Abstract
Time evolution, ergodic properties, and invariant measures of some classical fields described by second-order differential equations are discussed. The classical system is not ergodic. A class of Markovian random perturbations is considered. It is shown that the stochastic system with a fixed stochastic perturbation is uniquely ergodic. We discuss in detail the Euler equation in two dimensions and the wave equation in one dimension. In our models an arbitrarily small stochastic perturbation selects one of the invariant measures for the deterministic system.
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