Stochastic Equations for Complex Systems: Theoretical and Computational Topics

An Introduction to the Malliavin Calculus and Its Applications
     1 Introduction
          1.1 Finite Dimensional Case
          1.2 Malliavin Calculus on the Wiener Space
          1.3 Sobolev Spaces
          1.4 The Divergence as a Stochastic Integral
     2 Wiener Chaos and the Ornstein-Uhlenbeck Semigroup
          2.1 Multiple Stochastic Integrals
          2.2 The Ornstein-Uhlenbeck Semigroup
     3 Clark Ocone's Formula
     4 Application of Malliavin Calculus to Regularity  and Estimations of Densities
          4.1 First Density Formula
          4.2 Second Density Formula
          4.3 Existence and Smoothness of Densities
     5 Malliavin Calculus and Normal Approximation
          5.1 Stein's Method for Normal Approximation
          5.2 Normal Approximation on a Fixed Wiener Chaos
          5.3 Convergence of Densities
     References

Fractional Brownian Motion  and an Application to Fluids
     1 Introduction
          1.1 Fractional Brownian Motion
          1.2 Properties of Fractional Brownian Motion
          1.3 An Application of fBm to Fluids
     2 Some Preliminaries
          2.1 The Integrodifferential Equation
          2.2 Assumptions
          2.3 Functional Setting
     3 Some a Priori Estimates
          3.1 Fractional Integrals and Derivatives
          3.2 A Priori Estimates
     4 Stochastic Integrals with Respect to a fBm
     5 Main Results with Proofs
          5.1 Main Result
          5.2 Fixed Point Argument
          5.3 Proof of Theorem 1
     References

An Introduction to Large Deviations  and Equilibrium Statistical Mechanics  for Turbulent Flows
     1 Introduction
     2 Models of Turbulent Flows
          2.1 3D and 2D Hydrodynamics
          2.2 Global Invariants
          2.3 Geophysical Flows
          2.4 Discretized Form for 2D Euler Flows and Analogies  with Toy Models of Magnetic Systems
     3 Mean-Field Theory for 2D Flows
          3.1 Microcanonical Measure and Large Deviations  for the Energy and Vorticity Distribution
          3.2 Large Deviations for the Macrostates
          3.3 Thermodynamic Limit and Mean-Field Equation
          3.4 Non-equivalence of Ensembles
          3.5 Large Deviations for the Coarse-Grained Vorticity Field
          3.6 The Energy-Enstrophy Measure
     4 Conclusion
     References

Recent Developments on the Micropolar  and Magneto-Micropolar Fluid Systems: Deterministic and Stochastic Perspectives
     1 Introduction
     2 Deterministic Case
          2.1 Global Regularity of the 2D MMPF System with Zero Angular Viscosity
          2.2 Regularity Criterion of the 3D MMPF System in Terms  of Two Velocity Components
     3 Stochastic Case
          3.1 Existence of Weak Martingale Solution for the 3D MPF and the MMPF Systems with Non-Lipschitz Multiplicative Noise
          3.2 Unique Strong Solution for the 2D MPF and the MMPF Systems with Lipschitz Multiplicative Noise
     4 Conclusion
     References

Pathwise Sensitivity Analysis in Transient Regimes
     1 Introduction
     2 Time-Dependent Sensitivity Analysis
          2.1 Decomposition of the Pathwise Relative Entropy
          2.2 Sensitivity Analysis and Fisher Information Matrix
     3 Discrete-Time Markov Chains
     4 Continuous-Time Markov Chains
     5 Stochastic Differential Equations---Markov Processes
     6 Demonstration Example
          6.1 Continuous Time Markov Chains: An EGFR Model
     7 Conclusions
     References

The Langevin Approach: A Simple Stochastic Method for Complex Phenomena
     1 Introduction
     2 The Langevin Approach
          2.1 Processes in Scale
          2.2 Necessary Conditions: Stationarity and the Markov Property
          2.3 The Fokker-Planck Equation for Increments
          2.4 Langevin Processes in Scale
     3 Applying the Langevin Approach to Turbulence
          3.1 The Langevin Approach in Laboratory Turbulence
          3.2 The Langevin Approach in Simulated Turbulence
          3.3 Comparative Analysis
     4 Discussion and Conclusions
     References

Monte Carlo Simulations of Turbulent Non-premixed Combustion using  a Velocity Conditioned Mixing Model
     1 Introduction
     2 Stochastic Modeling of Turbulent Non-premixed Combustion
          2.1 Deterministic Conservation Equations
          2.2 Simplified Chemical Description
          2.3 Statistical Description
          2.4 Monte Carlo Method
          2.5 Hybrid Solution Algorithm
     3 Micro Mixing Models
          3.1 Description of IEM and IECM Models
          3.2 The Correct Local Scalar Isotropy
          3.3 Implementation of the IECM Micro-Mixing Model
     4 Test Case and Numerical Details
          4.1 Delft III Flame
          4.2 Numerical Setup
     5 Results and Discussion
          5.1 Velocity Statistics
          5.2 Results for the Mixture Fraction Fields
          5.3 Results for the Temperature
          5.4 Discussion on Parameters of the IECM Algorithm
     6 Conclusions
     References

Massively Parallel FDF Simulation  of Turbulent Reacting Flows
     1 Introduction
     2 Filtered Density Function
     3 Irregularly Portioned FDF Simulator
          3.1 Static Block Decomposition
          3.2 Irregularly Portioned Lagrangian Monte Carlo (IPLMC)
          3.3 Irregularly Portioned Lagrangian Monte Carlo Finite Difference (IPLMCFD)
     4 Concluding Remarks
     References