Variational exponential estimates and stationary large deviations for SPDEs
Abstract
We establish an abstract criterion for large deviations of invariant measures when the deterministic dynamics has a single globally attracting equilibrium. Deterministic regularisation and attraction yield an endpoint estimate for long controlled paths of bounded action, uniformly over each fixed bounded initial set. This allows the finite time Freidlin-Wentzell uniform LDP on bounded initial sets to be combined directly with stationary exponential norm estimates. The proof requires neither a separate exponential tightness argument nor a Dembo-Zeitouni uniform LDP, and imposes no decay condition on differences between controlled trajectories. Finite path approximation identifies the quasipotential as the good rate function. We also establish a variational criterion based on the Boué-Dupuis representation that derives exponential estimates from bounds for controlled outputs, without requiring their convergence. Uniform controlled terminal moments yield estimates uniform in time, which pass to Krylov-Bogoliubov invariant measures. Together, these results establish stationary LDPs in four regimes: two-dimensional Navier-Stokes equations with possibly unbounded, globally Lipschitz, uniformly positive scalar amplitudes and vanishing noise correlation; reaction-diffusion systems without the additional restriction linking noise growth to reaction strength; Allen-Cahn equations with opposite boundary values and globally Lipschitz, uniformly nondegenerate multiplicative space-time white noise on intervals of arbitrary length; and Burgers equations with fixed covariance (−Δ)α, 0≤α<1/2. For Burgers, we also obtain Gaussian norm estimates.
Related
- Berry--Esseen theorems, LLT, Edgeworth expansions, and large deviations for Locally Stationary Markov Chains
- Exponential Deviations and Hölder Continuity for Moving-Law Mixed Cocycles
- Bernstein type results for stationary points of variational integrals with subquadratic growth in two dimensions
- Exponential functionals of two random walks and stationary measures of discrete growth processes
- Stationary kinetic Krylov-Safonov estimates in space dimension one
BibTeX
@misc{bai2026variational,
title = {{Variational exponential estimates and stationary large deviations for SPDEs}},
author = {Rui Bai and Chunrong Feng and Huaizhong Zhao},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05830},
eprint = {2611.05830},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05830},
}
RIS
TY - UNPB
TI - Variational exponential estimates and stationary large deviations for SPDEs
AU - Rui Bai
AU - Chunrong Feng
AU - Huaizhong Zhao
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.05830
UR - https://arxiv.org/abs/2611.05830
AB - We establish an abstract criterion for large deviations of invariant measures when the deterministic dynamics has a single globally attracting equilibrium. Deterministic regularisation and attraction yield an endpoint estimate for long controlled paths of bounded action, uniformly over each fixed bounded initial set. This allows the finite time Freidlin-Wentzell uniform LDP on bounded initial sets to be combined directly with stationary exponential norm estimates. The proof requires neither a separate exponential tightness argument nor a Dembo-Zeitouni uniform LDP, and imposes no decay condition on differences between controlled trajectories. Finite path approximation identifies the quasipotential as the good rate function. We also establish a variational criterion based on the Boué-Dupuis representation that derives exponential estimates from bounds for controlled outputs, without requiring their convergence. Uniform controlled terminal moments yield estimates uniform in time, which pass to Krylov-Bogoliubov invariant measures. Together, these results establish stationary LDPs in four regimes: two-dimensional Navier-Stokes equations with possibly unbounded, globally Lipschitz, uniformly positive scalar amplitudes and vanishing noise correlation; reaction-diffusion systems without the additional restriction linking noise growth to reaction strength; Allen-Cahn equations with opposite boundary values and globally Lipschitz, uniformly nondegenerate multiplicative space-time white noise on intervals of arbitrary length; and Burgers equations with fixed covariance (−Δ)^α, 0≤α<1/2. For Burgers, we also obtain Gaussian norm estimates.
ER -