Probabilistic pointwise convergence problem of some dispersive equations

In this paper, we investigate the almost surely pointwise convergence problem of free KdV equation, free wave equation, free elliptic and non-elliptic Schr"odinger equation respectively. We firstly establish some estimates related to the Wiener decomposition of frequency spaces which are just Lemmas 2.1-2.6 in this paper. Secondly, by using Lemmas 2.1-2.6, 3.1, we establish the probabilistic estimates of some random series which are just Lemmas 3.2-3.11 in this paper. Finally, combining the density theorem in L${}^2$ with Lemmas 3.2-3.11, we obtain almost surely pointwise convergence of the solutions to corresponding equations with randomized initial data in $L^2$, which require much less regularity of the initial data than the rough data case. At the same time, we present the probabilistic density theorem, which is Lemma 3.11 in this paper.