Gamma Expansions of Level Two Large Deviation Rate Functionals for Metastable Diffusions
Abstract
Consider the diffusions dXtε=b(Xtε)dt+√(2ε)σ(Xtε)dBt on ℝd, where b and σ are smooth coefficients and Bt is an r-dimensional Brownian motion, with r≤ d. We allow nonreversible dynamics and a degenerate diffusion matrix. Let ℐε be the level-two large deviation rate functional of the empirical measure. Under our assumptions, including a complete metastable hierarchy described by resolvent convergence, we establish the full Γ-expansion ℐε=1/ε𝒥(−1)+𝒥(0)+∑p=1q1/θε(p)𝒥(p) in the sense of successive Γ-limits in the weak topology on probability measures. Here θε(p) are the metastable time scales, and each 𝒥(p), p≥1, is the lift of the level-two rate functional of the corresponding finite-state Markov chain. We establish the pre-metastable Γ-limits for generalized overdamped, underdamped, and third-order Langevin dynamics. For the generalized overdamped model, existing metastability results yield the full expansion under additional conditions. For the underdamped and third-order models, the full expansion remains conditional on verification of the complete metastable hierarchy.
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BibTeX
@misc{kim2026gamma,
title = {{Gamma Expansions of Level Two Large Deviation Rate Functionals for Metastable Diffusions}},
author = {Donghyun Kim and Insuk Seo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22117},
eprint = {2610.22117},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22117},
}
RIS
TY - UNPB
TI - Gamma Expansions of Level Two Large Deviation Rate Functionals for Metastable Diffusions
AU - Donghyun Kim
AU - Insuk Seo
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22117
UR - https://arxiv.org/abs/2610.22117
AB - Consider the diffusions dX_t^ε=b(X_t^ε)dt+√(2ε)σ(X_t^ε)dB_t on ℝ^d, where b and σ are smooth coefficients and B_t is an r-dimensional Brownian motion, with r≤ d. We allow nonreversible dynamics and a degenerate diffusion matrix. Let ℐ_ε be the level-two large deviation rate functional of the empirical measure. Under our assumptions, including a complete metastable hierarchy described by resolvent convergence, we establish the full Γ-expansion ℐ_ε=1/ε𝒥^(−1)+𝒥^(0)+∑_p=1^q1/θ_ε^(p)𝒥^(p) in the sense of successive Γ-limits in the weak topology on probability measures. Here θ_ε^(p) are the metastable time scales, and each 𝒥^(p), p≥1, is the lift of the level-two rate functional of the corresponding finite-state Markov chain. We establish the pre-metastable Γ-limits for generalized overdamped, underdamped, and third-order Langevin dynamics. For the generalized overdamped model, existing metastability results yield the full expansion under additional conditions. For the underdamped and third-order models, the full expansion remains conditional on verification of the complete metastable hierarchy.
ER -