Abstract
We consider the long time limit for the solutions of a discrete wave equation with weak stochastic forcing. The multiplicative noise conserves energy, and in the unpinned case also conserves momentum. We obtain a time-inhomogeneous Ornstein-Uhlenbeck equation for the limit wave function that holds for both 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.
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Communicated by L. Saint-Raymond
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Komorowski, T., Olla, S. & Ryzhik, L. Asymptotics of the Solutions of the Stochastic Lattice Wave Equation. Arch Rational Mech Anal 209, 455–494 (2013). https://doi.org/10.1007/s00205-013-0626-8
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DOI: https://doi.org/10.1007/s00205-013-0626-8