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We study the N → ∞ limit of the normalized largest component in some systems of N diffusive particles with mean-field interaction. By applying a universal time change, the interaction in noises is transferred to the drift terms, and the asymptotic behavior of the maximum becomes well-understood due to existing results in the literature. We expect that the normalized maximum in the original setting has the same limiting distribution as that of i.i.d copies of a solution to the corresponding McKean–Vlasov SDE and we present some results and numerical simulations that support this conjecture. [ABSTRACT FROM AUTHOR]

In this paper, a special sequence of controlled branching processes is considered. Using a martingale problem approach, we prove that under some mild conditions the limit of such processes is a kind of continuous branching process with dependent immigration constructed in Li (2019). The conditions are given in terms of probability generating functions and can be realized. [ABSTRACT FROM AUTHOR]