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We extend the convergence method introduced in our works [8–10] for almost sure global well-posedness of Gibbs measure evolutions of the nonlinear Schrödinger (NLS) and nonlinear wave (NLW) equations on the unit ball in to the case of the three dimensional NLS. This is the first probabilistic global well-posedness result for NLS with supercritical data on the unit ball in . The initial data is taken as a Gaussian random process lying in the support of the Gibbs measure associated to the equation,...
We also prove a long time existence result; more precisely we prove that for fixed there exists a set , such that any data , evolves up to time into a solution with , . In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space , that is in the supercritical scaling regime.
We give sufficient conditions for the existence of a matrix of probabilities such that a system of randomly chosen transformations , k = 1,...,N, with probabilities is asymptotically stable.
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