Singular random dynamics : Cetraro, Italy 2016 / Massimiliano Gubinelli, Panagiotis E. Souganidis, Nikolay Tzvetkov ; Franco Flandoli, Massimiliano Gubinelli, Martin Hairer, editors.

Gubinelli, Massimiliano [Browse]
  • Cham, Switzerland : Springer ; [Cetraro] : Fondazione CIME Roberto Conti, [2019]
  • ©2019
ix, 313 pages ; 24 cm.


  • Lecture notes in mathematics (Springer-Verlag) ; 2253. [More in this series]
  • Lecture notes in mathematics (Springer-Verlag). CIME Foundation subseries [More in this series]
  • Lecture notes in mathematics, 0075-8434 ; 2253
  • CIME Foundation series
Bibliographic references
Includes bibliographical references.
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Current copyright fee: GBP19.00 42\0.
  • Intro; Preface; General Remarks; The Courses; Final Remarks; Contents; 1 Introduction; References; 2 Lectures on Energy Solutions for the Stationary KPZ Equation; 2.1 Introduction; 2.1.1 Notations and Some Preliminaries; 2.1.2 White Noise; 2.2 The Ornstein-Uhlenbeck Process; 2.3 Gaussian Computations; 2.4 The Itô Trick; 2.5 An Approximation Scheme; 2.5.1 Time Reversal; 2.6 Controlled Processes and Energy Solutions; 2.6.1 Regularization by Noise for Controlled Processes; 2.7 Boltzmann-Gibbs Principle; 2.7.1 A First Computation; 2.8 The Hairer-Quastel Invariance Principle
  • 2.8.1 The Invariance Principle 2.9 Uniqueness of Energy Solutions; 2.9.1 Mapping to the SHE; 2.9.2 Convergence of the Remainder; References; 3 Pathwise Solutions for Fully Nonlinear First- and Second-Order Partial Differential Equations with Multiplicative Rough Time Dependence; 3.1 Introduction; 3.1.1 Organization of the Notes; 3.2 Motivation and Some Examples; 3.2.1 Motion of Interfaces; 3.2.2 A Stochastic Selection Principle; 3.2.3 Pathwise Stochastic Control Theory; 3.2.4 Mean Field Games; 3.3 The Main Difficulties and the Choice of Stochastic Calculus; 3.3.1 Difficulties
  • 3.3.2 The Choice of Stochastic Calculus: Stratonovich vs Itô 3.4 Single Versus Multiple Signals, the Method of Characteristics and Nonlinear pde with Linear Rough Dependence on Time; 3.4.1 Single Versus Multiple Signals; 3.4.2 Nonlinear pde with Linear Rough Dependence on Time; 3.4.3 Stochastic Characteristics; 3.5 Fully Nonlinear Equations with Semilinear Stochastic Dependence; 3.6 The Extension Operator for Spatially Homogeneous First-Order Problems; 3.6.1 A Summary of the General Strategy; 3.7 Pathwise Solutions for Equations with Non-smooth Hamiltonians; 3.7.1 Formulae for Solutions
  • 3.7.2 Pathwise Solutions for Nonsmooth Hamiltonians 3.7.3 Control of Cancellations for Spatially Dependent Hamiltonians; 3.7.4 The Interplay Between the Regularity of the Hamiltonians and the Paths; 3.8 Qualitative Properties; 3.8.1 Domain of Dependence and Finite Speed of Propagation; 3.8.2 Stochastic Intermittent Regularization; 3.8.3 Long Time Behavior of the "Rough" Viscosity Solutions; 3.9 Stochastic Viscosity Solutions; 3.10 Pathwise Solutions to Fully Nonlinear First and Second Order pde with Spatially Dependent Smooth Hamiltonians
  • 3.10.1 The General Problem, Strategy and Difficulties 3.10.2 Improvement of the Interval of Existence of Smooth Solutions; 3.10.3 The Necessity of the Assumptions; 3.10.4 Convex Hamiltonians and a Single Path; 3.10.5 Multiple Paths; 3.11 Perron's Method; 3.12 Approximation Schemes, Convergence and Error Estimates; 3.12.1 The Scheme Operator; 3.12.2 The Method of Proof; 3.12.3 The Main Examples; 3.12.4 The Need to Regularize the Paths; 3.13 Homogenization; 3.13.1 The Difficulties and General Strategy; 3.13.2 The Single-Noise Case; 3.13.3 The Multiple-Noise Case
  • 3.14 Stochastically Perturbed Reaction-Diffusion Equations and Front Propagation
  • 9783030295448 ((paperback))
  • 3030295443 ((paperback))
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