Analytical methods for Kolmogorov equations:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton, Fla.
CRC Press
[2017]
|
| Edition: | Second edition |
| Series: | Monographs and research notes in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturangaben S. 547 - 562 Enthält Index |
| Physical Description: | xxxix, 566 Seiten |
| ISBN: | 9781482243321 |
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| 100 | 1 | |a Lorenzi, Luca |e Verfasser |0 (DE-588)17394048X |4 aut | |
| 245 | 1 | 0 | |a Analytical methods for Kolmogorov equations |c Luca Lorenzi, University of Parma, Italy |
| 250 | |a Second edition | ||
| 264 | 1 | |a Boca Raton, Fla. |b CRC Press |c [2017] | |
| 264 | 4 | |c © 2017 | |
| 300 | |a xxxix, 566 Seiten | ||
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| 490 | 0 | |a Monographs and research notes in mathematics | |
| 500 | |a Literaturangaben S. 547 - 562 | ||
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Record in the Search Index
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|---|---|
| adam_text | MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS
Analytical
Methods for
Kolmogorov
Equations
Second Edition
Luca Lorenzi
University of Parma
Italy
CRC Press
Taylor amp; Francis Croup
Boca Raton London New York
CRC Press Is an Imprint of the
Taylor amp; Francis Croup, an Informa business
A CHAPMAN amp; HALL BOOK
Contents
Preface to the second edition xv
Preface to the first edition xvii
About the author xxi
Introduction xxiii
I Autonomous Kolmogorov equations 1
1 The elliptic equation and the Cauchy problem in Cb(RN)i the uniformly
elliptic case 3
1 0 Introduction 3
1 1 The elliptic equation and the resolvent R(A) 6
1 2 The Cauchy problem and the semigroup 8
1 3 The weak generator of {T(t)} 15
1 4 Analytic and non-analytic semigroups in Cb(RN) 19
1 5 The Markov process 24
1 6 The associated stochastic differential equation 26
2 One-dimensional theory 29
2 0 Introduction 29
2 1 The homogeneous equation 29
2 2 The nonhomogeneous equation 35
3 Uniqueness results, conservation of probability and maximum principles 43
3 0 Introduction 43
3 1 Conservation of probability and uniqueness 44
311 Maximum principles 45
312 The case when c=0 48
3 2 Non-uniqueness 50
4 Properties of T(t) in spaces of continuous functions 53
4 0 Introduction 53
4 1 Compactness of T(t) 54
411 The conservative case 54
412 The non-conservative case 63
4 2 On the inclusion T(t)(Cb{RN)) C Co(RN) 65
4 3 Invariance of Co(RN) 69
IX
X
Contents
5 Uniform estimates for the derivatives of the function T(t)f 73
5 0 Introduction 73
5 1 Uniform estimates 74
5 2 Some consequences 85
6 Pointwise estimates for the derivatives of the function T(t)f 91
6 0 Introduction 91
6 1 The first type of pointwise gradient estimates 92
6 2 The second type of pointwise gradient estimates 97
6 3 Further estimates when A=A+ (5, V) 104
7 Markov semigroups in Lp-spaces 107
7 0 Introduction 107
7 1 The general case 108
7 2 Schrödinger type operators: the case a 2 117
721 The semigroup (Tp(t)} and the spectrum of operator Ap 125
722 Concluding remarks 129
7 3 Schrödinger type operators: the case a 2 130
731 Concluding remarks 143
7 4 Some slightly more general operators 143
8 Estimates on the Green function 149
8 0 Introduction 149
8 1 The general case 151
811 Bibliographic remarks 158
8 2 The operator A = (1 + |x|a)A 159
8 3 The Schrödinger operator ^=A+c 168
831 The case when c is a decreasing radial potential 179
8 4 The Schrödinger operator A = (1 + |x|a)A + c when a € (0,2) 189
841A more general class of elliptic operators 195
9 The invariant measure /i and the semigroup in Lp(RN, ß) 197
9 0 Introduction 197
9 1 Existence, uniqueness and general properties 199
911 General properties and uniqueness 199
912 Existence by Khas’minskii theorem 206
913 Existence by compactness in Cb(RN) 210
914 Existence by symmetry 213
9 2 Regularity properties of invariant measures 216
921 Global L^-regularity of the density p 219
922 Global Sobolev regularity 224
9 3 Some consequences of the estimates in Chapter 6 230
9 4 Logarithmic Sobolev inequality, Poincare inequality, summability improving
properties 233
941 Concluding remarks 236
95A class of analytic semigroups in LP(RN, ß) 238
9 6 Non-analytic semigroups in LP(RN, //) 248
Contents
xi
10 The Ornstein-Uhlenbeck operator 251
10 0 Introduction 251
10 1 The formula for T(t)f 252
10 2 Properties of (T(i)} in Cb(RN) 254
10 3 The invariant measure ß and the semigroup in LP(RN: ß) 258
10 3 1 The domain of the realization of {T(t)} in LP(RN, ß) 266
10 3 2 The spectrum of Lp 273
10 3 3 Hermite polynomials 279
10 3 4 The sector of analyticity of Lp 282
10 4 The Ornstein-Uhlenbeck operator in LP(RN) 283
11 Degenerate Markov semigroups in R N 287
11 0 Introduction 287
11 1 Remarks on the assumptions and technical results 289
11 1 1 Ordering the derivatives of smooth functions 290
11 1 2 Technical results 290
11 2 Uniform estimates for Ae 295
11 3 Construction of the semigroup 304
11 3 1 Properties of the semigroup 309
11 4 Anisotropic Holder estimates 312
11 4 1 The case r=l 314
12 The Cauchy-Dirichlet problem 323
12 0 Introduction 323
12 1 Existence and uniqueness 324
12 2 Gradient estimates 328
12 21A priori gradient estimates 328
12 2 2 An auxiliary problem 331
12 2 3 Proof of Theorem 12 2 4 337
12 24A counterexample to the gradient estimates 338
13 The Cauchy-Neumann problem 341
13 0 Introduction 341
13 1 A maximum principle and a priori estimates 343
13 2 Existence and uniqueness of a classical solution to problem (13 0 1) 347
13 2 1 Proof of Theorem 13 2 5: the nonconvex case 348
13 211 Generation of analytic semigroups in Lp-spaces 351
13 212 Proof of Theorem 13 2 5 for smooth data 354
13 2 2 Proof of Theorem 13 2 5: the convex case 355
13 2 3 Proof of Theorem 13 2 5: exterior domains 356
13 3 Some properties of the semigroup (T(t)} 357
13 4 The weak generator of the semigroup and the elliptic equation (13 0 2) 359
13 5 Pointwise gradient estimates and their consequences 362
13 6 The invariant measure of the semigroup 369
Contents
xii
13 7 Final remarks 374
II Non-autonomous Kolmogorov equations 375
14 The evolution operator and the evolution semigroup in the space of
bounded and continuous functions 377
14 0 Introduction 377
14 1 The evolution operator and its continuity properties 379
14 2 Compactness of the evolution operator in Cb(M N) 385
14 3 Invariance of CbO^) 388
14 4 Gradient estimates 390
14 4 1 Schauder estimates for nonhomogeneous parabolic problems 394
14 5 The evolution semigroup {^(t)} 396
14 5 1 The weak generator G00 of ( T(£)} 397
14 5 2 An equivalent characterization of D(Goo) 398
14 5 3 The periodic case 401
15 Estimates for Green functions 403
15 0 Introduction 403
15 1 Integrability properties of Green functions 404
15 2 Kernel estimates 413
15 21A concrete application of Theorem 15 2 2 420
15 3 Concluding remarks 424
16 The evolution operator in Lp-spaces 425
16 0 Introduction 425
16 1 The evolution operator in LP(RN) 427
16 2 Evolution systems of measures 427
16 3 The evolution operator in spaces related to evolution systems of measures:
basic properties 436
16 4 Logarithmic Sobolev inequality, Poincare inequality and hypercontractivity 437
16 4 1 Logarithmic Sobolev inequality and consequences 438
16 4 2 Hypercontractivity 443
16 5 Supercontractivity and LSIe 444
16 6 Ultraboundedness 450
16 7 Ultracontractivity 453
17 The evolution semigroups and {«^(t)} in Xp-spaces 459
17 0 Introduction 459
17 1 The general case 461
17 1 1 Cores of the operator Gp 463
17 2 The periodic case 474
17 2 1 Cores 479
17 22A logarithmic Sobolev type inequality and compactness 481
18 The asymptotic behaviour of the evolution operator and the evolution
semigroup 485
18 0 Introduction 485
18 1 The general case 487
18 1 1 Exponential decay to zero 496
18 2 The periodic case 503
Contents
xiii
III Appendices 511
A Function spaces and smooth domains 513
A l Spaces of continuous or Holder continuous functions 513
A11 Isotropic spaces 513
A12 Anisotropic Holder spaces in RN 514
A 2 Parabolic Holder spaces 514
A 3 Lp- and Sobolev spaces 515
A31 The spaces J4?p,1((a,b) x R^) 516
A 4 Some properties of the distance function 519
B Basic notions of functional analysis in Banach spaces 521
B l Linear operators, spectrum and resolvent 521
B 2 Vector-valued Riemann integral 523
B 3 Some results from interpolation theory 524
C An overview of strongly continuous and analytic semigroups 529
C l Strongly continuous semigroups 529
C11 On the closure of the sum of generators of semigroups 532
C 2 Analytic semigroups 533
D PDE’s and analytic semigroups 537
DlA priori estimates 537
D 2 Classical maximum principles 542
D 3 Existence of classical solutions to PDE’s and analytic semigroups 542
Bibliography 547
Index
|
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| isbn | 9781482243321 |
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| spelling | Lorenzi, Luca Verfasser (DE-588)17394048X aut Analytical methods for Kolmogorov equations Luca Lorenzi, University of Parma, Italy Second edition Boca Raton, Fla. CRC Press [2017] © 2017 xxxix, 566 Seiten txt rdacontent n rdamedia nc rdacarrier Monographs and research notes in mathematics Literaturangaben S. 547 - 562 Enthält Index Kolmogorovsche Differentialgleichungen (DE-588)4164698-8 gnd rswk-swf Kolmogorovsche Differentialgleichungen (DE-588)4164698-8 s DE-604 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029306037&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lorenzi, Luca Analytical methods for Kolmogorov equations Kolmogorovsche Differentialgleichungen (DE-588)4164698-8 gnd |
| subject_GND | (DE-588)4164698-8 |
| title | Analytical methods for Kolmogorov equations |
| title_auth | Analytical methods for Kolmogorov equations |
| title_exact_search | Analytical methods for Kolmogorov equations |
| title_full | Analytical methods for Kolmogorov equations Luca Lorenzi, University of Parma, Italy |
| title_fullStr | Analytical methods for Kolmogorov equations Luca Lorenzi, University of Parma, Italy |
| title_full_unstemmed | Analytical methods for Kolmogorov equations Luca Lorenzi, University of Parma, Italy |
| title_short | Analytical methods for Kolmogorov equations |
| title_sort | analytical methods for kolmogorov equations |
| topic | Kolmogorovsche Differentialgleichungen (DE-588)4164698-8 gnd |
| topic_facet | Kolmogorovsche Differentialgleichungen |
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