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Author: Jie Xiong
Publisher: World Scientific
ISBN: 9814452378
Category : Mathematics
Languages : en
Pages : 176
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Book Description
The study of measure-valued processes in random environments has seen some intensive research activities in recent years whereby interesting nonlinear stochastic partial differential equations (SPDEs) were derived. Due to the nonlinearity and the non-Lipschitz continuity of their coefficients, new techniques and concepts have recently been developed for the study of such SPDEs. These include the conditional Laplace transform technique, the conditional mild solution, and the bridge between SPDEs and some kind of backward stochastic differential equations. This volume provides an introduction to these topics with the aim of attracting more researchers into this exciting and young area of research. It can be considered as the first book of its kind. The tools introduced and developed for the study of measure-valued processes in random environments can be used in a much broader area of nonlinear SPDEs. Contents:Introduction to SuperprocessesSuperprocesses in Random EnvironmentsLinear SPDEParticle Representations for a Class of Nonlinear SPDEsStochastic Log-Laplace EquationSPDEs for Density Fields of the Superprocesses in Random EnvironmentBackward Doubly Stochastic Differential EquationsFrom SPDE to BSDE Readership: Graduate students and researchers in the area of stochastic processes and applications. Keywords:Stochastic Partial Differential Equation;Superprocess in Random Environment;Backward Stochastic Differential EquationKey Features:Techniques are developed for specific SPDEs instead of for general SPDEs where the coefficients are not Lipschitz and the equations are highly nonlinearThe connection between SPDEs and backward stochastic differential equations are introducedFirst book in the area of measure-valued processes in random environmentsReviews: “The results presented in this monograph are due mainly to J. Xiong and his collaborators, but have been hitherto scattered in journal papers. Therefore, a book gathering them together and making them easily available is of interest for researchers in the field of measure-valued processes and/or stochastic partial differential equations.” Zentralblatt MATH “The book is based essentially on the various articles of Xiong on stochastic partial differential equations. The reader will profit from a tasteful selection of the material and from a focused and self-contained presentation.” Jahresber Dtsch Math
Author: Jie Xiong
Publisher: World Scientific
ISBN: 9814452378
Category : Mathematics
Languages : en
Pages : 176
Get Book
Book Description
The study of measure-valued processes in random environments has seen some intensive research activities in recent years whereby interesting nonlinear stochastic partial differential equations (SPDEs) were derived. Due to the nonlinearity and the non-Lipschitz continuity of their coefficients, new techniques and concepts have recently been developed for the study of such SPDEs. These include the conditional Laplace transform technique, the conditional mild solution, and the bridge between SPDEs and some kind of backward stochastic differential equations. This volume provides an introduction to these topics with the aim of attracting more researchers into this exciting and young area of research. It can be considered as the first book of its kind. The tools introduced and developed for the study of measure-valued processes in random environments can be used in a much broader area of nonlinear SPDEs. Contents:Introduction to SuperprocessesSuperprocesses in Random EnvironmentsLinear SPDEParticle Representations for a Class of Nonlinear SPDEsStochastic Log-Laplace EquationSPDEs for Density Fields of the Superprocesses in Random EnvironmentBackward Doubly Stochastic Differential EquationsFrom SPDE to BSDE Readership: Graduate students and researchers in the area of stochastic processes and applications. Keywords:Stochastic Partial Differential Equation;Superprocess in Random Environment;Backward Stochastic Differential EquationKey Features:Techniques are developed for specific SPDEs instead of for general SPDEs where the coefficients are not Lipschitz and the equations are highly nonlinearThe connection between SPDEs and backward stochastic differential equations are introducedFirst book in the area of measure-valued processes in random environmentsReviews: “The results presented in this monograph are due mainly to J. Xiong and his collaborators, but have been hitherto scattered in journal papers. Therefore, a book gathering them together and making them easily available is of interest for researchers in the field of measure-valued processes and/or stochastic partial differential equations.” Zentralblatt MATH “The book is based essentially on the various articles of Xiong on stochastic partial differential equations. The reader will profit from a tasteful selection of the material and from a focused and self-contained presentation.” Jahresber Dtsch Math
Author: Claudia Prévôt
Publisher: Springer
ISBN: 3540707816
Category : Mathematics
Languages : en
Pages : 148
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Book Description
These lectures concentrate on (nonlinear) stochastic partial differential equations (SPDE) of evolutionary type. There are three approaches to analyze SPDE: the "martingale measure approach", the "mild solution approach" and the "variational approach". The purpose of these notes is to give a concise and as self-contained as possible an introduction to the "variational approach". A large part of necessary background material is included in appendices.
Author: Jie Xiong
Publisher: World Scientific
ISBN: 981445236X
Category : Mathematics
Languages : en
Pages : 177
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Book Description
The study of measure-valued processes in random environments has seen some intensive research activities in recent years whereby interesting nonlinear stochastic partial differential equations (SPDEs) were derived. Due to the nonlinearity and the non-Lipschitz continuity of their coefficients, new techniques and concepts have recently been developed for the study of such SPDEs. These include the conditional Laplace transform technique, the conditional mild solution, and the bridge between SPDEs and some kind of backward stochastic differential equations. This volume provides an introduction to these topics with the aim of attracting more researchers into this exciting and young area of research. It can be considered as the first book of its kind. The tools introduced and developed for the study of measure-valued processes in random environments can be used in a much broader area of nonlinear SPDEs.
Author: Sergey V. Lototsky
Publisher: Springer
ISBN: 3319586475
Category : Mathematics
Languages : en
Pages : 508
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Book Description
Taking readers with a basic knowledge of probability and real analysis to the frontiers of a very active research discipline, this textbook provides all the necessary background from functional analysis and the theory of PDEs. It covers the main types of equations (elliptic, hyperbolic and parabolic) and discusses different types of random forcing. The objective is to give the reader the necessary tools to understand the proofs of existing theorems about SPDEs (from other sources) and perhaps even to formulate and prove a few new ones. Most of the material could be covered in about 40 hours of lectures, as long as not too much time is spent on the general discussion of stochastic analysis in infinite dimensions. As the subject of SPDEs is currently making the transition from the research level to that of a graduate or even undergraduate course, the book attempts to present enough exercise material to fill potential exams and homework assignments. Exercises appear throughout and are usually directly connected to the material discussed at a particular place in the text. The questions usually ask to verify something, so that the reader already knows the answer and, if pressed for time, can move on. Accordingly, no solutions are provided, but there are often hints on how to proceed. The book will be of interest to everybody working in the area of stochastic analysis, from beginning graduate students to experts in the field.
Author: George Adomian
Publisher: Academic Press
ISBN: 1483259099
Category : Science
Languages : en
Pages : 304
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Book Description
Nonlinear Stochastic Operator Equations deals with realistic solutions of the nonlinear stochastic equations arising from the modeling of frontier problems in many fields of science. This book also discusses a wide class of equations to provide modeling of problems concerning physics, engineering, operations research, systems analysis, biology, medicine. This text discusses operator equations and the decomposition method. This book also explains the limitations, restrictions and assumptions made in differential equations involving stochastic process coefficients (the stochastic operator case), which yield results very different from the needs of the actual physical problem. Real-world application of mathematics to actual physical problems, requires making a reasonable model that is both realistic and solvable. The decomposition approach or model is an approximation method to solve a wide range of problems. This book explains an inherent feature of real systems—known as nonlinear behavior—that occurs frequently in nuclear reactors, in physiological systems, or in cellular growth. This text also discusses stochastic operator equations with linear boundary conditions. This book is intended for students with a mathematics background, particularly senior undergraduate and graduate students of advanced mathematics, of the physical or engineering sciences.
Author: Robert C. Dalang
Publisher: Springer Science & Business Media
ISBN: 3540859934
Category : Mathematics
Languages : en
Pages : 230
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Book Description
This title contains lectures that offer an introduction to modern topics in stochastic partial differential equations and bring together experts whose research is centered on the interface between Gaussian analysis, stochastic analysis, and stochastic PDEs.
Author: Tomás Roubicek
Publisher: Springer Science & Business Media
ISBN: 3764373970
Category : Mathematics
Languages : en
Pages : 405
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Book Description
This book primarily concerns quasilinear and semilinear elliptic and parabolic partial differential equations, inequalities, and systems. The exposition quickly leads general theory to analysis of concrete equations, which have specific applications in such areas as electrically (semi-) conductive media, modeling of biological systems, and mechanical engineering. Methods of Galerkin or of Rothe are exposed in a large generality.
Author: István Gyöngy
Publisher: Springer
ISBN: 9783642165351
Category : Mathematics
Languages : en
Pages : 340
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Book Description
The $L_2$-theory of parabolic SPDEs is presented in this book. The development of the theory of SPDEs is motivated by problems arising in practice surrounding the numerical calculations of nonlinear filters for partially observed diffusion processes. To address these questions, the dependence of SPDEs on the driving semimartingales is investigated and new results on their numerical approximations are also given. In contrast to previous expositions, SPDEs driven by random measures and discontinuous semimartingales are also considered, and the theory of SPDEs driven by Levy processes are included as special cases. The author introduces a more general theory of SPDEs developing the theory of stochastic evolution equations in Banach spaces. He presents applications to large classes of linear and nonlinear SPDEs and , in particular, he developes a theory of SPDEs with unbounded coefficients in weighted Sobolev spaces. In this unique book regularity properties of the solutions are obtained via new results on dependence of the solutions on parameters, and existence and uniqueness theorems for parabolic SPDEs on smooth domains of $R^d$ are proven. Furthermore, the present book makes the theory more accessible for beginners, because initial linear parabolic SPDEs on the whole $R^d$ are considered, and the main existence and uniqueness results are obtained by elementary methods while exercises and applications are also provided
Author: Étienne Pardoux
Publisher: Springer Nature
ISBN: 3030890031
Category : Mathematics
Languages : en
Pages : 74
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Book Description
This book gives a concise introduction to the classical theory of stochastic partial differential equations (SPDEs). It begins by describing the classes of equations which are studied later in the book, together with a list of motivating examples of SPDEs which are used in physics, population dynamics, neurophysiology, finance and signal processing. The central part of the book studies SPDEs as infinite-dimensional SDEs, based on the variational approach to PDEs. This extends both the classical Itô formulation and the martingale problem approach due to Stroock and Varadhan. The final chapter considers the solution of a space-time white noise-driven SPDE as a real-valued function of time and (one-dimensional) space. The results of J. Walsh's St Flour notes on the existence, uniqueness and Hölder regularity of the solution are presented. In addition, conditions are given under which the solution remains nonnegative, and the Malliavin calculus is applied. Lastly, reflected SPDEs and their connection with super Brownian motion are considered. At a time when new sophisticated branches of the subject are being developed, this book will be a welcome reference on classical SPDEs for newcomers to the theory.
Author: Andreas Eberle
Publisher: Springer
ISBN: 3319749293
Category : Mathematics
Languages : en
Pages : 574
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Book Description
This Festschrift contains five research surveys and thirty-four shorter contributions by participants of the conference ''Stochastic Partial Differential Equations and Related Fields'' hosted by the Faculty of Mathematics at Bielefeld University, October 10–14, 2016. The conference, attended by more than 140 participants, including PostDocs and PhD students, was held both to honor Michael Röckner's contributions to the field on the occasion of his 60th birthday and to bring together leading scientists and young researchers to present the current state of the art and promising future developments. Each article introduces a well-described field related to Stochastic Partial Differential Equations and Stochastic Analysis in general. In particular, the longer surveys focus on Dirichlet forms and Potential theory, the analysis of Kolmogorov operators, Fokker–Planck equations in Hilbert spaces, the theory of variational solutions to stochastic partial differential equations, singular stochastic partial differential equations and their applications in mathematical physics, as well as on the theory of regularity structures and paracontrolled distributions. The numerous research surveys make the volume especially useful for graduate students and researchers who wish to start work in the above-mentioned areas, or who want to be informed about the current state of the art.
