Second Order Partial Differential Equations in Hilbert Spaces (London Mathematical Society Lecture Note Series, Series Number 293)

Second Order Linear Parabolic And Elliptic Equations Arise Frequently In Mathematics And Other Disciplines. For Example Parabolic Equations Are To Be Found In Statistical Mechanics And Solid State Theory, Their Infinite Dimensional Counterparts Are Important In Fluid Mechanics, Mathematical Finance And Population Biology, Whereas Nonlinear Parabolic Equations Arise In Control Theory. Here The Authors Present A State Of The Art Treatment Of The Subject From A New Perspective. The Main Tools Used Are Probability Measures In Hilbert And Banach Spaces And Stochastic Evolution Equations. There Is Then A Discussion Of How The Results In The Book Can Be Applied To Control Theory. This Area Is Developing Very Rapidly And There Are Numerous Notes And References That Point The Reader To More Specialised Results Not Covered In The Book. Coverage Of Some Essential Background Material Will Help Make The Book Self-contained And Increase Its Appeal To Those Entering The Subject. Giuseppe Da Prato, Jerzy Zabczyk. Includes Bibliographical References (p. 358-375) And Index. Contents......Page 7 Preface......Page 12 Part I THEORY IN SPACES OF CONTINUOUS FUNCTIONS......Page 19 1.1 Introduction and preliminaries......Page 21 1.2.1 Measures in metric spaces......Page 25 1.2.2 Gaussian measures......Page 26 1.2.3 Computation of some Gaussian integrals......Page 29 1.2.4 The reproducing kernel......Page 30 1.3 Absolute continuity of Gaussian measures......Page 35 1.3.1 Equivalence of product measures in.........Page 36 1.3.2 The Cameron-Martin formula......Page 40 1.3.3 The Feldman-Hajek theorem......Page 42 1.4 Brownian motion......Page 45 2.1 Preliminary results......Page 48 2.2 Approximation of continuous functions......Page 51 2.3.1 Interpolation between.........Page 54 2.3.2 Interpolatory estimates......Page 57 2.3.3 Additional interpolation results......Page 60 3.1 Preliminaries......Page 62 3.2 Strict solutions......Page 66 3.3.1 Q-derivatives......Page 72 3.3.2 Q-derivatives of generalized solutions......Page 75 3.4 Comments on the Gross Laplacian......Page 85 3.5 The heat semigroup and its generator......Page 87 4.1 Existence and uniqueness results......Page 94 4.2 Regularity of solutions......Page 96 4.3 The equation.........Page 101 4.3.1 The Liouville theorem......Page 105 5.1 Small perturbations......Page 108 5.2 Large perturbations......Page 111 6 Ornstein-Uhlenbeck equations......Page 117 6.1 Existence and uniqueness of strict solutions......Page 118 6.2 Classical solutions......Page 121 6.3 The Ornstein-Uhlenbeck semigroup......Page 129 6.3.1 Pi-Convergence......Page 130 6.3.2 Properties of the Pi-semigroup.........Page 131 6.3.3 The infinitesimal generator......Page 132 6.4 Elliptic equations......Page 134 6.4.1 Schauder estimates......Page 137 6.4.2 The Liouville theorem......Page 139 6.5 Perturbation results for parabolic equations......Page 140 6.6 Perturbation results for elliptic equations......Page 142 7 General parabolic equations......Page 145 7.1 Implicit function theorems......Page 146 7.2.1 Infinite dimensional Wiener processes......Page 149 7.2.2 Stochastic integration......Page 150 7.3.1 Convolution and evaluation maps......Page 151 7.3.2 Solutions of stochastic equations......Page 156 7.4 Space and time regularity of the generalized solutions......Page 157 7.5 Existence......Page 160 7.6 Uniqueness......Page 162 7.6.1 Uniqueness for the heat equation......Page 163 7.6.2 Uniqueness in the general case......Page 164 7.7 Strong Feller property......Page 168 8.1 Introduction......Page 174 8.2 Regularity of the generalized solution......Page 176 8.3 Existence theorems......Page 183 8.4 Uniqueness of the solutions......Page 196 Part II THEORY IN SOBOLEV SPACES......Page 203 9 L2 and Sobolev spaces......Page 205 9.1.1 Real Hermite polynomials......Page 206 9.1.2 Chaos expansions......Page 208 9.1.3 The space.........Page 211 9.2 Sobolev spaces......Page 212 9.2.1 The space.........Page 214 9.2.2 Some additional summability results......Page 215 9.2.3 Compactness of the embedding.........Page 216 9.2.4 The space.........Page 219 9.3 The Malliavin derivative......Page 221 10 Ornstein-Uhlenbeck semigroups on.........Page 223 10.1 Extension of.........Page 224 10.1.1 The adjoint of.........Page 229 10.2 The infinitesimal generator of.........Page 230 10.2.1 Characterization of the domain of.........Page 233 10.3 The case when... is strong Feller......Page 235 10.3.1 Additional regularity properties of.........Page 239 10.3.2 Hypercontractivity of.........Page 242 10.4.1 The second quantization operator......Page 246 10.5 Poincaré and log-Sobolev inequalities......Page 248 10.5.1 The case when M = 1 and Q = I......Page 250 10.5.2 A generalization......Page 253 10.6 Some additional regularity results when Q and A commute......Page 254 11 Perturbations of Ornstein-Uhlenbeck semigroups......Page 256 11.1 Bounded perturbations......Page 257 11.2 Lipschitz perturbations......Page 263 11.2.1 Some additional results on the Ornstein-Uhlenbeck semigroup......Page 269 11.2.2 The semigroup.........Page 274 11.2.3 The integration by parts formula......Page 278 11.2.4 Existence of a density......Page 281 12 Gradient systems......Page 285 12.1.1 Assumptions and setting of the problem......Page 286 12.1.2 The Sobolev space.........Page 289 12.1.3 Symmetry of the operator.........Page 290 12.1.4 The m-dissipativity of.........Page 292 12.2 The m-dissipativity.........Page 295 12.3 The case when U is convex......Page 299 12.3.1 Poincaré and log-Sobolev inequalities......Page 306 Part III APPLICATIONS TO CONTROL THEORY......Page 309 13 Second order Hamilton-Jacobi equations......Page 311 13.1 Assumptions and setting of the problem......Page 314 13.2 Hamilton-Jacobi equations with a Lipschitz Hamiltonian......Page 318 13.2.1 Stationary Hamilton-Jacobi equations......Page 320 13.3 Hamilton-Jacobi equation with a quadratic Hamiltonian......Page 323 13.3.1 Stationary equation......Page 326 13.4.1 Finite horizon......Page 328 13.4.2 Infinite horizon......Page 330 13.4.3 The limit as.........Page 332 14.1 Introduction......Page 334 14.2 Excessive weights and an existence result......Page 335 14.3 Weak solutions as value functions......Page 342 14.4 Excessive measures for Wiener processes......Page 346 Part IV APPENDICES......Page 351 A.1 The interpolation theorem......Page 353 A.2 Interpolation between a Banach space X and the domain of a linear operator in X......Page 354 B.1 Definition of null controllability......Page 356 B.2 Main results......Page 357 B.3 Minimal energy......Page 358 C.1 Continuity modulus......Page 365 C.2 Semiconcave and semiconvex functions......Page 366 C.3 The Hamilton-Jacobi semigroups......Page 369 Bibliography......Page 376 Index......Page 394 Second order linear parabolic and elliptic equations arise frequently in mathematical physics, biology and finance. Here the authors present a state of the art treatment of the subject from a new perspective. They then go on to discuss how the results in the book can be applied to control theory. This area is developing rapidly and there are numerous notes and references that point the reader to more specialized results not covered in the book. Coverage of some essential background material helps to make the book self contained. State of the art treatment of a subject which has applications in mathematical physics, biology and finance. Includes discussion of applications to control theory. There are numerous notes and references that point to further reading. Coverage of some essential background material helps to make the book self contained.