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Analysis of singular stochastic systems: two classes of examples / Matthias Liesenfeld ; angefertigt unter Betreuung durch Herrn Prof. Dr. Martin Kolb. Paderborn, 2021
Content
Abstract
Introduction
Stochastic spikes
Fleming Viot particle systems
One-dimensional time-homogeneous SDEs
Notions of solution and uniqueness
Scale function
Speed measure and Green function
Feller's Test for Explosions
Approach using Sturm-Liouville theory
h-transformation in the sense of Doob
Stochastic Spikes
Notation
Discussion of the parameter lambda
An embedded approximate Poisson process
A thinned renewal process
Poisson Limits in the high noise regime
Applications
Asymptotic linear stochastic differential equations
Homodyne detection of Rabi oscillation
Conclusion
Fleming-Viot particle Systems
Notation and basic properties
Problem formulation
Symmetry of the model
Self-similarity of generalized Bessel processes
Transition densities
Markov chains in general state space
(Non-)extinction criterion
Markov chain analysis
Integrability of the ergodic elements
Application of Birkhoff's ergodic theorem
Computation of the integrand ln R in Theorem 5.71
Three particles
Open problems
Properties of Hypergeometric Functions
Proof omitted in Remark 5.34
Source Codes
Large noise limits
Plot of Ephi0[ln R]
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