Path Integrals for Stochastic Processes

Author: Horacio S Wio
Publisher: World Scientific
ISBN: 9814449059
Format: PDF, Kindle
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This book provides an introductory albeit solid presentation of path integration techniques as applied to the field of stochastic processes. The subject began with the work of Wiener during the 1920's, corresponding to a sum over random trajectories, anticipating by two decades Feynman's famous work on the path integral representation of quantum mechanics. However, the true trigger for the application of these techniques within nonequilibrium statistical mechanics and stochastic processes was the work of Onsager and Machlup in the early 1950's. The last quarter of the 20th century has witnessed a growing interest in this technique and its application in several branches of research, even outside physics (for instance, in economy). The aim of this book is to offer a brief but complete presentation of the path integral approach to stochastic processes. It could be used as an advanced textbook for graduate students and even ambitious undergraduates in physics. It describes how to apply these techniques for both Markov and non-Markov processes. The path expansion (or semiclassical approximation) is discussed and adapted to the stochastic context. Also, some examples of nonlinear transformations and some applications are discussed, as well as examples of rather unusual applications. An extensive bibliography is included. The book is detailed enough to capture the interest of the curious reader, and complete enough to provide a solid background to explore the research literature and start exploiting the learned material in real situations. Contents:Stochastic Processes: A Short TourThe Path Integral for a Markov Stochastic ProcessGeneralized Path Expansion Scheme ISpace-Time Transformation IGeneralized Path Expansion Scheme IISpace-Time Transformation IINon-Markov Processes: Colored Noise CaseNon-Markov Processes: Non-Gaussian CaseNon-Markov Processes: Nonlinear CasesFractional Diffusion ProcessFeynman–Kac Formula, the Influence FunctionalOther Diffusion-Like ProblemsWhat was Left Out Readership: Advanced undergraduate and graduate students, researchers interested in stochastic analysis and statistical physics. Keywords:Path Integrals;Wiener Integrals;Stochastic Processes;Brownian Motion;Fractional MotionsKey Features:Offers an introductory presentation of path integral techniques focused on the realm of stochastic processesPresents the application of these techniques to the analysis of non-Markov and/or non-Gaussian process, as well as fractional motions discussed only in specialized articles, presented in a clear and didactic wayMost useful to become acquainted with these stochastic techniques for its application in real situations

Methods And Applications Of White Noise Analysis In Interdisciplinary Sciences

Author: Bernido Christopher C
Publisher: World Scientific
ISBN: 9814569135
Format: PDF, Mobi
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Analysis, modeling, and simulation for better understanding of diverse complex natural and social phenomena often require powerful tools and analytical methods. Tractable approaches, however, can be developed with mathematics beyond the common toolbox. This book presents the white noise stochastic calculus, originated by T Hida, as a novel and powerful tool in investigating physical and social systems. The calculus, when combined with Feynman's summation-over-all-histories, has opened new avenues for resolving cross-disciplinary problems. Applications to real-world complex phenomena are further enhanced by parametrizing non-Markovian evolution of a system with various types of memory functions. This book presents general methods and applications to problems encountered in complex systems, scaling in industry, neuroscience, polymer physics, biophysics, time series analysis, relativistic and nonrelativistic quantum systems.

Functional Analysis for Probability and Stochastic Processes

Author: Adam Bobrowski
Publisher: Cambridge University Press
ISBN: 9780521831666
Format: PDF, ePub, Docs
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This text is designed both for students of probability and stochastic processes, and for students of functional analysis. It presents some chosen parts of functional analysis that can help understand ideas from probability and stochastic processes. The subjects range from basic Hilbert and Banach spaces, through weak topologies and Banach algebras, to the theory of semigroups of bounded linear operators. Numerous standard and non-standard examples and exercises make the book suitable as a course textbook as well as for self-study.

Markov Processes

Author: Daniel T. Gillespie
Publisher: Gulf Professional Publishing
ISBN: 9780122839559
Format: PDF, ePub, Docs
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Markov process theory is basically an extension of ordinary calculus to accommodate functions whos time evolutions are not entirely deterministic. It is a subject that is becoming increasingly important for many fields of science. This book develops the single-variable theory of both continuous and jump Markov processes in a way that should appeal especially to physicists and chemists at the senior and graduate level. A self-contained, prgamatic exposition of the needed elements of random variable theory Logically integrated derviations of the Chapman-Kolmogorov equation, the Kramers-Moyal equations, the Fokker-Planck equations, the Langevin equation, the master equations, and the moment equations Detailed exposition of Monte Carlo simulation methods, with plots of many numerical examples Clear treatments of first passages, first exits, and stable state fluctuations and transitions Carefully drawn applications to Brownian motion, molecular diffusion, and chemical kinetics

Continuous Time Markov Processes

Author: Thomas Milton Liggett
Publisher: American Mathematical Soc.
ISBN: 0821849492
Format: PDF, Kindle
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Markov processes are among the most important stochastic processes for both theory and applications. This book develops the general theory of these processes, and applies this theory to various special examples. The initial chapter is devoted to the most important classical example - one dimensional Brownian motion. This, together with a chapter on continuous time Markov chains, provides the motivation for the general setup based on semigroups and generators. Chapters on stochastic calculus and probabilistic potential theory give an introduction to some of the key areas of application of Brownian motion and its relatives. A chapter on interacting particle systems treats a more recently developed class of Markov processes that have as their origin problems in physics and biology. This is a textbook for a graduate course that can follow one that covers basic probabilistic limit theorems and discrete time processes.

Multiparameter Processes

Author: Davar Khoshnevisan
Publisher: Springer Science & Business Media
ISBN: 0387216316
Format: PDF, Kindle
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Self-contained presentation: from elementary material to state-of-the-art research; Much of the theory in book-form for the first time; Connections are made between probability and other areas of mathematics, engineering and mathematical physics

Simple Brownian Diffusion

Author: Daniel Thomas Gillespie
Publisher: OUP Oxford
ISBN: 0191641537
Format: PDF
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Brownian diffusion is the motion of one or more solute molecules in a sea of very many, much smaller solvent molecules. Its importance today owes mainly to cellular chemistry, since Brownian diffusion is one of the ways in which key reactant molecules move about inside a living cell. This book focuses on the four simplest models of Brownian diffusion: the classical Fickian model, the Einstein model, the discrete-stochastic (cell-jumping) model, and the Langevin model. The authors carefully develop the theories underlying these models, assess their relative advantages, and clarify their conditions of applicability. Special attention is given to the stochastic simulation of diffusion, and to showing how simulation can complement theory and experiment. Two self-contained tutorial chapters, one on the mathematics of random variables and the other on the mathematics of continuous Markov processes (stochastic differential equations), make the book accessible to researchers from a broad spectrum of technical backgrounds.

Stochastic Integrals

Author: Heinrich von Weizsäcker
Publisher: Vieweg+Teubner Verlag
ISBN: 9783528063108
Format: PDF, ePub
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This text introduces at a moderate speed and in a thorough way the basic concepts of the theory of stochastic integrals and Ito calculus for sem i martingales. There are many reasons to study this subject. We are fascinated by the contrast between general measure theoretic arguments and concrete probabilistic problems, and by the own flavour of a new differential calculus. For the beginner, a lot of work is necessary to go through this text in detail. As areward it should enable her or hirn to study more advanced literature and to become at ease with a couple of seemingly frightening concepts. Already in this introduction, many enjoyable and useful facets of stochastic analysis show up. We start out having a glance at several elementary predecessors of the stochastic integral and sketching some ideas behind the abstract theory of semimartingale integration. Having introduced martingales and local martingales in chapters 2 - 4, the stochastic integral is defined for locally uniform limits of elementary processes in chapter S. This corresponds to the Riemann integral in one-dimensional analysis and it suffices for the study of Brownian motion and diffusion processes in the later chapters 9 and 12.