Exponential mixing for the white-forced damped nonlinear wave equation

Abstract

The paper is devoted to studying the stochastic nonlinear wave (NLW) equationThe equation is supplemented with the Dirichlet boundary condition. Here f is a nonlinear term, h(x) is a function in H 1 0 (D) and η(t, x) is a non-degenerate white noise. We show that the Markov process associated with the flow ξu(t) = [u(t),u(t)] has a unique stationary measure µ, and the law of any solution converges to µ with exponential rate in the dual-Lipschitz norm.

Large Deviations for Stationary Measures of Stochastic Nonlinear Wave Equations\\with Smooth White Noise

“…Indeed, in view of inequality (1.3) and proposition 3.2 of [27], the invariant measure D " of this process is mixing in b 0 .H /, and we have inclusion (2.2). Moreover, by the very definition of " 1 , we have u.…”

“…Once again using proposition 4.1 in [27], but this time on the interval OE0; T , we see that there is N N 1 such that for all N N we have (A. 19).…”

“…To handle this new problem, we use the estimates obtained for . " / together with the mixing property of " established in [27] to construct an auxiliary finitely additive measure y " defined on Borel subsets of H that satisfies…”

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

“…where F is a primitive of f , ν is a positive number less than (λ 1 ∧ γ)/8. Let us note that inequality (1.2) is slightly more restrictive than the one used in [Mar14]; this hypothesis allows us to establish the exponential tightness property (see Section 5.1). We consider the NLW equation in the phase space H = H 1 × L 2 endowed with the norm which will play the role of the weight function.…”

“…We denote by (u t , P u ), u t = [u t ,u t ] the Markov family associated with this stochastic NLW equation and parametrised by the initial condition u = [u 0 , u 1 ]. The exponential ergodicity for this family is established in [Mar14], this result is recalled below in Theorem 1.1.…”