Stochastic Partial Differential Equations: Analysis and Numerical Approximations

297 Pages·2015·1.39 MB·English

Stochastic Partial Differential Equations:
Analysis and Numerical Approximations
Arnulf Jentzen
September 14, 2015
2
Preface
These lecture notes have been written for the course “401-4606-00L Numerical Anal-
ysis of Stochastic Partial Differential Equations” in the spring semester 2014 and
in the spring semester 2015. These lecture notes are far away from being complete
and remain under construction. In particular, these lecture notes do not yet con-
tain a suitable comparison of the presented material with existing results, arguments
and notions in the literature. This will be the subject of a future version of these
lecture notes. Furthermore, these lecture notes do not contain a number of proofs,
arguments and intuitions. For most of this additional material, the reader is re-
ferred to the lectures of the course “401-4606-00L Numerical Analysis of Stochastic
Partial Differential Equations” in the spring semester 2014. Sonja Cox and Ryan
Kurniawan are gratefully acknowledged for their very helpful advice and assistance,
especially for their help with the Matlab programs. Daniel Conus is also gratefully
acknowledged for several comments that helped to improve the presentation of the
results. In addition, we thank Antti Knowles for fruitful discussions on white noise.
The students of the course “401-4606-00L Numerical Analysis of Stochastic Partial
Differential Equations” in the spring semester 2014 are gratefully acknowledged for
pointing out a number of misprints to me. Special thanks are due to Timo Welti for
bringing a number of misprints to my notice.
Zu¨rich, February 2015
Arnulf Jentzen
3
Exercises
Solutions to the exercises can be turned in the designated mailbox in the anteroom
HG G 53.x.
Exercise Exercises Deadline
sheet
1 Exercises 1.1.8, 1.1.9, 2.2.6, and 2.4.4 05.03.2015, 10:15 AM
2 Exercises 2.5.17, 2.5.20, 3.3.23, 3.5.2, and 3.5.3 19.03.2015, 10:15 AM
3 Exercises 3.5.4, 3.5.15, 3.5.21, and 4.2.5 01.04.2015, 10:15 AM
4 Exercises 4.3.4, 4.7.8, 5.3.22, and 6.1.6 15.04.2015, 10:15 AM
5 Exercises 6.2.9, 6.2.11, 6.2.13, 6.2.14, and 6.2.21 24.04.2015, 10:15 AM
6 Exercises 7.1.15, 8.1.14, and 8.1.15 07.05.2015, 10:15 AM
7 Exercises 7.1.16, 7.2.2, and 8.2.5 14.05.2015, 10:15 AM
8 Exercises 8.1.6, 9.1.6, and 9.4.1 21.05.2015, 10:15 AM
4
Contents
I Foundations in mathematical analysis 13
1 Gronwall-type inequalities 15
1.1 Properties of the beta and the gamma function . . . . . . . . . . . . 15
1.1.1 Functional equation of the gamma function . . . . . . . . . . . 16
1.1.2 Monotonicity properties of the gamma and the beta function . 16
1.1.3 Upper bounds for sums containing the beta and the gamma
function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2 Integral operators related to the beta function . . . . . . . . . . . . . 19
1.3 Generalized exponential-type functions . . . . . . . . . . . . . . . . . 21
1.4 Generalized Gronwall-type inequalities . . . . . . . . . . . . . . . . . 21
1.4.1 Gronwall-type inequalities with a singularity at the initial time 24
1.4.2 Gronwall-type inequalities without a singularity at the initial
time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Regularity of nonlinear functions 27
2.1 General functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.1 Nonlinear characterization of the Borel sigma-algebra . . . . . 28
2.2.2 Pointwise limits of measurable functions . . . . . . . . . . . . 29
2.2.3 Lˆp-sets of measurable functions for p P r0,8q . . . . . . . . . . 31
2.3 Simple functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Strongly measurable functions . . . . . . . . . . . . . . . . . . . . . . 32
2.4.1 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.2 Strongly measurable functions . . . . . . . . . . . . . . . . . . 33
2.4.3 Pointwise approximations of strongly measurable functions . . 34
2.4.4 Sums of strongly measurable functions . . . . . . . . . . . . . 36
2.4.5 Lp-spaces of strongly measurable functions for p P r0,8q . . . 37
2.5 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5
6 CONTENTS
2.5.1 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5.2 Semi-metric spaces . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5.3 Continuity properties of functions . . . . . . . . . . . . . . . . 41
2.5.4 Modulus of continuity . . . . . . . . . . . . . . . . . . . . . . 42
2.5.5 Extensions of uniformly continuous functions . . . . . . . . . . 42
3 Linear functions 45
3.1 Linear spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 An intermezzo on sums over possibly uncountable index sets . . . . . 46
3.2.1 Fubini’s theorem in the case of non-sigma-finite measure spaces 46
3.2.2 Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.2.1 Confinal sequences . . . . . . . . . . . . . . . . . . . 47
3.2.3 Sums over possibly uncountable index sets . . . . . . . . . . . 49
3.2.4 Fubini for sums . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3.1 Best approximations and projections in Hilbert spaces . . . . 53
3.3.2 Examples of orthonormal bases . . . . . . . . . . . . . . . . . 54
3.3.2.1 Trigonometric functions . . . . . . . . . . . . . . . . 54
3.3.2.2 Orthonormal basis in L2pBorel ;|¨| q . . . . . . . . 55
p0,1q R
3.3.2.3 Transformations of orthonormal bases . . . . . . . . 61
3.4 Linear functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.4.1 Continuous linear functions on normed vector spaces . . . . . 63
3.4.2 Compact operators on Banach spaces . . . . . . . . . . . . . . 65
3.4.3 Nuclear operators on Banach spaces . . . . . . . . . . . . . . . 65
3.4.3.1 Definition of Nuclear operators . . . . . . . . . . . . 65
3.4.3.2 Relation of bounded linear operators and nuclear op-
erators . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4.3.3 Structure of the space of nuclear operators . . . . . . 68
3.4.3.4 Ideal property of the set of nuclear operators . . . . 69
3.4.3.5 Characterization of nuclear operators . . . . . . . . . 70
3.4.4 Hilbert-Schmidt operators on Hilbert spaces . . . . . . . . . . 70
3.4.4.1 Independence of the orthonormal basis . . . . . . . . 70
3.4.4.2 The Hilbert space of Hilbert-Schmidt operators . . . 71
3.4.4.3 Hilbert-Schmidt embeddings . . . . . . . . . . . . . . 72
3.5 Diagonal linear operators on Hilbert spaces . . . . . . . . . . . . . . . 73
3.5.1 Laplace operators on bounded domains . . . . . . . . . . . . . 74
3.5.1.1 Laplace operators with Dirichlet boundary conditions 75
3.5.1.2 Laplace operators with Neumann boundary conditions 76
CONTENTS 7
3.5.1.3 Laplace operators with periodic boundary conditions 77
3.5.2 Spectral decomposition for a diagonal linear operator . . . . . 78
3.5.3 Fractional powers of a diagonal linear operator . . . . . . . . . 80
3.5.4 Domain Hilbert space associated to a diagonal linear operator 81
3.5.5 Interpolation spaces associated to a diagonal linear operator . 82
3.6 The Bochner integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.6.1 Existence and uniqueness of the Bochner integral . . . . . . . 84
3.6.2 Definition of the Bochner integral . . . . . . . . . . . . . . . . 85
4 Semigroups of bounded linear operators 87
4.1 Definition of a semigroup of bounded linear operators . . . . . . . . . 87
4.2 Types of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.3 The generator of a semigroup . . . . . . . . . . . . . . . . . . . . . . 88
4.4 A global a priori bound for semigroups . . . . . . . . . . . . . . . . . 90
4.5 Strongly continuous semigroups . . . . . . . . . . . . . . . . . . . . . 90
4.5.1 A priori bounds for strongly continuous semigroups . . . . . . 90
4.5.2 Pointwise convergence in the space of bounded linear operators 92
4.5.3 Existence of solutions of linear ordinary differential equations
in Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.5.4 Domains of generators of strongly continuous semigroups . . . 94
4.5.5 Generators of strongly continuous semigroups . . . . . . . . . 95
4.5.6 A generalization of matrix exponentials to infinite dimensions 97
4.5.7 A characterization of strongly continuous semigroups . . . . . 98
4.6 Uniformly continuous semigroups . . . . . . . . . . . . . . . . . . . . 98
4.6.1 Matrix exponential in Banach spaces . . . . . . . . . . . . . . 99
4.6.2 Continuous invertibility of bounded linear operators in Banach
spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.6.3 Generators of uniformly continuous semigroup . . . . . . . . . 102
4.6.4 A characterization result for uniformly continuous semigroups 103
4.6.5 An a priori bound for uniformly continuous semigroups . . . . 104
4.7 Semigroups generated by diagonal operators . . . . . . . . . . . . . . 105
4.7.1 Semigroup generated by the Laplace operator . . . . . . . . . 107
4.7.2 Smoothing effect of the semigroup . . . . . . . . . . . . . . . . 108
II Foundations in probability theory 111
5 Random variables with values in infinite dimensional spaces 113
5.1 Borel sigma-algebras on normed vector spaces . . . . . . . . . . . . . 113
8 CONTENTS
5.1.1 The Hahn-Banach theorem . . . . . . . . . . . . . . . . . . . . 113
5.1.2 Norm representations in normed vector spaces . . . . . . . . . 114
5.1.3 Linear characterization of the Borel sigma-algebra . . . . . . . 115
5.2 Measures on normed vector spaces . . . . . . . . . . . . . . . . . . . . 116
5.2.1 Uniqueness theorem for measures . . . . . . . . . . . . . . . . 116
5.2.2 Fourier transform of a measure . . . . . . . . . . . . . . . . . 117
5.2.2.1 Characteristic functionals . . . . . . . . . . . . . . . 117
5.2.2.2 Fourier transform on separable normed vector spaces 118
5.2.2.3 Almost surely separably supported . . . . . . . . . . 119
5.2.2.4 Trace set . . . . . . . . . . . . . . . . . . . . . . . . 120
5.2.2.5 Fourier transform on normed vector spaces . . . . . . 123
5.2.3 Covariance of a measure . . . . . . . . . . . . . . . . . . . . . 125
5.2.3.1 The Baire category theorem on complete metric spaces125
5.2.3.2 Regularities for correlations on normed vector spaces 125
5.2.3.3 Covariances of measures and random variables . . . . 128
5.2.4 Gaussian measures on normed vector spaces . . . . . . . . . . 129
5.2.4.1 Fourier transform of a Gaussian measure . . . . . . . 131
5.3 Probability measures on Hilbert spaces . . . . . . . . . . . . . . . . . 132
5.3.1 Nuclear operators on Hilbert spaces . . . . . . . . . . . . . . . 132
5.3.2 Expectation and covariance operator . . . . . . . . . . . . . . 136
5.3.3 Karhunen-Lo`eve expansion . . . . . . . . . . . . . . . . . . . . 138
5.3.4 Gaussian measures on Hilbert spaces . . . . . . . . . . . . . . 139
5.3.4.1 Karhunen-Lo`eve expansion . . . . . . . . . . . . . . 139
5.3.4.2 Construction of Gaussian measures on Hilbert spaces 140
5.3.4.3 Karhunen-Lo`eve expansion for Brownian motion . . 144
6 Stochastic processes 153
6.1 Hilbert space valued stochastic processes . . . . . . . . . . . . . . . . 153
6.1.1 Standard Wiener processes . . . . . . . . . . . . . . . . . . . . 153
6.1.2 Pseudo inverse . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.2 Stochastic Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.2.1 Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.2.2 Lenglart’s inequality . . . . . . . . . . . . . . . . . . . . . . . 158
6.2.3 Modifications and indistinguishability . . . . . . . . . . . . . . 162
6.2.4 Predictability . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.2.5 Construction of the stochastic integral . . . . . . . . . . . . . 165
6.2.6 Elementary processes revisited . . . . . . . . . . . . . . . . . . 174
6.2.7 Cylindrical Wiener process . . . . . . . . . . . . . . . . . . . . 176
CONTENTS 9
III Stochastic Partial Differential Equations (SPDEs) 179
7 Solutions of SPDEs 181
7.1 Existence, uniqueness and properties of mild solutions of SPDEs . . . 181
7.1.1 Mild solutions of SPDEs . . . . . . . . . . . . . . . . . . . . . 181
7.1.2 A setting for SPDEs with globally Lipschitz continuous non-
linearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.1.3 A strong perturbation estimate for SPDEs . . . . . . . . . . . 184
7.1.4 Uniqueness of mild solutions of SPDEs . . . . . . . . . . . . . 188
7.1.4.1 UniquenessofpredictablemildsolutionsofSEEswith
globally Lipschitz continuous coefficients . . . . . . . 188
7.1.4.2 Uniqueness of left-continuous mild solutions of SEEs
with semi-globally Lipschitz continuous coefficients . 188
7.1.5 Existence and regularity of mild solutions of SPDEs . . . . . . 192
7.1.6 A priori bounds for mild solutions of SPDEs . . . . . . . . . . 192
7.1.7 Temporal-regularity of solution processes of SPDEs . . . . . . 197
7.1.8 Existence of continuous solutions . . . . . . . . . . . . . . . . 198
7.2 Examples of SPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
7.2.1 Second order SPDEs . . . . . . . . . . . . . . . . . . . . . . . 200
IV Numerical Analysis of SPDEs 205
8 Strong numerical approximations for SPDEs 207
8.1 Spatial spectral Galerkin approximations for SPDEs . . . . . . . . . . 207
8.1.1 Galerkin projections . . . . . . . . . . . . . . . . . . . . . . . 207
8.1.2 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
8.1.3 A strong numerical approximation result for spectral Galerkin
approximations of SPDEs . . . . . . . . . . . . . . . . . . . . 214
8.2 Temporal numerical approximations for SPDEs . . . . . . . . . . . . 218
8.2.1 Euler type approximations for SPDEs . . . . . . . . . . . . . . 219
8.2.1.1 Exponential Euler method . . . . . . . . . . . . . . . 219
8.2.1.2 Accelerated exponential Euler method . . . . . . . . 220
8.2.1.3 Linear-implicit Euler method . . . . . . . . . . . . . 221
8.2.1.4 Linear-implicit Crank-Nicolson-Euler method . . . . 222
8.2.2 Nonlinearity-stopped Euler type approximations for SPDEs . . 223
8.2.2.1 Nonlinearity-stopped exponential Euler method . . . 224
8.2.2.2 Nonlinearity-stopped linear-implicit Euler method . . 225
8.2.3 Milstein type approximations for SPDEs . . . . . . . . . . . . 226
10 CONTENTS
8.2.3.1 Exponential Milstein method . . . . . . . . . . . . . 226
8.2.3.2 Linear-implicit Milstein method . . . . . . . . . . . . 228
8.2.3.3 Linear-implicit Crank-Nicolson-Milstein method . . . 229
8.2.4 Strong convergence analysis for exponential Euler approxima-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
8.3 Noise approximations for SPDEs . . . . . . . . . . . . . . . . . . . . 241
8.3.1 Noise perturbation estimates . . . . . . . . . . . . . . . . . . . 241
8.3.2 Noise approximations for SPDEs . . . . . . . . . . . . . . . . 242
8.4 Full discretizations for SPDEs . . . . . . . . . . . . . . . . . . . . . . 245
8.4.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
8.4.2 Full-discrete spectral Galerkin exponential Euler method for
SPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
8.4.3 Full-discrete spectral Galerkin linear-implicit Euler method for
SPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
8.4.4 Full-discrete spectral Galerkin nonlinearity-stopped exponen-
tial Euler method for SPDEs . . . . . . . . . . . . . . . . . . . 250
8.4.5 Full-discretespectralGalerkinnonlinearity-stoppedlinear-implicit
Euler method for SPDEs . . . . . . . . . . . . . . . . . . . . . 251
9 Weak numerical approximations for SPDEs 255
9.1 An Itˆo type formula for SPDEs . . . . . . . . . . . . . . . . . . . . . 255
9.1.1 A setting for mild stochastic calculus . . . . . . . . . . . . . . 255
9.1.2 Mild stochastic processes . . . . . . . . . . . . . . . . . . . . . 256
9.1.3 Mild Itoˆ formula . . . . . . . . . . . . . . . . . . . . . . . . . 257
9.2 Solution processes of SPDEs . . . . . . . . . . . . . . . . . . . . . . . 260
9.3 Transformations of semigroups of solutions of SPDEs . . . . . . . . . 261
9.4 Weak convergence for temporal numerical approximations for SPDEs 263
9.5 Weak convergence of Galerkin projections for SPDEs . . . . . . . . . 263
9.5.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
9.5.2 Weak convergence for spatial spectral Galerkin projections . . 264
10 Additional material 269
10.1 Egorov’s theorem on almost uniform convergence . . . . . . . . . . . 269
10.1.1 General measure spaces . . . . . . . . . . . . . . . . . . . . . 269
10.1.1.1 Almost sure convergence . . . . . . . . . . . . . . . . 269
10.1.1.2 Luzin uniform-type convergence . . . . . . . . . . . . 271
10.1.1.3 Almost uniform convergence . . . . . . . . . . . . . . 271
10.1.2 Finite measure spaces . . . . . . . . . . . . . . . . . . . . . . 272