Contents -- Preface -- IRIMA -- Participants -- Sergio Albeverio and Yana Belopolskaya Probabilistic Approach to Hydrodynamic Equations -- 1. Diffusion process and the Navier-Stokes equations -- 2. Convergence of successive approximations -- Acknowledgements -- References -- Hakima Bessaih and Franco Flandoli A Mean Field Result for 3D Vortex Filaments -- 1. Introduction -- 2. Abstract mean field result -- 3. Proofs -- 3.1. Uniform bound on the marginals densities -- 3.2. Variational characterization of hN and known results -- 3.3. Weak limit of hNk -- 3.4. Convergence of the variational problems -- 3.5. Properties of the limit variational problem -- 4. Application to vortex filaments -- 4.1. Brownian vortex filaments -- 4.1.1. Introduction -- 4.1.2. Space of configurations -- 4.1.3. Cross-section and its random selection -- 4.1.4. Reference measure -- 4.1.5. The energy -- 4.2. Other models -- 4.2.1. Brownian semimartingales -- 4.2.2. Processes with finite p variation -- References -- Bjorn Bottcher and Niels Jacob Remarks on Meixner-type Processes -- 1. Introduction -- Acknowledgement: -- 2. Meixner Processes -- 3. Symbols of Meixner-type -- 4. Meixner-type Processes -- References -- Zdzistaw Brzeinaak Some Remarks on Ito and Stratonovich Integration in 2-smooth Banach Spaces -- 1. Introduction -- 2. Ito integral in 2-smooth Banach spaces -- 3. Burkholder inequality -- 4. An Example -- 5. Relationship with the approach of Mikulevicius and Rozovskii -- 6. Approximation of the Stratonovich integral -- 6.1. The result -- 6.2. The proof -- Acknowledgments -- References -- Tomas Caraballo The Long-time Behaviour of Stochastic 2D-Navier-Stokes Equations -- 1. Introduction -- 2. The exponential stability of solutions -- 3. Exponential stabilizability and stabilization -- 4. Pathwise stability and stabilizability with general decay rate.

5. Conclusions, comments and open problems -- Acknowledgments -- References -- Pao-Liu Chow Semilinear Stochastic Wave Equations -- 1. Introduction -- 2. Linear Stochastic Hyperbolic Equation -- 3. Semilinear Stochastic Hyperbolic Equations -- 4. Applications -- Acknowledgments -- References -- Nigel J. Cutland Stochastic Navier-Stokes Equations: Loeb Space Techniques & Attractors -- 1. Introduction -- 2. Existence for the stochastic Navier-Stokes equations -- 2.1. Mathematical formulation -- 2.1.1. Definition of solutions to the stochastic Navier-Stokes equations -- 2.2. Loeb space methods -- 3. Attractors for stochastic Navier-Stokes equations -- 3.1. Measure attractors -- 3.2. Stochastic attractors -- 3.2.1. Existence of a stochastic attractor for the Navier-Stokes equations -- 3.3. Process attractors -- Appendix -- A . l . The hyperreals -- A.2. Construction of a Loeb space -- References -- Arnaud Debussche The 2D-Navier-Stokes Equations Perturbed by a Delta Correlated Noise -- 1. Introduction -- 2. Notations -- 3. Preliminaries -- 4. Existence and uniqueness -- 5 . Ergodicity -- References -- Sergio Albeverio and Benedetta Ferrario Invariant Measures of Levy-Khinchine Type for 2D Fluids -- 1. Introduction -- 2. Invariant measures of LQvy-Khinchine type -- 3. Deterministic dynamics -- 4. Stochastic dynamics -- 5 . Final remarks -- Acknowledgments -- References -- Franco Flandoli Some Remarks on a Statistical Theory of Turbulent Flows -- 1. Introduction -- 2. Vortex filaments -- 2.1. Random 1 -currents -- 2.2. Back to vortex filaments i n 3D fluids -- 3. 3D stochastic Navier-Stokes equation: weak stationary solutions -- 4. The viewpoint of random dynamical systems -- RSB theory. -- Fokker-Planck equation. -- RSB theory for random dynamical systems. -- 4.1. Random attractors -- 4.2. Random invariant measures.

4.3. RSB properties of the random invariant measures -- 5 . Conclusions -- References -- Christophe Giraud Some Properties of Burgers Turbulence with White Noise Initial Conditions -- 1. Introduction -- 2. Some background on Burgers equation -- 3. Parabolic hull of a Brownian motion -- 4. Burgers turbulence with white noise initial velocity -- 4.1. State at a fixed time t > 0 -- 4.2. Time evolution of the turbulence -- 5. Burgers turbulence with some other initial velocities of white noise type -- 5.1. The one-sided white noise case -- 5.2. The periodic white noise case -- 6. Some open problems -- References -- Yuri E. Gliklikh Deterministic Viscous Hydrodynamics via Stochastic Processes on Groups of Diffeomorphisms -- 1. Preliminaries and Introduction -- 2. Basic notion from the geometry of groups of diffeomorphisms -- 3. Description of viscous hydrodynamics -- Acknowledgments -- References -- Niels Jacob and Aubrey Truman Further Classes of Pseudo-differential Operators Applicable to Modelling in Finance and Turbulence -- 1. Introduction -- 2. Some basic facts on Feller processes -- 3. Hoh's symbolic calculus -- 4. Baldus' Weyl calculus approach -- 5. Relations to subordination in the sense of Bochner and operators of variable order of differentiability -- References -- Benjamin Jourdain and Tony Lelievre Mathematical Analysis of a Stochastic Differential Equation Arising in the Micro-Macro Modelling of Polymeric Fluids -- 1. Introduction -- 2. Existence and uniqueness -- 2.1. Trajectorial uniqueness for solutions with values in B -- 2.2. Existence in the case g Lt -- 2.3. Existence in the case g L1toc (R+) -- 3. Does the solution reach the boundary ? -- 3.1. Necessary and suflcient conditions -- 3.2. Non-uniqueness in case b < 2 -- 4. Invariant probability measure in case g = 0 and b 2 -- Acknowledgments -- Bibliography.

Hannelore Lisei and Michael Scheutzow On the Dispersion of Sets under the Action of an Isotropic Brownian Flow -- 1. Introduction -- 2. Isotropic Brownian Flows -- 3. The Upper Bound -- 4. The Lower Bound -- 5. Open Problems -- Appendix -- References -- Aubrey Truman, Chris N. Reynolds and David Williams Stochastic Burgers Equation in d-dimensions - A One-dimensional Analysis: Hot and Cool Caustics and Intermittence of Stochastic Turbulence -- 1. Introduction -- 1.1. When does Ht have cusps? -- 2. Some Geometrical Results ( = 1) -- 3. Intermittence of Stochastic Turbulence -- 4. Some Analytical Results (Small ) -- 5. Some Applications -- 5.1. Hot and Cool Parts of the Caustic -- 5.2. Intermittence of Stochastic Turbulence - a simple example in two dimensions -- 5.3. process for small noise in 2 dimensions -- Acknowledgement -- References -- Armen Shirikyan A Version of the Law of Large Numbers and Applications -- 1. Introduction -- Notation -- 2. Strong law of large numbers for mixing-type Markov chains -- 2.1. Formulation of the result -- 2.2. Proof of Theorem 2.1 -- 3. Applications -- 3.1. Dissipative PDE's perturbed by a bounded kick force -- 3.2. The Navier-Stokes system perturbed by an unbounded kick force -- References -- Marian Slodicka Comprehensive Models for Wells -- 1. Introduction -- 2. Point sources -- 3. Wells with a non-negligible radius -- 4. Robin type boundary condition -- 4.1. Variational formulation, well-posedness -- 4.2. Numerical scheme -- 4.3. Convergence of the scheme -- References -- Enrique Thomann and Mina Ossiander Stochastic Cascades Applied to the Navier-Stokes Equations -- 1. Introduction -- 2. Applications to the Navier-Stokes Equations -- 2.1. Stochastic Recursion -- 2.2. Successive Iterations of a Contraction Map -- 3. Application to KPP Equations -- 4. Damped Burgers Equation, Some Open Problems -- Acknowledgments.

References -- Aubrey Truman and Jiang-Lun Wu Stochastic Burgers Equation with Levy Space-Time White Noise -- 1. Introduction -- 2. Poisson White Noise and Stochastic Integration -- 3. Burgers Equation Driven by Levy Space-Time White Noise -- Acknowledgements -- References -- Tusheng Zhang A Comparison Theorem for Solutions of Backward Stochastic Differential Equations with Two Reflecting Barriers and Its Applications -- 1. Introduction -- 2. Backward SDE With Two Reflecting Barrier Processes -- 3. Existence of Solutions of A Backward SDE with Two Reflecting Barriers -- Acknowledgements: -- References -- Aubrey Truman and Huaizhong Zhao Burgers Equation and the WKB-Langer Asymptotic L2 Approximation of Eigenfunctions and Their Derivatives -- 1. Introduction -- 2. WKB-Langer asymptotic expansions -- 3. Semi-classical approximation of eigenfunctions and their derivatives in L2 -- Acknowledgement -- References.