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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.22510 |
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| _version_ | 1866911621453971456 |
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| author | Hong, Wei Hu, Shanshan Liu, Wei Yang, Shiyuan |
| author_facet | Hong, Wei Hu, Shanshan Liu, Wei Yang, Shiyuan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_22510 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotics of Multi-Scale McKean--Vlasov Diffusions with Super-Linear Kernels: a Lifted Semigroup Approach Hong, Wei Hu, Shanshan Liu, Wei Yang, Shiyuan Probability 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. |
| title | Asymptotics of Multi-Scale McKean--Vlasov Diffusions with Super-Linear Kernels: a Lifted Semigroup Approach |
| topic | Probability |
| url | https://arxiv.org/abs/2604.22510 |