For the 1D Schrödinger map equation with zero Neumann data, non-trivial statistically stationary solutions exist, constructed by vanishing-viscosity limits of invariant measures of the stochastic Landau-Lifshitz-Gilbert equation.
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Statistical solutions to the Schr\"odinger map equation in 1D, via the randomly forced Landau-Lifschitz-Gilbert equation
For the 1D Schrödinger map equation with zero Neumann data, non-trivial statistically stationary solutions exist, constructed by vanishing-viscosity limits of invariant measures of the stochastic Landau-Lifshitz-Gilbert equation.