Unique Ergodicity for a Class of Stochastic Hyperbolic Equations with Additive Space-Time White Noise

Abstract

AbstractIn this paper, we consider a certain class of second order nonlinear PDEs with damping and space-time white noise forcing, posed on the d-dimensional torus. This class includes the wave equation for $$d=1$$ d = 1 and the beam equation for $$d\le 3$$ d ≤ 3 . We show that the Gibbs measure is the unique invariant measure for this system. Since the flow does not satisfy the strong Feller property, we introduce a new technique for showing unique ergodicity. This approach may be also useful in situations in which finite-time blowup is possible.