Unique Ergodicity for a Class of Stochastic Hyperbolic Equations with Additive Space-Time White Noise
Leonardo Tolomeo
Abstract
Abstract In 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 $$d=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and the beam equation for $$d\le 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> . 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.