From Lévy-Type Processes to Parabolic SPDEs

212 Pages·2016·2.673 MB·English

by  Davar Khoshnevisan, René Schilling (auth.), Frederic Utzet, Lluis Quer-Sardanyons (eds.)

Advanced Courses in Mathematics CRM Barcelona
Davar Khoshnevisan
René Schilling
From Lévy-Type
Processes to
Parabolic SPDEs

Advanced Courses in Mathematics
CRM Barcelona
Centre de Recerca Matemàtica
Managing Editor:
Enric Ventura

More information about this series at http://www.springer.com/series/5038
Davar Khoshnevisan • René Schilling
From Lévy-Type Processes
to Parabolic SPDEs
Editors for this volume:
Lluís Quer-Sardanyons, Universitat Autònoma de Barcelona
Frederic Utzet, Universitat Autònoma de Barcelo na
Davar Khoshnevisan René Schilling
Department of Mathematics Institut für Mathematische Stochastik
University of Utah Fachrichtung Mathematik
Salt Lake City, Utah, USA TU Dresden
Dresden, Germany

ISSN 2297-0304 ISSN2 297-0312 (electronic)
Advanced Courses in Mathematics - CRM Barcelona
ISBN 978-3-319-34119-4 ISBN 978-3-319-34120-0 (eBook)
DOI 10.1007/978-3-319-34120-0

Library of Congress Control Number: 2016962054

Mathematics Subject Classification (2010): 60H15, 60G51, 35K57, 60H20, 60J25, 60J35

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Contents
Part I An Introduction to L´evy and Feller Processes
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Symbols and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 L´evy Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 On the Markov Property . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5 A Digression: Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6 The Generator of a L´evy Process . . . . . . . . . . . . . . . . . . . . . . 41
7 Construction of L´evy Processes . . . . . . . . . . . . . . . . . . . . . . . 49
8 Two Special L´evy Processes . . . . . . . . . . . . . . . . . . . . . . . . . 55
9 Random Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
10 A Digression: Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . 73
11 From L´evy to Feller Processes . . . . . . . . . . . . . . . . . . . . . . . 87
12 Symbols and Semimartingales . . . . . . . . . . . . . . . . . . . . . . . . 99
13 D´enouement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Appendix: Some Classical Results . . . . . . . . . . . . . . . . . . . . . . . 115
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
v
vi Contents
Part II Invariance and Comparison Principles for Parabolic
Stochastic Partial Differential Equations
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
14 White Noise
14.1 Some heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
14.2 LCA groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
14.3 White noise on G . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
14.4 Space-time white noise . . . . . . . . . . . . . . . . . . . . . . . . 139
14.5 The Walsh stochastic integral . . . . . . . . . . . . . . . . . . . . 140
14.5.1 Simple random fields . . . . . . . . . . . . . . . . . . . . . 141
14.5.2 Elementary random fields . . . . . . . . . . . . . . . . . . 142
14.5.3 Walsh-integrable random fields . . . . . . . . . . . . . . . 142
14.6 Moment inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 144
14.7 Examples of Walsh-integrable random fields . . . . . . . . . . . . 145
14.7.1 Integral kernels . . . . . . . . . . . . . . . . . . . . . . . . 145
14.7.2 Stochastic convolutions . . . . . . . . . . . . . . . . . . . 147
14.7.3 Relation to Itˆo integrals . . . . . . . . . . . . . . . . . . . 148
15 L´evy Processes
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
15.1.1 L´evy processes on . . . . . . . . . . . . . . . . . . . . . 152
(cid:0)
15.1.2 L´evy processes on . . . . . . . . . . . . . . . . . . . . . 155
(cid:2)
15.1.3 L´evy processes on . . . . . . . . . . . . . . . . . . . . . 156
(cid:3)
15.1.4 L´evy processes on /n . . . . . . . . . . . . . . . . . . . 156
(cid:3) (cid:3)
15.2 The semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
15.3 The Kolmogorov–Fokker–Planckequation . . . . . . . . . . . . . 158
15.3.1 L´evy processes on . . . . . . . . . . . . . . . . . . . . . 158
(cid:0)
16 SPDEs
16.1 A heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
16.2 A parabolic SPDE . . . . . . . . . . . . . . . . . . . . . . . . . . 169
16.2.1 L´evy processes on . . . . . . . . . . . . . . . . . . . . . 171
(cid:0)
16.2.2 L´evy processes on a denumerable LCA group . . . . . . . 172
16.2.3 Proof of Theorem 16.2.2 . . . . . . . . . . . . . . . . . . . 173
16.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
16.3.1 The trivial group . . . . . . . . . . . . . . . . . . . . . . . 175
16.3.2 The cyclic group on two elements . . . . . . . . . . . . . . 176
16.3.3 The integer group . . . . . . . . . . . . . . . . . . . . . . 178
16.3.4 The additive reals . . . . . . . . . . . . . . . . . . . . . . 179
16.3.5 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . 180
Contents vii
17 An Invariance Principle for Parabolic SPDEs
17.1 A central limit theorem . . . . . . . . . . . . . . . . . . . . . . . 183
17.2 A local central limit theorem . . . . . . . . . . . . . . . . . . . . 185
17.3 Particle systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
18 Comparison Theorems
18.1 Positivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
18.2 The Cox–Fleischmann–Greveninequality . . . . . . . . . . . . . . 196
18.3 Slepian’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 199
19 A Dash of Color
19.1 Reproducing kernel Hilbert spaces . . . . . . . . . . . . . . . . . 203
19.2 Colored noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
19.2.1 Example: white noise . . . . . . . . . . . . . . . . . . . . . 206
19.2.2 Example: Hilbert–Schmidt covariance . . . . . . . . . . . 206
19.2.3 Example: spatially-homogeneouscovariance . . . . . . . . 207
19.2.4 Example: tensor-product covariance . . . . . . . . . . . . 208
19.3 Linear SPDEs with colored-noise forcing . . . . . . . . . . . . . . 210
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
In fond memory of Don Burkholder (1927–2013),
Wenbo V. Li (1963–2013), and Marc Yor (1949–2014).
Part I
An Introduction to L´evy
and Feller Processes
By Ren´e L. Schilling