Stochastic Partial Differential Equations and Related Fields: In Honor of Michael Rockner, Spderf, Bielefeld, Germany, October 10 -14, 2016

Preface......Page 6
Acknowledgements......Page 9
Contents......Page 10
Organization......Page 14
List of Participants......Page 16
Part I Longer Contributions......Page 20
 1 Introduction......Page 21
 2 The Case of a Non-differentiable Diffusion Matrix: Existence and Higher Integrability of Densities......Page 24
 3 The Case of a Sobolev Differentiable Diffusion Matrix......Page 30
 4 Harnack\'s Inequality and Lower and Upper Bounds......Page 31
 5 Existence of Probability Solutions......Page 32
 6 Uniqueness Problems......Page 34
 7 The Infinite-Dimensional Case......Page 37
 References......Page 39
 1 Introduction......Page 43
 2 Energy Form calEρ and Weighted Beppo Levi Space calGρ......Page 47
 3 Liouville Property of Rotation Invariant Energy Forms on mathbbRn......Page 49
 4 Strongly Local Transient Dirichlet Form calE and a Time Change  of the Associated Diffusion......Page 52
 5 One-Point Reflection of  at......Page 54
 6 Liouville Property of calE and Uniqueness of a Symmetric Conservative Diffusion Extension of......Page 56
 References......Page 59
  1.1 Finite Dimensional Case......Page 61
  1.2 Linear SPDE......Page 66
 2 Nonlinear SPDE......Page 68
  2.1 Scalar Conservation Laws......Page 70
  2.2 Well-Posedness by Noise for Stochastic Inhomogeneous Scalar Conservation Laws......Page 72
  2.3 Regularization by Noise for Stochastic Scalar Conservation Laws......Page 76
  2.4 Open Interfaces and Porous Media Equations......Page 77
 References......Page 81
 1 Introduction......Page 86
 2 Paraproducts......Page 91
 3 Paracontrolled Analysis......Page 93
 4 Ambiguities and Renormalisation......Page 100
 5 Higher Order Expansions......Page 103
 6 Weak Universality......Page 108
 7 Anderson Hamiltonian......Page 111
 8 Singular Martingale Problem......Page 114
 References......Page 115
 1 Introduction and Setting of the Problem......Page 117
 2 Preliminaries on the Ornstein–Uhlenbeck Semigroup......Page 119
  3.1 Basic Assumptions......Page 122
  3.2 Tightness......Page 125
  3.3 Other Assumptions......Page 131
  4.1 The Rank Condition......Page 138
  4.2 The Semigroup Associated to a Non Autonomous Problem......Page 139
  4.3 The Case When C-1 is Bounded......Page 141
  4.4 The Case When Tr C References......Page 145
Part II Stochastic Partial Differential Equations and Regularity Structures......Page 146
 1 Introduction......Page 147
 2 A Geometric Approach......Page 148
 3 Constrained ``Heat\'\' Equation......Page 150
 4 Local Existence and Invariance......Page 154
  5.1 Reaction Diffusion Equation......Page 156
  5.2 Navier–Stokes Equations on a Torus mathbbT2......Page 157
 6 Generalisation to Stochastic PDEs......Page 158
 References......Page 159
 1 Introduction......Page 161
 2 SPDEs with Additive Volterra Noise......Page 164
 3 SPDEs with Multiplicative Gaussian Volterra Noise......Page 168
 References......Page 172
 1 Introduction......Page 173
 2 Benchmark Results for Gaussian Random Fields......Page 174
  2.1 First Example: The Brownian Sheet......Page 175
  2.2 Anisotropic Gaussian Random Fields......Page 176
  2.3 Funaki\'s Random String......Page 177
  3.1 Systems of Nonlinear Wave Equations in Spatial Dimension 1......Page 178
  3.2 Other Non-linear Systems of SPDEs......Page 179
  3.4 Proving the Lower Bound......Page 181
  3.5 Results for Systems of Non-linear Equations......Page 185
 4 Polarity of Points in Critical Dimensions......Page 186
 References......Page 189
Curvature Motion Perturbed  by a Direction-Dependent Colored Noise......Page 191
  1.1 A Brief Overview......Page 192
  1.2 The MMC Perturbed by a Direction-Dependent Noise......Page 194
 2 Main Result......Page 197
 3 Uniform Moment Estimates on κ(n)......Page 200
  3.1 Bounds for Φ1n, Φ2n, and Φ3n,i for ninmathbbN......Page 201
  3.2 Estimates on Ψ1n,i and Ψ2n,i,j......Page 202
  3.3 Estimates Based on the Mixing Property......Page 203
  3.4 Proof of Proposition 3......Page 204
  4.1 Martingale Problem Corresponding to the Limit Process (κ,WQ)......Page 205
  4.2 Proof of Theorem 2......Page 207
 References......Page 213
 1 General Setting. Main Results......Page 215
 2 Main Idea......Page 220
 3 Generalizations......Page 221
 4 Extensions to Non-local Operators......Page 223
 References......Page 224
Dynamics of SPDEs Driven by a Small Fractional Brownian Motion with Hurst Parameter Larger than 1/2......Page 226
 1 Introduction......Page 227
 2 Preliminaires......Page 228
 3 Evolution Equations Driven by an fBm with Hin(1/2,1)......Page 229
 4 Exponential Stability of SPDEs Driven by fBm  with Hin(1/2,1)......Page 231
 References......Page 236
 1 Introduction and Main Result......Page 238
 2 Well-Posedness of an Auxiliary Equation......Page 240
 3 Proof of Theorem 1......Page 247
 References......Page 248
 1 Introduction......Page 249
  2.1 Setting......Page 251
  2.2 Lower Bounds for the Squared Norm......Page 252
  2.3 Lower Bounds for the Weak Error of a Particular Regular Test Function......Page 258
 References......Page 259
 1 Introduction......Page 261
 2 Preliminaries......Page 263
  2.1 Commutator Estimates......Page 266
 3 An Approximating Equation......Page 267
 4 A Priori Estimates......Page 269
  5.2 Approach via Curvature-Dimension Conditions......Page 272
 References......Page 273
 1 Random Data Cauchy Problem for a Schrödinger-Type Equation......Page 275
 2 Random Data Cauchy Problem for a KdV-Type Equation......Page 277
 3 Random Data Cauchy Problem for a Wave Equation......Page 278
 References......Page 279
SPDEs and Renormalisation......Page 282
 References......Page 287
 1 Introduction......Page 289
 2 Rescaling Approach......Page 291
  3.1 Global Well-Posedness......Page 292
  3.2 Noise Effect on Blow-Up......Page 293
  3.3 Optimal Bilinear Control......Page 294
 4 SNLS with Logarithmic Nonlinearity......Page 296
 References......Page 298
Part III Stochastic Analysis Including Geometric Aspects......Page 300
 1 The Problem......Page 301
 2 The Existence and Uniqueness of a Generalized Solution......Page 303
 3 Long-Time Dynamical Behavior......Page 306
 4 The H-Theorem......Page 307
 References......Page 309
 1 Introduction......Page 311
 2 Magnetic Field Blow-Up......Page 312
 3 Rough Lead-Lag Process......Page 319
 References......Page 325
 1 Introduction......Page 326
 2 Diffusion Operators......Page 327
  3.1 Horizontal Lifts, Semi-connections,  and the Decomposition......Page 328
  3.2 The Vertical Component  mathcalB V......Page 329
  4.1 Associated Bundles......Page 330
  4.2 Induced Operators on Sections of Associated Bundles......Page 331
  4.3 Conditioning and the Semigroup on Sections of F*......Page 333
  4.4 The Classical Case: Hodge–Kodaira Operators......Page 335
  5.2 Baudoin Connections......Page 337
 References......Page 338
 1 Introduction......Page 339
  2.1 Notation......Page 341
  2.2 Rough Drivers......Page 342
 3 A Priori Estimates for Rough Partial Differential Equations......Page 345
 4 Uniqueness for Reflected Rough Differential Equations......Page 347
 References......Page 349
 1 Introduction......Page 351
 2 Summary of Results......Page 355
 3 Key Ingredient......Page 356
 References......Page 362
Synchronization, Lyapunov Exponents  and Stable Manifolds for Random Dynamical Systems......Page 364
 2 Preliminaries and Notation......Page 365
 3 Top Lyapunov Exponent and Synchronization......Page 366
 4 Examples......Page 369
 References......Page 371
 1 Introduction......Page 372
 2 Linear Fokker–Planck–Kolmogorov Equations......Page 374
 3 Nonlinear Fokker–Planck–Kolmogorov Equations: Existence......Page 376
 4 Nonlinear Fokker–Planck–Kolmogorov Equations: Uniqueness......Page 381
 References......Page 384
 1 Coupling Method for Harnack Inequality......Page 385
 2 Hypercontractivity Using Coupling and Harnack Inequality......Page 387
 3 Hypercontractivity for Degenerate SDEs......Page 389
 4 Hypercontractivity for Functional SPDEs......Page 390
 References......Page 392
 1 Introduction......Page 394
 2 General Settings and Useful Tools......Page 396
 3 Multidimensional Singular Stochastic Flows......Page 399
 4 Singular Stochastic Hamiltonian Flows......Page 401
 References......Page 405
Part IV Dirichlet Forms, Markov Processes and Potential Theory......Page 407
Invariant, Super and Quasi-martingale Functions of a Markov Process......Page 408
 1 Introduction......Page 409
 2 Differences of Excessive Functions and Quasimartingales of Markov Processes......Page 410
  2.1 Criteria for Quasimartingale Functions......Page 412
  2.2 Applications to Semi-Dirichlet Forms......Page 414
 3 Excessive and Invariant Functions on Lp-spaces......Page 416
  3.1 Martingale Functions with Respect to the Dual Markov Process......Page 417
 4 L1-harmonic Functions and Invariant Probability Measures......Page 418
 References......Page 420
Mean Value Inequalities for Jump Processes......Page 422
 1 Introduction......Page 423
  2.1 Setting......Page 425
  2.2 Heat Kernel Estimates......Page 427
  2.3 Parabolic Harnack Inequalities......Page 430
 3 Lp-Mean Value Inequality......Page 433
 References......Page 437
 1 Introduction......Page 439
 2 Quasi-regular Positivity Preserving Coercive Forms......Page 440
 3 h-Transform of Positivity Preserving Semigroups......Page 442
 4 Distribution Flows Associated with Positivity Preserving Coercive Forms......Page 444
 References......Page 449
Some Thoughts and Investigations on Densities of One-Parameter Operator Semi-groups......Page 450
 References......Page 458
 1 Introduction......Page 460
 2 Framework and Results......Page 462
 3 An Open Problem......Page 469
 References......Page 470
 1 Introduction......Page 471
 2 Setting and the Main Result......Page 474
 3 Proofs......Page 477
 References......Page 486
 1 Introduction......Page 487
 2 Remarks on Tightness (III)......Page 488
 3 Uniform Large Deviations......Page 489
 4 Compactness of Symmetric Markov Semigroups......Page 492
 5 Boundedness of Eigenfunctions......Page 495
 References......Page 496
Part V Applications Including Mathematical Physics......Page 498
From Non-symmetric Particle Systems  to Non-linear PDEs on Fractals......Page 499
 References......Page 507
 1 Introduction......Page 510
 2 Energy Solutions to the Stochastic Burgers Equation......Page 514
 References......Page 521
 1 Generalities......Page 523
  1.1 The Anharmonic Crystal......Page 524
  1.2 The Path Measures......Page 526
  2.1 Phase Transitions and Quantum Stabilization......Page 528
  2.2 Local Dynamics......Page 530
 References......Page 531
 1 Kolmogorov Representation Theorem......Page 533
 2 Continuous Coding......Page 537
 References......Page 541
 1 Introduction......Page 542
 2 Unlabeled Dynamics: Quasi-Gibbs Property......Page 543
 3 Labeled Dynamics: A Scheme of Dirichlet Spaces......Page 545
 4 ISDE-Representation: Logarithmic Derivative......Page 546
 5 Strong Solutions of ISDEs and Pathwise Uniqueness......Page 547
 6 Tail Triviality: Application to Interacting Brownian Motions......Page 548
 7 Examples Arising from Random Matrix Theory......Page 549
 8 Algebraic Construction and Finite Particle Approxiations......Page 550
 References......Page 551
 1 Introduction......Page 553
  2.1 Wick Power......Page 555
  2.2 Relations Between the Two Solutions......Page 556
  2.3 Markov Uniqueness in the Restricted Sense......Page 558
  2.4 Ergodicity......Page 559
 3 Three Dimensional Case......Page 561
 References......Page 562
Author Index......Page 564