Ling, Chengcheng: Stochastic differential equations with singular drifts and multiplicative noises. 2020
Inhalt
- Introduction
- Background and Motivation
- Well-posedness of SDEs
- Non-explosion of solutions to SDEs driven by continuous noise
- Density of the solutions to SDEs driven by jump noise
- Main results
- Structure of this thesis
- Outlook
- Preliminaries
- Mixed-norm Lebesgue spaces
- Lévy processes and non-local pseudo-differential operators
- Strong solutions, weak solutions and martingale solutions to SDEs
- Estimates of the fundamental solutions to second order parabolic equations
- Main methods
- Existence and Uniqueness of a global strong solution to an SDE driven by continuous noise in mixed-norm Lebesgue spaces on Q=[0,)Rd
- Preliminaries and main results
- Regularity estimates for parabolic type partial differential equations
- Krylov estimates and existence of weak solutions
- Itô's formula for functions in Sobolve spaces with mixed-norm
- Pathwise uniqueness of strong solutions
- Existence and uniqueness of a maximally defined local strong solution to an SDE driven by continuous noise in mixed-norm Lebesgue spaces on a general space time domain Q[0,)Rd
- Non-explosion of the solutions to SDEs driven by continuous noise in mixed-norm Lebesgue spaces
- Preliminaries and main result
- Probabilistic representation of solutions to parabolic partial differential equations
- Some auxiliary proofs
- Proof of Theorem 5.2
- Diffusions in random media
- M-particle systems with gradient dynamics
- Existence and uniqueness of weak solutions to SDEs with distributional valued drifts and jump type noise
- Preliminaries and main results
- Preparations
- Schauder estimates for (6.1)
- Martingale solutions and weak solutions
- Regularity of densities of weak solutions
- Appendix
- Khasminskii's lemma
- Non-explosion lemma
- Girsanov transformation
- Urysohn Lemma
- Equivalence between martingale solutions and weak solutions
- The Sobolev embedding theorem in mixed-norm spaces
- References