The general area of stochastic PDEs is interesting to mathematicians because it contains an enormous number of challenging open problems. There is also a great deal of interest in this topic because it has deep applications in disciplines that range from applied mathematics, statistical mechanics, and theoretical physics, to theoretical neuroscience, theory of complex chemical reactions [including polymer science], fluid dynamics, and mathematical finance. The stochastic PDEs that are studied in this book are similar to the familiar PDE for heat in a thin rod, but with the additional restriction that the external forcing density is a two-parameter stochastic process, or what is more commonly the case, the forcing is a "random noise," also known as a "generalized random field." At several points in the lectures, there are examples that highlight the phenomenon that stochastic PDEs are not a subset of PDEs. In fact, the introduction of noise in some partial differential equations can bring about not a small perturbation, but truly fundamental changes to the system that the underlying PDE is attempting to describe. The topics covered include a brief introduction to the stochastic heat equation, structure theory for the linear stochastic heat equation, and an in-depth look at intermittency properties of the solution to semilinear stochastic heat equations. Specific topics include stochastic integrals à la Norbert Wiener, an infinite-dimensional Itô-type stochastic integral, an example of a parabolic Anderson model, and intermittency fronts. There are many possible approaches to stochastic PDEs. The selection of topics and techniques presented here are informed by the guiding example of the stochastic heat equation. A co-publication of the AMS and CBMS.
Author: Frederi Viens
Publisher: Springer Science & Business Media
Release Date: 2013-02-15
The stochastic calculus of variations of Paul Malliavin (1925 - 2010), known today as the Malliavin Calculus, has found many applications, within and beyond the core mathematical discipline. Stochastic analysis provides a fruitful interpretation of this calculus, particularly as described by David Nualart and the scores of mathematicians he influences and with whom he collaborates. Many of these, including leading stochastic analysts and junior researchers, presented their cutting-edge research at an international conference in honor of David Nualart's career, on March 19-21, 2011, at the University of Kansas, USA. These scholars and other top-level mathematicians have kindly contributed research articles for this refereed volume.
This book contains original research papers by leading experts in the fields of probability theory, stochastic analysis, potential theory and mathematical physics. There is also a historical account on Masatoshi Fukushima's contribution to mathematics, as well as authoritative surveys on the state of the art in the field. Contents:Professor Fukushima's Work:The Mathematical Work of Masatoshi Fukushima — An Essay (Zhen-Qing Chen, Niels Jacob, Masayoshi Takeda and Toshihiro Uemura)Bibliography of Masatoshi FukushimaContributions:Quasi Regular Dirichlet Forms and the Stochastic Quantization Problem (Sergio Albeverio, Zhi-Ming Ma and Michael Röckner)Comparison of Quenched and Annealed Invariance Principles for Random Conductance Model: Part II (Martin Barlow, Krzysztof Burdzy and Adám Timár)Some Historical Aspects of Error Calculus by Dirichlet Forms (Nicolas Bouleau)Stein's Method, Malliavin Calculus, Dirichlet Forms and the Fourth Moment Theorem (Louis H Y Chen and Guillaume Poly)Progress on Hardy-Type Inequalities (Mu-Fa Chen)Functional Inequalities for Pure-Jump Dirichlet Forms (Xin Chen, Feng-Yu Wang and Jian Wang)Additive Functionals and Push Forward Measures Under Veretennikov's Flow (Shizan Fang and Andrey Pilipenko)On a Result of D W Stroock (Patrick J Fitzsimmons)Consistent Risk Measures and a Non-Linear Extension of Backwards Martingale Convergence (Hans Föllmer and Irina Penner)Unavoidable Collections of Balls for Processes with Isotropic Unimodal Green Function (Wolfhard Hansen)Functions of Locally Bounded Variation on Wiener Spaces (Masanori Hino)A Dirichlet Space on Ends of Tree and Superposition of Nodewise Given Dirichlet Forms with Tier Linkage (Hiroshi Kaneko)Dirichlet Forms in Quantum Theory (Witold Karwowski and Ludwig Streit)On a Stability of Heat Kernel Estimates under Generalized Non-Local Feynman-Kac Perturbations for Stable-Like Processes (Daehong Kim and Kazuhiro Kuwae)Martin Boundary for Some Symmetric Lévy Processes (Panki Kim, Renming Song and Zoran Vondraček)Level Statistics of One-Dimensional Schrödinger Operators with Random Decaying Potential (Shinichi Kotani and Fumihiko Nakano)Perturbation of the Loop Measure (Yves Le Jan and Jay Rosen)Regular Subspaces of Dirichlet Forms (Liping Li and Jiangang Ying)Quasi-Regular Semi-Dirichlet Forms and Beyond (Zhi-Ming Ma, Wei Sun and Li-Fei Wang)Large Deviation Estimates for Controlled Semi-Martingales (Hideo Nagai)A Comparison Theorem for Backward SPDEs with Jumps (Bernt Øksendal, Agnès Sulem and Tusheng Zhang)On a Construction of a Space-Time Diffusion Process with Boundary Condition (Yoichi Oshima)Lower Bounded Semi-Dirichlet Forms Associated with Lévy Type Operators (René L Schilling and Jian Wang)Ultracontractivity for Non-Symmetric Markovian Semigroups (Ichiro Shigekawa)Metric Measure Spaces with Variable Ricci Bounds and Couplings of Brownian Motions (Karl-Theodor Sturm)Intrinsic Ultracontractivity and Semi-Small Perturbation for Skew Product Diffusion Operators (Matsuyo Tomisaki) Readership: Researchers in probability, stochastic analysis and mathematical physics. Key Features:Research papers by leading expertsHistorical account of M Fukushima's contribution to mathematicsAuthoritative surveys on the state of the art in the fieldKeywords:Probability Theory;Markov Processes;Dirichlet Forms;Potential Theory;Mathematical Physics
This volume contains twenty contributions in the area of mathematical physics where Fritz Gesztesy made profound contributions. There are three survey papers in spectral theory, differential equations, and mathematical physics, which highlight, in particu
Author: Peter K. Friz
Publisher: Cambridge University Press
Release Date: 2010-02-04
Rough path analysis provides a fresh perspective on Ito's important theory of stochastic differential equations. Key theorems of modern stochastic analysis (existence and limit theorems for stochastic flows, Freidlin-Wentzell theory, the Stroock-Varadhan support description) can be obtained with dramatic simplifications. Classical approximation results and their limitations (Wong-Zakai, McShane's counterexample) receive 'obvious' rough path explanations. Evidence is building that rough paths will play an important role in the future analysis of stochastic partial differential equations and the authors include some first results in this direction. They also emphasize interactions with other parts of mathematics, including Caratheodory geometry, Dirichlet forms and Malliavin calculus. Based on successful courses at the graduate level, this up-to-date introduction presents the theory of rough paths and its applications to stochastic analysis. Examples, explanations and exercises make the book accessible to graduate students and researchers from a variety of fields.
This work considers a small random perturbation of alpha-stable jump type nonlinear reaction-diffusion equations with Dirichlet boundary conditions over an interval. It has two stable points whose domains of attraction meet in a separating manifold with several saddle points. Extending a method developed by Imkeller and Pavlyukevich it proves that in contrast to a Gaussian perturbation, the expected exit and transition times between the domains of attraction depend polynomially on the noise intensity in the small intensity limit. Moreover the solution exhibits metastable behavior: there is a polynomial time scale along which the solution dynamics correspond asymptotically to the dynamic behavior of a finite-state Markov chain switching between the stable states.
Author: Peter K. Friz
Release Date: 2014-08-26
Lyons’ rough path analysis has provided new insights in the analysis of stochastic differential equations and stochastic partial differential equations, such as the KPZ equation. This textbook presents the first thorough and easily accessible introduction to rough path analysis. When applied to stochastic systems, rough path analysis provides a means to construct a pathwise solution theory which, in many respects, behaves much like the theory of deterministic differential equations and provides a clean break between analytical and probabilistic arguments. It provides a toolbox allowing to recover many classical results without using specific probabilistic properties such as predictability or the martingale property. The study of stochastic PDEs has recently led to a significant extension – the theory of regularity structures – and the last parts of this book are devoted to a gentle introduction. Most of this course is written as an essentially self-contained textbook, with an emphasis on ideas and short arguments, rather than pushing for the strongest possible statements. A typical reader will have been exposed to upper undergraduate analysis courses and has some interest in stochastic analysis. For a large part of the text, little more than Itô integration against Brownian motion is required as background.
Author: Nicholas J. Higham
Publisher: Princeton University Press
Release Date: 2015-09-15
This is the most authoritative and accessible single-volume reference book on applied mathematics. Featuring numerous entries by leading experts and organized thematically, it introduces readers to applied mathematics and its uses; explains key concepts; describes important equations, laws, and functions; looks at exciting areas of research; covers modeling and simulation; explores areas of application; and more. Modeled on the popular Princeton Companion to Mathematics, this volume is an indispensable resource for undergraduate and graduate students, researchers, and practitioners in other disciplines seeking a user-friendly reference book on applied mathematics. Features nearly 200 entries organized thematically and written by an international team of distinguished contributors Presents the major ideas and branches of applied mathematics in a clear and accessible way Explains important mathematical concepts, methods, equations, and applications Introduces the language of applied mathematics and the goals of applied mathematical research Gives a wide range of examples of mathematical modeling Covers continuum mechanics, dynamical systems, numerical analysis, discrete and combinatorial mathematics, mathematical physics, and much more Explores the connections between applied mathematics and other disciplines Includes suggestions for further reading, cross-references, and a comprehensive index
Originally published in 2 volumes, this text develops the theory of systems of stochastic differential equations and presents applications in probability, partial differential equations, and stochastic control problems. 1975 edition.
Author: Gennadiy Zhegunov
Publisher: Springer Science & Business Media
Release Date: 2005-07-25
Life is a diverse and ubiquitous phenomenon on Earth, characterized by fundamental features distinguishing living bodies from nonliving material. Yet it is also so complex that it has long defied precise definition. This book from a seasoned biologist offers new insights into the nature of life by illuminating a fascinating architecture of dualities inherent in its existence and propagation. Life is connected with individual living beings, yet it is also a collective and inherently global phenomenon of the material world. It embodies a dual existence of cycles of phenotypic life, and their unseen driver — an uninterrupted march of genetic information whose collective immortality is guaranteed by individual mortality. Although evolution propagates and tunes species of organisms, the beings produced can be regarded merely as tools for the survival and cloning of genomes written in an unchanging code. What are the physical versus informational bases and driving forces of life, and how do they unite as an integrated system? What does time mean for individuals, life on the global scale, and the underlying information? This accessible examination of principles and evidence shows that a network of dualities lies at the heart of biological puzzles that have engaged the human mind for millennia.
Author: Giuseppe Da Prato
Publisher: Cambridge University Press
Release Date: 2002-07-25
Second order linear parabolic and elliptic equations arise frequently in mathematics and other disciplines. For example parabolic equations are to be found in statistical mechanics and solid state theory, their infinite dimensional counterparts are important in fluid mechanics, mathematical finance and population biology, whereas nonlinear parabolic equations arise in control theory. Here the authors present a state of the art treatment of the subject from a new perspective. The main tools used are probability measures in Hilbert and Banach spaces and stochastic evolution equations. There is then a discussion of how the results in the book can be applied to control theory. This area is developing very rapidly and there are numerous notes and references that point the reader to more specialised results not covered in the book. Coverage of some essential background material will help make the book self-contained and increase its appeal to those entering the subject.