Markov Operators, Positive Semigroups and Approximation Processes

326 Pages·2014·1.32 MB·English

by  Altomare

Francesco Altomare, Mirella Cappelletti Montano, Vita Leonessa, Ioan Raşa
Markov Operators, Positive Semigroups and Approximation Processes
De Gruyter Studies
in Mathematics
Edited by
Carsten Carstensen, Berlin, Germany
Nicola Fusco, Napoli, Italy
Fritz Gesztesy, Columbia, Missouri, USA
Niels Jacob, Swansea, United Kingdom
Karl-Hermann Neeb, Erlangen, Germany
Volume 61
Francesco Altomare, Mirella Cappelletti
Montano, Vita Leonessa, Ioan Raşa
Markov Operators,
Positive Semigroups
and Approximation
Processes
DE GRUYTER
MathematicsSubjectClassification2010
35A35,35B40,35K20,35K65,41-02,41A36,41A63,46-02,46E15,47-02,47B65,47D07,47F05,
60J60.
Authors
Prof.FrancescoAltomare Dr.VitaLeonessa
UniversitàdegliStudidiBariAldoMoro UniversitàdegliStudidellaBasilicata
DipartimentodiMatematica DipartimentodiMatematica,Informaticaed
ViaE.Orabona4 Economia
70125Bari CampusdiMacchiaRomana
Italy Vialedell’AteneoLucano10
[email protected] 85100Potenza
Italy
Dr.MirellaCappellettiMontano [email protected]
UniversitàdegliStudidiBariAldoMoro
DipartimentodiMatematica Prof.IoanRaşa
ViaE.Orabona4 TechnicalUniversityofCluj-Napoca
70125Bari DepartmentofMathematics
Italy Str.Memorandumului28
[email protected] 400114Cluj-Napoca
Romania
[email protected]
ISBN978-3-11-037274-8
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To our parents and families

Preface
Ifyouhavebuiltcastlesintheair,yourworkneednotbelost;thatiswheretheyshould
be.Nowputthefoundationunderthem.
Henry David Thoreau
Duringthelasttwentyyearsimportantprogresseshavebeenmade,fromthepoint
ofviewofconstructiveapproximationtheory,inthestudyofinitial-boundaryvalue
differential problems of parabolic type governed by positive -semigroups of op-
erators. The main aim of this approach is to construct suitable positive approxi-
0
mation processes whose iterates strongly converge to the sem𝐶igroups which, as it
is well-known, in principle furnish the solutions to the relevant initial-boundary
valuedifferentialproblems.Bymeansofsuchkindofapproximationitisthenpos-
sible to investigate, among other things, preservation properties and the asymp-
toticbehaviorofthesemigroups,i.e.,spatialregularitypropertiesandasymptotic
behavior of the solutions to the differential problems.
This series of researches, for a survey on which we refer to [25] and the ref-
erences therein, has its roots in several studies developed between 1989 and 1994
which are documented in Chapter 6 of the monograph [18]. These studies were
concerned with special classes of second-order elliptic differential operators act-
ing on spaces of smooth functions on finite dimensional compact convex subsets,
which are generated by a positive projection. The projections themselves offer
the tools to construct an approximation process whose iterates converge to the
relevant semigroup, making possible the development of a qualitative analysis as
above.
Thistheorydisclosedseveralinterestingapplicationsbystressingtherelation-
ship among positive semigroups, initial-boundary value problems, Markov pro-
cesses and approximation theory, and by offering, among other things, a unifying
approachtothestudyofdiversedifferentialproblems.Nevertheless,overthesub-
sequentyears,ithasnaturallyarisentheneedtoextendthetheorybydeveloping
a parallel one for positive operators rather than for positive projections and by
trying to include in the same project of investigations more general differential
operators having a first-order term.
Theaimofthisresearchmonographistoaccomplishsuchanattemptbycon-
sidering complete second-order (degenerate) elliptic differential operators whose
leading coefficients are generated by a positive operator, by means of which it
is possible to construct suitable approximation processes which approximate the
relevantsemigroups.Someaspectsofthetheoryaretreatedalsoininfinitedimen-
sional settings.
viii Preface
The above described more general framework guarantees to notably enlarge
the class of differential operators by including, in particular, those which can be
obtainedbyusualoperationswithpositiveoperatorssuchasconvexcombinations,
compositions,tensorproductsandsoon.Moreover,thisnon-trivialgeneralization
discloses new challenging problems as well.
However, the approximation processes which we construct in terms of the
given positive operator seem to have an interest in their own also for the approx-
imation of continuous functions. For these reasons, a special emphasis is placed
upon various aspects of this theme as well.
Wearehappytothankseveralfriendsandcolleaguesand,particularly,Michele
Campiti and Rainer Nagel for their critical reading of the manuscript.
We also wish to express our appreciation to Niels Jacob for his interest in
this work. We thank him and Walter de Gruyter & Co. Publishing House for
accepting to publish this monograph in the Series Studies in Mathematics as well
as for producing it according with their usual high standards.
September 2014
Francesco Altomare
Mirella Cappelletti Montano
Vita Leonessa
Ioan Raşa
Contents
Preface vii
Introduction 1
Guide to the reader and interdependence of sections 5
Notation 7
1 Positive linear operators and approximation problems 13
1.1 Positive linear functionals and operators 13
1.1.1 Positive Radon measures 15
1.1.2 Choquet boundaries 21
1.1.3 Bauer simplices 23
1.2 Korovkin-type approximation theorems 26
1.3 Further convergence criteria for nets of positive linear operators 30
1.4 Asymptotic behaviour of Lipschitz contracting Markov
semigroups 38
1.5 Asymptotic formulae for positive linear operators 45
1.6 Moduli of smoothness and degree of approximation by positive linear
operators 54
1.7 Notes and comments 59
2 -semigroups of operators and linear evolution equations 63
2.1 -semigroups of operators and abstract Cauchy problems 63
0
2.1.1 𝐶 -semigroups and their generators 63
0
2.1.2 G𝐶eneration theorems and abstract Cauchy problems 70
0
2.2 A𝐶pproximation of -semigroups 77
2.3 Feller and Markov semigroups of operators 88
0
2.3.1 Basic properties 𝐶 88
2.3.2 Markov Processes 92
2.3.3 Second-order differential operators on real intervals and Feller
theory 95
2.3.4 Multidimensional second-order differential operators and Markov
semigroups 98
2.4 Notes and comments 103
3 Bernstein-Schnabl operators associated with Markov operators 105
3.1 Generalities, definitions and examples 106
3.1.1 Bernstein-Schnabl operators on 108
[0,1]
x Contents
3.1.2 Bernstein-Schnabl operators on Bauer simplices 109
3.1.3 Bernstein operators on polytopes 110
3.1.4 Bernstein-Schnabl operators associated with strictly elliptic differential
operators 110
3.1.5 Bernstein-Schnabl operators associated with tensor products of Markov
operators 111
3.1.6 Bernstein-Schnabl operators associated with convex combinations of
Markov operators 112
3.1.7 Bernstein-Schnabl operators associated with convex convolution
products of Markov operators 113
3.2 Approximation properties and rate of convergence 115
3.3 Preservation of Hölder continuity 121
3.3.1 Smallest Lipschitz constants and triangles 127
3.3.2 Smallest Lipschitz constants and parallelograms 128
3.4 Bernstein-Schnabl operators and convexity 131
3.5 Monotonicity properties 143
3.6 Notes and comments 153
4 Differential operators and Markov semigroups associated with Markov
operators 155
4.1 Asymptotic formulae for Bernstein-Schnabl operators 156
4.2 Differential operators associated with Markov operators 162
4.3 Markov semigroups generated by differential operators associated with
Markov operators 169
4.4 Preservation properties and asymptotic behaviour 181
4.5 The special case of the unit interval 189
4.5.1 Degenerate differential operators on 189
4.5.2 Approximation properties by means of Bernstein-Schnabl
operators 194 [0,1]
4.5.3 Preservation properties and asymptotic behaviour 196
4.5.4 The saturation class of Bernstein-Schnabl operators and the Favard
class of their limit semigroups 199
4.6 Notes and comments 205
5 Perturbed differential operators and modified Bernstein-Schnabl
operators 209
5.1 Lototsky-Schnabl operators 210
5.2 A modification of Bernstein-Schnabl operators 216
5.3 Approximation properties 220
5.4 Preservation properties 224
5.5 Asymptotic formulae 227