A self-avoiding walk is a path on a lattice that does not visit the same site more than once. In spite of this simple definition, many of the most basic questions about this model are difficult to resolve in a mathematically rigorous fashion. In particular, we do not know much about how far an n- step self-avoiding walk typically travels from its starting point, or even how many such walks there are. These and other important questions about the self-avoiding walk remain unsolved in the rigorous mathematical sense, although the physics and chemistry communities have reached consensus on the...
A self-avoiding walk is a path on a lattice that does not visit the same site more than once. In spite of this simple definition, many of the most bas...
This book is based on research that, to a large extent, started around 1990, when a research project on fluid flow in stochastic reservoirs was initiated by a group including some of us with the support of VISTA, a research coopera- tion between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap A.S. (Statoil). The purpose of the project was to use stochastic partial differential equations (SPDEs) to describe the flow of fluid in a medium where some of the parameters, e.g., the permeability, were stochastic or "noisy." We soon realized that the theory of SPDEs at...
This book is based on research that, to a large extent, started around 1990, when a research project on fluid flow in stochastic reservoirs was initia...
A hundred years ago it became known that deterministic systems can exhibit very complex behavior. By proving that ordinary differential equations can exhibit strange behavior, Poincare undermined the founda- tions of Newtonian physics and opened a window to the modern theory of nonlinear dynamics and chaos. Although in the 1930s and 1940s strange behavior was observed in many physical systems, the notion that this phenomenon was inherent in deterministic systems was never suggested. Even with the powerful results of S. Smale in the 1960s, complicated be- havior of deterministic systems...
A hundred years ago it became known that deterministic systems can exhibit very complex behavior. By proving that ordinary differential equations can ...
This book studies the foundations of the theory of linear and nonlinear forms in single and multiple random variables including the single and multiple random series and stochastic integrals, both Gaussian and non-Gaussian. This subject is intimately connected with a number of classical problems of probability theory such as the summation of independent random variables, martingale theory, and Wiener's theory of polynomial chaos. The book contains a number of older results as well as more recent, or previously unpublished, results. The emphasis is on domination principles for comparison of...
This book studies the foundations of the theory of linear and nonlinear forms in single and multiple random variables including the single and multipl...
The book aims at presenting a detailed and mathematically rigorous exposition of the theory and applications of a class of point processes and piecewise deterministic p- cesses. The framework is suf?ciently general to unify the treatment of several classes of stochastic phenomena: point processes, Markov chains and other Markov processes in continuous time, semi-Markov processes, queueing and storage models, and li- lihood processes. There are applications to ?nance, insurance and risk, population models, survival analysis, and congestion models. A major aim has been to show the versatility...
The book aims at presenting a detailed and mathematically rigorous exposition of the theory and applications of a class of point processes and piecewi...
Safety critical and high-integrity systems, such as industrial plants and economic systems can be subject to abrupt changes - for instance due to component or interconnection failure, and sudden environment changes etc.
Combining probability and operator theory, Discrete-Time Markov Jump Linear Systems provides a unified and rigorous treatment of recent results for the control theory of discrete jump linear systems, which are used in these areas of application.
The book is designed for experts in linear systems with Markov jump parameters, but is also of...
Safety critical and high-integrity systems, such as industrial plants and economic systems can be subject to abrupt changes - for instance due to c...
There have been ten years since the publication of the ?rst edition of this book. Since then, new applications and developments of the Malliavin c- culus have appeared. In preparing this second edition we have taken into account some of these new applications, and in this spirit, the book has two additional chapters that deal with the following two topics: Fractional Brownian motion and Mathematical Finance. The presentation of the Malliavin calculus has been slightly modi?ed at some points, where we have taken advantage of the material from the lecturesgiveninSaintFlourin1995(seereference...
There have been ten years since the publication of the ?rst edition of this book. Since then, new applications and developments of the Malliavin c- cu...
The purpose of this book is to present a comprehensive account of the different definitions of stochastic integration for fBm, and to give applications of the resulting theory. Particular emphasis is placed on studying the relations between the different approaches. Readers are assumed to be familiar with probability theory and stochastic analysis, although the mathematical techniques used in the book are thoroughly exposed and some of the necessary prerequisites, such as classical white noise theory and fractional calculus, are recalled in the appendices. This book will be a valuable...
The purpose of this book is to present a comprehensive account of the different definitions of stochastic integration for fBm, and to give applicat...
Limit theorems for stochastic processes are an important part of probability theory and mathematical statistics and one model that has attracted the attention of many researchers working in the area is that of limit theorems for randomly stopped stochastic processes.This volume is the first to present a state-of-the-art overview of this field, with many of the results published for the first time. It covers the general conditions as well as the basic applications of the theory, and it covers and demystifies the vast, and technically demanding, Russian literature in detail. A survey of the...
Limit theorems for stochastic processes are an important part of probability theory and mathematical statistics and one model that has attracted the a...
A beautiful interplay between probability theory (Markov processes, martingale theory) on the one hand and operator and spectral theory on the other yields a uniform treatment of several kinds of Hamiltonians such as the Laplace operator, relativistic Hamiltonian, Laplace-Beltrami operator, and generators of Ornstein-Uhlenbeck processes. For such operators regular and singular perturbations of order zero and their spectral properties are investigated. A complete treatment of the Feynman-Kac formula is given. The theory is applied to such topics as compactness or trace class properties...
A beautiful interplay between probability theory (Markov processes, martingale theory) on the one hand and operator and spectral theory on the othe...
