Probabilities and potential: C Potential theory for discrete and continuous semigroups
Gespeichert in:
| Beteiligte Personen: | , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch Französisch |
| Veröffentlicht: |
Amsterdam [u.a.]
North-Holland
1988
|
| Schriftenreihe: | North-Holland mathematics studies
151 |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000497833&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XIV, 416 S. |
| ISBN: | 0444557075 072040701X |
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| 240 | 1 | 0 | |a Probabilités et potentiel |
| 245 | 1 | 0 | |a Probabilities and potential |n C |p Potential theory for discrete and continuous semigroups |c Claude Dellacherie and Paul-André Meyer |
| 264 | 1 | |a Amsterdam [u.a.] |b North-Holland |c 1988 | |
| 300 | |a XIV, 416 S. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a North-Holland mathematics studies |v 151 | |
| 490 | 0 | |a North-Holland mathematics studies |v ... | |
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Datensatz im Suchindex
| _version_ | 1819309210315259904 |
|---|---|
| adam_text | ix
(
Contents
CHAPTER IX. KERNELS AND EXCESSIVE FUNCTIONS 1
1. Kernels
Kernels (1 2). Extension to functions 3nd measures (3 5). Composi¬
tion (6). Examples (7). Feller properties (8). Construction of kernels
(9 10). Regularization of pseudo kernels (11 14). Improving regularity
by composition (15 13). An extension of Egorov s theorem (19 20). Formal
calculations with kernels (21 23).
2. B(£esj^ve jFujctions_ with respect to a kernel 19
Excessive and invariant functions (24 25). The potential kernel (26).
The Riesz decomposition (27 29). Approximation by potentials (30). Reduc¬
tion of an excessive function on a set (31 32). The principles of
potential theory (33 37). Reductions II: reduction on a function (33 41).
Some applications (42), the strong order (43 52). Probabilistic inter¬
pretations (54 57).
3. Excessive measures and sweeping 43
Excessive measures (58 62). Sweeping (63 65). The filling scheme
(66 67). Rost s theorems (68 70). The Skorokhod problem (71 74). Pointwise
ergodic theory: Hopf s lemma (75 76), Brunei s lemma (77), Chacon Ornstein
theorem (73 80), identification of the limit (81 83).
4. Complementary results^ 65
Study of excessive functions without restriction of finitude (84 87)
and application to the strong order (33). Good potentials and pure pot¬
entials (89). Characterization of invariant functions (90). What is a
potential theoretic notion? (91). A complement to no. 30 (92).
x Contents
CHAPTER X. THEORY OF REDUCTIONS AND SWEEPING 72
1. Gambling houses, reductions^
Gambling houses (1 3). J supermedian functions (4) and sweeping of
measures (5). Examples (6). Extension to measures of mass 1 (8). Reduc¬
tions (9 13). Analytic functions and anaiyticity of reductions (14 16).
Approximation lemmas (17 20) and application to sweeping (21 23). Use of
stationary markovian strategies (24). Probabilistic interpretation (25 27).
2. Sweeping defined by a convex cone of continuous functions 90
Definitions (28), and examples (29). A version of the Hahn Banach
theorem (30 31). Strassen s decomposition theorem (32 33). Characteriza¬
tion of the sweepings of a measure (34 36). The Dini Cartan lemma (37 38).
Applications to sweeping (39 40). Maximal sweepings (41) and the Choquet
boundary (42 43). A maximum principle (44). Existence of maximal sweep¬
ings: study of the non metrizable case (45 46). A method of compactifica
tion (47). Adapted cones (48).
3. Convex compact sets 108
Convex, concave and affine functions (50), and their elementary prop¬
erties (51 54). Medial limits (55 57). Translation of the results of
Section 2 (58 59). Choquet s theorem on integral representation: existence
(60 61) and uniqueness (62 64). Extension to certain cones without compact
base (65 67). Uniqueness in Loomis s sense (68 70). Examples:
harmonic functions (71), completely monotone functions on R+(72 73) or a
commutative semigroup (74). Interpretation as a property of positive type
(75). Convolution semigroups on F+and Le vy functions (76 78). Elementary
theory of kernels of positive and negative type (79 86). Functions of
positive type on a group and Bochner s theorem (87 91). Functions of
negative type on a commutative group (92 99) and the Levy Khinchin
formula (100 103).
CHAPTER XI (APPENDIX TO CHAPTER X): NEW METHODS IN CAPACITY THEORY,
APPLICATIONS TO GAMBLING HOUSES 163
1. Multi capacities
Definition of multicapacities (1). Examples (2). The general
Contents xi
capacitability theorem (3 5). Passage to functional arguments (6 7).
Examples (8). A theorem of Novikov and its applications (9 11).
2. Cjpacitary and analytic operatqrjs 173
Capacitary operators (12), composition (13), preservation of
analyticity (14). Analytic operators and functionals (15). The general
separation theorem (16 17). Privileged Souslin schemes and the precise
capacitability theorem (18 19). A capacitary Fubini s theorem (20).
3. Application to gambling houses 185
Some examples of capacitary and analytic operators (21 24). Existence
of Borel supermedian functions (25). Cones of continuous functions and
associated gambling houses (26 29). Construction of Borel supermedian
functions (30 31). Structure of analytic gambling houses (32). A theorem
of functional analysis (33 37). Analyticity of the saturation of an
analytic gambling house (38 39). An extension of Strassen s theorem (40).
4. j/a_rious applications and complements 206
Application to the regularization of pseudo kernels (41). Thickness
of a set (42 46). Application to thin sets (47) and to the domination of
sets of measures (48 49). A characterization of basic kernels (50).
Application to random sets (51 52). Analytic derivations (53 55). Applica¬
tion to translation operators (56).
CHAPTER XII. SEMIGROUPS AND RESOLVENTS
1. Th_e fundamental definitions 219
Semigroups (1 2). Supermedian and excessive functions (3 4).
Potential kernel, resolvent (5 6). General definition of resolvents (7 8).
Supermedian and excessive functions for a resolvent (9 10). Sets of
potential zero (11), regularisation of supermedian functions (12).
Invariant functions (13) and the Riesz decomposition (14). Approximation
of excessive functions by potentials (15 17). Excessive functions for a
semigroup and its resolvent (18). Absorbing sets (19). Some calculations
with a resolvent (20 21).
xii Contents
2. Elements^ of potential theory 234
Existence and first properties of the reduction (22 23). Reductions
and the strong order (24 26). The principles of potential theory (27 29).
Reduction of a difference of two excessive functions (30 31).
Characterisations of supermedian functions by their domination properties
(32). Properties of the order and the strong order (33 35).
Excessive measures (36 37). Representation as an increasing limit of
potentials (38). Mass of an excessive function with respect to an
excessive measure (39 40). The hypothesis of absolute continuity (41 43).
3. Ergodic jthgor.y _fcr a resolvent 250
General notations (44). Resolvents on a Banach space (45). Strict
positivity of a resolvent on L (46 47). Local convergence in norm (48 49)
and application to potentials (50). Ergodic theory at infinity: Brunei s
lemma (51 52) and the Chacon Ornstein theorem (53).
A maximal lemma (54 55). Application to the local ergodic theorem in
L (56) and to within sets of potential zero (57). Entrance and exit laws
(58 59). Local ergodic theorem for an exit law in L (60 62) and to
within sets of potential zero (63). Extension to super exit laws (64 66).
Appendix: proof of the derivation theorem by means of reductions (67).
4. Resolvents ij duality 276
Resolvents in duality (68 70). Excessive measures and coexcessive
functions (71). Construction of kernel functions (72 74). Transformations
of resolvents in duality (75 78). Construction of a coresolvent (79).
5. Corm2actificatioji_methods 290
Ray resolvents (80) and Feller semigroups (81). Branch points (82).
Construction of right (83) and left (84) Ray semigroups. Ray compactifica
tion (85 87). Construction of a resolvent starting from a bounded
potential kernel (88). Transformation of a basic resolvent into a compact
resolvent (89), and semi continuity of excessive functions (90). Integral
representation of excessive functions (92 96).
Contents xiii
CHAPTER XIII. CONSTRUCTION OF RESOLVENTS AND SEMIGROUPS
1. The Hi 11 e Yosida_ theory 309
Semigroups and resolvents on a Banach space (1 3). First part of the
Hiiie Yosida theorem: passage from resolvent to semigroup (4 5). Generator
and cogenerator (6 7). Elementary properties the generator (8) and the
cogenerator (9). The case of a closed resolvent (10). Passage from the
generator to the resolvent (11). Dissipative and codissipative operators
(12). Second part of the Hille Yosida theorem: passage from generator to
resolvent (13). Passage from cogenerator to resolvent (14). Relations
with the principles of potential theory (15 17).
2. Applications to Hunt s theorem 323
Feller semigroups (18 20). Passage from the submarkovian to the
markovian case (21). Construction of a resolvent starting from a potential:
Hunt s theorem (22 24).
3. jome_ examples 3 29
Semigroups on a finite set (25) or a countable set (26). Convolution
semigroups (27), subordinators (28), semigroups of uniform translation (29),
Poisson semigroups (29), one sided stable semigroups (29). Application of
the stable 1/2 semigroup to functional analysis (30). The general sense
of the word stable (31). Brownian motion in IR (32). Symmetric stable
and Cauchy semigroups (33). Details for the case d = 1 (34) and extension
to d dimensions (35). The general notion of a product semigroup (36). A
general principle of passing to the quotient (37), application to reflect¬
ing brownian motions and to Bessel processes. Another interpretation of
Bes, (38). Brownian motion on the circle (39). The Cauchy semigroup on
the sphere (40). Ornstein Uhlenbeck semigroup (41) and Hermite polynomials
(42).
4. ^pjie^j^niarki^r^jriergy 354
Approximate energy forms, the energy form and the energy norm (43 44).
Almost symmetric resolvents (44). Example: convolution semigroups (45).
The symmetric case and spectral theory (46). Elementary properties of the
energy forms (47) and characterization of the Dirichlet space (48). The
xiv Contents
projection theorem (52). Reconstruction of the resolvent (53). The
enlarged Dirichlet space (54 57). Excessive functions of finite energy
(58 61). Regular Dirichlet spaces (62 63).
COMMENTARIES AND HISTORICAL NOTES 377
BIBLIOGRAPHY 387
INDEX OF NOTATIONS 408
INDEX OF TERMINOLOGY 411
ADDITIONS AND CORRECTIONS TO THE PRECEDING VOLUMES 415
|
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| author | Dellacherie, Claude Meyer, Paul André 1934-2003 |
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| id | DE-604.BV000794462 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T07:26:48Z |
| institution | BVB |
| isbn | 0444557075 072040701X |
| language | English French |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-000497833 |
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| physical | XIV, 416 S. |
| publishDate | 1988 |
| publishDateSearch | 1988 |
| publishDateSort | 1988 |
| publisher | North-Holland |
| record_format | marc |
| series | North-Holland mathematics studies |
| series2 | North-Holland mathematics studies |
| spellingShingle | Dellacherie, Claude Meyer, Paul André 1934-2003 Probabilities and potential North-Holland mathematics studies |
| title | Probabilities and potential |
| title_alt | Probabilités et potentiel |
| title_auth | Probabilities and potential |
| title_exact_search | Probabilities and potential |
| title_full | Probabilities and potential C Potential theory for discrete and continuous semigroups Claude Dellacherie and Paul-André Meyer |
| title_fullStr | Probabilities and potential C Potential theory for discrete and continuous semigroups Claude Dellacherie and Paul-André Meyer |
| title_full_unstemmed | Probabilities and potential C Potential theory for discrete and continuous semigroups Claude Dellacherie and Paul-André Meyer |
| title_short | Probabilities and potential |
| title_sort | probabilities and potential potential theory for discrete and continuous semigroups |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000497833&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000094852 (DE-604)BV000003247 |
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