A study of interacting stochastic networks :large scale, long time behavior and fluctuations

Wen Sun 1
1 MAMBA - Modelling and Analysis for Medical and Biological Applications
Inria de Paris, LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
Abstract : This PhD document is devoted to the analyses of large stochastic networks used to study mathematical models in communication networks and in biology. The first part consists of the analysis of three models used to evaluate the efficiency of duplication and placement algorithms in large distributed systems. These models are represented by large stochastic networks under different scaling regimes assumptions. In Chapter 2, the dynamic of the system can be described with the empirical measure associated to a multi-dimensional transient Markov process. We provide a detailed study of these processes on several fast time scales. Stochastic averaging principles with multiple time scales are investigated in particular. In Chapter 3 and Chapter 4, the interactions have unbounded jump sizes and occur within a limited random set of nodes. We develop a convenient mean field analysis in despite that the associated empirical measures do not have autonomous evolution equations. We also study the long time behavior of the corresponding limiting nonlinear jump diffusions. The second part analyzes two models used to study the variability in the polyme- rization phenomena occurring in a biological context. In Chapter 5, we investigate the polymerization and fragmentation processes with an assumption of critical nucleus size. A scaling analysis of these stochastic models show that the sharp phase transition and, especially the large variance, observed in the experiments can be explained by these models. In Chapter 6, we provide a functional central limit theorem in the classical (infinite dimensional) stochastic Becker-Döring model.
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Wen Sun. A study of interacting stochastic networks :large scale, long time behavior and fluctuations. Probability [math.PR]. Université Pierre et Marie Curie, 2018. English. ⟨tel-01813434⟩

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