Analytic Theory of It?Stochastic Differential Equations with Non-Smooth Coefficients (Paperback, 2022)

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)

Outlook

Wilhelm Stannat (지은이)    정보 더보기

Haesung Lee (지은이)    정보 더보기

Gerald Trutnau (지은이)    정보 더보기

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