Diffusions And Elliptic Operators
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Diffusions And Elliptic Operators
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Author : Richard F. Bass
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-05-11
Diffusions And Elliptic Operators written by Richard F. Bass and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-05-11 with Mathematics categories.
A discussion of the interplay of diffusion processes and partial differential equations with an emphasis on probabilistic methods. It begins with stochastic differential equations, the probabilistic machinery needed to study PDE, and moves on to probabilistic representations of solutions for PDE, regularity of solutions and one dimensional diffusions. The author discusses in depth two main types of second order linear differential operators: non-divergence operators and divergence operators, including topics such as the Harnack inequality of Krylov-Safonov for non-divergence operators and heat kernel estimates for divergence form operators, as well as Martingale problems and the Malliavin calculus. While serving as a textbook for a graduate course on diffusion theory with applications to PDE, this will also be a valuable reference to researchers in probability who are interested in PDE, as well as for analysts interested in probabilistic methods.
Lectures On Probability Theory And Statistics
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Author : Amir Dembo
language : en
Publisher: Springer Science & Business Media
Release Date : 2005-11-03
Lectures On Probability Theory And Statistics written by Amir Dembo and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-11-03 with Mathematics categories.
This volume contains two of the three lectures that were given at the 33rd Probability Summer School in Saint-Flour (July 6-23, 2003). Amir Dembo’s course is devoted to recent studies of the fractal nature of random sets, focusing on some fine properties of the sample path of random walk and Brownian motion. In particular, the cover time for Markov chains, the dimension of discrete limsup random fractals, the multi-scale truncated second moment and the Ciesielski-Taylor identities are explored. Tadahisa Funaki’s course reviews recent developments of the mathematical theory on stochastic interface models, mostly on the so-called \nabla \varphi interface model. The results are formulated as classical limit theorems in probability theory, and the text serves with good applications of basic probability techniques.
Stochastic Flows And Jump Diffusions
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Author : Hiroshi Kunita
language : en
Publisher: Springer
Release Date : 2019-03-26
Stochastic Flows And Jump Diffusions written by Hiroshi Kunita and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-03-26 with Mathematics categories.
This monograph presents a modern treatment of (1) stochastic differential equations and (2) diffusion and jump-diffusion processes. The simultaneous treatment of diffusion processes and jump processes in this book is unique: Each chapter starts from continuous processes and then proceeds to processes with jumps.In the first part of the book, it is shown that solutions of stochastic differential equations define stochastic flows of diffeomorphisms. Then, the relation between stochastic flows and heat equations is discussed. The latter part investigates fundamental solutions of these heat equations (heat kernels) through the study of the Malliavin calculus. The author obtains smooth densities for transition functions of various types of diffusions and jump-diffusions and shows that these density functions are fundamental solutions for various types of heat equations and backward heat equations. Thus, in this book fundamental solutions for heat equations and backward heatequations are constructed independently of the theory of partial differential equations.Researchers and graduate student in probability theory will find this book very useful.
Ergodic Control Of Diffusion Processes
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Author : Ari Arapostathis
language : en
Publisher: Cambridge University Press
Release Date : 2012
Ergodic Control Of Diffusion Processes written by Ari Arapostathis and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Mathematics categories.
The first comprehensive account of controlled diffusions with a focus on ergodic or 'long run average' control.
Stochastic Analysis And Diffusion Processes
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Author : Gopinath Kallianpur
language : en
Publisher: OUP Oxford
Release Date : 2014-01-09
Stochastic Analysis And Diffusion Processes written by Gopinath Kallianpur and has been published by OUP Oxford this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-01-09 with Mathematics categories.
Stochastic Analysis and Diffusion Processes presents a simple, mathematical introduction to Stochastic Calculus and its applications. The book builds the basic theory and offers a careful account of important research directions in Stochastic Analysis. The breadth and power of Stochastic Analysis, and probabilistic behavior of diffusion processes are told without compromising on the mathematical details. Starting with the construction of stochastic processes, the book introduces Brownian motion and martingales. The book proceeds to construct stochastic integrals, establish the Itô formula, and discuss its applications. Next, attention is focused on stochastic differential equations (SDEs) which arise in modeling physical phenomena, perturbed by random forces. Diffusion processes are solutions of SDEs and form the main theme of this book. The Stroock-Varadhan martingale problem, the connection between diffusion processes and partial differential equations, Gaussian solutions of SDEs, and Markov processes with jumps are presented in successive chapters. The book culminates with a careful treatment of important research topics such as invariant measures, ergodic behavior, and large deviation principle for diffusions. Examples are given throughout the book to illustrate concepts and results. In addition, exercises are given at the end of each chapter that will help the reader to understand the concepts better. The book is written for graduate students, young researchers and applied scientists who are interested in stochastic processes and their applications. The reader is assumed to be familiar with probability theory at graduate level. The book can be used as a text for a graduate course on Stochastic Analysis.
Degenerate Diffusion Operators Arising In Population Biology
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Author : Charles L. Epstein
language : en
Publisher: Princeton University Press
Release Date : 2013-04-04
Degenerate Diffusion Operators Arising In Population Biology written by Charles L. Epstein and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-04 with Mathematics categories.
This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the Martingale problem and therefore the existence of the associated Markov process. Charles Epstein and Rafe Mazzeo use an "integral kernel method" to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high codimensional strata of the boundary. Epstein and Mazzeo establish the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Hölder spaces. They show that the semigroups defined by these operators have holomorphic extensions to the right half-plane. Epstein and Mazzeo also demonstrate precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations.
Functional Analytic Techniques For Diffusion Processes
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Author : Kazuaki Taira
language : en
Publisher: Springer Nature
Release Date : 2022-05-28
Functional Analytic Techniques For Diffusion Processes written by Kazuaki Taira and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-05-28 with Mathematics categories.
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.
Nonlocal And Nonlinear Diffusions And Interactions New Methods And Directions
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Author : José Antonio Carrillo
language : en
Publisher: Springer
Release Date : 2017-10-03
Nonlocal And Nonlinear Diffusions And Interactions New Methods And Directions written by José Antonio Carrillo and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-10-03 with Mathematics categories.
Presenting a selection of topics in the area of nonlocal and nonlinear diffusions, this book places a particular emphasis on new emerging subjects such as nonlocal operators in stationary and evolutionary problems and their applications, swarming models and applications to biology and mathematical physics, and nonlocal variational problems. The authors are some of the most well-known mathematicians in this innovative field, which is presently undergoing rapid development. The intended audience includes experts in elliptic and parabolic equations who are interested in extending their expertise to the nonlinear setting, as well as Ph.D. or postdoctoral students who want to enter into the most promising research topics in the field.
Convolution Like Structures Differential Operators And Diffusion Processes
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Author : Rúben Sousa
language : en
Publisher: Springer Nature
Release Date : 2022-07-27
Convolution Like Structures Differential Operators And Diffusion Processes written by Rúben Sousa and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-07-27 with Mathematics categories.
This book provides an introduction to recent developments in the theory of generalized harmonic analysis and its applications. It is well known that convolutions, differential operators and diffusion processes are interconnected: the ordinary convolution commutes with the Laplacian, and the law of Brownian motion has a convolution semigroup property with respect to the ordinary convolution. Seeking to generalize this useful connection, and also motivated by its probabilistic applications, the book focuses on the following question: given a diffusion process Xt on a metric space E, can we construct a convolution-like operator * on the space of probability measures on E with respect to which the law of Xt has the *-convolution semigroup property? A detailed analysis highlights the connection between the construction of convolution-like structures and disciplines such as stochastic processes, ordinary and partial differential equations, spectral theory, special functions and integral transforms. The book will be valuable for graduate students and researchers interested in the intersections between harmonic analysis, probability theory and differential equations.
Fokker Planck Kolmogorov Equations
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Author : Vladimir I. Bogachev
language : en
Publisher: American Mathematical Society
Release Date : 2022-02-10
Fokker Planck Kolmogorov Equations written by Vladimir I. Bogachev and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-02-10 with Mathematics categories.
This book gives an exposition of the principal concepts and results related to second order elliptic and parabolic equations for measures, the main examples of which are Fokker–Planck–Kolmogorov equations for stationary and transition probabilities of diffusion processes. Existence and uniqueness of solutions are studied along with existence and Sobolev regularity of their densities and upper and lower bounds for the latter. The target readership includes mathematicians and physicists whose research is related to diffusion processes as well as elliptic and parabolic equations.