Mathematical Analysis of Random Phenomena : proceedings of the international conference, Hammamet, Tunisia, 12-17 September 2005

This volume highlights recent developments of stochastic analysis with a wide spectrum of applications, including stochastic differential equations, stochastic geometry, and nonlinear partial differential equations. While modern stochastic analysis may appear to be an abstract mixture of classical analysis and probability theory, this book shows that, in fact, it can provide versatile tools useful in many areas of applied mathematics where the phenomena being described are random. The geometrical aspects of stochastic analysis, often regarded as the most promising for applications, are specially investigated by various contributors to the volume. Contents......Page 8 Preface HBLENE AIRAULT......Page 6 0 . Introduction......Page 10 1. Bracket, metric and structure constants......Page 13 2. Tensor fields on diff(S1), their expressions in the trigonometrical basis......Page 14 4. The Levi-Civita connection on H \ Diff (S1)......Page 17 5. Commuting with the Hilbert transform, torsionless and ant isymmetry......Page 19 Part 11. Stochastics on H \ Diff (Sl) and integration by parts formula......Page 25 References......Page 30 1. The Ornstein-Uhlenbeck operator on a Berezinian space......Page 32 2. The classical one-dimensional Ornstein-Uhlenbeck process......Page 33 3. The Ornstein-Uhlenbeck process on an Euclidean space E......Page 34 4. Invariant measures under a commutator hypothesis......Page 35 References......Page 38 1. Introduction......Page 40 2. Basic elements of an infinite dimensional martingale and stochastic integration......Page 42 3. Main results......Page 45 References......Page 52 1. Introduction......Page 54 2. Generalized Fourier-Gauss transforms......Page 55 3. Unitarity......Page 57 4. White Noise Operators......Page 58 5. The Main Results......Page 59 References......Page 63 1. Introduction......Page 66 2. Notation and preliminaries......Page 67 3. Large deviation for Gaussian measures on S’(R)......Page 69 4.1. Generalized Gross heat equation......Page 76 4.3. Ventcel and Freidlin’s estimate......Page 78 References......Page 79 1. Introduction......Page 82 2.1. Test and generalized functions spaces......Page 83 2.2. The Convolution Product *......Page 86 3. Solution of the n-dimensional convolution equation......Page 88 References......Page 92 1. Introduction......Page 94 2. Diffusions on the homeomorphism group of the torus......Page 95 3. The variational principle and the associated dynamics......Page 97 4. The minimal entropy principle......Page 99 References......Page 100 1. Introduction......Page 102 2. A non-probabilistic variational principle......Page 103 3. Conservation of Energy......Page 106 References......Page 108 Convolution calculus on white noise spaces and Feynman graph representation of generalized renormalization flows TAKEYUKI HIDA, SI SI......Page 110 References......Page 118 2. Invariance of the noises......Page 120 2.2. (Gaussian) white noise......Page 121 2.3. Two-dimensional valued white noise.......Page 123 2.4. Gauge transformations......Page 124 3. Poisson noise......Page 125 4. Application to information sociology......Page 127 References......Page 128 1. Introduction......Page 130 2. a-stable white noise functionals......Page 131 3. Chaos decomposition of a-stable white noise functionals......Page 134 4. The Segal-Bargmann transform of square-integrable a-stable white noise functionals......Page 136 5 . Test and generalized functionals......Page 138 6. Annihilation, creation, and conservation operators......Page 144 7. A quantum decomposition of stable processes......Page 145 References......Page 149 0. Introduction......Page 150 1. Preliminaries......Page 151 2. Kirillov's construction of an action of Diff(S1) on a space of univalent functions......Page 154 3. The Neretin polynomials and the representation p......Page 156 4. Definition of an unitarizing measure and a non-existence result......Page 158 References......Page 161 1. Introduction......Page 164 2. Analysis on the Wiener space......Page 165 3. FKG inequality on the Wiener space......Page 169 4. The discrete case......Page 173 References......Page 175 1. Introduction......Page 176 2. The ground state path integral representation......Page 177 3. A finite-dimensional example......Page 180 4. Conclusions......Page 181 References......Page 183 1. Introduction......Page 186 2. Framework......Page 187 3. A representation theorem for functionals a, a class o diffusions......Page 192 4. Application to sensitivity with respect to the starting point......Page 196 References......Page 197 1. Introduction......Page 200 2. Preliminaries.......Page 202 3. Admissible monomials......Page 207 4. Wick’s Theorem......Page 211 5. Duality......Page 216 6. Meyer's formula......Page 218 References......Page 221 1. Introduction......Page 222 2. On classical integrability......Page 223 3. Quantum integrability......Page 228 4. On stochastic integrability......Page 229 5. Examples, open problems and prospects......Page 235 Acknowledgments......Page 238 References......Page 239 List of participants......Page 240 Geometry and integration by parts on H \ Diff(S1) -- Invariant measures for Ornstein-Uhlenbeck operators -- Backward stochastic differential equations with respect to martingales -- Partial unitarity arising from quadratic quantum white noise -- Schilder's theorem for Gaussian white noise distributions -- A nonlinear stochastic equation of convolution type -- Variational principle for diffusions on the diffeomorphism group with the H2 metric -- On a variational principle for the Navier-Stokes equation -- Convolution calculus on white noise spaces and Feynman graph representation of generalized renormalization flows -- Characterizations of standard noises and applications -- Analysis of stable white noise functionals -- Unitarizing measures for a representation of the Virasoro algebra, according to Kirillov and Malliavin: state of the problem -- FKG inequality on the Wiener space via predictable representation -- Path-integral estimates of ground-state functionals -- A representation theorem and a sensitivity result for functionals of jump diffusions -- Creation and annihilation operators on locally compact spaces -- From the geometry of parabolic PDE to the geometry of SDE

this Volume Highlights Recent Developments Of Stochastic Analysis With A Wide Spectrum Of Applications, Including Stochastic Differential Equations, Stochastic Geometry, And Nonlinear Partial Differential Equations.

while Modern Stochastic Analysis May Appear To Be An Abstract Mixture Of Classical Analysis And Probability Theory, This Book Shows That, In Fact, It Can Provide Versatile Tools Useful In Many Areas Of Applied Mathematics Where The Phenomena Being Described Are Random. The Geometrical Aspects Of Stochastic Analysis, Often Regarded As The Most Promising For Applications, Are Specially Investigated By Various Contributors To The Volume.

Highlights developments of stochastic analysis with applications, including stochastic differential equations, stochastic geometry, and nonlinear partial differential equations. This book shows that modern stochastic analysis can provide tools useful in areas of applied mathematics where the phenomena being described are random.