Items related to Analytic Theory of Itô-Stochastic Differential Equations...
Synopsis
Introduction
- main questions to be answered
- our approach
- what is meant with analytical
- why Lp-measure spaces with weights? Lebesgue measure too restrictive (... from the perspective of stochastics), e.g. there are unique invariant measures different to Lebesgue measure
- orientation towards weighted measure spaces (pre-invariant measures)
1. The Cauchy problem in Lp-spaces with weights
1.1 The abstract setting, existence
1.2 Existence and regularity of pre-invariant densities (class of admissible coefficients)
1.3 Uniqueness (Lp-uniqueness), regularity and analytic irreducibility of solutions to the CP
2. Stochastic Differential Equations
2.1 Existence
2.1.1 Construction of a Markov process corresponding to a regularized version of the solution to the Cauchy problem
2.1.2. Main tools: Krylov type estimate of additive functionals $\mathbb{E}_x[\int_0^t f(X_s)ds]$
2.1.3. Identification of weak solutions to SDEs (or identification of the SDE weakly solved by ...)
2.2.1 Non-explosion and moment inequalities
2.2.2 Irreducibility, transience and recurrence
2.2.3 Long time behavior: Ergodicity, existence and uniqueness of invariant measures, examples/counterexamples
2.3 Uniqueness
2.3.1 Pathwise uniqueness and strong solutions
2.3.2 Uniqueness in law (via the martingale problem)
2.4 Further topics (convergence, approximation)
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