Some Asymptotic Results for Weakly Interacting Particle Systems Public Deposited

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Last Modified
  • March 20, 2019
Creator
  • Wu, Ruoyu
    • Affiliation: College of Arts and Sciences, Department of Statistics and Operations Research
Abstract
  • Weakly interacting particle systems have been widely used as models in many areas, including, but not limited to, communication systems, mathematical finance, chemical and biological systems, and social sciences. In this dissertation, we establish law of large numbers (LLN), central limit theorems (CLT), large deviation principles (LDP) and moderate deviation principles (MDP) for several types of such systems. The dissertation consists of two parts. In the first part, we are mainly concerned with LLN and CLT. We begin by studying weakly interacting multi-type particle systems that arise in neurosciences to model a network of interacting spiking neurons. We prove a CLT showing the centered and suitably normalized empirical measures converge in distribution to a Gaussian random eld. This result can in particular be applied to single-type systems to characterize the joint asymptotic behavior of large disjoint subpopulations. We then establish a CLT for a multi-type model where each particle is affected by a common source of noise. Here the limit is not Gaussian but rather described through a suitable Gaussian mixture. Next, we consider weakly interacting particle systems in a setting where not every pair of particles interacts, but rather particle interactions are governed by Erdos--Renyi random graphs and an interaction between a pair of particles occurs only when there is a corresponding edge in the graph. Edges can form and break down independently as time evolves. We prove a LLN and CLT under conditions on the edge probabilities. The second part of this dissertation concerns MDP and LDP for certain weakly interacting particle systems. We study interacting systems of both diffusions and of pure jump Markov processes with a countable state space. We are interested in estimating probabilities of moderate deviations of empirical measure processes, from the LLN limit. For both systems a MDP is established which is formulated in terms of a LDP with an appropriate speed function, for suitably centered and normalized empirical measure processes. Finally, we study particle approximations for certain nonlinear heat equations using a system of Brownian motions with killing. A LLN and LDP for sub-probability measure valued processes given as the empirical measure of the alive Brownian particles are proved. We also give, as a byproduct, a convenient variational representation for expectations of nonnegative functionals of Brownian motions along with an i.i.d. sequence of random variables.
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  • In Copyright
Advisor
  • Zhang, Kai
  • Bhamidi, Shankar
  • Carlstein, Edward
  • Budhiraja, Amarjit
  • Pipiras, Vladas
Degree
  • Doctor of Philosophy
Degree granting institution
  • University of North Carolina at Chapel Hill Graduate School
Graduation year
  • 2016
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