From Lévy-Type Processes to Parabolic SPDEs / by Davar Khoshnevisan, René Schilling ; edited by Frederic Utzet, Lluis Quer-Sardanyons.

Format:

Book

Author/Creator:

Khoshnevisan, Davar., Author.

Schilling, René., Author.

Contributor:

Utzet, Frederic., Editor.

Quer-Sardanyons, Lluis., Editor.

Series:

Advanced Courses in Mathematics - CRM Barcelona, 2297-0304

Language:

English

Subjects (All):

Probabilities.

Differential equations, Partial.

Probability Theory and Stochastic Processes.

Partial Differential Equations.

Local Subjects:

Probability Theory and Stochastic Processes.

Partial Differential Equations.

Physical Description:

1 online resource (VIII, 219 p.)

Edition:

1st ed. 2016.

Place of Publication:

Cham : Springer International Publishing : Imprint: Birkhäuser, 2016.

Summary:

This volume presents the lecture notes from two courses given by Davar Khoshnevisan and René Schilling, respectively, at the second Barcelona Summer School on Stochastic Analysis. René Schilling’s notes are an expanded version of his course on Lévy and Lévy-type processes, the purpose of which is two-fold: on the one hand, the course presents in detail selected properties of the Lévy processes, mainly as Markov processes, and their different constructions, eventually leading to the celebrated Lévy-Itô decomposition. On the other, it identifies the infinitesimal generator of the Lévy process as a pseudo-differential operator whose symbol is the characteristic exponent of the process, making it possible to study the properties of Feller processes as space inhomogeneous processes that locally behave like Lévy processes. The presentation is self-contained, and includes dedicated chapters that review Markov processes, operator semigroups, random measures, etc. In turn, Davar Khoshnevisan’s course investigates selected problems in the field of stochastic partial differential equations of parabolic type. More precisely, the main objective is to establish an Invariance Principle for those equations in a rather general setting, and to deduce, as an application, comparison-type results. The framework in which these problems are addressed goes beyond the classical setting, in the sense that the driving noise is assumed to be a multiplicative space-time white noise on a group, and the underlying elliptic operator corresponds to a generator of a Lévy process on that group. This implies that stochastic integration with respect to the above noise, as well as the existence and uniqueness of a solution for the corresponding equation, become relevant in their own right. These aspects are also developed and supplemented by a wealth of illustrative examples.

Contents:

Intro

Contents

Part I: An Introduction to Lévy and Feller Processes

Preface

Symbols and Notation

Chapter 1: Orientation

Chapter 2: Lévy Processes

Chapter 3: Examples

Chapter 4: On the Markov Property

Chapter 5: A Digression: Semigroups

Chapter 6: The Generator of a Lévy Process

Chapter 7: Construction of Lévy Processes

Chapter 8: Two Special Lévy Processes

Chapter 9: Random Measures

Chapter 10: A Digression: Stochastic Integrals

Chapter 11: From Lévy to Feller Processes

Chapter 12: Symbols and Semimartingales

Chapter 13: Dénouement

Appendix: Some Classical Results

The Cauchy-Abel functional equation

Characteristic functions and moments

Vague and weak convergence of measures

Convergence in distribution

The predictable σ-algebra

The structure of translation invariant operators

Bibliography

Part II: Invariance and Comparison Principles for Parabolic Stochastic Partial Differential Equations

Chapter 14: White Noise

14.1 Some heuristics

14.2 LCA groups

14.3 White noise on G

14.4 Space-time white noise

14.5 The Walsh stochastic integral

14.5.1 Simple random fields

14.5.2 Elementary random fields

14.5.3 Walsh-integrable random fields

14.6 Moment inequalities

14.7 Examples of Walsh-integrable random fields

14.7.1 Integral kernels

14.7.2 Stochastic convolutions

14.7.3 Relation to Itô integrals

Chapter 15: Lévy Processes

15.1 Introduction

15.1.1 Lévy processes on R

15.1.2 Lévy processes on T

15.1.3 Lévy processes on Z

15.1.4 Lévy processes on Z/nZ

15.2 The semigroup

15.3 The Kolmogorov-Fokker-Planck equation

15.3.1 Lévy processes on R

Chapter 16: SPDEs

16.1 A heat equation

16.2 A parabolic SPDE

16.2.1 Lévy processes on R

16.2.2 Lévy processes on a denumerable LCA group.

16.2.3 Proof of Theorem 16.2.2

16.3 Examples

16.3.1 The trivial group

16.3.2 The cyclic group on two elements

16.3.3 The integer group

16.3.4 The additive reals

16.3.5 Higher dimensions

Chapter 17: An Invariance Principle for Parabolic SPDEs

17.1 A central limit theorem

17.2 A local central limit theorem

17.3 Particle systems

Chapter 18: Comparison Theorems

18.1 Positivity

18.2 The Cox-Fleischmann-Greven inequality

18.3 Slepian's inequality

Chapter 19: A Dash of Color

19.1 Reproducing kernel Hilbert spaces

19.2 Colored noise

19.2.1 Example: white noise

19.2.2 Example: Hilbert-Schmidt covariance

19.2.3 Example: spatially-homogeneous covariance

19.2.4 Example: tensor-product covariance

19.3 Linear SPDEs with colored-noise forcing

Index.

Notes:

Includes index.

Description based on publisher supplied metadata and other sources.

ISBN:

3-319-34120-0

OCLC:

974295863