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 $(-Δ)^α$, $0\leqα<1/2$. For Burgers, we also obtain Gaussian norm estimates.