Titlebook: Stochastic Systems with Time Delay; Probabilistic and Th Sarah A.M. Loos Book 2021 The Editor(s) (if applicable) and The Author(s), under e

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Stochastic Systems with Time Delay978-3-030-80771-9Series ISSN 2190-5053 Series E-ISSN 2190-5061

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Infinite Fokker-Planck Hierarchyderstand the non-Markovian dynamics. Later in Chap. ., we will present a new derivation of the FP description using a Markovian embedding technique [.,.,.,.,.,.]. To better understand the technical and conceptual difference of our approach, we will here also review some earlier work, in particular, Refs. [.,.,.].

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Approximation for the Two-time Probability density the Fokker-Planck hierarchy. In particular, we have discussed the perturbation theory and the force-linearization closure (Chap. .). While these approaches render quite accurate descriptions of the intrawell dynamics, they do not yield any information about the .-time PDF.

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Summaryis a colloidal (Brownian) particle subject to a position-dependent, time-delayed feedback force and nonlinear static forces stemming from a potential. We describe its noisy motion by a Langevin equation, which is, due to the delay, non-local in time and infinite-dimensional.

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Force-Linearization ClosureIn this chapter, we turn to the problem of finding (approximate) probabilistic solutions of systems with delay. In particular, we discuss an approximation scheme for the steady-state one-time PDF, which we have introduced in Ref. [.], called . (FLC).

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Outlook—Open Questions and Further PerspectivesWe close with some comments putting the findings presented in this thesis into a broader perspective and giving an outlook on future research. We begin with formulating a few obvious follow-up questions and take a wider view towards the end.