Positive harmonic functions and diffusion:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge University Press
[1995]
|
| Ausgabe: | First published |
| Schriftenreihe: | Cambridge studies in advanced mathematics
45 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xvi, 474 Seiten |
| ISBN: | 0521470145 9780521470148 9780521059831 9780511526244 |
Internformat
MARC
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| 100 | 1 | |a Pinsky, Ross G. |d 1955- |0 (DE-588)141663146 |4 aut | |
| 245 | 1 | 0 | |a Positive harmonic functions and diffusion |c Ross G. Pinsky |
| 250 | |a First published | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge University Press |c [1995] | |
| 264 | 4 | |c © 1995 | |
| 300 | |a xvi, 474 Seiten | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Cambridge studies in advanced mathematics |v 45 | |
| 650 | 4 | |a Elliptischer Differentialoperator - Harmonische Funktion - Diffusionsprozess | |
| 650 | 0 | 7 | |a Harmonische Funktion |0 (DE-588)4159122-7 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Diffusionsprozess |0 (DE-588)4274463-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Diffusion |0 (DE-588)4012277-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Positive Funktion |0 (DE-588)4175428-1 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Harmonische Funktion |0 (DE-588)4159122-7 |D s |
| 689 | 0 | 1 | |a Diffusion |0 (DE-588)4012277-3 |D s |
| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a Diffusionsprozess |0 (DE-588)4274463-5 |D s |
| 689 | 1 | 1 | |a Harmonische Funktion |0 (DE-588)4159122-7 |D s |
| 689 | 1 | 2 | |a Positive Funktion |0 (DE-588)4175428-1 |D s |
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| 830 | 0 | |a Cambridge studies in advanced mathematics |v 45 |w (DE-604)BV000003678 |9 45 | |
| 856 | 4 | 2 | |m HBZ Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006754785&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-006754785 | |
Datensatz im Suchindex
| _version_ | 1816050205100867584 |
|---|---|
| adam_text |
Contents
Preface xi
List of notation xv
1 Existence and uniqueness for diffusion processes 1
1.1 Stochastic processes and filiations 1
1.2 Conditional expectation, martingales and stopping times 2
1.3 Markov processes and semigroups 4
1.4 Brownian motion 6
1.5 Ito processes 7
1.6 Stochastic integrals 11
1.7 Stochastic differential equations 17
1.8 The formulation of the martingale problem on Rd 25
1.9 The Cameron Martin Girsanov transformation and the mar¬
tingale problem on Rd 27
1.10 The Stroock Varadhan solution to the martingale problem on
Rd 29
1.11 The generalized martingale problem on Rd 36
1.12 The martingale problem on D C Rd 39
1.13 The generalized martingale problem on D C Rd 41
1.14 The canonical construction and the definition of a diffusion
process 43
1.15 The lifetime of a diffusion process and the definition of
explosion 43
Exercises 44
2 The basic properties of diffusion processes 47
2.0 Introduction 47
2.1 Stochastic representation for solutions of elliptic equations and
elliptic inequalities 48
2.2 The behavior of the exit time from a small ball 50
2.3 The Blumenthal 0 1 law and the regularity of boundary points 53
2.4 The Feynman Kac formula 59
2.5 Proof of Theorem 1.5.1 61
2.6 The Stroock Varadhan support theorem 65
viii Contents
2.7 Transience and recurrence for diffusions on Rd 68
2.8 Transience and recurrence for diffusions on D C Rd 74
Exercises 75
3 The spectral theory of elliptic operators on smooth bounded domains 77
3.1 The spectral theory of compact operators 77
3.2 Maximum principles and the Schauder estimates 79
3.3 Existence and uniqueness for the Dirichlet problem 86
3.4 The semigroup and its generator on (Bo = {u e C(D): u = 0 on
3D} 88
3.5 Thespectramof(L,'Z)a)onl3a={u6Ca(5):tt = 0on3D} 91
3.6 The existence and the behavior of the principal eigenvalue 94
3.7 A mini max variational formula for the principal eigenvalue 102
Exercises 119
4 Generalized spectral theory for elliptic operators on arbitrary
domains 123
4.0 Introduction 123
4.1 The fc transform 125
4.2 The Green's measure and the Green's function and criteria
for their existence 129
4.3 Criticality and the generalized principal eigenvalue 144
4.4 The generalized principal eigenvalue: continuity properties
and mini max principles 156
4.5 Criteria for the finiteness or infiniteness of the generalized
principal eigenvalue 163
4.6 Perturbation theory for critical and subcritical operators 164
4.7 L — kc is critical if D is bounded and L is uniformly elliptic
with bounded coefficients 176
4.8 Invariant measures and invariant functions for transition
measures 179
4.9 The product L1 property and its connection to the asymptotic
behavior of the transition measure; positive and null recur¬
rence 183
4.10 The product L1 property and the existence of eigenfunctions
in the symmetric case 193
4.11 Extension to the case where D is a non compact open mani¬
fold or a compact manifold without boundary 196
Exercises 200
5 Applications to the one dimensional case and the radially symmetric
multi dimensional case 207
5.0 Introduction 207
5.1 Criticality properties 208
Contents ix
5.2 The Laplacian perturbed by radially symmetric and compactly
supported functions 219
Exercises 231
6 Criteria for transience or recurrence and explosion or non explosion
of diffusion processes 235
6.0 Introduction 235
6.1 The Liapunov method for transience, recurrence and positive
recurrence 236
6.2 Application of the Liapunov method Case I: uniformity in
the non radial variables 238
6.3 Application of the Liapunov method Case II: non uniformity
in the non radial variables 241
6.4 Transience and recurrence for diffusions corresponding to sym¬
metric operators via the Dirichlet principle: a variational ap¬
proach 255
6.5 A generalized Dirichlet principle for non symmetric operators 261
6.6 Transience and recurrence for diffusions via the generalized
Dirichlet principle: a mini max approach 270
6.7 The Liapunov method and its application for explosion and
non explosion criteria 277
Exercises 279
7 Positive harmonic functions and the Martin boundary: general theory 283
7.0 Introduction 283
7.1 The Martin boundary 283
7.2 Conditioned diffusions and h transforms and the behavior of a
transient diffusion as it exits its domain 286
7.3 Positive solutions of minimal growth and applications to
Green's functions and ground states 291
7.4 Harmonic measure for exterior regions 309
7.5 The exterior harmonic measure boundary 318
7.6 A characterization of the Martin boundary in terms of the
exterior harmonic measure boundary 321
Exercises 330
8 Positive harmonic functions and the Martin boundary: applications to
certain classes of operator 333
8.0 Introduction 333
8.1 The Martin boundary in the case of a Lipschitz Euclidean
boundary 333
8.2 Periodic operators on Rd: positive harmonic functions and
criticality, Martin boundary, behavior of /^ under perturba¬
tions 347
x Content
8.3 Fuchsian operators and their adjoints 371
8.4 Auxiliary results I: explosion inward from the boundary 376
8.5 Auxiliary results II: operators in skew product form 380
8.6 The Martin boundary for a class of operators in skew product
form: statement of general theorem, applications to particular
cases and comparison principles 384
8.7 The Martin boundary for a class of operators in skew product
form: proof of general theorem 405
8.8 Some remarks 419
8.9 Survey of other results 422
Exercises 427
9 Bounded harmonic functions and applications to Brownian motion
and the Laplacian on a manifold of non positive curvature 435
9.1 Probabilistic characterization of bounded harmonic functions 435
9.2 The connection between bounded harmonic functions and
bounded solutions in exterior domains 439
9.3 Examples 442
9.4 Brownian motion on a manifold of non positive curvature 444
9.5 Brownian motion on a model manifold of non positive cur¬
vature: transience, recurrence, bounded harmonic functions,
Martin boundary 447
9.6 Bounded harmonic functions on manifolds with curvature
bounded between two negative constants 452
Exercises 456
References 461
Index 471 |
| any_adam_object | 1 |
| author | Pinsky, Ross G. 1955- |
| author_GND | (DE-588)141663146 |
| author_facet | Pinsky, Ross G. 1955- |
| author_role | aut |
| author_sort | Pinsky, Ross G. 1955- |
| author_variant | r g p rg rgp |
| building | Verbundindex |
| bvnumber | BV010170156 |
| classification_rvk | SK 820 |
| classification_tum | MAT 315f MAT 606f |
| ctrlnum | (OCoLC)246865379 (DE-599)BVBBV010170156 |
| dewey-full | 519.233 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.233 |
| dewey-search | 519.233 |
| dewey-sort | 3519.233 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | First published |
| format | Book |
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| id | DE-604.BV010170156 |
| illustrated | Not Illustrated |
| indexdate | 2024-11-18T09:00:37Z |
| institution | BVB |
| isbn | 0521470145 9780521470148 9780521059831 9780511526244 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006754785 |
| oclc_num | 246865379 |
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| physical | xvi, 474 Seiten |
| publishDate | 1995 |
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| series | Cambridge studies in advanced mathematics |
| series2 | Cambridge studies in advanced mathematics |
| spelling | Pinsky, Ross G. 1955- (DE-588)141663146 aut Positive harmonic functions and diffusion Ross G. Pinsky First published Cambridge [u.a.] Cambridge University Press [1995] © 1995 xvi, 474 Seiten txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 45 Elliptischer Differentialoperator - Harmonische Funktion - Diffusionsprozess Harmonische Funktion (DE-588)4159122-7 gnd rswk-swf Diffusionsprozess (DE-588)4274463-5 gnd rswk-swf Diffusion (DE-588)4012277-3 gnd rswk-swf Positive Funktion (DE-588)4175428-1 gnd rswk-swf Harmonische Funktion (DE-588)4159122-7 s Diffusion (DE-588)4012277-3 s DE-604 Diffusionsprozess (DE-588)4274463-5 s Positive Funktion (DE-588)4175428-1 s DE-188 Erscheint auch als Online-Ausgabe 978-0-511-52624-4 Cambridge studies in advanced mathematics 45 (DE-604)BV000003678 45 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006754785&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Pinsky, Ross G. 1955- Positive harmonic functions and diffusion Cambridge studies in advanced mathematics Elliptischer Differentialoperator - Harmonische Funktion - Diffusionsprozess Harmonische Funktion (DE-588)4159122-7 gnd Diffusionsprozess (DE-588)4274463-5 gnd Diffusion (DE-588)4012277-3 gnd Positive Funktion (DE-588)4175428-1 gnd |
| subject_GND | (DE-588)4159122-7 (DE-588)4274463-5 (DE-588)4012277-3 (DE-588)4175428-1 |
| title | Positive harmonic functions and diffusion |
| title_auth | Positive harmonic functions and diffusion |
| title_exact_search | Positive harmonic functions and diffusion |
| title_full | Positive harmonic functions and diffusion Ross G. Pinsky |
| title_fullStr | Positive harmonic functions and diffusion Ross G. Pinsky |
| title_full_unstemmed | Positive harmonic functions and diffusion Ross G. Pinsky |
| title_short | Positive harmonic functions and diffusion |
| title_sort | positive harmonic functions and diffusion |
| topic | Elliptischer Differentialoperator - Harmonische Funktion - Diffusionsprozess Harmonische Funktion (DE-588)4159122-7 gnd Diffusionsprozess (DE-588)4274463-5 gnd Diffusion (DE-588)4012277-3 gnd Positive Funktion (DE-588)4175428-1 gnd |
| topic_facet | Elliptischer Differentialoperator - Harmonische Funktion - Diffusionsprozess Harmonische Funktion Diffusionsprozess Diffusion Positive Funktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006754785&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000003678 |
| work_keys_str_mv | AT pinskyrossg positiveharmonicfunctionsanddiffusion |