  In this paper, we consider the pointwise convergence problem of free
Ostrovsky equation with rough data and random data. Firstly, we show the almost
everywhere pointwise convergence of free Ostrovsky equation in
$H^{s}(\mathbb{R})$ with $s\geq \frac{1}{4}$ with rough data. Secondly, we
present counterexamples showing that the maximal function estimate related to
the free Ostrovsky equation can fail if $s<\frac{1}{4}$. Finally, for every
$x\in \mathbb{R}$, we show the almost surely pointwise convergence of free
Ostrovsky equation in $L^{2}(\mathbb{R})$ with random data. The main tools are
the density theorem, high-low frequency idea, Wiener decomposition and Lemmas
2.1-2.6 as well as the probabilistic estimates of some random series which are
just Lemmas 3.2-3.4 in this paper. The main difficulty is that zero is the
singular point of the phase functions of free Ostrovsky equation. We use
high-low frequency idea to conquer the difficulties.
