 529

ISBN-13: 1107569915

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This is a biography of the great scientist, Erwin Schrödinger (author of What is Life?), which draws upon recollections of his family and friends, as well as on contemporary records, diaries and letters. It aims to reveal the fundamental motives that drove him.

Schrödinger
Schrödinger

Author: Walter J. Moore

Publisher: Cambridge University Press

Published: 1992-05-29

Total Pages: 532

ISBN-13: 1139935771

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Erwin Schrödinger was a brilliant and charming Austrian, a great scientist, and a man with a passionate interest in people and ideas. In this, the first comprehensive biography of Schrödinger, Walter Moore draws upon recollections of Schrödinger's friends, family and colleagues, and on contemporary records, letters and diaries. Schrödinger's life is portrayed against the backdrop of Europe at a time of change and unrest. His best-known scientific work was the discovery of wave mechanics, for which he was awarded the Nobel prize in 1933. However, Erwin was also an enthusiastic explorer of the ideas of Hindu mysticism, and in the mountains of his beloved Tyrol he sought a philosophic unity of Mind and Nature. Although not Jewish, he left his prestigious position at Berlin University as soon as the Nazis seized power. After a short time in Oxford he moved to Graz, but barely escaped from Austria after the Anschluss. He then helped Eamon de Valera establish an Institute for Advanced Studies in Dublin. It was here that he spent the happiest years of his life, and also where he wrote his most famous and influential book What is Life?, which attracted some of the brightest minds of his generation into molecular biology. Schrodinger enjoyed a close friendship with Einstein, and the two maintained a prolific correspondence all their lives. Schrödinger led a very intense life, both in his scientific research and in his personal life. Walter Moore has written a highly readable biography of this fascinating and complex man which will appeal not only to scientists but to anyone interested in the history of our times, and in the life and thought of one of the great men of twentieth-century science.