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London Mathematical Society Lecture Note Series. 293
Second Order Partial Differential
Equations in Hilbert Spaces
Giuseppe Da Prato
Scuola Normale Superiore di Pisa
Jerzy Zabczyk
Polish Academy of Sciences, Warsaw
The Pitt Building, Trumpington Street, Cambridge, United Kingdom
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LONDONMATHEMATICALSOCIETYLECTURENOTESERIES
ManagingEditor:ProfessorN.J.Hitchin,MathematicalInstitute,
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109 Diophantineanalysis, J.LOXTON&A.VANDERPOORTEN(eds)
113 Lecturesontheasymptotictheoryofideals, D.REES
116 Representationsofalgebras, P.J.WEBB(ed)
119 Triangulatedcategoriesintherepresentationtheoryoffinite-dimensionalalgebras, D.HAPPEL
121 ProceedingsofGroups–StAndrews1985, E.ROBERTSON&C.CAMPBELL(eds)
128 Descriptivesettheoryandthestructureofsetsofuniqueness, A.S.KECHRIS&A.LOUVEAU
130 Modeltheoryandmodules, M.PREST
131 Algebraic,extremal&metriccombinatorics, M.-M.DEZA,P.FRANKL&I.G.ROSENBERG(eds)
138 AnalysisatUrbana,II, E.BERKSON,T.PECK,&J.UHL(eds)
139 Advancesinhomotopytheory, S.SALAMON,B.STEER&W.SUTHERLAND(eds)
140 GeometricaspectsofBanachspaces, E.M.PEINADOR&A.RODES(eds)
141 Surveysincombinatorics1989, J.SIEMONS(ed)
144 Introductiontouniformspaces, I.M.JAMES
146 Cohen-MacaulaymodulesoverCohen-Macaulayrings, Y.YOSHINO
148 Helicesandvectorbundles, A.N.RUDAKOVetal
149 Solitons,nonlinearevolutionequationsandinversescattering, M.ABLOWITZ&P.CLARKSON
150 Geometryoflow-dimensionalmanifolds1, S.DONALDSON&C.B.THOMAS(eds)
151 Geometryoflow-dimensionalmanifolds2, S.DONALDSON&C.B.THOMAS(eds)
152 Oligomorphicpermutationgroups, P.CAMERON
153 L-functionsandarithmetic, J.COATES&M.J.TAYLOR(eds)
155 Classificationtheoriesofpolarizedvarieties, TAKAOFUJITA
158 GeometryofBanachspaces, P.F.X.MULLER&W.SCHACHERMAYER(eds)
159 GroupsStAndrews1989volume1, C.M.CAMPBELL&E.F.ROBERTSON(eds)
160 GroupsStAndrews1989volume2, C.M.CAMPBELL&E.F.ROBERTSON(eds)
161 Lecturesonblocktheory, BURKHARDKU¨LSHAMMER
163 Topicsinvarietiesofgrouprepresentations, S.M.VOVSI
164 Quasi-symmetricdesigns, M.S.SHRIKANDE&S.S.SANE
166 Surveysincombinatorics,1991, A.D.KEEDWELL(ed)
168 Representationsofalgebras, H.TACHIKAWA&S.BRENNER(eds)
169 Booleanfunctioncomplexity, M.S.PATERSON(ed)
170 ManifoldswithsingularitiesandtheAdams-Novikovspectralsequence, B.BOTVINNIK
171 Squares, A.R.RAJWADE
172 Algebraicvarieties, GEORGER.KEMPF
173 Discretegroupsandgeometry, W.J.HARVEY&C.MACLACHLAN(eds)
174 Lecturesonmechanics, J.E.MARSDEN
175 Adamsmemorialsymposiumonalgebraictopology1, N.RAY&G.WALKER(eds)
176 Adamsmemorialsymposiumonalgebraictopology2, N.RAY&G.WALKER(eds)
177 Applicationsofcategoriesincomputerscience, M.FOURMAN,P.JOHNSTONE&A.PITTS(eds)
178 LowerK-andL-theory, A.RANICKI
179 Complexprojectivegeometry, G.ELLINGSRUDetal
180 LecturesonergodictheoryandPesintheoryoncompactmanifolds, M.POLLICOTT
181 GeometricgrouptheoryI, G.A.NIBLO&M.A.ROLLER(eds)
182 GeometricgrouptheoryII, G.A.NIBLO&M.A.ROLLER(eds)
183 Shintanizetafunctions, A.YUKIE
184 Arithmeticalfunctions, W.SCHWARZ&J.SPILKER
185 Representationsofsolvablegroups, O.MANZ&T.R.WOLF
186 Complexity:knots,colouringsandcounting, D.J.A.WELSH
187 Surveysincombinatorics,1993, K.WALKER(ed)
188 Localanalysisfortheoddordertheorem, H.BENDER&G.GLAUBERMAN
189 Locallypresentableandaccessiblecategories, J.ADAMEK&J.ROSICKY
190 Polynomialinvariantsoffinitegroups, D.J.BENSON
191 Finitegeometryandcombinatorics, F.DECLERCKetal
192 Symplecticgeometry, D.SALAMON(ed)
194 Independentrandomvariablesandrearrangementinvariantspaces, M.BRAVERMAN
195 Arithmeticofblowupalgebras, WOLMERVASCONCELOS
196 Microlocalanalysisfordifferentialoperators, A.GRIGIS&J.SJO¨STRAND
197 Two-dimensionalhomotopyandcombinatorialgrouptheory, C.HOG-ANGELONIetal
198 Thealgebraiccharacterizationofgeometric4-manifolds, J.A.HILLMAN
199 InvariantpotentialtheoryintheunitballofCn, MANFREDSTOLL
200 TheGrothendiecktheoryofdessinsd’enfant, L.SCHNEPS(ed)
201 Singularities, JEAN-PAULBRASSELET(ed)
202 Thetechniqueofpseudodifferentialoperators, H.O.CORDES
203 HochschildcohomologyofvonNeumannalgebras, A.SINCLAIR&R.SMITH
204 Combinatorialandgeometricgrouptheory, A.J.DUNCAN,N.D.GILBERT&J.HOWIE(eds)
205 Ergodictheoryanditsconnectionswithharmonicanalysis, K.PETERSEN&I.SALAMA(eds)
207 GroupsofLietypeandtheirgeometries, W.M.KANTOR&L.DIMARTINO(eds)
208 Vectorbundlesinalgebraicgeometry, N.J.HITCHIN,P.NEWSTEAD&W.M.OXBURY(eds)
209 Arithmeticofdiagonalhypersurfacesoverfinitefields, F.Q.GOUVEA&N.YUI
210 HilbertC*-modules, E.C.LANCE
211 Groups93Galway/StAndrewsI, C.M.CAMPBELLetal(eds)
212 Groups93Galway/StAndrewsII, C.M.CAMPBELLetal(eds)
214 GeneralisedEuler-Jacobiinversionformulaandasymptoticsbeyondallorders, V.KOWALENKOetal
215 Numbertheory1992–93, S.DAVID(ed)
216 Stochasticpartialdifferentialequations, A.ETHERIDGE(ed)
217 Quadraticformswithapplicationstoalgebraicgeometryandtopology, A.PFISTER
218 Surveysincombinatorics,1995, PETERROWLINSON(ed)
220 Algebraicsettheory, A.JOYAL&I.MOERDIJK
221 Harmonicapproximation, S.J.GARDINER
222 Advancesinlinearlogic, J.-Y.GIRARD,Y.LAFONT&L.REGNIER(eds)
223 Analyticsemigroupsandsemilinearinitialboundaryvalueproblems, KAZUAKITAIRA
224 Computability,enumerability,unsolvability, S.B.COOPER,T.A.SLAMAN&S.S.WAINER(eds)
225 Amathematicalintroductiontostringtheory, S.ALBEVERIO,J.JOST,S.PAYCHA,S.SCARLATTI
226 Novikovconjectures,indextheoremsandrigidityI, S.FERRY,A.RANICKI&J.ROSENBERG(eds)
227 Novikovconjectures,indextheoremsandrigidityII, S.FERRY,A.RANICKI&J.ROSENBERG(eds)
228 ErgodictheoryofZdactions, M.POLLICOTT&K.SCHMIDT(eds)
229 Ergodicityforinfinitedimensionalsystems, G.DAPRATO&J.ZABCZYK
230 Prolegomenatoamiddlebrowarithmeticofcurvesofgenus2, J.W.S.CASSELS&E.V.FLYNN
231 Semigrouptheoryanditsapplications, K.H.HOFMANN&M.W.MISLOVE(eds)
232 ThedescriptivesettheoryofPolishgroupactions, H.BECKER&A.S.KECHRIS
233 Finitefieldsandapplications, S.COHEN&H.NIEDERREITER(eds)
234 Introductiontosubfactors, V.JONES&V.S.SUNDER
235 Numbertheory1993-94, S.DAVID(ed)
236 TheJamesforest, H.FETTER&B.GAMBOADEBUEN
237 Sievemethods,exponentialsums,andtheirapplicationsinnumbertheory, G.R.H.GREAVESetal
238 Representationtheoryandalgebraicgeometry, A.MARTSINKOVSKY&G.TODOROV(eds)
239 Cliffordalgebrasandspinors, P.LOUNESTO
240 Stablegroups, FRANKO.WAGNER
241 Surveysincombinatorics,1997, R.A.BAILEY(ed)
242 GeometricGaloisactionsI, L.SCHNEPS&P.LOCHAK(eds)
243 GeometricGaloisactionsII, L.SCHNEPS&P.LOCHAK(eds)
244 Modeltheoryofgroupsandautomorphismgroups, D.EVANS(ed)
245 Geometry,combinatorialdesignsandrelatedstructures, J.W.P.HIRSCHFELDetal
246 p-Automorphismsoffinitep-groups, E.I.KHUKHRO
247 Analyticnumbertheory, Y.MOTOHASHI(ed)
248 Tametopologyando-minimalstructures, LOUVANDENDRIES
249 Theatlasoffinitegroups:tenyearson, ROBERTCURTIS&ROBERTWILSON(eds)
250 Charactersandblocksoffinitegroups, G.NAVARRO
251 Gro¨bnerbasesandapplications, B.BUCHBERGER&F.WINKLER(eds)
252 Geometryandcohomologyingrouptheory, P.KROPHOLLER,G.NIBLO,R.STO¨HR(eds)
253 Theq-Schuralgebra, S.DONKIN
254 Galoisrepresentationsinarithmeticalgebraicgeometry, A.J.SCHOLL&R.L.TAYLOR(eds)
255 Symmetriesandintegrabilityofdifferenceequations, P.A.CLARKSON&F.W.NIJHOFF(eds)
256 AspectsofGaloistheory, HELMUTVO¨LKLEINetal
257 Anintroductiontononcommutativedifferentialgeometryanditsphysicalapplications2ed, J.MADORE
258 Setsandproofs, S.B.COOPER&J.TRUSS(eds)
259 Modelsandcomputability, S.B.COOPER&J.TRUSS(eds)
260 GroupsStAndrews1997inBath,I, C.M.CAMPBELLetal
261 GroupsStAndrews1997inBath,II, C.M.CAMPBELLetal
263 Singularitytheory, BILLBRUCE&DAVIDMOND(eds)
264 Newtrendsinalgebraicgeometry, K.HULEK,F.CATANESE,C.PETERS&M.REID(eds)
265 Ellipticcurvesincryptography, I.BLAKE,G.SEROUSSI&N.SMART
267 Surveysincombinatorics,1999, J.D.LAMB&D.A.PREECE(eds)
268 Spectralasymptoticsinthesemi-classicallimit, M.DIMASSI&J.SJO¨STRAND
269 Ergodictheoryandtopologicaldynamics, M.B.BEKKA&M.MAYER
270 AnalysisonLieGroups, N.T.VAROPOULOS&S.MUSTAPHA
271 Singularperturbationsofdifferentialoperators, S.ALBEVERIO&P.KURASOV
272 Charactertheoryfortheoddorderfunction, T.PETERFALVI
273 Spectraltheoryandgeometry, E.B.DAVIES&Y.SAFAROV(eds)
274 TheMandelbrotset,themeandvariations, TANLEI(ed)
275 Computatoinalandgeometricaspectsofmodernalgebra, M.D.ATKINSONetal(eds)
276 Singularitiesofplanecurves, E.CASAS-ALVERO
277 Descriptivesettheoryanddynamicalsystems, M.FOREMANetal(eds)
278 Globalattractorsinabstractparabolicproblems, J.W.CHOLEWA&T.DLOTKO
279 Topicsinsymbolicdynamicsandapplications, F.BLANCHARD,A.MAASS&A.NOGUEIRA(eds)
280 CharactersandAutomorphismGroupsofCompactRiemannSurfaces, T.BREUER
281 Explicitbirationalgeometryof3-folds, ALESSIOCORTI&MILESREID(eds)
282 Auslander-Buchweitzapproximationsofequivariantmodules, M.HASHIMOTO
283 Nonlinearelasticity, R.OGDEN&Y.FU(eds)
284 Foundationsofcomputationalmathematics, R.DEVORE,A.ISERLES&E.SULI(eds)
285 Rationalpointsoncurvesoverfinitefields:TheoryandApplications, H.NIEDERREITER&C.XING
286 Cliffordalgebrasandspinors2ndedn, P.LOUNESTO
287 TopicsonRiemannsurfacesandFuchsiangroups, E.BUJALANCE,A.F.COSTA&E.MARTINEZ(eds)
288 Surveysincombinatorics,2001, J.W.P.HIRSCHFELD(ed)
289 AspectsofSobolev-typeinequalities, L.SALOFF-COSTE
290 QuantumgroupsandLietheory, A.PRESSLEY
291 Titsbuildingsandthemodeltheoryofgroups, K.TENT
292 Aquantumgroupsprimer, S.MAJID
Contents
Preface x
I THEORY IN SPACES OF CONTINUOUS
FUNCTIONS 1
1 Gaussian measures 3
1.1 Introduction and preliminaries . . . . . . . . . . . . . . . . . 3
1.2 Definition and first properties of Gaussian measures . . . . . 7
1.2.1 Measures in metric spaces . . . . . . . . . . . . . . . . 7
1.2.2 Gaussian measures . . . . . . . . . . . . . . . . . . . . 8
1.2.3 Computation of some Gaussian integrals . . . . . . . . 11
1.2.4 The reproducing kernel . . . . . . . . . . . . . . . . . 12
1.3 Absolute continuity of Gaussian measures . . . . . . . . . . . 17
1.3.1 Equivalence of product measures in R∞ . . . . . . . . 18
1.3.2 The Cameron-Martin formula . . . . . . . . . . . . . . 22
1.3.3 The Feldman-Hajek theorem . . . . . . . . . . . . . . 24
1.4 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 Spaces of continuous functions 30
2.1 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2 Approximation of continuous functions . . . . . . . . . . . . . 33
2.3 Interpolation spaces . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.1 Interpolation between UC (H) and UC1(H) . . . . . 36
b b
2.3.2 Interpolatory estimates . . . . . . . . . . . . . . . . . 39
2.3.3 Additional interpolation results . . . . . . . . . . . . . 42
3 The heat equation 44
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 Strict solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 48
v
vi Contents
3.3 Regularity of generalized solutions . . . . . . . . . . . . . . . 54
3.3.1 Q-derivatives . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.2 Q-derivatives of generalized solutions . . . . . . . . . . 57
3.4 Comments on the Gross Laplacian . . . . . . . . . . . . . . . 67
3.5 The heat semigroup and its generator . . . . . . . . . . . . . 69
4 Poisson’s equation 76
4.1 Existence and uniqueness results . . . . . . . . . . . . . . . . 76
4.2 Regularity of solutions . . . . . . . . . . . . . . . . . . . . . . 78
4.3 The equation ∆ u = g . . . . . . . . . . . . . . . . . . . . . . 83
Q
4.3.1 The Liouville theorem . . . . . . . . . . . . . . . . . . 87
5 Elliptic equations with variable coefficients 90
5.1 Small perturbations . . . . . . . . . . . . . . . . . . . . . . . 90
5.2 Large perturbations . . . . . . . . . . . . . . . . . . . . . . . 93
6 Ornstein-Uhlenbeck equations 99
6.1 Existence and uniqueness of strict solutions . . . . . . . . . . 100
6.2 Classical solutions . . . . . . . . . . . . . . . . . . . . . . . . 103
6.3 The Ornstein-Uhlenbeck semigroup . . . . . . . . . . . . . . . 111
6.3.1 π-Convergence . . . . . . . . . . . . . . . . . . . . . . 112
6.3.2 Properties of the π-semigroup (R ) . . . . . . . . . . . 113
t
6.3.3 The infinitesimal generator . . . . . . . . . . . . . . . 114
6.4 Elliptic equations . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.4.1 Schauder estimates . . . . . . . . . . . . . . . . . . . . 119
6.4.2 The Liouville theorem . . . . . . . . . . . . . . . . . . 121
6.5 Perturbation results for parabolic equations . . . . . . . . . . 122
6.6 Perturbation results for elliptic equations . . . . . . . . . . . 124
7 General parabolic equations 127
7.1 Implicit function theorems . . . . . . . . . . . . . . . . . . . . 128
7.2 Wiener processes and stochastic equations . . . . . . . . . . . 131
7.2.1 Infinite dimensional Wiener processes . . . . . . . . . 131
7.2.2 Stochastic integration . . . . . . . . . . . . . . . . . . 132
7.3 Dependence of the solutions to stochastic equations on initial
data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.3.1 Convolution and evaluation maps . . . . . . . . . . . . 133
7.3.2 Solutions of stochastic equations . . . . . . . . . . . . 138
7.4 Space and time regularity of the generalized solutions . . . . 139
7.5 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Contents vii
7.6 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.6.1 Uniqueness for the heat equation . . . . . . . . . . . . 145
7.6.2 Uniqueness in the general case . . . . . . . . . . . . . 146
7.7 Strong Feller property . . . . . . . . . . . . . . . . . . . . . . 150
8 Parabolic equations in open sets 156
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
8.2 Regularity of the generalized solution . . . . . . . . . . . . . . 158
8.3 Existence theorems . . . . . . . . . . . . . . . . . . . . . . . . 165
8.4 Uniqueness of the solutions . . . . . . . . . . . . . . . . . . . 178
II THEORY IN SOBOLEV SPACES 185
9 L2 and Sobolev spaces 187
9.1 Itoˆ-Wiener decomposition . . . . . . . . . . . . . . . . . . . . 188
9.1.1 Real Hermite polynomials . . . . . . . . . . . . . . . . 188
9.1.2 Chaos expansions . . . . . . . . . . . . . . . . . . . . . 190
9.1.3 The space L2(H,µ;H) . . . . . . . . . . . . . . . . . . 193
9.2 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 194
9.2.1 The space W1,2(H,µ) . . . . . . . . . . . . . . . . . . 196
9.2.2 Some additional summability results . . . . . . . . . . 197
9.2.3 Compactness of the embedding W1,2(H,µ) ⊂ L2(H,µ) 198
9.2.4 The space W2,2(H,µ) . . . . . . . . . . . . . . . . . . 201
9.3 The Malliavin derivative . . . . . . . . . . . . . . . . . . . . . 203
10 Ornstein-Uhlenbeck semigroups on Lp(H,µ) 205
10.1 Extension of (R ) to Lp(H,µ) . . . . . . . . . . . . . . . . . 206
t
10.1.1 The adjoint of (R ) in L2(H,µ) . . . . . . . . . . . . . 211
t
10.2 The infinitesimal generator of (R ) . . . . . . . . . . . . . . . 212
t
10.2.1 Characterization of the domain of L . . . . . . . . . . 215
2
10.3 The case when (R ) is strong Feller . . . . . . . . . . . . . . . 217
t
10.3.1 Additional regularity properties of (R ) . . . . . . . . 221
t
10.3.2 Hypercontractivity of (R ) . . . . . . . . . . . . . . . . 224
t
10.4 Arepresentationformulafor(R )intermsofthesecondquan-
t
tization operator . . . . . . . . . . . . . . . . . . . . . . . . . 228
10.4.1 The second quantization operator. . . . . . . . . . . . 228
10.4.2 The adjoint of (R ) . . . . . . . . . . . . . . . . . . . . 230
t
10.5 Poincar´e and log-Sobolev inequalities . . . . . . . . . . . . . . 230
10.5.1 The case when M = 1 and Q = I . . . . . . . . . . . . 232
viii Contents
10.5.2 A generalization . . . . . . . . . . . . . . . . . . . . . 235
10.6 Some additional regularity results when Q and A commute . 236
11 Perturbations of Ornstein-Uhlenbeck semigroups 238
11.1 Bounded perturbations . . . . . . . . . . . . . . . . . . . . . . 239
11.2 Lipschitz perturbations. . . . . . . . . . . . . . . . . . . . . . 245
11.2.1 Some additional results on the Ornstein-Uhlenbeck
semigroup . . . . . . . . . . . . . . . . . . . . . . . . . 251
11.2.2 The semigroup (P ) in Lp(H,ν) . . . . . . . . . . . . . 256
t
11.2.3 The integration by parts formula . . . . . . . . . . . . 260
11.2.4 Existence of a density . . . . . . . . . . . . . . . . . . 263
12 Gradient systems 267
12.1 General results . . . . . . . . . . . . . . . . . . . . . . . . . . 268
12.1.1 Assumptions and setting of the problem . . . . . . . . 268
12.1.2 The Sobolev space W1,2(H,ν). . . . . . . . . . . . . . 271
12.1.3 Symmetry of the operator N . . . . . . . . . . . . . . 272
0
12.1.4 The m-dissipativity of N on L1(H,ν). . . . . . . . . . 274
1
12.2 The m-dissipativity of N on L2(H,ν) . . . . . . . . . . . . . 277
2
12.3 The case when U is convex . . . . . . . . . . . . . . . . . . . 281
12.3.1 Poincar´e and log-Sobolev inequalities . . . . . . . . . . 288
III APPLICATIONS TO CONTROL THEORY 291
13 Second order Hamilton-Jacobi equations 293
13.1 Assumptions and setting of the problem . . . . . . . . . . . . 296
13.2 Hamilton-Jacobi equations with a Lipschitz Hamiltonian . . . 300
13.2.1 Stationary Hamilton-Jacobi equations . . . . . . . . . 302
13.3 Hamilton-Jacobi equation with a quadratic Hamiltonian . . . 305
13.3.1 Stationary equation . . . . . . . . . . . . . . . . . . . 308
13.4 Solution of the control problem . . . . . . . . . . . . . . . . . 310
13.4.1 Finite horizon . . . . . . . . . . . . . . . . . . . . . . . 310
13.4.2 Infinite horizon . . . . . . . . . . . . . . . . . . . . . . 312
13.4.3 The limit as ε → 0 . . . . . . . . . . . . . . . . . . . . 314
14 Hamilton-Jacobi inclusions 316
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
14.2 Excessive weights and an existence result . . . . . . . . . . . 317
14.3 Weak solutions as value functions . . . . . . . . . . . . . . . . 324
