From Particle Systems to Partial Differential Equations III: Particle Systems and PDEs III, Braga, Portugal, December 2014 (Springer Proceedings in Mathematics & Statistics Book 162)

The main focus of this book is on different topics in probability theory, partial differential equations and kinetic theory, presenting some of the latest developments in these fields. It addresses mathematical problems concerning applications in physics, engineering, chemistry and biology that were presented at the Third International Conference on Particle Systems and Partial Differential Equations, held at the University of Minho, Braga, Portugal in December 2014. The purpose of the conference was to bring together prominent researchers working in the fields of particle systems and partial differential equations, providing a venue for them to present their latest findings and discuss their areas of expertise. Further, it was intended to introduce a vast and varied public, including young researchers, to the subject of interacting particle systems, its underlying motivation, and its relation to partial differential equations. This book will appeal to probabilists, analysts and those mathematicians whose work involves topics in mathematical physics, stochastic processes and differential equations in general, as well as those physicists whose work centers on statistical mechanics and kinetic theory. Preface 6 Contents 7 On Linear Hypocoercive BGK Models 9 1 Introduction 9 2 Discrete Coercive BGK Models 14 2.1 Multi-velocity BGK Models 19 3 A Discrete Hypocoercive BGK Model 22 3.1 Lyapunov's Direct Method 24 4 Space-Inhomogeneous BGK Models 27 4.1 A Two Velocity BGK Model 28 4.2 A Multi-velocity BGK Model 31 4.3 A Continuous Velocity BGK Model 34 4.4 Linearized BGK Equation 38 4.5 Local Asymptotic Stability for the BGK Equation 42 References 45 Hydrodynamic Limit of Quantum Random Walks 46 1 Introduction 46 2 The State Space of the QRW 48 3 The Dynamics of a Single QRW 49 4 Hydrodynamic Limit for a System of Independent Quantum Random Walks 52 5 Local Equilibrium 53 References 57 Sub-shock Formation in Reacting Gas Mixtures 58 1 Introduction 58 2 Governing Equations 61 3 Shock-Wave Structure Solutions 63 3.1 Existence and Uniqueness of Upstream Equilibrium State 66 3.2 Singular Barriers and Critical Mach Numbers 70 4 Chemical Reactions and Concluding Remarks 73 References 78 Compactness of Linearized Kinetic Operators 80 1 Introduction 80 2 The Classical Boltzmann Equation Case 81 2.1 Boltzmann's Equation 81 2.2 Earlier Compactness Results 85 2.3 Grad's Procedure 85 2.4 Extensions in the Cut-off Case 90 2.5 Extensions to Kernels Without Cut-off 92 3 The Compactness Properties for the Linearized Kinetic Operators for Mixtures 96 3.1 Building the Linearized Collision Operator for Mixtures 96 3.2 Elements of Proof for the Compactness 99 References 103 Asymptotics for FBSDES with Jumps and Connections with Partial Integral Differential Equations 105 1 Motivation and Some Words About FBSDEs 105 2 Functional Setting and Preliminary Results for the FBSDE System 111 3 The Asymptotic Study 117 4 Statement of a Large Deviations Principle 122 References 125 Entropy Dissipation Estimates for the Landau Equation: General Cross Sections 127 1 Introduction and Main Result 127 1.1 Description of the Landau Operator and Equation 127 2 Proofs of the Theorems 138 References 148 The Boltzmann Equation over mathbbRD: Dispersion Versus Dissipation 150 1 The Boltzmann Dynamics over mathbbRD: Two Competing Processes 150 1.1 Dissipation Effect of Collisions 151 1.2 Dispersion Effect of Advection in mathbbRD 154 1.3 Two Competing Mechanisms 154 2 Global Maxwellians 155 3 The Cauchy Problem for the Boltzmann Equation in mathbbRD 157 3.1 The Boltzmann Collision Integral 157 3.2 The Time-Averaged Collision Frequency 159 3.3 Existence and Uniqueness for the Cauchy Problem 159 4 Large Time Dynamics for the Boltzmann Equation in mathbbRD 161 5 Scattering Theory for the Boltzmann Equation in mathbbRD 164 6 Some Open Problems 166 6.1 Problem 1: The BGK Model 166 6.2 Problem 2: Scattering and Cercignani's Conjecture 167 6.3 Problem 3: Non Cutoff Molecular Interactions 169 6.4 Problem 4: Scattering and Boltzmann-Grad Limit 170 References 170 The Gradient Flow Approach to Hydrodynamic Limits for the Simple Exclusion Process 172 1 Introduction 172 2 Gradient Flow Structure for Reversible Markov Chains 173 2.1 Framework 173 2.2 Continuity Equation 175 3 Scaling Limits and Gradient Flows 177 4 Symmetric Simple Exclusion Process (SSEP) 180 4.1 Model: Definitions and Notations 180 4.2 The Gradient Flow Approach to Theorem 4.2 183 4.3 Bounds and Convergence 184 References 188 Symmetries and Martingales in a Stochastic Model for the Navier-Stokes Equation 190 1 The Weak Stochastic Euler-Lagrange Condition 192 1.1 Admissible Trajectories 192 1.2 A Weak Description of Random Trajectories 192 1.3 A Weak Euler-Lagrange Condition 193 2 Navier-Stokes Equation and the Weak Euler-Lagrange Condition 194 2.1 Description of Dissipative Flows by Laws of Semi-martingales 194 2.2 Stochastic Action and Relative Entropy 195 2.3 Least Action Principle and Relative Entropy 195 3 Invariances and the Stochastic Noether Theorem 196 3.1 Symmetry by Translation and the Momentum Process 197 3.2 Symmetry by Rotation and the Kinetic Momentum Process 198 References 198 Convergence of Diffusion-Drift Many Particle Systems in Probability Under a Sobolev Norm 200 1 Introduction 200 2 Convergence in Linfty(L2)capL2(H1) Norm 202 3 Discussion About Higher Order Sobolev Norm 223 References 228 From Market Data to Agent-Based Models and Stochastic Differential Equations 229 1 Introduction 229 2 A Data-Reconstructed Market Model [3] 232 3 Mathematical Consistency of the Fractional Volatility Model: Arbitrage and Completeness [5] 234 3.1 No-Arbitrage 234 3.2 Incompleteness 236 3.3 Leverage, a Modified Model and Completeness [16,5] 237 3.4 A Remark on Long Memory and Fractional Brownian Motion 239 4 Agent-Based Interpretation of the Fractional Volatility Model 240 4.1 A Market Model with Self-adapted or Fixed Strategies 241 4.2 A Limit-Order Book Market Model 241 5 Further Properties of the Fractional Volatility Model 243 References 246 Global Asymptotic Stability of a General Nonautonomous Cohen-Grossberg Model with Unbounded Amplification Functions 247 1 Introduction 247 2 Preliminaries 249 3 Asymptotic Stability 252 4 Applications 257 References 266 Phase Transitions and Coarse-Graining for a System of Particles in the Continuum 267 1 Introduction 267 2 Model 268 2.1 Mean-Field Model 270 3 Contour Model 272 4 The Main Results 274 5 Outline of the Proof 277 5.1 Coarse-Graining 279 5.2 Cluster Expansion 282 References 287 Modelling of Systems with a Dispersed Phase: ``Measuring'' Small Sets in the Presence of Elliptic Operators 288 1 Introduction 288 2 Capacity and Correctors 289 3 Homogeneous Boundary Conditions 298 4 Non-homogeneous Boundary Conditions 300 5 Adding Constraints on the Solutions 302 6 Conclusion 302 References 303 Derivation of the Boltzmann Equation: Hard Spheres, Short-Range Potentials and Beyond 304 1 Introduction 304 2 Hard-Sphere Interaction 310 3 Short-Range Interactions 316 4 Beyond the Short Range 321 References 323 Duality Relations for the Periodic ASEP Conditioned on a Low Current 325 1 Introduction 325 2 Definitions and Notation 327 2.1 State Space and Configurations 327 2.2 Definition of the ASEP 328 2.3 Representation of the Generator in the Natural Tensor Basis 331 2.4 The Quantum Algebra Uq[mathfrakgl(2)] 334 2.5 Duality in the Quantum Hamiltonian Formalism 337 2.6 Shock/Antishock Measures 338 3 Results 341 4 Proofs 343 4.1 Proof of Theorem 1 343 4.2 Proof of Theorem 2 344 4.3 Proofs of Theorems 3 and 4 345 References 351 Front Matter....Pages i-viii On Linear Hypocoercive BGK Models....Pages 1-37 Hydrodynamic Limit of Quantum Random Walks....Pages 39-50 Sub-shock Formation in Reacting Gas Mixtures....Pages 51-72 Compactness of Linearized Kinetic Operators....Pages 73-97 Asymptotics for FBSDES with Jumps and Connections with Partial Integral Differential Equations....Pages 99-120 Entropy Dissipation Estimates for the Landau Equation: General Cross Sections....Pages 121-143 The Boltzmann Equation over \({{\mathbb R}^{{\mathrm {D}}}}\) : Dispersion Versus Dissipation....Pages 145-166 The Gradient Flow Approach to Hydrodynamic Limits for the Simple Exclusion Process....Pages 167-184 Symmetries and Martingales in a Stochastic Model for the Navier-Stokes Equation....Pages 185-194 Convergence of Diffusion-Drift Many Particle Systems in Probability Under a Sobolev Norm....Pages 195-223 From Market Data to Agent-Based Models and Stochastic Differential Equations....Pages 225-242 Global Asymptotic Stability of a General Nonautonomous Cohen-Grossberg Model with Unbounded Amplification Functions....Pages 243-262 Phase Transitions and Coarse-Graining for a System of Particles in the Continuum....Pages 263-283 Modelling of Systems with a Dispersed Phase: “Measuring” Small Sets in the Presence of Elliptic Operators....Pages 285-300 Derivation of the Boltzmann Equation: Hard Spheres, Short-Range Potentials and Beyond....Pages 301-321 Duality Relations for the Periodic ASEP Conditioned on a Low Current....Pages 323-350