Long-Time Large Deviation Asymptotics for Mean-Field Granular Media Equations
Abstract
We study long-time mean-field large deviations for granular-media diffusions. Under a dissipativity assumption, for arbitrary TN→∞, the occupation measure of the N-particle empirical measure satisfies a large deviation principle on 𝒫(𝒫(ℝd)) with speed NTN and rate function ℐ(Γ)=∫𝒫(ℝd)i(μ)Γ(dμ). Here i(μ) is one quarter of the free-energy dissipation: i(μ)=1/4|∇δℱ/δμ(μ)|L2(μ)2, where ℱ is the free energy of the model. The local cost i(μ) can be interpreted as the per-unit-time Dawson--Gärtner action of the constant path at μ, and ℐ averages these costs according to Γ. The main difficulty in the upper bound is that empirical measures are atomic, so their free energy is infinite. We address this by analyzing a regularized free-energy functional along the empirical-measure flow of controlled granular-media diffusions. The lower bound uses approximation by finitely supported occupation measures whose atoms have smooth densities with Gaussian tails. For quadratic potentials, we show that the variational problem associated with an atypically large overall space-time variance exhibits a transition between a Dirac minimizer at a Gaussian profile and non-Dirac minimizers supported on Gaussian profiles with different centers.
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BibTeX
@misc{budhiraja2026longtime,
title = {{Long-Time Large Deviation Asymptotics for Mean-Field Granular Media Equations}},
author = {Amarjit Budhiraja and Jack Sullivan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09038},
eprint = {2610.09038},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09038},
}
RIS
TY - UNPB
TI - Long-Time Large Deviation Asymptotics for Mean-Field Granular Media Equations
AU - Amarjit Budhiraja
AU - Jack Sullivan
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.09038
UR - https://arxiv.org/abs/2610.09038
AB - We study long-time mean-field large deviations for granular-media diffusions. Under a dissipativity assumption, for arbitrary T_N→∞, the occupation measure of the N-particle empirical measure satisfies a large deviation principle on 𝒫(𝒫(ℝ^d)) with speed NT_N and rate function ℐ(Γ)=∫_𝒫(ℝ^d)i(μ)Γ(dμ). Here i(μ) is one quarter of the free-energy dissipation: i(μ)=1/4|∇δℱ/δμ(μ)|_L^2(μ)^2, where ℱ is the free energy of the model. The local cost i(μ) can be interpreted as the per-unit-time Dawson--Gärtner action of the constant path at μ, and ℐ averages these costs according to Γ. The main difficulty in the upper bound is that empirical measures are atomic, so their free energy is infinite. We address this by analyzing a regularized free-energy functional along the empirical-measure flow of controlled granular-media diffusions. The lower bound uses approximation by finitely supported occupation measures whose atoms have smooth densities with Gaussian tails. For quadratic potentials, we show that the variational problem associated with an atypically large overall space-time variance exhibits a transition between a Dirac minimizer at a Gaussian profile and non-Dirac minimizers supported on Gaussian profiles with different centers.
ER -