Local large deviations principle for occupation measures of the stochastic damped nonlinear wave equation

Abstract

We consider the damped nonlinear wave (NLW) equation driven by a spatially regular white noise. Assuming that the noise is non-degenerate in all Fourier modes, we establish a large deviations principle (LDP) for the occupation measures of the trajectories. The lower bound in the LDP is of a local type, which is related to the weakly dissipative nature of the equation and seems to be new in the context of randomly forced PDE's. The proof is based on an extension of methods developed in [JNPS] and [JNPS14] in the case of kick forced dissipative PDE's with parabolic regularisation property such as, for example, the Navier-Stokes system and the complex Ginzburg-Landau equations. We also show that a high concentration towards the stationary measure is impossible, by proving that the rate function that governs the LDP cannot have the trivial form (i.e., vanish on the stationary measure and be infinite elsewhere). P t (v, Γ)σ(dv), t ≥ 0, 1 Some estimates for the H s -norm of the solutions are given in Section 5.2.

Large deviations for the Navier–Stokes equations driven by a white-in-time noise

“…where P N is the orthogonal projection in H onto the space H N defined by (1.11). The following result is a version of the Foiaş-Prodi estimate obtained in [FP67]; see also [KS12, Section 2.1.8] for a similar result for the Navier-Stokes system (with different equation instead of (6.2)) and [MN18a,Section 7.3] for the damped nonlinear wave equation.…”

“…The proofs of these papers are based on a Kifer type criterion for LDP and a study of the large-time behaviour of generalised Markov semigroups. These results have been later extended in [MN18a] to the case of the stochastic damped nonlinear wave equation driven by a spatially regular white noise. The main result of that paper is an LDP of local type.…”

“…See the papers [JNPS15, JNPS18, MN18b] for similar results in the case of a discrete-time random dynamical system and [MN18a] for the case of the stochastic damped nonlinear wave equation. The main difficulty in the proof of Theorem 4.1 comes from the fact that the oscillation of the potential V can be arbitrarily large.…”

“…The coupling processes are constructed following the arguments of [MN18a]. Let us take any z, z ∈ H and denote by u t and u t the solutions of (1.5) issued from z and z .…”

Rigorous results in space-periodic two-dimensional turbulence

“…This type of results were first obtained by Donsker and Varadhan in the case of finite-dimensional diffusion processes [DV75] and later extended to many other situations. In the context of the Navier-Stokes equations, the theory was developed in the case of random kicks [JNPS15, JNPS17] (see also the recent papers [MN17,Ner17] devoted to spatially regular white noise), and we now describe the main achievements.…”