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In this paper, we prove convergence in distribution of Langevin processes in the overdamped asymptotics. The proof relies on the classical perturbed test function (or corrector) method, which is used both to show tightness in path space, and to identify the extracted limit with a martingale problem. The result holds assuming the continuity of the gradient of the potential energy, and a mild control of the initial kinetic energy.Date: December 1, 2018. 1991

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In this paper, we study the unconditional uniqueness of solution for the Cauchy problem of Ḣsc (0 ≤ s c < 1) critical nonlinear Schrödinger equations (NLS). By employing paraproduct estimates and Strichartz estimates, we prove that unconditional uniqueness of solution holds in C t (I; Ḣsc (R d )) for d ≥ 6. This extends earlier results by Yin Yin Su Win and Y. Tsutsumi [19] and Cazenave [3].