Second Order Pde S In Finite And Infinite Dimension

Author : Sandra Cerrai
Publisher : Springer
Page : 332 pages
File Size : 44,6 Mb
Release : 2003-07-01
Category : Mathematics
ISBN : 9783540451471

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Second Order PDE's in Finite and Infinite Dimension by Sandra Cerrai Pdf

The main objective of this monograph is the study of a class of stochastic differential systems having unbounded coefficients, both in finite and in infinite dimension. We focus our attention on the regularity properties of the solutions and hence on the smoothing effect of the corresponding transition semigroups in the space of bounded and uniformly continuous functions. As an application of these results, we study the associated Kolmogorov equations, the large-time behaviour of the solutions and some stochastic optimal control problems together with the corresponding Hamilton- Jacobi-Bellman equations. In the literature there exists a large number of works (mostly in finite dimen sion) dealing with these arguments in the case of bounded Lipschitz-continuous coefficients and some of them concern the case of coefficients having linear growth. Few papers concern the case of non-Lipschitz coefficients, but they are mainly re lated to the study of the existence and the uniqueness of solutions for the stochastic system. Actually, the study of any further properties of those systems, such as their regularizing properties or their ergodicity, seems not to be developed widely enough. With these notes we try to cover this gap.

Author : Sandra Cerrai
Publisher : Unknown
Page : 344 pages
File Size : 43,5 Mb
Release : 2014-09-01
Category : Electronic
ISBN : 3662200376

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Second Order Pde's in Finite and Infinite Dimension by Sandra Cerrai Pdf

Author : Fred Espen Benth,Giulia Di Nunno,Tom Lindstrom,Bernt Øksendal,Tusheng Zhang
Publisher : Springer Science & Business Media
Page : 672 pages
File Size : 43,8 Mb
Release : 2007-04-24
Category : Mathematics
ISBN : 9783540708476

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Stochastic Analysis and Applications by Fred Espen Benth,Giulia Di Nunno,Tom Lindstrom,Bernt Øksendal,Tusheng Zhang Pdf

The Abel Symposium 2005 was organized as a tribute to the work of Kiyosi Ito on the occasion of his 90th birthday. Distinguished researchers from all over presented the newest developments within the exciting and fast growing field of stochastic analysis. This volume combines both papers from the invited speakers and contributions by the presenting lecturers. In addition, it includes the Memoirs that Kiyoshi Ito wrote for this occasion.

Author : Giorgio Fabbri,Fausto Gozzi,Andrzej Święch
Publisher : Springer
Page : 916 pages
File Size : 42,7 Mb
Release : 2017-06-22
Category : Mathematics
ISBN : 9783319530673

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Stochastic Optimal Control in Infinite Dimension by Giorgio Fabbri,Fausto Gozzi,Andrzej Święch Pdf

Providing an introduction to stochastic optimal control in infinite dimension, this book gives a complete account of the theory of second-order HJB equations in infinite-dimensional Hilbert spaces, focusing on its applicability to associated stochastic optimal control problems. It features a general introduction to optimal stochastic control, including basic results (e.g. the dynamic programming principle) with proofs, and provides examples of applications. A complete and up-to-date exposition of the existing theory of viscosity solutions and regular solutions of second-order HJB equations in Hilbert spaces is given, together with an extensive survey of other methods, with a full bibliography. In particular, Chapter 6, written by M. Fuhrman and G. Tessitore, surveys the theory of regular solutions of HJB equations arising in infinite-dimensional stochastic control, via BSDEs. The book is of interest to both pure and applied researchers working in the control theory of stochastic PDEs, and in PDEs in infinite dimension. Readers from other fields who want to learn the basic theory will also find it useful. The prerequisites are: standard functional analysis, the theory of semigroups of operators and its use in the study of PDEs, some knowledge of the dynamic programming approach to stochastic optimal control problems in finite dimension, and the basics of stochastic analysis and stochastic equations in infinite-dimensional spaces.

Author : Maria Giovanna Garroni,Jose Luis Menaldi
Publisher : CRC Press
Page : 240 pages
File Size : 54,5 Mb
Release : 2002-02-20
Category : Mathematics
ISBN : 9781420035797

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Second Order Elliptic Integro-Differential Problems by Maria Giovanna Garroni,Jose Luis Menaldi Pdf

The Green function has played a key role in the analytical approach that in recent years has led to important developments in the study of stochastic processes with jumps. In this Research Note, the authors-both regarded as leading experts in the field- collect several useful results derived from the construction of the Green function and its estim

Author : Lorenzo Zambotti
Publisher : Springer
Page : 162 pages
File Size : 49,7 Mb
Release : 2017-02-27
Category : Mathematics
ISBN : 9783319520964

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Random Obstacle Problems by Lorenzo Zambotti Pdf

Studying the fine properties of solutions to Stochastic (Partial) Differential Equations with reflection at a boundary, this book begins with a discussion of classical one-dimensional diffusions as the reflecting Brownian motion, devoting a chapter to Bessel processes, and moves on to function-valued solutions to SPDEs. Inspired by the classical stochastic calculus for diffusions, which is unfortunately still unavailable in infinite dimensions, it uses integration by parts formulae on convex sets of paths in order to describe the behaviour of the solutions at the boundary and the contact set between the solution and the obstacle. The text may serve as an introduction to space-time white noise, SPDEs and monotone gradient systems. Numerous open research problems in both classical and new topics are proposed.

Author : Haesung Lee,Wilhelm Stannat,Gerald Trutnau
Publisher : Springer Nature
Page : 139 pages
File Size : 46,6 Mb
Release : 2022-08-27
Category : Mathematics
ISBN : 9789811938313

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Analytic Theory of Itô-Stochastic Differential Equations with Non-smooth Coefficients by Haesung Lee,Wilhelm Stannat,Gerald Trutnau Pdf

This book provides analytic tools to describe local and global behavior of solutions to Itô-stochastic differential equations with non-degenerate Sobolev diffusion coefficients and locally integrable drift. Regularity theory of partial differential equations is applied to construct such solutions and to obtain strong Feller properties, irreducibility, Krylov-type estimates, moment inequalities, various types of non-explosion criteria, and long time behavior, e.g., transience, recurrence, and convergence to stationarity. The approach is based on the realization of the transition semigroup associated with the solution of a stochastic differential equation as a strongly continuous semigroup in the Lp-space with respect to a weight that plays the role of a sub-stationary or stationary density. This way we obtain in particular a rigorous functional analytic description of the generator of the solution of a stochastic differential equation and its full domain. The existence of such a weight is shown under broad assumptions on the coefficients. A remarkable fact is that although the weight may not be unique, many important results are independent of it. Given such a weight and semigroup, one can construct and further analyze in detail a weak solution to the stochastic differential equation combining variational techniques, regularity theory for partial differential equations, potential, and generalized Dirichlet form theory. Under classical-like or various other criteria for non-explosion we obtain as one of our main applications the existence of a pathwise unique and strong solution with an infinite lifetime. These results substantially supplement the classical case of locally Lipschitz or monotone coefficients.We further treat other types of uniqueness and non-uniqueness questions, such as uniqueness and non-uniqueness of the mentioned weights and uniqueness in law, in a certain sense, of the solution.

Author : Andreas Eberle,Martin Grothaus,Walter Hoh,Moritz Kassmann,Wilhelm Stannat,Gerald Trutnau
Publisher : Springer
Page : 574 pages
File Size : 43,5 Mb
Release : 2018-07-03
Category : Mathematics
ISBN : 9783319749297

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Stochastic Partial Differential Equations and Related Fields by Andreas Eberle,Martin Grothaus,Walter Hoh,Moritz Kassmann,Wilhelm Stannat,Gerald Trutnau Pdf

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.

Author : François Delarue
Publisher : American Mathematical Society
Page : 284 pages
File Size : 51,6 Mb
Release : 2021-12-14
Category : Mathematics
ISBN : 9781470455866

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Mean Field Games by François Delarue Pdf

This volume is based on lectures delivered at the 2020 AMS Short Course “Mean Field Games: Agent Based Models to Nash Equilibria,” held January 13–14, 2020, in Denver, Colorado. Mean field game theory offers a robust methodology for studying large systems of interacting rational agents. It has been extraordinarily successful and has continued to develop since its inception. The six chapters that make up this volume provide an overview of the subject, from the foundations of the theory to applications in economics and finance, including computational aspects. The reader will find a pedagogical introduction to the main ingredients, from the forward-backward mean field game system to the master equation. Also included are two detailed chapters on the connection between finite games and mean field games, with a pedestrian description of the different methods available to solve the convergence problem. The volume concludes with two contributions on applications of mean field games and on existing numerical methods, with an opening to machine learning techniques.

Author : Gregory Budzban,Harry Randolph Hughes,Henri Schurz
Publisher : American Mathematical Soc.
Page : 236 pages
File Size : 46,9 Mb
Release : 2016-06-29
Category : Combinatorial geometry
ISBN : 9781470419455

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Probability on Algebraic and Geometric Structures by Gregory Budzban,Harry Randolph Hughes,Henri Schurz Pdf

This volume contains the proceedings of the International Research Conference “Probability on Algebraic and Geometric Structures”, held from June 5–7, 2014, at Southern Illinois University, Carbondale, IL, celebrating the careers of Philip Feinsilver, Salah-Eldin A. Mohammed, and Arunava Mukherjea. These proceedings include survey papers and new research on a variety of topics such as probability measures and the behavior of stochastic processes on groups, semigroups, and Clifford algebras; algebraic methods for analyzing Markov chains and products of random matrices; stochastic integrals and stochastic ordinary, partial, and functional differential equations.

Author : Franco Flandoli
Publisher : Springer
Page : 182 pages
File Size : 41,7 Mb
Release : 2011-03-02
Category : Mathematics
ISBN : 9783642182310

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Random Perturbation of PDEs and Fluid Dynamic Models by Franco Flandoli Pdf

The book deals with the random perturbation of PDEs which lack well-posedness, mainly because of their non-uniqueness, in some cases because of blow-up. The aim is to show that noise may restore uniqueness or prevent blow-up. This is not a general or easy-to-apply rule, and the theory presented in the book is in fact a series of examples with a few unifying ideas. The role of additive and bilinear multiplicative noise is described and a variety of examples are included, from abstract parabolic evolution equations with non-Lipschitz nonlinearities to particular fluid dynamic models, like the dyadic model, linear transport equations and motion of point vortices.

Author : Peter H. Baxendale,Sergey V. Lototsky
Publisher : World Scientific
Page : 416 pages
File Size : 47,5 Mb
Release : 2007
Category : Mathematics
ISBN : 9789812770639

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Stochastic Differential Equations by Peter H. Baxendale,Sergey V. Lototsky Pdf

This volume consists of 15 articles written by experts in stochastic analysis. The first paper in the volume, Stochastic Evolution Equations by N V Krylov and B L Rozovskii, was originally published in Russian in 1979. After more than a quarter-century, this paper remains a standard reference in the field of stochastic partial differential equations (SPDEs) and continues to attract the attention of mathematicians of all generations. Together with a short but thorough introduction to SPDEs, it presents a number of optimal, and essentially unimprovable, results about solvability for a large class of both linear and non-linear equations. The other papers in this volume were specially written for the occasion of Prof RozovskiiOCOs 60th birthday. They tackle a wide range of topics in the theory and applications of stochastic differential equations, both ordinary and with partial derivatives."

Author : Dirk Blomker
Publisher : World Scientific
Page : 137 pages
File Size : 50,8 Mb
Release : 2007
Category : Mathematics
ISBN : 9789812770608

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Amplitude Equations for Stochastic Partial Differential Equations by Dirk Blomker Pdf

Rigorous error estimates for amplitude equations are well known for deterministic PDEs, and there is a large body of literature over the past two decades. However, there seems to be a lack of literature for stochastic equations, although the theory is being successfully used in the applied community, such as for convective instabilities, without reliable error estimates at hand. This book is the first step in closing this gap. The author provides details about the reduction of dynamics to more simpler equations via amplitude or modulation equations, which relies on the natural separation of time-scales present near a change of stability. For students, the book provides a lucid introduction to the subject highlighting the new tools necessary for stochastic equations, while serving as an excellent guide to recent research.

Author : Horst Reinhard Beyer
Publisher : Springer
Page : 291 pages
File Size : 51,9 Mb
Release : 2007-04-10
Category : Mathematics
ISBN : 9783540711292

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Beyond Partial Differential Equations by Horst Reinhard Beyer Pdf

This book introduces the treatment of linear and nonlinear (quasi-linear) abstract evolution equations by methods from the theory of strongly continuous semigroups. The theoretical part is accessible to graduate students with basic knowledge in functional analysis, with only some examples requiring more specialized knowledge from the spectral theory of linear, self-adjoint operators in Hilbert spaces. Emphasis is placed on equations of the hyperbolic type which are less often treated in the literature.

Author : Luca Lorenzi
Publisher : CRC Press
Page : 607 pages
File Size : 46,5 Mb
Release : 2016-10-04
Category : Mathematics
ISBN : 9781482243345

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Analytical Methods for Kolmogorov Equations by Luca Lorenzi Pdf

The second edition of this book has a new title that more accurately reflects the table of contents. Over the past few years, many new results have been proven in the field of partial differential equations. This edition takes those new results into account, in particular the study of nonautonomous operators with unbounded coefficients, which has received great attention. Additionally, this edition is the first to use a unified approach to contain the new results in a singular place.