Large deviations for small noise diffusions over long time
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- by Amarjit Budhiraja and Pavlos Zoubouloglou
- Trans. Amer. Math. Soc. Ser. B 11 (2024), 1-63
- DOI: https://doi.org/10.1090/btran/172
- Published electronically: January 2, 2024
Abstract:
We study two problems. First, we consider the large deviation behavior of empirical measures of certain diffusion processes as, simultaneously, the time horizon becomes large and noise becomes vanishingly small. The law of large numbers (LLN) of the empirical measure in this asymptotic regime is given by the unique equilibrium of the noiseless dynamics. Due to degeneracy of the noise in the limit, the methods of Donsker and Varadhan [Comm. Pure Appl. Math. 29 (1976), pp. 389–461] are not directly applicable and new ideas are needed. Second, we study a system of slow-fast diffusions where both the slow and the fast components have vanishing noise on their natural time scales. This time the LLN is governed by a degenerate averaging principle in which local equilibria of the noiseless system obtained from the fast dynamics describe the asymptotic evolution of the slow component. We establish a large deviation principle that describes probabilities of divergence from this behavior. On the one hand our methods require stronger assumptions than the nondegenerate settings, while on the other hand the rate functions take simple and explicit forms that have striking differences from their nondegenerate counterparts.References
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Bibliographic Information
- Amarjit Budhiraja
- Affiliation: Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, North Carolina 27599
- MR Author ID: 602759
- ORCID: 0000-0002-7912-5433
- Email: budhiraj@email.unc.edu
- Pavlos Zoubouloglou
- Affiliation: Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, North Carolina 27599
- Email: pavlos@live.unc.edu
- Received by editor(s): July 14, 2022
- Received by editor(s) in revised form: May 16, 2023
- Published electronically: January 2, 2024
- Additional Notes: The first author was supported in part by the NSF (DMS-1814894, DMS-1853968, DMS-2134107 and DMS-2152577). The second author was partly supported by the 2022 Summer Fellowship awarded through UNC’s Graduate School.
- © Copyright 2024 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 11 (2024), 1-63
- MSC (2020): Primary 60F10, 60J60, 60J25, 60H10
- DOI: https://doi.org/10.1090/btran/172
- MathSciNet review: 4683868