Asymptotics of the solutions of the stochastic lattice wave equation

Tomasz Komorowski (1) , Stefano Olla (2) , Lenya Ryzhik (3)

1 Instytut Matematyki = Institute of Mathematics [Lublin]
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 Department of Mathematics [Stanford]

We consider the long time limit theorems for the solutions of a discrete wave equation with a weak stochastic forcing. The multiplicative noise conserves the energy and the momentum. We obtain a time-inhomogeneous Ornstein-Uhlenbeck equation for the limit wave function that holds both for square integrable and statistically homogeneous initial data. The limit is understood in the point-wise sense in the former case, and in the weak sense in the latter. On the other hand, the weak limit for square integrable initial data is deterministic.

Domaines: Probabilités [math.PR]

Tomasz Komorowski, Stefano Olla, Lenya Ryzhik. Asymptotics of the solutions of the stochastic lattice wave equation. Archive for Rational Mechanics and Analysis, 2013, 209, pp.455-494. ⟨10.1007/s00205-013-0626-8⟩. ⟨hal-00701588⟩