Asymptotics of Multi-Scale McKean–Vlasov Diffusions with Super-Linear Kernels: a Lifted Semigroup Approach

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  • Authors: Wei Hong, Shanshan Hu, Wei Liu, Shiyuan Yang
  • Preprint year: 2026
  • First public date: 2026-04-24
  • arXiv: 2604.22510

Abstract

In this work, we establish the small-noise asymptotic behaviour (namely, the functional law of large numbers and the large deviation principle) for multi-scale McKean–Vlasov diffusions with super-linear kernels. In this setting, the interaction depends on the laws of both the slow component and the fast oscillating process. Consequently, the frozen (parameterized) system exhibits McKean–Vlasov dynamics, forming a nonlinear Markov process and thereby rendering the analysis more complex compared to existing works. We develop a lifted semigroup argument and employ a generalized Khasminskii time discretization scheme to derive the small-noise limit of the slow variable, providing explicit convergence rates. Furthermore, we introduce the notion of a lifted viable pair and utilize a generalized functional occupation measure approach to establish the Laplace principle, which is equivalent to the large deviation principle. The main results of this work find broad applications in multi-scale models arising in fields such as machine learning and optimization theory. In particular, our results can be employed to analyze the dynamics of multi-scale consensus-based methods for multilevel optimization, where the coefficients typically satisfy local Lipschitz continuity on the interaction kernels.

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BibTeX

@article{arxiv260422510,
  title = {Asymptotics of Multi-Scale McKean--Vlasov Diffusions with Super-Linear Kernels: a Lifted Semigroup Approach},
  author = {Wei Hong and Shanshan Hu and Wei Liu and Shiyuan Yang},
  year = {2026},
  eprint = {2604.22510},
  archivePrefix = {arXiv},
  primaryClass = {math.PR}
}

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