Diffusion Processes with Random Interactions.
In this thesis we study the asymptotic behavior of certain systems of n randomly interacting diffusion processes in the limit when n goes to infinity. The first main objective of this thesis is to obtain a McKean-Vlasov type limit for the empirical measures associated to such systems in the almost sure sense with respect to the random interactions. The second main objective of this thesis is to study the equilibrium behaviors of such random systems in the limit when n goes to infinity. We establish the asymptotic behaviors of certain random Gibbs measures, which can be viewed as a mathematical rigorous version of the van Hemmen et al spin glass model. Finally, we consider a system in which the random interactions are assumed to be independent Brownian motions. Under proper conditions the limiting process has a unique invariant measure which is a zero mean Gaussian measure such that its variance solves the famous Sherrington-Kirkpatrick spin glass fixed point equation.
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- Mathematics; Physics: General