Almost Sure Global Well-posedness for Fractional Cubic Schrödinger equation on torus
In [12], we proved that $1$-d periodic fractional Schrödinger equation with cubic nonlinearity is locally well-posed in $H^s$ for $s>\frac{1-α}{2}$ and globally well-posed for $s>\frac{5α-1}{6}$. In this paper we define an invariant probability measure $μ$ on $H^s$ for $s<α-\frac{1}{2}$, so that for any $ε>0$ there is a set $Ω\subset H^s$ such that $μ(Ω^c)<ε$ and the equation is globally well-posed for initial data in $Ω$. We see that this fills the gap between the local well-posedness and the global well-posedness range in almost sure sense for $\frac{1-α}{2}<α-\frac{1}{2}$, i.e. $α>\frac{2}{3}$ in almost sure sense.
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