Trends In Stochastic Analysis: Festschrift In Honour Of Heinrich Von Weizsäcker London Mathematical Society

Presenting important trends in the field of stochastic analysis, this collection of thirteen articles provides an overview of recent developments and new results. Written by leading experts in the field, the articles cover a wide range of topics, ranging from an alternative set-up of rigorous probability to the sampling of conditioned diffusions. Applications in physics and biology are treated, with discussion of Feynman formulas, intermittency of Anderson models and genetic inference. A large number of the articles are topical surveys of probabilistic tools such as chaining techniques, and of research fields within stochastic analysis, including stochastic dynamics and multifractal analysis. Showcasing the diversity of research activities in the field, this book is essential reading for any student or researcher looking for a guide to modern trends in stochastic analysis and neighbouring fields. Title......Page 4 Copyright......Page 5 Contents......Page 6 Preface......Page 8 Heinrich von Weizsäcker’s students ......Page 10 Heinrich von Weizsäcker’s publications ......Page 12 I. Foundations and techniques in stochastic analysis......Page 18 1.1 Introduction......Page 20 1.2 Events and random variables – an outline......Page 24 1.3 Spaces with denumerable separation......Page 26 1.4 The axioms for random variables......Page 29 1.5 Events......Page 32 1.6 Equality and a.s. equality......Page 36 Bibliography......Page 41 2.1 Introduction......Page 42 2.2 The competitors......Page 44 2.3 Chaining at work......Page 47 2.4 Examples and complements......Page 53 2.5 Dispersion of sets: upper bounds......Page 59 2.6 Appendix: Proofs of the chaining lemmas......Page 64 Bibliography......Page 69 3.1 Introduction......Page 72 3.1.1 Notation......Page 74 3.2 Skew-products......Page 75 3.2.1 Admissible measures......Page 77 3.2.2 A simple example......Page 78 3.3 Skew-products of Markov processes versus random dynamical systems......Page 79 3.4 Ergodicity criteria for Markov semigroups......Page 82 3.4.1 Off-white noise systems......Page 85 3.4.2 Another quasi-Markov property......Page 87 3.5 The Gaussian case......Page 89 3.5.1 The quasi-Markov property......Page 91 3.5.2 The strong Feller property......Page 93 3.5.3 The off-white noise case......Page 97 3.6 Appendix A: Equivalence of the strong and ultra Feller properties......Page 98 3.7 Appendix B: Some Gaussian measure theory......Page 101 Bibliography......Page 103 4.1 Introduction......Page 106 4.2 The speed of fragmentation in stick-breaking......Page 108 4.3 A polymer model in a random environment......Page 114 4.4 The speed of emergence in Kingman’s coalescent......Page 121 4.5 Conclusion......Page 126 Bibliography......Page 127 II. Construction, simulation, discretization of stochastic processes......Page 128 Abstract......Page 130 5.1 Introduction......Page 131 5.2 Example: confinement to a curved planar wire......Page 133 5.2.1 Geometry: an embedded curved wire......Page 135 5.2.2 Rescaling......Page 136 5.2.3 Renormalization......Page 137 5.2.4 Epiconvergence......Page 143 5.2.5 Strong convergence of the generators......Page 146 5.2.6 The result for the induced metric......Page 149 5.3 Conditioned Brownian motion......Page 151 5.4 The case of Riemannian submanifolds......Page 153 5.5 Two limits......Page 156 5.6 Two open problems......Page 162 5.6.1 Tubes with fibres of variable shape......Page 163 Bibliography......Page 164 6.1 Introduction......Page 166 6.2.1 The Metropolis–Hastings algorithm......Page 168 6.2.2 Langevin sampling......Page 170 6.3.1 Linear equations......Page 171 6.3.2 Semilinear equations......Page 173 6.4 Conditioned diffusions......Page 177 6.5.1 Construction of the smoothing SPDE......Page 182 6.5.2 Some remarks about smoothing......Page 187 6.6 Metropolis–Hastings algorithm on path space......Page 188 6.7 Conclusion......Page 191 Bibliography......Page 192 7.1 Introduction......Page 194 7.1.1 The information constraints......Page 197 7.1.2 Synopsis......Page 199 7.2.1 Classical setting......Page 200 7.2.2 Orlicz-norm distortion......Page 204 7.3 Gaussian signals......Page 208 7.3.1 Asymptotic estimates......Page 209 7.3.2 Some examples......Page 210 7.3.3 A particular random coding strategy......Page 211 7.3.4 The fractional Brownian motion......Page 212 7.4.1 The mutual information and Shannon’s Source Coding Theorem......Page 214 7.4.2 The derivation of the distortion rate function D(r, 2)......Page 216 7.4.3 Generalizations to other coding quantities (first approach)......Page 220 7.4.4 Generalizations to other coding quantities (second approach)......Page 221 7.4.5 A particular random quantization procedure......Page 223 7.5 Diffusions......Page 225 7.5.1 The decoupling method......Page 227 7.5.2 The corresponding rate allocation problem......Page 230 7.6.1 Estimates based on moment conditions on the increments......Page 233 Bibliography......Page 236 III. Stochastic analysis in mathematical physics......Page 240 Abstract......Page 242 8.1.1 Motivation......Page 243 8.1.2 Intermittency......Page 244 8.2.1 Model......Page 245 8.2.2 Main theorems......Page 246 8.2.3 Discussion......Page 248 8.3.1 Model......Page 249 8.3.2 Main theorems......Page 250 8.4 Symmetric voter model......Page 251 8.4.2 Main theorems......Page 252 8.4.3 Discussion......Page 253 Bibliography......Page 254 9.1 Introduction......Page 256 9.2 What is a stochastic dynamical system?......Page 257 9.3 Spectral theory of linear cocycles: hyperbolicity......Page 258 9.4 The local stable manifold theorem......Page 263 9.5 Stochastic systems with finite memory......Page 268 9.5.1 Existence of cocycles for regular sfde’s......Page 269 9.6 Semilinear see’s......Page 273 9.6.1 Smooth cocycles for semilinear see’s and spde’s......Page 274 9.7 Examples: semilinear spde’s......Page 280 9.8.1 Anticipating semilinear sfde’s......Page 282 9.8.2 Anticipating semilinear see’s......Page 286 Bibliography......Page 287 10.1 Introduction......Page 290 10.2 Feynman formulae, Feynman pseudo-measures and Feynman integrals......Page 292 10.3 Feynman formulae via the Chernoff theorem......Page 297 10.4 Representation of solutions to the Schr ̈odinger equation by the Feynman integral over trajectories in phase space......Page 298 10.5 Feynman formulae for the Cauchy-Dirichlet problem [20]......Page 300 10.6 Stochastic Schr ̈odinger–Ito equation (Belavkin equation) with two-dimensional white noise [9]......Page 301 10.7 Equations on manifolds of mappings ([21])......Page 302 10.8 Feynman formulae for the diffusion equation with coordinate dependent diffusion coefficient and for the Schrödinger equation with coordinate dependent mass......Page 306 Bibliography......Page 308 11.1 Introduction......Page 310 11.2 Wiener space and Fock space......Page 314 11.3 Quantization and the Malliavin Calculus......Page 319 11.4 Quantization and white noise analysis......Page 322 11.5 Fedosov quantization for Taubes limit model......Page 326 Bibliography......Page 330 IV. Stochastic analysis in mathematical biology......Page 334 12.1 Introduction......Page 336 12.2 Population genetic models with neutral types......Page 338 12.2.1 “Classical” limit results in the finite variance regime......Page 339 12.2.2 Beyond finite variance: occasional extremer eproduction events......Page 342 12.2.3 Introducing mutation......Page 345 12.2.4 Lookdown......Page 350 12.3 Neutral genealogies: beyond Kingman’s coalescent......Page 353 12.3.1 Genealogies and coalescent processes......Page 354 12.4 Population genetic inference......Page 359 12.4.1 Finite-alleles recursion I: Using the lookdown construction......Page 360 12.4.2 Finite-alleles recursion II: Generator approach......Page 362 12.4.3 A Monte Carlo scheme for sampling probabilities......Page 365 12.4.4 Simulating samples......Page 366 Bibliography......Page 368 Abstract......Page 372 13.1 Introduction......Page 373 13.2 The discrete ratchet – Haigh’s approach......Page 376 13.3 The Fleming–Viot diffusion......Page 379 13.4 The infinite population limit......Page 380 13.5 One-dimensional diffusion approximations......Page 384 13.6 Simulations......Page 390 13.7 Discussion......Page 394 Bibliography......Page 396 Presenting Important Trends In The Field Of Stochastic Analysis, This Collection Of Thirteen Articles Provides An Overview Of Recent Developments And New Results. The Articles Cover A Wide Range Of Topics, From An Alternative Set-up Of Rigorous Probability To The Sampling Of Conditioned Diffusions. Random Variables -- Without Basic Space / Götz Kersting -- Chaining Techniques And Their Application To Stochastic Flows / Michael Scheutzow -- Ergodic Properties Of A Class Of Non-markovian Processes / Martin Hairer -- Why Study Multifractal Spectra? / Peter Mörters -- Construction Of Surface Measures For Brownian Motion / Nadia Sidorova, Olaf Wittich -- Sampling Conditioned Diffusions / Martin Hairer, Andrew Stuart, Jochen Voss -- Asymptotic Formulae For Coding Problems : A Review / Steffen Dereich -- Intermittency On Catalysts / Jürgen Gärtner, Frank Den Hollander, Gregory Maillard -- Stochastic Dynamical Systems In Infinite Dimensions / Salah-eldin A. Mohammed -- Feynman Formulae For Evolutionary Equations / Oleg G. Smolyanov -- Deformation Quantization In Infinite Dimensional Analysis / Remi Léandre -- Measure-valued Diffusions, Coalescents And Genetic Inference / Matthias Birkner, Jochen Blath -- How Often Does The Ratchet Click? : Facts, Heuristics, Asymptotics / Alison M. Etheridge, Peter Pfaffelhuber, Anton Wakolbinger. Edited By Jochen Blath, Peter Mörters And Michael Scheutzow. Includes Bibliographical References. This outstanding collection of thirteen articles provides an overview of recent developments of significant trends in the field of stochastic analysis. Written by leading experts in the field, the articles cover a wide range of topics, ranging from an alternative set-up of rigorous probability to the sampling of conditioned diffusions. Applications in physics and biology are treated, with discussion of Feynman formulas, intermittency of Anderson models and genetic inference. A large number of the articles are topical surveys of probabilistic tools such as chaining techniques, and of research fields within stochastic analysis. Showcasing the diversity of research activities in the field, this book is essential reading for any student or researcher looking for a guide to modern trends in stochastic analysis and neighboring fields. Presenting important trends in the field of stochastic analysis, this collection of thirteen articles provides an overview of recent developments and new results. Leading experts discuss a wide range of topics, ranging from an alternative set-up of rigorous probability to sampling of conditioned diffusions, from Feynmann formulas to genetic inference.