Seminaire de Probabilites XLIX

Preface......Page 6
Contents......Page 7
 1.1 Introduction......Page 9
  1.2.1 The Case of One Centered Ball, i.e. y=0......Page 15
  1.2.2 A First Estimate for a General y Using Perturbation......Page 17
 1.3 Using Lyapunov Functions......Page 19
  1.3.1 Two Useful Lemmas on Lyapunov Function Method......Page 20
  1.3.2 Localizing Around the Obstacles......Page 22
  1.3.3 Localizing Away from the Obstacles and the Origin......Page 24
  1.3.4 Localizing Away from the Origin for the Far Enough Obstacles......Page 26
  1.3.5 Localizing Around the Origin for a Far Enough Single Obstacle......Page 28
  1.3.6 A General Result for a Single Obstacle with Small Radius......Page 29
  1.3.7 The Case of Infinitely Many Obstacles......Page 30
 1.4 General Spherical Obstacles Using Stochastic Calculus......Page 31
  1.4.1 The Rate of Rotation......Page 32
  1.4.2 The Poincaré Inequality in the Shell S......Page 35
  1.4.3 A New Estimate for an Obstacle Which Is Not too Close to the Origin......Page 37
  1.4.4 Finiteness of the Poincaré Constant for an Infinite Number of Spherical Obstacles......Page 38
  1.5.1 Lower Bounds for Hypercubes via Stochastic Calculus......Page 39
  1.5.2 An Isoperimetric Approach for Hypercubes......Page 40
   1.5.2.1 Squared Obstacle......Page 41
   1.5.2.2 Hypercubes......Page 42
  1.5.3 Back to Spherical Obstacles: Another Lower Bound......Page 43
  1.5.4 How to Kill the Poincaré Constant with Far But Small Non Convex Obstacles......Page 46
 Appendix 1: Existence and Uniqueness of the Process......Page 48
  Finite Number of Obstacles......Page 49
  Infinite Number of Obstacles......Page 50
 Appendix 2: Useful Estimates for Exponential Moments of Hitting Times......Page 52
 Appendix 3: The Case N=1: Another Estimate for a General y Using Decomposition of Variance......Page 54
  A Bound for C1......Page 55
  Controlling df/du......Page 57
 References......Page 62
 2.1 Introduction......Page 64
 2.2 Balance Between Growth and Fragmentation......Page 67
 2.3 Tails of the Stationary Distribution......Page 74
 References......Page 80
3 Iterated Proportional Fitting Procedure and Infinite Products of Stochastic Matrices......Page 82
  3.1.1 The Iterative Proportional Fitting Procedure......Page 83
  3.1.2 Probabilistic Interpretations and Tools......Page 85
 3.2 Old and New Results......Page 86
  3.3.1 Proof of Theorem 3.7......Page 95
  3.3.2 Proof of Theorem 3.8......Page 102
  3.4.1 Condition for the Non-existence of a Solution with Support Included in Supp(X0)......Page 105
  3.4.2 Condition for the Existence of Additional Zeroes Shared by Every Solution in Γ(X0)......Page 106
  3.5.1 Results on the Quantities Ri(X2n) and Cj(X2n+1)......Page 109
  3.5.2 A Function Associated to Each Element of Γ1......Page 111
 3.6 Proof of Theorem 3.2......Page 112
  3.7.1 Asymptotic Behavior of the Sequences (R(Xn)) and (C(Xn))......Page 116
  3.7.2 Convergence and Limit of (Xn)......Page 118
   3.8.1.2 Relations Between the Components of the Limits, and Block Structure......Page 119
   3.8.1.4 Formula for λk......Page 120
  3.8.2 Proof of Theorem 3.5......Page 121
  3.8.3 Proof of Theorem 3.6......Page 122
 References......Page 124
 4.1 Introduction......Page 125
 4.2 Sketch of the Proof......Page 129
 4.3 Proof of Proposition 4.2......Page 131
  4.4.1 Matricial Master Equation......Page 141
  4.4.2 Variance Estimates......Page 145
  4.4.3 Estimates of Resolvent Entries......Page 148
 4.5 Proof of Theorem 4.3......Page 151
 Appendix 1......Page 158
 Appendix 2......Page 167
 References......Page 169
 5.1 Introduction......Page 171
 5.2 Quasi-Stationary Behavior Under Two-Sided Estimates......Page 174
  5.3.1 A General Result......Page 176
  5.3.2 The Case of Diffusions in Compact Riemannian Manifolds......Page 177
  5.3.3 Proof of Theorem 5.2......Page 180
   5.3.3.1 Return to a Compact Conditionally on Non-absorption......Page 181
   5.3.3.2 Proof of (A1')......Page 182
   5.3.3.3 Proof of (A2')......Page 184
 Appendix: Proof of (5.2)......Page 186
 References......Page 187
 6.1 Introduction......Page 189
 6.2 Classical Bismut-Elworthy-Li Formula for One-Dimensional Diffusions......Page 191
 6.3 Bessel Processes: Notations and Basic Facts......Page 195
 6.4 Derivative in Space of the Bessel Semi-group......Page 196
 6.5 Differentiability of the Flow......Page 200
 6.6 Properties of η......Page 202
  6.6.1 Regularity of the Sample Paths of η......Page 203
  6.6.3 Continuity of (Dt)t ≥0......Page 206
  6.6.4 Martingale Property of (Dt)t ≥0......Page 209
  6.6.5 Moment Estimates for the Martingale (Dt)t ≥0......Page 212
 6.7 A Bismut-Elworthy-Li Formula for the Bessel Processes......Page 214
 Appendix......Page 220
 References......Page 226
 7.1 Introduction......Page 227
 7.2 Set-Up......Page 231
  7.2.1 Cramer's Theorem for the Poisson Distribution......Page 233
  7.2.2 The Legendre-Fenchel Transform and the Rate Function......Page 234
  7.2.3 Assumptions on the Process ZN......Page 235
 7.3 Law of Large Numbers......Page 240
  7.4.1 Properties of the Legendre Fenchel Transform......Page 244
  7.4.2 Equality of L and L......Page 252
  7.4.3 Further Properties of the Legendre Fenchel Transform......Page 254
  7.4.4 The Rate Function......Page 257
  7.4.5 I is a Good Rate Function......Page 260
  7.4.6 A Property of Non-exponential Equivalence......Page 264
  7.5.1 LDP Lower Bound If the Rates Are Bounded Away from Zero......Page 266
  7.5.2 LDP Lower Bound with Vanishing Rates......Page 275
  7.6.1 Piecewise Linear Approximation......Page 297
  7.6.2 The Modified Rate Function Iδ......Page 298
  7.6.3 Distance of YN to Φδ......Page 304
  7.6.4 Main Results......Page 313
 7.7 Exit Time from a Domain......Page 316
  7.7.1 Auxiliary Results......Page 317
  7.7.2 Main Results......Page 323
  7.7.3 The Case of a Characteristic Boundary......Page 327
 7.8 Example: The SIRS Model......Page 328
 Appendix: Change of Measure......Page 332
 References......Page 333
 8.1 Introduction......Page 334
 8.2 Brownian Motion and Flows......Page 341
  8.3.1 Differentiability of the Flows......Page 343
  8.3.2 The Heat Semi-group......Page 344
  8.3.3 The Transport Semi-group......Page 345
 8.4 The Euler Scheme......Page 346
 8.5 An Intermediary Convergence Result......Page 348
 8.6 The Splitting Procedure......Page 349
 8.7 The Weights and Their Limits: The Girsanov Theorem......Page 350
  8.7.1 Exponential Representation of the One-Step Euler Scheme......Page 351
  8.7.2 The Weights......Page 352
  8.7.3 Uniform Integrability of the Weights......Page 353
  8.7.4 Identification of the Limit of the Weights: Stochastic Calculus At last......Page 354
 8.8 The Infinitesimal Generator of the Semi-group (Xt)t≥0......Page 355
  8.9.1 The Feynman-Kac Formula......Page 358
  8.9.2 The Infinitesimal Generator of a Semi-group Under a h-Transform......Page 359
  8.9.3 An Alternative Formulation for the Girsanov Weights for Some Special Form of the Drift......Page 360
 8.10 Complement: On the Regularity of the Drift and the Difference Between Classical and Stochastic Analysis......Page 361
  The Heat Semi-group......Page 362
  The Transport Semi-group......Page 363
 References......Page 364
 9.1 Introduction......Page 367
 9.2 Proof of Proposition 9.1......Page 369
 9.3 Applications......Page 372
 9.4 Examples......Page 374
 References......Page 376
 10.1 Introduction......Page 378
 10.2 The Finite Non-transient Setting......Page 384
 10.3 On the Finite Transient Setting......Page 388
 10.4 Comparisons of Mixing Speeds......Page 397
 10.5 On Infinite State Spaces......Page 403
 References......Page 406
 11.1 Introduction......Page 407
  11.2.1 Main Results......Page 414
  11.2.2 How to Use the Extended Lp-Dual Quantization Error Modulus?......Page 415
  11.3.1 Definition and First Properties......Page 418
  11.3.2 Local Properties of the Dual Quantization Functional......Page 421
  11.4.1 Extended Pierce Lemma......Page 422
  11.4.2 A d-Dimensional Non-asymptotic Upper-Bound for the Dual Quantization Error......Page 432
 11.5 Proof of the Sharp Rate Theorem......Page 433
 11.6 Concluding Remarks and Prospects......Page 453
 Appendix: Numerical Results for dn,2(X)2......Page 454
 References......Page 455
 12.1 Introduction......Page 457
 12.2 Setting and Results......Page 459
 12.3 Proof of Theorem 12.1......Page 462
 12.4 Proofs of Theorems 12.2 and 12.3......Page 464
 12.5 Proof of Theorem 12.4......Page 468
 12.6 Proof of Theorem 12.5......Page 471
 References......Page 473
 13.1 Introduction......Page 476
 13.2 Preliminaries......Page 478
 13.3 Martingale Chaoses......Page 481
 13.4 Kernels Associated with Chaos Expansions......Page 484
 13.5 Integral Representations......Page 487
 13.6 Change of Measure......Page 492
 References......Page 494
 14.1 Introduction......Page 496
 14.2 Statement of the Results......Page 498
  14.2.2 Consequences on the Sequence of Records......Page 499
  14.2.3 Invariance Principle for the Record Times, Conditionally on the Signs......Page 500
  14.2.4 Invariance Principle for the Record Times of the SPRW......Page 501
  14.2.6 Number of Visited Points for the SPRW......Page 502
  14.2.7 Open Questions......Page 503
 14.3 Preliminary Lemmas......Page 504
 14.4 Signs of Records of the PRW......Page 508
 14.5 Record Values of the PRW......Page 510
 14.6 Record Times of the PRW: Conditional Case......Page 513
 14.7 Record Times of the PRW: Unconditional Case......Page 515
 14.8 Positive Recurrence......Page 519
 References......Page 520
 15.1 Introduction......Page 521
 15.2 Preliminaries......Page 524
 15.3 Renormalized Zero Resolvent......Page 529
 15.4 Tanaka Formula......Page 532
 15.5 Examples......Page 535
 References......Page 542