Abstract

This paper studies the quintic nonlinear Schrödinger equation on ${ℝ}^d$ with randomized initial data below the critical regularity $H^{\left(d-1\right)∕2}$ for $d\ge 3$. The main result is a proof of almost sure local well-posedness given a Wiener randomization of the data in $H^s$ for $s\in \left(\frac{1}{2}\left(d-2\right),\frac{1}{2}\left(d-1\right)\right)$. The argument further develops the techniques introduced in the work of Á. Bényi, T. Oh and O.  Pocovnicu on the cubic problem. The paper concludes with a condition for almost sure global well-posedness.

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Mathematical Subject Classification 2010

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