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We consider the defocusing nonlinear Schrödinger equations on the two-dimensional compact Riemannian manifold without boundary or a bounded domain in ℝ^2. In particular, we discuss the Wick renormalization in terms of the Hermite polynomials and the Laguerre polynomials and construct the Gibbs measures corresponding to the Wick ordered Hamiltonian. Then, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure.
|Number of pages||49|
|Journal||Stochastics and Partial Differential Equations: Analysis and Computations|
|Early online date||26 Mar 2018|
|Publication status||Published - Sep 2018|
- nonlinear Schrödinger equation
- Gibbs measure
- Wick ordering
- Hermite polynomial
- Laguerre polynomial
- white noise functional
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- 1 Finished
ProbDynDispEq - Probabilistic and Dynamical Study of Nonlinear Dispersive Equations
1/03/15 → 29/02/20