Rehmeier, Marco: Nonuniqueness of Laws on State and Path Space: Flow Selections and Superposition for Fokker-Planck-Kolmogorov Equations and Convex Integration for Stochastic [...]. 2021
Inhalt
- General notation and basic facts
- I Solution flows for Fokker–Planck–Kolmogorov equations
- Introduction
- Introduction to Fokker–Planck–Kolmogorov equations
- Selection theorems for stochastic systems
- Main results: linear case
- Nonlinear and infinite-dimensional equations
- Solution flows for linear equations
- Linear FPK equations
- Solution flows
- Proofs of main results
- Applications and examples
- Comparison to Markovian semigroups
- Solution flows for nonlinear equations
- Solution flows for FPK equations for measures on infinite-dimensional spaces
- Auxiliary results on FPK equations
- II Superposition principle for nonlinear Fokker–Planck–Kolmogorov equations
- Introduction
- Superposition principle for finite-dimensional equations
- Superposition principle for (stochastic) nonlinear FPK equations
- Superposition principle for deterministic nonlinear FPK equations
- Nonlinear FPK equations
- Geometry on SP
- Proof of main result
- Consequences: Existence, uniqueness and an application
- Superposition principle for stochastic nonlinear FPK equations
- From finite-dimensional differential to continuity equations
- III Nonuniqueness in law for stochastic hypodissipative Navier–Stokes equations
- Introduction
- (Fractional) Navier–Stokes equations
- A brief history of convex integration
- Stochastic PDEs
- Main result
- Preliminaries
- Fourier analysis on T3, Sobolev spaces, fractional Laplacian
- Notation
- Martingale solutions
- Extension of local martingale solutions
- Proof of the main result
- Outline of the proof
- Decomposition of the equation
- Analytically weak local solutions
- From analytically weak to local martingale solutions
- Conclusion of the proof
- Convex integration for stochastic hypodissipative NSE
- Regularity for the stochastic linear equation
- IV General appendices