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2017 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Stochastic Partial Differential Equations

verfasst von: Sergey V. Lototsky, Boris L. Rozovsky

Verlag: Springer International Publishing

Buchreihe : Universitext

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
We use the same notation x for a point in the real line \(\mathbb{R}\) or in a d-dimensional Euclidean space \(\mathbb{R}^{\mathrm{d}}\). For \(x = (x_{1},\ldots,x_{\mathrm{d}}) \in \mathbb{R}^{\mathrm{d}}\), \(\vert x\vert = \sqrt{x_{1 }^{2 } +\ldots +x_{\mathrm{d} }^{2}}\); for \(x,y \in \mathbb{R}^{\mathrm{d}}\), xy = x 1 y 1 + … + x d y d. Integral over the real line can be written either as \(\int _{\mathbb{R}}\) or as ∫ −∞ +∞ . Sometimes, when there is no danger of confusion, the domain of integration, in any number of dimensions, is omitted altogether.
Sergey V. Lototsky, Boris L. Rozovsky
Chapter 2. Basic Ideas
Abstract
Given a probability space \((\varOmega,\mathcal{F}, \mathbb{P})\) and two measurable spaces, \((A,\mathcal{A})\) and \((B,\mathcal{B})\), a random function X is a (measurable) mapping from \((A\times \varOmega,\mathcal{A}\times \mathcal{F})\) to \((B,\mathcal{Y})\). In the traditional terminology, the random process corresponds to \(A,B \subset \mathbb{R}\); a random field corresponds to \(A = \mathbb{R}^{\mathrm{d}}\), \(B = \mathbb{R}\). A sample path , or sample trajectory of X is the function X(⋅ , ω) for fixed ω ∈ Ω. A modification of X is a random function \(\bar{X}\) such that, for every a ∈ A, \(\mathbb{P}\big(X(a) =\bar{ X}(a)\big) = 1\); note that this, in general, DOES NOT mean that \(\mathbb{P}\big(X(a) =\bar{ X}(a)\ \text{for all}\ a \in A\big) = 1\) (although it does if both X and \(\bar{X}\) are continuous).
Sergey V. Lototsky, Boris L. Rozovsky
Chapter 3. Stochastic Analysis in Infinite Dimensions
Abstract
This chapter contains somewhat abstract but necessary, material on functional analysis and stochastic calculus. To save time, one can move on to the following chapters and come back as necessary.
Sergey V. Lototsky, Boris L. Rozovsky
Chapter 4. Linear Equations: Square-Integrable Solutions
Abstract
There are many standard references on SODEs and even more standard references on deterministic PDEs. Here are a few of each, listed in a non-decreasing order of difficulty:
Sergey V. Lototsky, Boris L. Rozovsky
Chapter 5. The Polynomial Chaos Method
Abstract
Separation of variables is a powerful idea in the study of partial differential equations, and the polynomial chaos method is a particular implementation of this idea for stochastic equations. While the elementary outcome ω is typically never mentioned explicitly in the notation of random objects, it is a variable that can potentially be separated from other variables, and the objective of this chapter is to outline a systematic approach to doing just that. Along the way, it quickly becomes clear that many ideas are closely connected to another modern branch of stochastic analysis, namely, Malliavin Calculus, and we explore these connections throughout.
Sergey V. Lototsky, Boris L. Rozovsky
Chapter 6. Parameter Estimation for Diagonal SPDEs
Abstract
Let U = U(t, x) be the temperature of the top layer of a body of water such as lake, sea, or ocean. Various historical data provide information about the long-time average value \(\bar{U}\) of U.
Sergey V. Lototsky, Boris L. Rozovsky
Backmatter
Metadaten
Titel
Stochastic Partial Differential Equations
verfasst von
Sergey V. Lototsky
Boris L. Rozovsky
Copyright-Jahr
2017
Electronic ISBN
978-3-319-58647-2
Print ISBN
978-3-319-58645-8
DOI
https://doi.org/10.1007/978-3-319-58647-2