Singular random dynamics : Cetraro, Italy 2016 /: Cetraro, Italy 2016. ([2019])

• Record Type: Book
• Further Information: Note: Massimiliano Gubinelli, Panagiotis E. Souganidis, Nikolay Tzvetkov ; Franco Flandoli, Massimiliano Gubinelli, Martin Hairer, editors.
• Editors: Souganidis, Panagiotis E; Tzvetkov, Nikolay; Flandoli, Franco; Hairer, Martin
• Contents: 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 Principle2.9 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 Itô3.4 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 Hamiltonians3.7.3 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 Difficulties3.10.2 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
• Publisher Details: Cham, Switzerland : Springer
• Publication Date: 2019
• Extent: 1 online resource (324 pages)
• Subjects: 519.2/2; Stochastic partial differential equations; Electronic books; Electronic books
• Languages: English
• ISBNs: 9783030295455; 3030295451
• Related ISBNs: 9783030295448
• Notes: Note: Description based on online resource; title from digital title page (viewed on December 05, 2019).
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.